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/Fin/Tuple/Basic.lean
|
Fin.preimage_insertNth_Icc_of_mem
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nβ : Type v\ninst✝ : (i : Fin (n + 1)) → Preorder (α i)\ni : Fin (n + 1)\nx : α i\nq₁ q₂ : (j : Fin (n + 1)) → α j\nhx : x ∈ Icc (q₁ i) (q₂ i)\np : (j : Fin n) → α (↑(succAbove i) j)\n⊢ p ∈ insertNth i x ⁻¹' Icc q₁ q₂ ↔ p ∈ Icc (fun j => q₁ (↑(succAbove i) j)) fun j => q₂ (↑(succAbove i) j)",
"tactic": "simp only [mem_preimage, insertNth_mem_Icc, hx, true_and_iff]"
}
] |
[
824,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
821,
1
] |
Mathlib/Analysis/Convex/Topology.lean
|
Convex.subset_interior_image_homothety_of_one_lt
|
[] |
[
338,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/LinearAlgebra/Contraction.lean
|
dualTensorHomEquivOfBasis_symm_cancel_left
|
[
{
"state_after": "no goals",
"state_before": "ι : Type w\nR : Type u\nM : Type v₁\nN : Type v₂\nP : Type v₃\nQ : Type v₄\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : AddCommGroup P\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R M\ninst✝⁴ : Module R N\ninst✝³ : Module R P\ninst✝² : Module R Q\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nb : Basis ι R M\nx : Dual R M ⊗[R] N\n⊢ ↑(LinearEquiv.symm (dualTensorHomEquivOfBasis b)) (↑(dualTensorHom R M N) x) = x",
"tactic": "rw [← dualTensorHomEquivOfBasis_apply b,\n LinearEquiv.symm_apply_apply <| dualTensorHomEquivOfBasis (N := N) b]"
}
] |
[
197,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.algebraMap_apply
|
[
{
"state_after": "σ : Type u_1\nR : Type u_3\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nr : R\n⊢ ↑(RingHom.comp (map σ (algebraMap R A)) (C σ R)) r = ↑(C σ A) (↑(algebraMap R A) r)",
"state_before": "σ : Type u_1\nR : Type u_3\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nr : R\n⊢ ↑(algebraMap R (MvPowerSeries σ A)) r = ↑(C σ A) (↑(algebraMap R A) r)",
"tactic": "change (MvPowerSeries.map σ (algebraMap R A)).comp (C σ R) r = _"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type u_3\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nr : R\n⊢ ↑(RingHom.comp (map σ (algebraMap R A)) (C σ R)) r = ↑(C σ A) (↑(algebraMap R A) r)",
"tactic": "simp"
}
] |
[
653,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
650,
1
] |
Mathlib/Data/List/Nodup.lean
|
List.Nodup.filter
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\np : α → Bool\nl : List α\n⊢ Nodup l → Nodup (List.filter p l)",
"tactic": "simpa using Pairwise.filter (fun a ↦ p a)"
}
] |
[
295,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Order/Max.lean
|
not_isMax_of_lt
|
[] |
[
324,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
|
BoxIntegral.TaggedPrepartition.disjUnion_tag_of_mem_left
|
[] |
[
375,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.div_le_of_le_mul'
|
[
{
"state_after": "m n k l : ℕ\nh : m ≤ k * n\nk0 : k = 0\n⊢ 0 ≤ n",
"state_before": "m n k l : ℕ\nh : m ≤ k * n\nk0 : k = 0\n⊢ m / k ≤ n",
"tactic": "rw [k0, Nat.div_zero]"
},
{
"state_after": "no goals",
"state_before": "m n k l : ℕ\nh : m ≤ k * n\nk0 : k = 0\n⊢ 0 ≤ n",
"tactic": "apply zero_le"
}
] |
[
386,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
380,
11
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.toFinsupp_X_pow
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ (X ^ n).toFinsupp = Finsupp.single n 1",
"tactic": "rw [X_pow_eq_monomial, toFinsupp_monomial]"
}
] |
[
895,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
894,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.map_erase
|
[
{
"state_after": "case empty\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.127295\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx : α\n⊢ map f (erase 0 x) = erase (map f 0) (f x)\n\ncase cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.127295\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx y : α\ns : Multiset α\nih : map f (erase s x) = erase (map f s) (f x)\n⊢ map f (erase (y ::ₘ s) x) = erase (map f (y ::ₘ s)) (f x)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.127295\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx : α\ns : Multiset α\n⊢ map f (erase s x) = erase (map f s) (f x)",
"tactic": "induction' s using Multiset.induction_on with y s ih"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.127295\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx y : α\ns : Multiset α\nih : map f (erase s x) = erase (map f s) (f x)\nhxy : y = x\n⊢ map f (erase (y ::ₘ s) x) = erase (map f (y ::ₘ s)) (f x)\n\ncase neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.127295\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx y : α\ns : Multiset α\nih : map f (erase s x) = erase (map f s) (f x)\nhxy : ¬y = x\n⊢ map f (erase (y ::ₘ s) x) = erase (map f (y ::ₘ s)) (f x)",
"state_before": "case cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.127295\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx y : α\ns : Multiset α\nih : map f (erase s x) = erase (map f s) (f x)\n⊢ map f (erase (y ::ₘ s) x) = erase (map f (y ::ₘ s)) (f x)",
"tactic": "by_cases hxy : y = x"
},
{
"state_after": "no goals",
"state_before": "case empty\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.127295\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx : α\n⊢ map f (erase 0 x) = erase (map f 0) (f x)",
"tactic": "simp"
},
{
"state_after": "case pos.refl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.127295\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx : α\ns : Multiset α\nih : map f (erase s x) = erase (map f s) (f x)\n⊢ map f (erase (x ::ₘ s) x) = erase (map f (x ::ₘ s)) (f x)",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.127295\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx y : α\ns : Multiset α\nih : map f (erase s x) = erase (map f s) (f x)\nhxy : y = x\n⊢ map f (erase (y ::ₘ s) x) = erase (map f (y ::ₘ s)) (f x)",
"tactic": "cases hxy"
},
{
"state_after": "no goals",
"state_before": "case pos.refl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.127295\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx : α\ns : Multiset α\nih : map f (erase s x) = erase (map f s) (f x)\n⊢ map f (erase (x ::ₘ s) x) = erase (map f (x ::ₘ s)) (f x)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.127295\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nhf : Injective f\nx y : α\ns : Multiset α\nih : map f (erase s x) = erase (map f s) (f x)\nhxy : ¬y = x\n⊢ map f (erase (y ::ₘ s) x) = erase (map f (y ::ₘ s)) (f x)",
"tactic": "rw [s.erase_cons_tail hxy, map_cons, map_cons, (s.map f).erase_cons_tail (hf.ne hxy), ih]"
}
] |
[
1338,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1331,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.rpow_zero_pos
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ 0 < x ^ 0",
"tactic": "simp"
}
] |
[
103,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Analysis/Convex/Strict.lean
|
strictConvex_iff_div
|
[
{
"state_after": "𝕜 : Type u_1\n𝕝 : Type ?u.221625\nE : Type u_2\nF : Type ?u.221631\nβ : Type ?u.221634\ninst✝⁵ : LinearOrderedField 𝕜\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nx✝ : E\nh : StrictConvex 𝕜 s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\n⊢ a / (a + b) + b / (a + b) = 1",
"state_before": "𝕜 : Type u_1\n𝕝 : Type ?u.221625\nE : Type u_2\nF : Type ?u.221631\nβ : Type ?u.221634\ninst✝⁵ : LinearOrderedField 𝕜\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nx✝ : E\nh : StrictConvex 𝕜 s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\n⊢ (a / (a + b)) • x + (b / (a + b)) • y ∈ interior s",
"tactic": "apply h hx hy hxy (div_pos ha <| add_pos ha hb) (div_pos hb <| add_pos ha hb)"
},
{
"state_after": "𝕜 : Type u_1\n𝕝 : Type ?u.221625\nE : Type u_2\nF : Type ?u.221631\nβ : Type ?u.221634\ninst✝⁵ : LinearOrderedField 𝕜\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nx✝ : E\nh : StrictConvex 𝕜 s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\n⊢ (a + b) / (a + b) = 1",
"state_before": "𝕜 : Type u_1\n𝕝 : Type ?u.221625\nE : Type u_2\nF : Type ?u.221631\nβ : Type ?u.221634\ninst✝⁵ : LinearOrderedField 𝕜\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nx✝ : E\nh : StrictConvex 𝕜 s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\n⊢ a / (a + b) + b / (a + b) = 1",
"tactic": "rw [← add_div]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕝 : Type ?u.221625\nE : Type u_2\nF : Type ?u.221631\nβ : Type ?u.221634\ninst✝⁵ : LinearOrderedField 𝕜\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nx✝ : E\nh : StrictConvex 𝕜 s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\n⊢ (a + b) / (a + b) = 1",
"tactic": "exact div_self (add_pos ha hb).ne'"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕝 : Type ?u.221625\nE : Type u_2\nF : Type ?u.221631\nβ : Type ?u.221634\ninst✝⁵ : LinearOrderedField 𝕜\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nx✝ : E\nh : Set.Pairwise s fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → (a / (a + b)) • x + (b / (a + b)) • y ∈ interior s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • x + b • y ∈ interior s",
"tactic": "convert h hx hy hxy ha hb <;> rw [hab, div_one]"
}
] |
[
415,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/Data/Subtype.lean
|
Subtype.exists'
|
[] |
[
64,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
11
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean
|
Submonoid.closure_inv
|
[
{
"state_after": "case a\nα : Type ?u.59382\nG : Type u_1\nM : Type ?u.59388\nR : Type ?u.59391\nA : Type ?u.59394\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ns✝ t u : Set M\ninst✝ : Group G\ns : Set G\n⊢ closure s⁻¹ ≤ (closure s)⁻¹\n\ncase a\nα : Type ?u.59382\nG : Type u_1\nM : Type ?u.59388\nR : Type ?u.59391\nA : Type ?u.59394\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ns✝ t u : Set M\ninst✝ : Group G\ns : Set G\n⊢ (closure s)⁻¹ ≤ closure s⁻¹",
"state_before": "α : Type ?u.59382\nG : Type u_1\nM : Type ?u.59388\nR : Type ?u.59391\nA : Type ?u.59394\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ns✝ t u : Set M\ninst✝ : Group G\ns : Set G\n⊢ closure s⁻¹ = (closure s)⁻¹",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a\nα : Type ?u.59382\nG : Type u_1\nM : Type ?u.59388\nR : Type ?u.59391\nA : Type ?u.59394\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ns✝ t u : Set M\ninst✝ : Group G\ns : Set G\n⊢ s ⊆ ↑(closure s)",
"state_before": "case a\nα : Type ?u.59382\nG : Type u_1\nM : Type ?u.59388\nR : Type ?u.59391\nA : Type ?u.59394\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ns✝ t u : Set M\ninst✝ : Group G\ns : Set G\n⊢ closure s⁻¹ ≤ (closure s)⁻¹",
"tactic": "rw [closure_le, coe_inv, ← Set.inv_subset, inv_inv]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type ?u.59382\nG : Type u_1\nM : Type ?u.59388\nR : Type ?u.59391\nA : Type ?u.59394\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ns✝ t u : Set M\ninst✝ : Group G\ns : Set G\n⊢ s ⊆ ↑(closure s)",
"tactic": "exact subset_closure"
},
{
"state_after": "case a\nα : Type ?u.59382\nG : Type u_1\nM : Type ?u.59388\nR : Type ?u.59391\nA : Type ?u.59394\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ns✝ t u : Set M\ninst✝ : Group G\ns : Set G\n⊢ s⁻¹ ⊆ ↑(closure s⁻¹)",
"state_before": "case a\nα : Type ?u.59382\nG : Type u_1\nM : Type ?u.59388\nR : Type ?u.59391\nA : Type ?u.59394\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ns✝ t u : Set M\ninst✝ : Group G\ns : Set G\n⊢ (closure s)⁻¹ ≤ closure s⁻¹",
"tactic": "rw [inv_le, closure_le, coe_inv, ← Set.inv_subset]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type ?u.59382\nG : Type u_1\nM : Type ?u.59388\nR : Type ?u.59391\nA : Type ?u.59394\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ns✝ t u : Set M\ninst✝ : Group G\ns : Set G\n⊢ s⁻¹ ⊆ ↑(closure s⁻¹)",
"tactic": "exact subset_closure"
}
] |
[
174,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Data/Fin/Tuple/Monotone.lean
|
Monotone.vecCons
|
[] |
[
81,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Fintype.univ_Prop
|
[
{
"state_after": "α : Type ?u.126304\nβ : Type ?u.126307\nγ : Type ?u.126310\n⊢ univ.val = True ::ₘ {False}",
"state_before": "α : Type ?u.126304\nβ : Type ?u.126307\nγ : Type ?u.126310\n⊢ univ.val = {True, False}.val",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.126304\nβ : Type ?u.126307\nγ : Type ?u.126310\n⊢ univ.val = True ::ₘ {False}",
"tactic": "rfl"
}
] |
[
996,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
995,
1
] |
Mathlib/Data/Option/Basic.lean
|
Option.pmap_some
|
[] |
[
174,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.mul_bot_of_neg
|
[] |
[
969,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
966,
1
] |
Mathlib/Data/Option/Basic.lean
|
Option.map_injective
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.798\nδ : Type ?u.801\nf : α → β\nHf : Function.Injective f\na₁ a₂ : α\nH : Option.map f (some a₁) = Option.map f (some a₂)\n⊢ some a₁ = some a₂",
"tactic": "rw [Hf (Option.some.inj H)]"
}
] |
[
71,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Order.le_cof
|
[
{
"state_after": "α : Type u_1\nr✝ r : α → α → Prop\ninst✝ : IsRefl α r\nc : Cardinal\n⊢ (∀ (b : Cardinal), b ∈ {c | ∃ S, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) ∧ (#↑S) = c} → c ≤ b) ↔\n ∀ {S : Set α}, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) → c ≤ (#↑S)",
"state_before": "α : Type u_1\nr✝ r : α → α → Prop\ninst✝ : IsRefl α r\nc : Cardinal\n⊢ c ≤ cof r ↔ ∀ {S : Set α}, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) → c ≤ (#↑S)",
"tactic": "rw [cof, le_csInf_iff'' (cof_nonempty r)]"
},
{
"state_after": "α : Type u_1\nr✝ r : α → α → Prop\ninst✝ : IsRefl α r\nc : Cardinal\n⊢ (∀ {S : Set α}, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) → c ≤ (#↑S)) →\n ∀ (b : Cardinal), b ∈ {c | ∃ S, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) ∧ (#↑S) = c} → c ≤ b",
"state_before": "α : Type u_1\nr✝ r : α → α → Prop\ninst✝ : IsRefl α r\nc : Cardinal\n⊢ (∀ (b : Cardinal), b ∈ {c | ∃ S, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) ∧ (#↑S) = c} → c ≤ b) ↔\n ∀ {S : Set α}, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) → c ≤ (#↑S)",
"tactic": "use fun H S h => H _ ⟨S, h, rfl⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ninst✝ : IsRefl α r\nc : Cardinal\nH : ∀ {S : Set α}, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) → c ≤ (#↑S)\nS : Set α\nh : ∀ (a : α), ∃ b, b ∈ S ∧ r a b\n⊢ c ≤ (#↑S)",
"state_before": "α : Type u_1\nr✝ r : α → α → Prop\ninst✝ : IsRefl α r\nc : Cardinal\n⊢ (∀ {S : Set α}, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) → c ≤ (#↑S)) →\n ∀ (b : Cardinal), b ∈ {c | ∃ S, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) ∧ (#↑S) = c} → c ≤ b",
"tactic": "rintro H d ⟨S, h, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nr✝ r : α → α → Prop\ninst✝ : IsRefl α r\nc : Cardinal\nH : ∀ {S : Set α}, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) → c ≤ (#↑S)\nS : Set α\nh : ∀ (a : α), ∃ b, b ∈ S ∧ r a b\n⊢ c ≤ (#↑S)",
"tactic": "exact H h"
}
] |
[
87,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/CategoryTheory/Subobject/Comma.lean
|
CategoryTheory.StructuredArrow.projectSubobject_factors
|
[
{
"state_after": "C : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nS : D\nT : C ⥤ D\ninst✝¹ : HasLimits C\ninst✝ : PreservesLimits T\nA P : StructuredArrow S T\nf : P ⟶ A\nhf : Mono f\n⊢ (P.hom ≫ T.map (Subobject.underlyingIso f.right).inv) ≫ T.map (Subobject.arrow (Subobject.mk f.right)) = A.hom",
"state_before": "C : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nS : D\nT : C ⥤ D\ninst✝¹ : HasLimits C\ninst✝ : PreservesLimits T\nA P : StructuredArrow S T\nf : P ⟶ A\nhf : Mono f\n⊢ (P.hom ≫ T.map (Subobject.underlyingIso (MonoOver.arrow (MonoOver.mk' f)).right).inv) ≫\n T.map (Subobject.arrow (projectSubobject (Subobject.mk f))) =\n A.hom",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nS : D\nT : C ⥤ D\ninst✝¹ : HasLimits C\ninst✝ : PreservesLimits T\nA P : StructuredArrow S T\nf : P ⟶ A\nhf : Mono f\n⊢ (P.hom ≫ T.map (Subobject.underlyingIso f.right).inv) ≫ T.map (Subobject.arrow (Subobject.mk f.right)) = A.hom",
"tactic": "simp [← T.map_comp]"
}
] |
[
71,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Order/InitialSeg.lean
|
InitialSeg.acc
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.24005\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : r ≼i s\na✝ : α\nh : Acc r a✝\na : α\nx✝ : ∀ (y : α), r y a → Acc r y\nha : ∀ (y : α), r y a → Acc s (↑f y)\nb : β\nhb : s b (↑f a)\n⊢ Acc s b",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.24005\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : r ≼i s\na : α\n⊢ Acc r a → Acc s (↑f a)",
"tactic": "refine' fun h => Acc.recOn h fun a _ ha => Acc.intro _ fun b hb => _"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.24005\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : r ≼i s\na✝ : α\nh : Acc r a✝\na : α\nx✝ : ∀ (y : α), r y a → Acc r y\nha : ∀ (y : α), r y a → Acc s (↑f y)\na' : α\nhb : s (↑f a') (↑f a)\n⊢ Acc s (↑f a')",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.24005\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : r ≼i s\na✝ : α\nh : Acc r a✝\na : α\nx✝ : ∀ (y : α), r y a → Acc r y\nha : ∀ (y : α), r y a → Acc s (↑f y)\nb : β\nhb : s b (↑f a)\n⊢ Acc s b",
"tactic": "obtain ⟨a', rfl⟩ := f.init hb"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.24005\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : r ≼i s\na✝ : α\nh : Acc r a✝\na : α\nx✝ : ∀ (y : α), r y a → Acc r y\nha : ∀ (y : α), r y a → Acc s (↑f y)\na' : α\nhb : s (↑f a') (↑f a)\n⊢ Acc s (↑f a')",
"tactic": "exact ha _ (f.map_rel_iff.mp hb)"
}
] |
[
219,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
11
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.preimage_sphere
|
[] |
[
1039,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1038,
1
] |
Mathlib/GroupTheory/Abelianization.lean
|
Abelianization.map_comp
|
[] |
[
193,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieHom.idealRange_eq_map
|
[
{
"state_after": "case h\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nm✝ : L'\n⊢ m✝ ∈ idealRange f ↔ m✝ ∈ LieIdeal.map f ⊤",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n⊢ idealRange f = LieIdeal.map f ⊤",
"tactic": "ext"
},
{
"state_after": "case h\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nm✝ : L'\n⊢ m✝ ∈ LieSubmodule.lieSpan R L' ↑(LieSubalgebra.map f ⊤) ↔ m✝ ∈ LieIdeal.map f ⊤",
"state_before": "case h\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nm✝ : L'\n⊢ m✝ ∈ idealRange f ↔ m✝ ∈ LieIdeal.map f ⊤",
"tactic": "simp only [idealRange, range_eq_map]"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nm✝ : L'\n⊢ m✝ ∈ LieSubmodule.lieSpan R L' ↑(LieSubalgebra.map f ⊤) ↔ m✝ ∈ LieIdeal.map f ⊤",
"tactic": "rfl"
}
] |
[
913,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
910,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.preimage
|
[] |
[
892,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
890,
1
] |
Mathlib/GroupTheory/Finiteness.lean
|
Monoid.fg_def
|
[] |
[
103,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Infinite.prod_right
|
[] |
[
930,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
929,
11
] |
Std/Data/Int/Lemmas.lean
|
Int.le_add_of_neg_add_le_left
|
[
{
"state_after": "a b c : Int\nh : a + -b ≤ c\n⊢ a ≤ b + c",
"state_before": "a b c : Int\nh : -b + a ≤ c\n⊢ a ≤ b + c",
"tactic": "rw [Int.add_comm] at h"
},
{
"state_after": "no goals",
"state_before": "a b c : Int\nh : a + -b ≤ c\n⊢ a ≤ b + c",
"tactic": "exact Int.le_add_of_sub_left_le h"
}
] |
[
1000,
36
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
998,
11
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.exp_sum
|
[] |
[
537,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/Algebra/Ring/BooleanRing.lean
|
mul_add_mul
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.4614\nγ : Type ?u.4617\ninst✝ : BooleanRing α\na b : α\nthis : a + b = a + b + (a * b + b * a)\n⊢ a * b + b * a = 0",
"state_before": "α : Type u_1\nβ : Type ?u.4614\nγ : Type ?u.4617\ninst✝ : BooleanRing α\na b : α\n⊢ a * b + b * a = 0",
"tactic": "have : a + b = a + b + (a * b + b * a) :=\n calc\n a + b = (a + b) * (a + b) := by rw [mul_self]\n _ = a * a + a * b + (b * a + b * b) := by rw [add_mul, mul_add, mul_add]\n _ = a + a * b + (b * a + b) := by simp only [mul_self]\n _ = a + b + (a * b + b * a) := by abel"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.4614\nγ : Type ?u.4617\ninst✝ : BooleanRing α\na b : α\nthis : a + b = a + b + (a * b + b * a)\n⊢ a * b + b * a = 0",
"tactic": "rwa [self_eq_add_right] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.4614\nγ : Type ?u.4617\ninst✝ : BooleanRing α\na b : α\n⊢ a + b = (a + b) * (a + b)",
"tactic": "rw [mul_self]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.4614\nγ : Type ?u.4617\ninst✝ : BooleanRing α\na b : α\n⊢ (a + b) * (a + b) = a * a + a * b + (b * a + b * b)",
"tactic": "rw [add_mul, mul_add, mul_add]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.4614\nγ : Type ?u.4617\ninst✝ : BooleanRing α\na b : α\n⊢ a * a + a * b + (b * a + b * b) = a + a * b + (b * a + b)",
"tactic": "simp only [mul_self]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.4614\nγ : Type ?u.4617\ninst✝ : BooleanRing α\na b : α\n⊢ a + a * b + (b * a + b) = a + b + (a * b + b * a)",
"tactic": "abel"
}
] |
[
103,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Data/Int/ModEq.lean
|
Int.modEq_comm
|
[] |
[
78,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
HasFDerivWithinAt.sub_const
|
[] |
[
539,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
537,
8
] |
Mathlib/Analysis/BoxIntegral/Partition/Filter.lean
|
BoxIntegral.IntegrationParams.MemBaseSet.mono'
|
[] |
[
365,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
360,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.le_nfpBFamily
|
[] |
[
274,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/AlgebraicTopology/SimplicialObject.lean
|
CategoryTheory.SimplicialObject.δ_comp_δ''
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : i ≤ ↑Fin.castSucc j\n⊢ X.map (SimplexCategory.δ (Fin.succ j)).op ≫ X.map (SimplexCategory.δ (Fin.castLT i (_ : ↑i < n + 2))).op =\n X.map (SimplexCategory.δ i).op ≫ X.map (SimplexCategory.δ j).op",
"state_before": "C : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : i ≤ ↑Fin.castSucc j\n⊢ δ X (Fin.succ j) ≫ δ X (Fin.castLT i (_ : ↑i < n + 2)) = δ X i ≫ δ X j",
"tactic": "dsimp [δ]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : i ≤ ↑Fin.castSucc j\n⊢ X.map (SimplexCategory.δ (Fin.succ j)).op ≫ X.map (SimplexCategory.δ (Fin.castLT i (_ : ↑i < n + 2))).op =\n X.map (SimplexCategory.δ i).op ≫ X.map (SimplexCategory.δ j).op",
"tactic": "simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_δ'' H]"
}
] |
[
129,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Analysis/Normed/Field/Basic.lean
|
one_le_nnnorm_one
|
[] |
[
210,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.map_add
|
[] |
[
780,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
779,
1
] |
Mathlib/Analysis/ODE/PicardLindelof.lean
|
PicardLindelof.tMin_le_tMax
|
[] |
[
98,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPullback.of_horiz_isIso
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : IsIso fst\ninst✝ : IsIso g\nsq : CommSq fst snd f g\ns : PullbackCone f g\n⊢ (fun s => PullbackCone.fst s ≫ inv fst) s ≫ snd = PullbackCone.snd s",
"state_before": "C : Type u₁\ninst✝² : Category C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : IsIso fst\ninst✝ : IsIso g\nsq : CommSq fst snd f g\n⊢ IsLimit (CommSq.cone sq)",
"tactic": "refine'\n PullbackCone.IsLimit.mk _ (fun s => s.fst ≫ inv fst) (by aesop_cat)\n (fun s => _) (by aesop_cat)"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : IsIso fst\ninst✝ : IsIso g\nsq : CommSq fst snd f g\ns : PullbackCone f g\n⊢ (fun s => PullbackCone.fst s ≫ inv fst) s ≫ snd = PullbackCone.snd s",
"tactic": "simp only [← cancel_mono g, Category.assoc, ← sq.w, IsIso.inv_hom_id_assoc, s.condition]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : IsIso fst\ninst✝ : IsIso g\nsq : CommSq fst snd f g\n⊢ ∀ (s : PullbackCone f g), (fun s => PullbackCone.fst s ≫ inv fst) s ≫ fst = PullbackCone.fst s",
"tactic": "aesop_cat"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : IsIso fst\ninst✝ : IsIso g\nsq : CommSq fst snd f g\n⊢ ∀ (s : PullbackCone f g) (m : s.pt ⟶ P),\n m ≫ fst = PullbackCone.fst s → m ≫ snd = PullbackCone.snd s → m = (fun s => PullbackCone.fst s ≫ inv fst) s",
"tactic": "aesop_cat"
}
] |
[
324,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/Order/Filter/Archimedean.lean
|
atTop_hasAntitoneBasis_of_archimedean
|
[] |
[
107,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/Data/Part.lean
|
Part.ext
|
[] |
[
120,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
mem_nhds_uniformity_iff_left
|
[
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.58164\ninst✝ : UniformSpace α\nx : α\ns : Set α\n⊢ {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α ↔ {p | p.snd = x → p.fst ∈ s} ∈ Prod.swap <$> 𝓤 α",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.58164\ninst✝ : UniformSpace α\nx : α\ns : Set α\n⊢ s ∈ 𝓝 x ↔ {p | p.snd = x → p.fst ∈ s} ∈ 𝓤 α",
"tactic": "rw [uniformity_eq_symm, mem_nhds_uniformity_iff_right]"
},
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.58164\ninst✝ : UniformSpace α\nx : α\ns : Set α\n⊢ {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α ↔ {p | p.snd = x → p.fst ∈ s} ∈ Prod.swap <$> 𝓤 α",
"tactic": "rfl"
}
] |
[
709,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
706,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.intervalIntegral_pos_of_pos
|
[] |
[
1320,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1317,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.ofDual_lt_iff
|
[] |
[
901,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
899,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_eq_nnreal
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\n⊢ (⨆ (g : α →ₛ ℝ≥0∞) (_ : ↑g ≤ fun a => f a), SimpleFunc.lintegral g μ) =\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"state_before": "α : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\n⊢ (∫⁻ (a : α), f a ∂μ) =\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"tactic": "rw [lintegral]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\n⊢ SimpleFunc.lintegral φ μ ≤\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"state_before": "α : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\n⊢ (⨆ (g : α →ₛ ℝ≥0∞) (_ : ↑g ≤ fun a => f a), SimpleFunc.lintegral g μ) =\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"tactic": "refine'\n le_antisymm (iSup₂_le fun φ hφ => _) (iSup_mono' fun φ => ⟨φ.map ((↑) : ℝ≥0 → ℝ≥0∞), le_rfl⟩)"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\n⊢ SimpleFunc.lintegral φ μ ≤\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ\n\ncase neg\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\n⊢ SimpleFunc.lintegral φ μ ≤\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"state_before": "α : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\n⊢ SimpleFunc.lintegral φ μ ≤\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"tactic": "by_cases h : ∀ᵐ a ∂μ, φ a ≠ ∞"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nψ : α →ₛ ℝ≥0 := SimpleFunc.map ENNReal.toNNReal φ\n⊢ SimpleFunc.lintegral φ μ ≤\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\n⊢ SimpleFunc.lintegral φ μ ≤\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"tactic": "let ψ := φ.map ENNReal.toNNReal"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nψ : α →ₛ ℝ≥0 := SimpleFunc.map ENNReal.toNNReal φ\nh : ↑(SimpleFunc.map ENNReal.some ψ) =ᵐ[μ] ↑φ\n⊢ SimpleFunc.lintegral φ μ ≤\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nψ : α →ₛ ℝ≥0 := SimpleFunc.map ENNReal.toNNReal φ\n⊢ SimpleFunc.lintegral φ μ ≤\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"tactic": "replace h : ψ.map ((↑) : ℝ≥0 → ℝ≥0∞) =ᵐ[μ] φ := h.mono fun a => ENNReal.coe_toNNReal"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nψ : α →ₛ ℝ≥0 := SimpleFunc.map ENNReal.toNNReal φ\nh : ↑(SimpleFunc.map ENNReal.some ψ) =ᵐ[μ] ↑φ\nthis : ∀ (x : α), ↑(↑ψ x) ≤ f x\n⊢ SimpleFunc.lintegral φ μ ≤\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nψ : α →ₛ ℝ≥0 := SimpleFunc.map ENNReal.toNNReal φ\nh : ↑(SimpleFunc.map ENNReal.some ψ) =ᵐ[μ] ↑φ\n⊢ SimpleFunc.lintegral φ μ ≤\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"tactic": "have : ∀ x, ↑(ψ x) ≤ f x := fun x => le_trans ENNReal.coe_toNNReal_le_self (hφ x)"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nψ : α →ₛ ℝ≥0 := SimpleFunc.map ENNReal.toNNReal φ\nh : ↑(SimpleFunc.map ENNReal.some ψ) =ᵐ[μ] ↑φ\nthis : ∀ (x : α), ↑(↑ψ x) ≤ f x\n⊢ SimpleFunc.lintegral φ μ ≤\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"tactic": "exact\n le_iSup_of_le (φ.map ENNReal.toNNReal) (le_iSup_of_le this (ge_of_eq <| lintegral_congr h))"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\n⊢ SimpleFunc.lintegral φ μ ≤\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\n⊢ SimpleFunc.lintegral φ μ ≤\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"tactic": "have h_meas : μ (φ ⁻¹' {∞}) ≠ 0 := mt measure_zero_iff_ae_nmem.1 h"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\n⊢ ∃ i, b < ⨆ (_ : ∀ (x : α), ↑(↑i x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\n⊢ SimpleFunc.lintegral φ μ ≤\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"tactic": "refine' le_trans le_top (ge_of_eq <| (iSup_eq_top _).2 fun b hb => _)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\n⊢ ∃ n, b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})\n\ncase neg.intro\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\nn : ℕ\nhn : b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})\n⊢ ∃ i, b < ⨆ (_ : ∀ (x : α), ↑(↑i x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\n⊢ ∃ i, b < ⨆ (_ : ∀ (x : α), ↑(↑i x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ",
"tactic": "obtain ⟨n, hn⟩ : ∃ n : ℕ, b < n * μ (φ ⁻¹' {∞})"
},
{
"state_after": "case neg.intro\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\nn : ℕ\nhn : b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})\n⊢ ∃ i, b < ⨆ (_ : ∀ (x : α), ↑(↑i x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ",
"state_before": "α : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\n⊢ ∃ n, b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})\n\ncase neg.intro\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\nn : ℕ\nhn : b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})\n⊢ ∃ i, b < ⨆ (_ : ∀ (x : α), ↑(↑i x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ",
"tactic": "exact exists_nat_mul_gt h_meas (ne_of_lt hb)"
},
{
"state_after": "case neg.intro\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\nn : ℕ\nhn : b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})\n⊢ b <\n ⨆ (_ : ∀ (x : α), ↑(↑(restrict (const α ↑n) (↑φ ⁻¹' {⊤})) x) ≤ f x),\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (restrict (const α ↑n) (↑φ ⁻¹' {⊤}))) μ",
"state_before": "case neg.intro\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\nn : ℕ\nhn : b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})\n⊢ ∃ i, b < ⨆ (_ : ∀ (x : α), ↑(↑i x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ",
"tactic": "use (const α (n : ℝ≥0)).restrict (φ ⁻¹' {∞})"
},
{
"state_after": "case neg.intro\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\nn : ℕ\nhn : b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})\n⊢ (∀ (x : α), indicator (↑φ ⁻¹' {⊤}) (fun x => ↑(Function.const α (↑n) x)) x ≤ f x) ∧ b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})",
"state_before": "case neg.intro\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\nn : ℕ\nhn : b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})\n⊢ b <\n ⨆ (_ : ∀ (x : α), ↑(↑(restrict (const α ↑n) (↑φ ⁻¹' {⊤})) x) ≤ f x),\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (restrict (const α ↑n) (↑φ ⁻¹' {⊤}))) μ",
"tactic": "simp only [lt_iSup_iff, exists_prop, coe_restrict, φ.measurableSet_preimage, coe_const,\n ENNReal.coe_indicator, map_coe_ennreal_restrict, SimpleFunc.map_const, ENNReal.coe_nat,\n restrict_const_lintegral]"
},
{
"state_after": "case neg.intro\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\nn : ℕ\nhn : b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})\nx : α\nhx : x ∈ ↑φ ⁻¹' {⊤}\n⊢ ↑(Function.const α (↑n) x) ≤ ↑φ x",
"state_before": "case neg.intro\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\nn : ℕ\nhn : b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})\n⊢ (∀ (x : α), indicator (↑φ ⁻¹' {⊤}) (fun x => ↑(Function.const α (↑n) x)) x ≤ f x) ∧ b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})",
"tactic": "refine' ⟨indicator_le fun x hx => le_trans _ (hφ _), hn⟩"
},
{
"state_after": "case neg.intro\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\nn : ℕ\nhn : b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})\nx : α\nhx : ↑φ x = ⊤\n⊢ ↑(Function.const α (↑n) x) ≤ ↑φ x",
"state_before": "case neg.intro\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\nn : ℕ\nhn : b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})\nx : α\nhx : x ∈ ↑φ ⁻¹' {⊤}\n⊢ ↑(Function.const α (↑n) x) ≤ ↑φ x",
"tactic": "simp only [mem_preimage, mem_singleton_iff] at hx"
},
{
"state_after": "no goals",
"state_before": "case neg.intro\nα : Type u_1\nβ : Type ?u.80536\nγ : Type ?u.80539\nδ : Type ?u.80542\nm✝ : MeasurableSpace α\nμ✝ ν : Measure α\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ↑φ ≤ fun a => f a\nh : ¬∀ᵐ (a : α) ∂μ, ↑φ a ≠ ⊤\nh_meas : ↑↑μ (↑φ ⁻¹' {⊤}) ≠ 0\nb : ℝ≥0∞\nhb : b < ⊤\nn : ℕ\nhn : b < ↑n * ↑↑μ (↑φ ⁻¹' {⊤})\nx : α\nhx : ↑φ x = ⊤\n⊢ ↑(Function.const α (↑n) x) ≤ ↑φ x",
"tactic": "simp only [hx, le_top]"
}
] |
[
249,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.map_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_3\nR₁ : Type ?u.482742\nR₂ : Type u_2\nR₃ : Type ?u.482748\nR₄ : Type ?u.482751\nS : Type ?u.482754\nK : Type ?u.482757\nK₂ : Type ?u.482760\nM : Type u_4\nM' : Type ?u.482766\nM₁ : Type ?u.482769\nM₂ : Type u_1\nM₃ : Type ?u.482775\nM₄ : Type ?u.482778\nN : Type ?u.482781\nN₂ : Type ?u.482784\nι : Type ?u.482787\nV : Type ?u.482790\nV₂ : Type ?u.482793\ninst✝¹⁴ : Semiring R\ninst✝¹³ : Semiring R₂\ninst✝¹² : Semiring R₃\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M'\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\ninst✝⁵ : Module R₂ M₂\ninst✝⁴ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₂₁ : R₂ →+* R\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\np p' : Submodule R M\nq q' : Submodule R₂ M₂\nq₁ q₁' : Submodule R M'\nr : R\nx y : M\ninst✝ : RingHomSurjective σ₁₂\nF : Type ?u.483374\nsc : SemilinearMapClass F σ₁₂ M M₂\nthis : ∃ x, x ∈ p\n⊢ ∀ (x : M₂), x ∈ map 0 p ↔ x ∈ ⊥",
"tactic": "simp [this, eq_comm]"
}
] |
[
741,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
739,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.inf_id_set_eq_sInter
|
[] |
[
750,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
749,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.two_mul_odd_div_two
|
[
{
"state_after": "no goals",
"state_before": "m n k l : ℕ\nhn : n % 2 = 1\n⊢ 2 * (n / 2) = n - 1",
"tactic": "conv =>\n rhs\n rw [← Nat.mod_add_div n 2, hn, @add_tsub_cancel_left]"
}
] |
[
456,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
453,
1
] |
Mathlib/Data/Fintype/Card.lean
|
Fintype.card_lt_of_injective_not_surjective
|
[] |
[
482,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
479,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.not_mem_bot
|
[] |
[
728,
5
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
727,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.compl_def
|
[] |
[
1624,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1623,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.map_lowerCentralSeries_le
|
[
{
"state_after": "case zero\nR : Type u\nL : Type v\nM : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\nM₂ : Type w₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nf : M →ₗ⁅R,L⁆ M₂\n⊢ LieSubmodule.map f (lowerCentralSeries R L M Nat.zero) ≤ lowerCentralSeries R L M₂ Nat.zero\n\ncase succ\nR : Type u\nL : Type v\nM : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nM₂ : Type w₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nf : M →ₗ⁅R,L⁆ M₂\nk : ℕ\nih : LieSubmodule.map f (lowerCentralSeries R L M k) ≤ lowerCentralSeries R L M₂ k\n⊢ LieSubmodule.map f (lowerCentralSeries R L M (Nat.succ k)) ≤ lowerCentralSeries R L M₂ (Nat.succ k)",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nM₂ : Type w₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nk : ℕ\nf : M →ₗ⁅R,L⁆ M₂\n⊢ LieSubmodule.map f (lowerCentralSeries R L M k) ≤ lowerCentralSeries R L M₂ k",
"tactic": "induction' k with k ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\nL : Type v\nM : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\nM₂ : Type w₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nf : M →ₗ⁅R,L⁆ M₂\n⊢ LieSubmodule.map f (lowerCentralSeries R L M Nat.zero) ≤ lowerCentralSeries R L M₂ Nat.zero",
"tactic": "simp only [Nat.zero_eq, lowerCentralSeries_zero, le_top]"
},
{
"state_after": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nM₂ : Type w₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nf : M →ₗ⁅R,L⁆ M₂\nk : ℕ\nih : LieSubmodule.map f (lowerCentralSeries R L M k) ≤ lowerCentralSeries R L M₂ k\n⊢ ⁅⊤, LieSubmodule.map f (lowerCentralSeries R L M k)⁆ ≤ ⁅⊤, lowerCentralSeries R L M₂ k⁆",
"state_before": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nM₂ : Type w₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nf : M →ₗ⁅R,L⁆ M₂\nk : ℕ\nih : LieSubmodule.map f (lowerCentralSeries R L M k) ≤ lowerCentralSeries R L M₂ k\n⊢ LieSubmodule.map f (lowerCentralSeries R L M (Nat.succ k)) ≤ lowerCentralSeries R L M₂ (Nat.succ k)",
"tactic": "simp only [LieModule.lowerCentralSeries_succ, LieSubmodule.map_bracket_eq]"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nM₂ : Type w₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nf : M →ₗ⁅R,L⁆ M₂\nk : ℕ\nih : LieSubmodule.map f (lowerCentralSeries R L M k) ≤ lowerCentralSeries R L M₂ k\n⊢ ⁅⊤, LieSubmodule.map f (lowerCentralSeries R L M k)⁆ ≤ ⁅⊤, lowerCentralSeries R L M₂ k⁆",
"tactic": "exact LieSubmodule.mono_lie_right _ _ ⊤ ih"
}
] |
[
168,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.natDegree_eq_of_degree_eq
|
[
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\np q✝ r : R[X]\ninst✝ : Semiring S\nq : S[X]\nh : degree p = degree q\n⊢ WithBot.unbot' 0 (degree p) = WithBot.unbot' 0 (degree q)",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\np q✝ r : R[X]\ninst✝ : Semiring S\nq : S[X]\nh : degree p = degree q\n⊢ natDegree p = natDegree q",
"tactic": "unfold natDegree"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\np q✝ r : R[X]\ninst✝ : Semiring S\nq : S[X]\nh : degree p = degree q\n⊢ WithBot.unbot' 0 (degree p) = WithBot.unbot' 0 (degree q)",
"tactic": "rw [h]"
}
] |
[
166,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tsum_sigma'
|
[] |
[
811,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
809,
11
] |
Mathlib/Order/Heyting/Hom.lean
|
CoheytingHom.comp_assoc
|
[] |
[
470,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
468,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
HasFDerivAtFilter.const_sub
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.531341\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.531436\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nhf : HasFDerivAtFilter f f' x L\nc : F\n⊢ HasFDerivAtFilter (fun x => c - f x) (-f') x L",
"tactic": "simpa only [sub_eq_add_neg] using hf.neg.const_add c"
}
] |
[
605,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
603,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
LinearMap.mkContinuous₂_norm_le
|
[] |
[
768,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
766,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Basic.lean
|
Pmf.tsum_coe_indicator_ne_top
|
[] |
[
83,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Topology/Sets/Opens.lean
|
TopologicalSpace.Opens.nonempty_coe
|
[] |
[
108,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
11
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
csSup_image2_eq_csInf_csSup
|
[] |
[
1400,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1397,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.coe_fn_injective
|
[] |
[
113,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean
|
CategoryTheory.MonoidalCategory.unitors_inv_equal
|
[
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\n⊢ (λ_ (𝟙_ C)).inv = (ρ_ (𝟙_ C)).inv",
"tactic": "coherence"
}
] |
[
76,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.comap_neg_atTop
|
[] |
[
844,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
843,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
mem_vectorSpan_pair
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type u_1\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.372847\np₁ p₂ : P\nv : V\n⊢ v ∈ vectorSpan k {p₁, p₂} ↔ ∃ r, r • (p₁ -ᵥ p₂) = v",
"tactic": "rw [vectorSpan_pair, Submodule.mem_span_singleton]"
}
] |
[
1282,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1280,
1
] |
Mathlib/Algebra/Parity.lean
|
Even.isSquare_zpow
|
[
{
"state_after": "case intro\nF : Type ?u.50615\nα : Type u_1\nβ : Type ?u.50621\nR : Type ?u.50624\ninst✝ : Group α\nn : ℤ\na : α\n⊢ IsSquare (a ^ (n + n))",
"state_before": "F : Type ?u.50615\nα : Type u_1\nβ : Type ?u.50621\nR : Type ?u.50624\ninst✝ : Group α\nn : ℤ\n⊢ Even n → ∀ (a : α), IsSquare (a ^ n)",
"tactic": "rintro ⟨n, rfl⟩ a"
},
{
"state_after": "no goals",
"state_before": "case intro\nF : Type ?u.50615\nα : Type u_1\nβ : Type ?u.50621\nR : Type ?u.50624\ninst✝ : Group α\nn : ℤ\na : α\n⊢ IsSquare (a ^ (n + n))",
"tactic": "exact ⟨a ^ n, zpow_add _ _ _⟩"
}
] |
[
218,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.multiset_prod_mem
|
[] |
[
175,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
11
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.aleph0_le_mul_iff
|
[
{
"state_after": "α β : Type u\na b : Cardinal\nh : ¬a * b < ℵ₀ ↔ ¬(a = 0 ∨ b = 0 ∨ a < ℵ₀ ∧ b < ℵ₀) := Iff.not mul_lt_aleph0_iff\n⊢ ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ b ≠ 0 ∧ (ℵ₀ ≤ a ∨ ℵ₀ ≤ b)",
"state_before": "α β : Type u\na b : Cardinal\n⊢ ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ b ≠ 0 ∧ (ℵ₀ ≤ a ∨ ℵ₀ ≤ b)",
"tactic": "let h := (@mul_lt_aleph0_iff a b).not"
},
{
"state_after": "no goals",
"state_before": "α β : Type u\na b : Cardinal\nh : ¬a * b < ℵ₀ ↔ ¬(a = 0 ∨ b = 0 ∨ a < ℵ₀ ∧ b < ℵ₀) := Iff.not mul_lt_aleph0_iff\n⊢ ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ b ≠ 0 ∧ (ℵ₀ ≤ a ∨ ℵ₀ ≤ b)",
"tactic": "rwa [not_lt, not_or, not_or, not_and_or, not_lt, not_lt] at h"
}
] |
[
1579,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1577,
1
] |
Mathlib/Logic/Equiv/TransferInstance.lean
|
Equiv.ringEquiv_apply
|
[] |
[
183,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Init/Logic.lean
|
cast_proof_irrel
|
[] |
[
50,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.pow_ne_one_of_pos_of_lt
|
[] |
[
373,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
NNReal.tsum_le_of_sum_range_le
|
[] |
[
1156,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1154,
1
] |
Mathlib/Data/Set/Sigma.lean
|
Set.preimage_image_sigmaMk_of_ne
|
[
{
"state_after": "case h\nι : Type u_1\nι' : Type ?u.678\nα : ι → Type u_2\nβ : ι → Type ?u.688\ns✝ s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\nu : Set ((i : ι) × α i)\nx✝ : (i : ι) × α i\ni j : ι\na : α i\nh : i ≠ j\ns : Set (α j)\nx : α i\n⊢ x ∈ Sigma.mk i ⁻¹' (Sigma.mk j '' s) ↔ x ∈ ∅",
"state_before": "ι : Type u_1\nι' : Type ?u.678\nα : ι → Type u_2\nβ : ι → Type ?u.688\ns✝ s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\nu : Set ((i : ι) × α i)\nx : (i : ι) × α i\ni j : ι\na : α i\nh : i ≠ j\ns : Set (α j)\n⊢ Sigma.mk i ⁻¹' (Sigma.mk j '' s) = ∅",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u_1\nι' : Type ?u.678\nα : ι → Type u_2\nβ : ι → Type ?u.688\ns✝ s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\nu : Set ((i : ι) × α i)\nx✝ : (i : ι) × α i\ni j : ι\na : α i\nh : i ≠ j\ns : Set (α j)\nx : α i\n⊢ x ∈ Sigma.mk i ⁻¹' (Sigma.mk j '' s) ↔ x ∈ ∅",
"tactic": "simp [h.symm]"
}
] |
[
36,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
33,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.update_eq_erase_add_single
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.300562\nγ : Type ?u.300565\nι : Type ?u.300568\nM : Type u_2\nM' : Type ?u.300574\nN : Type ?u.300577\nP : Type ?u.300580\nG : Type ?u.300583\nH : Type ?u.300586\nR : Type ?u.300589\nS : Type ?u.300592\ninst✝ : AddZeroClass M\nf : α →₀ M\na : α\nb : M\nj : α\n⊢ ↑(update f a b) j = ↑(erase a f + single a b) j",
"state_before": "α : Type u_1\nβ : Type ?u.300562\nγ : Type ?u.300565\nι : Type ?u.300568\nM : Type u_2\nM' : Type ?u.300574\nN : Type ?u.300577\nP : Type ?u.300580\nG : Type ?u.300583\nH : Type ?u.300586\nR : Type ?u.300589\nS : Type ?u.300592\ninst✝ : AddZeroClass M\nf : α →₀ M\na : α\nb : M\n⊢ update f a b = erase a f + single a b",
"tactic": "ext j"
},
{
"state_after": "case h.inl\nα : Type u_1\nβ : Type ?u.300562\nγ : Type ?u.300565\nι : Type ?u.300568\nM : Type u_2\nM' : Type ?u.300574\nN : Type ?u.300577\nP : Type ?u.300580\nG : Type ?u.300583\nH : Type ?u.300586\nR : Type ?u.300589\nS : Type ?u.300592\ninst✝ : AddZeroClass M\nf : α →₀ M\na : α\nb : M\n⊢ ↑(update f a b) a = ↑(erase a f + single a b) a\n\ncase h.inr\nα : Type u_1\nβ : Type ?u.300562\nγ : Type ?u.300565\nι : Type ?u.300568\nM : Type u_2\nM' : Type ?u.300574\nN : Type ?u.300577\nP : Type ?u.300580\nG : Type ?u.300583\nH : Type ?u.300586\nR : Type ?u.300589\nS : Type ?u.300592\ninst✝ : AddZeroClass M\nf : α →₀ M\na : α\nb : M\nj : α\nh : a ≠ j\n⊢ ↑(update f a b) j = ↑(erase a f + single a b) j",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.300562\nγ : Type ?u.300565\nι : Type ?u.300568\nM : Type u_2\nM' : Type ?u.300574\nN : Type ?u.300577\nP : Type ?u.300580\nG : Type ?u.300583\nH : Type ?u.300586\nR : Type ?u.300589\nS : Type ?u.300592\ninst✝ : AddZeroClass M\nf : α →₀ M\na : α\nb : M\nj : α\n⊢ ↑(update f a b) j = ↑(erase a f + single a b) j",
"tactic": "rcases eq_or_ne a j with (rfl | h)"
},
{
"state_after": "no goals",
"state_before": "case h.inl\nα : Type u_1\nβ : Type ?u.300562\nγ : Type ?u.300565\nι : Type ?u.300568\nM : Type u_2\nM' : Type ?u.300574\nN : Type ?u.300577\nP : Type ?u.300580\nG : Type ?u.300583\nH : Type ?u.300586\nR : Type ?u.300589\nS : Type ?u.300592\ninst✝ : AddZeroClass M\nf : α →₀ M\na : α\nb : M\n⊢ ↑(update f a b) a = ↑(erase a f + single a b) a",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h.inr\nα : Type u_1\nβ : Type ?u.300562\nγ : Type ?u.300565\nι : Type ?u.300568\nM : Type u_2\nM' : Type ?u.300574\nN : Type ?u.300577\nP : Type ?u.300580\nG : Type ?u.300583\nH : Type ?u.300586\nR : Type ?u.300589\nS : Type ?u.300592\ninst✝ : AddZeroClass M\nf : α →₀ M\na : α\nb : M\nj : α\nh : a ≠ j\n⊢ ↑(update f a b) j = ↑(erase a f + single a b) j",
"tactic": "simp [Function.update_noteq h.symm, single_apply, h, erase_ne, h.symm]"
}
] |
[
1056,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1050,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
|
Complex.cpow_neg_one
|
[
{
"state_after": "no goals",
"state_before": "x : ℂ\n⊢ x ^ (-1) = x⁻¹",
"tactic": "simpa using cpow_neg x 1"
}
] |
[
114,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/FieldTheory/Minpoly/Field.lean
|
minpoly.dvd
|
[
{
"state_after": "case pos\nA : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : p = 0\n⊢ minpoly A x ∣ p\n\ncase neg\nA : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : ¬p = 0\n⊢ minpoly A x ∣ p",
"state_before": "A : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\n⊢ minpoly A x ∣ p",
"tactic": "by_cases hp0 : p = 0"
},
{
"state_after": "case neg\nA : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : ¬p = 0\nhx : IsIntegral A x\n⊢ minpoly A x ∣ p",
"state_before": "case neg\nA : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : ¬p = 0\n⊢ minpoly A x ∣ p",
"tactic": "have hx : IsIntegral A x := by\n rw [← isAlgebraic_iff_isIntegral]\n exact ⟨p, hp0, hp⟩"
},
{
"state_after": "case neg\nA : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : ¬p = 0\nhx : IsIntegral A x\n⊢ p %ₘ minpoly A x = 0",
"state_before": "case neg\nA : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : ¬p = 0\nhx : IsIntegral A x\n⊢ minpoly A x ∣ p",
"tactic": "rw [← dvd_iff_modByMonic_eq_zero (monic hx)]"
},
{
"state_after": "case neg\nA : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : ¬p = 0\nhx : IsIntegral A x\nhnz : ¬p %ₘ minpoly A x = 0\n⊢ False",
"state_before": "case neg\nA : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : ¬p = 0\nhx : IsIntegral A x\n⊢ p %ₘ minpoly A x = 0",
"tactic": "by_contra hnz"
},
{
"state_after": "case neg\nA : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : ¬p = 0\nhx : IsIntegral A x\nhnz : ¬p %ₘ minpoly A x = 0\nhd : degree (minpoly A x) ≤ degree (p %ₘ minpoly A x)\n⊢ False",
"state_before": "case neg\nA : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : ¬p = 0\nhx : IsIntegral A x\nhnz : ¬p %ₘ minpoly A x = 0\n⊢ False",
"tactic": "have hd := degree_le_of_ne_zero A x hnz\n ((aeval_modByMonic_eq_self_of_root (monic hx) (aeval _ _)).trans hp)"
},
{
"state_after": "case neg\nA : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : ¬p = 0\nhx : IsIntegral A x\nhnz : ¬p %ₘ minpoly A x = 0\nhd : ¬False\n⊢ degree (p %ₘ minpoly A x) < degree (minpoly A x)",
"state_before": "case neg\nA : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : ¬p = 0\nhx : IsIntegral A x\nhnz : ¬p %ₘ minpoly A x = 0\nhd : degree (minpoly A x) ≤ degree (p %ₘ minpoly A x)\n⊢ False",
"tactic": "contrapose! hd"
},
{
"state_after": "no goals",
"state_before": "case neg\nA : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : ¬p = 0\nhx : IsIntegral A x\nhnz : ¬p %ₘ minpoly A x = 0\nhd : ¬False\n⊢ degree (p %ₘ minpoly A x) < degree (minpoly A x)",
"tactic": "exact degree_modByMonic_lt _ (monic hx)"
},
{
"state_after": "no goals",
"state_before": "case pos\nA : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : p = 0\n⊢ minpoly A x ∣ p",
"tactic": "simp only [hp0, dvd_zero]"
},
{
"state_after": "A : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : ¬p = 0\n⊢ IsAlgebraic A x",
"state_before": "A : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : ¬p = 0\n⊢ IsIntegral A x",
"tactic": "rw [← isAlgebraic_iff_isIntegral]"
},
{
"state_after": "no goals",
"state_before": "A : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nhp : ↑(Polynomial.aeval x) p = 0\nhp0 : ¬p = 0\n⊢ IsAlgebraic A x",
"tactic": "exact ⟨p, hp0, hp⟩"
}
] |
[
83,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
mul_le_mul_of_nonpos_of_nonpos
|
[] |
[
385,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.drop_drop
|
[
{
"state_after": "case a\nα : Type u\nβ : Type v\nδ : Type w\nn m : ℕ\ns : Stream' α\nn✝ : ℕ\n⊢ nth (drop n (drop m s)) n✝ = nth (drop (n + m) s) n✝",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\nn m : ℕ\ns : Stream' α\n⊢ drop n (drop m s) = drop (n + m) s",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\nδ : Type w\nn m : ℕ\ns : Stream' α\nn✝ : ℕ\n⊢ nth (drop n (drop m s)) n✝ = nth (drop (n + m) s) n✝",
"tactic": "simp [Nat.add_assoc]"
}
] |
[
68,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Topology/Maps.lean
|
OpenEmbedding.map_nhds_eq
|
[] |
[
558,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
556,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.pointed_zero
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.98911\nG : Type ?u.98914\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\n⊢ Pointed 0",
"tactic": "rw [Pointed, mem_zero]"
}
] |
[
460,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
460,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Complex.sin_add_nat_mul_two_pi
|
[] |
[
1177,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1176,
1
] |
Mathlib/Order/Partition/Finpartition.lean
|
Finpartition.exists_le_of_le
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nH : ¬∃ c, c ∈ P.parts ∧ c ≤ b\n⊢ False",
"state_before": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\n⊢ ∃ c, c ∈ P.parts ∧ c ≤ b",
"tactic": "by_contra H"
},
{
"state_after": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nH : ¬∃ c, c ∈ P.parts ∧ c ≤ b\n⊢ Disjoint b a",
"state_before": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nH : ¬∃ c, c ∈ P.parts ∧ c ≤ b\n⊢ False",
"tactic": "refine' Q.ne_bot hb (disjoint_self.1 <| Disjoint.mono_right (Q.le hb) _)"
},
{
"state_after": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nH : ¬∃ c, c ∈ P.parts ∧ c ≤ b\n⊢ ∀ ⦃i : α⦄, i ∈ P.parts → Disjoint b (id i)",
"state_before": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nH : ¬∃ c, c ∈ P.parts ∧ c ≤ b\n⊢ Disjoint b a",
"tactic": "rw [← P.supParts, Finset.disjoint_sup_right]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nH : ¬∃ c, c ∈ P.parts ∧ c ≤ b\nc : α\nhc : c ∈ P.parts\n⊢ Disjoint b (id c)",
"state_before": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nH : ¬∃ c, c ∈ P.parts ∧ c ≤ b\n⊢ ∀ ⦃i : α⦄, i ∈ P.parts → Disjoint b (id i)",
"tactic": "rintro c hc"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nH : ¬∃ c, c ∈ P.parts ∧ c ≤ b\nc : α\nhc : c ∈ P.parts\nd : α\nhd : d ∈ Q.parts\nhcd : c ≤ d\n⊢ Disjoint b (id c)",
"state_before": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nH : ¬∃ c, c ∈ P.parts ∧ c ≤ b\nc : α\nhc : c ∈ P.parts\n⊢ Disjoint b (id c)",
"tactic": "obtain ⟨d, hd, hcd⟩ := h hc"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nH : ¬∃ c, c ∈ P.parts ∧ c ≤ b\nc : α\nhc : c ∈ P.parts\nd : α\nhd : d ∈ Q.parts\nhcd : c ≤ d\n⊢ b ≠ d",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nH : ¬∃ c, c ∈ P.parts ∧ c ≤ b\nc : α\nhc : c ∈ P.parts\nd : α\nhd : d ∈ Q.parts\nhcd : c ≤ d\n⊢ Disjoint b (id c)",
"tactic": "refine' (Q.disjoint hb hd _).mono_right hcd"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nH : ¬∃ c, c ∈ P.parts ∧ c ≤ b\nc : α\nhc : c ∈ P.parts\nhd : b ∈ Q.parts\nhcd : c ≤ b\n⊢ False",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nH : ¬∃ c, c ∈ P.parts ∧ c ≤ b\nc : α\nhc : c ∈ P.parts\nd : α\nhd : d ∈ Q.parts\nhcd : c ≤ d\n⊢ b ≠ d",
"tactic": "rintro rfl"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nc : α\nhc : c ∈ P.parts\nhd : b ∈ Q.parts\nhcd : c ≤ b\nH : ∀ (x : α), x ∈ P.parts → ¬x ≤ b\n⊢ False",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nH : ¬∃ c, c ∈ P.parts ∧ c ≤ b\nc : α\nhc : c ∈ P.parts\nhd : b ∈ Q.parts\nhcd : c ≤ b\n⊢ False",
"tactic": "simp only [not_exists, not_and] at H"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b : α\nP Q : Finpartition a\nh : P ≤ Q\nhb : b ∈ Q.parts\nc : α\nhc : c ∈ P.parts\nhd : b ∈ Q.parts\nhcd : c ≤ b\nH : ∀ (x : α), x ∈ P.parts → ¬x ≤ b\n⊢ False",
"tactic": "exact H _ hc hcd"
}
] |
[
340,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/Data/List/Basic.lean
|
List.Sublist.map
|
[] |
[
3412,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3411,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.toDual_min'
|
[
{
"state_after": "F : Type ?u.354773\nα : Type u_1\nβ : Type ?u.354779\nγ : Type ?u.354782\nι : Type ?u.354785\nκ : Type ?u.354788\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\nhs : Finset.Nonempty s\n⊢ ↑(↑toDual (min' s hs)) = ↑(max' (image (↑toDual) s) (_ : Finset.Nonempty (image (↑toDual) s)))",
"state_before": "F : Type ?u.354773\nα : Type u_1\nβ : Type ?u.354779\nγ : Type ?u.354782\nι : Type ?u.354785\nκ : Type ?u.354788\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\nhs : Finset.Nonempty s\n⊢ ↑toDual (min' s hs) = max' (image (↑toDual) s) (_ : Finset.Nonempty (image (↑toDual) s))",
"tactic": "rw [← WithBot.coe_eq_coe]"
},
{
"state_after": "F : Type ?u.354773\nα : Type u_1\nβ : Type ?u.354779\nγ : Type ?u.354782\nι : Type ?u.354785\nκ : Type ?u.354788\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\nhs : Finset.Nonempty s\n⊢ sup s (WithBot.some ∘ fun x => ↑toDual x) = sup s ((WithBot.some ∘ fun x => x) ∘ ↑toDual)",
"state_before": "F : Type ?u.354773\nα : Type u_1\nβ : Type ?u.354779\nγ : Type ?u.354782\nι : Type ?u.354785\nκ : Type ?u.354788\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\nhs : Finset.Nonempty s\n⊢ ↑(↑toDual (min' s hs)) = ↑(max' (image (↑toDual) s) (_ : Finset.Nonempty (image (↑toDual) s)))",
"tactic": "simp only [min'_eq_inf', id_eq, toDual_inf', Function.comp_apply, coe_sup', max'_eq_sup',\n sup_image]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.354773\nα : Type u_1\nβ : Type ?u.354779\nγ : Type ?u.354782\nι : Type ?u.354785\nκ : Type ?u.354788\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\nhs : Finset.Nonempty s\n⊢ sup s (WithBot.some ∘ fun x => ↑toDual x) = sup s ((WithBot.some ∘ fun x => x) ∘ ↑toDual)",
"tactic": "rfl"
}
] |
[
1452,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1447,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
|
EuclideanGeometry.sin_eq_zero_iff_angle_eq_zero_or_angle_eq_pi
|
[] |
[
501,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
499,
8
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.ofReal_le_iff_le_toReal
|
[
{
"state_after": "case intro\nα : Type ?u.809256\nβ : Type ?u.809259\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na : ℝ\nb : ℝ≥0\n⊢ ENNReal.ofReal a ≤ ↑b ↔ a ≤ ENNReal.toReal ↑b",
"state_before": "α : Type ?u.809256\nβ : Type ?u.809259\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na : ℝ\nb : ℝ≥0∞\nhb : b ≠ ⊤\n⊢ ENNReal.ofReal a ≤ b ↔ a ≤ ENNReal.toReal b",
"tactic": "lift b to ℝ≥0 using hb"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type ?u.809256\nβ : Type ?u.809259\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na : ℝ\nb : ℝ≥0\n⊢ ENNReal.ofReal a ≤ ↑b ↔ a ≤ ENNReal.toReal ↑b",
"tactic": "simpa [ENNReal.ofReal, ENNReal.toReal] using Real.toNNReal_le_iff_le_coe"
}
] |
[
2140,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2137,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.apply_lineMap
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type ?u.494433\nP3 : Type ?u.494436\nV4 : Type ?u.494439\nP4 : Type ?u.494442\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : P1 →ᵃ[k] P2\np₀ p₁ : P1\nc : k\n⊢ ↑f (↑(lineMap p₀ p₁) c) = ↑(lineMap (↑f p₀) (↑f p₁)) c",
"tactic": "simp [(lineMap_apply), (map_vadd), (linearMap_vsub)]"
}
] |
[
603,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
600,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.limsInf_principal
|
[] |
[
567,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
566,
1
] |
Mathlib/MeasureTheory/PiSystem.lean
|
MeasurableSpace.DynkinSystem.has_compl_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nd : DynkinSystem α\na : Set α\nh : Has d (aᶜ)\n⊢ Has d a",
"tactic": "simpa using d.has_compl h"
}
] |
[
569,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
568,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.le_toMeasure_apply
|
[] |
[
676,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
674,
1
] |
Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
|
DiscreteValuationRing.addVal_zero
|
[] |
[
422,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean
|
FormalMultilinearSeries.order_eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\n𝕜' : Type u'\nE : Type v\nF : Type w\nG : Type x\ninst✝¹⁰ : CommRing 𝕜\nn : ℕ\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : Module 𝕜 E\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : TopologicalAddGroup E\ninst✝⁵ : ContinuousConstSMul 𝕜 E\ninst✝⁴ : AddCommGroup F\ninst✝³ : Module 𝕜 F\ninst✝² : TopologicalSpace F\ninst✝¹ : TopologicalAddGroup F\ninst✝ : ContinuousConstSMul 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nhp : p ≠ 0\n⊢ order p = 0 ↔ p 0 ≠ 0",
"tactic": "simp [order_eq_zero_iff', hp]"
}
] |
[
244,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/CategoryTheory/Filtered.lean
|
CategoryTheory.IsCofiltered.of_equivalence
|
[] |
[
746,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
745,
1
] |
Mathlib/FieldTheory/Finite/Basic.lean
|
ZMod.pow_totient
|
[
{
"state_after": "case zero\nK : Type ?u.893582\nR : Type ?u.893585\nx : (ZMod Nat.zero)ˣ\n⊢ x ^ φ Nat.zero = 1\n\ncase succ\nK : Type ?u.893582\nR : Type ?u.893585\nn✝ : ℕ\nx : (ZMod (Nat.succ n✝))ˣ\n⊢ x ^ φ (Nat.succ n✝) = 1",
"state_before": "K : Type ?u.893582\nR : Type ?u.893585\nn : ℕ\nx : (ZMod n)ˣ\n⊢ x ^ φ n = 1",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case zero\nK : Type ?u.893582\nR : Type ?u.893585\nx : (ZMod Nat.zero)ˣ\n⊢ x ^ φ Nat.zero = 1",
"tactic": "rw [Nat.totient_zero, pow_zero]"
},
{
"state_after": "no goals",
"state_before": "case succ\nK : Type ?u.893582\nR : Type ?u.893585\nn✝ : ℕ\nx : (ZMod (Nat.succ n✝))ˣ\n⊢ x ^ φ (Nat.succ n✝) = 1",
"tactic": "rw [← card_units_eq_totient, pow_card_eq_one]"
}
] |
[
371,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
368,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_neg_Ioo
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ -Ioo a b = Ioo (-b) (-a)",
"tactic": "simp [← Ioi_inter_Iio, inter_comm]"
}
] |
[
167,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.le_eq_subset
|
[] |
[
123,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Algebra/Lie/OfAssociative.lean
|
LieSubmodule.toEndomorphism_restrict_eq_toEndomorphism
|
[
{
"state_after": "case h.a\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nx : L\nh :\n optParam\n (∀ (m : M) (hm : m ∈ ↑N),\n ↑(LinearMap.comp (↑(toEndomorphism R L M) x) (Submodule.subtype ↑N)) { val := m, property := hm } ∈ ↑N)\n (_ :\n ∀ (m : M) (hm : m ∈ ↑N),\n ↑(LinearMap.comp (↑(toEndomorphism R L M) x) (Submodule.subtype ↑N)) { val := m, property := hm } ∈ ↑N)\nx✝ : { x // x ∈ ↑N }\n⊢ ↑(↑(LinearMap.restrict (↑(toEndomorphism R L M) x) h) x✝) = ↑(↑(↑(toEndomorphism R L { x // x ∈ ↑N }) x) x✝)",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nx : L\nh :\n optParam\n (∀ (m : M) (hm : m ∈ ↑N),\n ↑(LinearMap.comp (↑(toEndomorphism R L M) x) (Submodule.subtype ↑N)) { val := m, property := hm } ∈ ↑N)\n (_ :\n ∀ (m : M) (hm : m ∈ ↑N),\n ↑(LinearMap.comp (↑(toEndomorphism R L M) x) (Submodule.subtype ↑N)) { val := m, property := hm } ∈ ↑N)\n⊢ LinearMap.restrict (↑(toEndomorphism R L M) x) h = ↑(toEndomorphism R L { x // x ∈ ↑N }) x",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.a\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nx : L\nh :\n optParam\n (∀ (m : M) (hm : m ∈ ↑N),\n ↑(LinearMap.comp (↑(toEndomorphism R L M) x) (Submodule.subtype ↑N)) { val := m, property := hm } ∈ ↑N)\n (_ :\n ∀ (m : M) (hm : m ∈ ↑N),\n ↑(LinearMap.comp (↑(toEndomorphism R L M) x) (Submodule.subtype ↑N)) { val := m, property := hm } ∈ ↑N)\nx✝ : { x // x ∈ ↑N }\n⊢ ↑(↑(LinearMap.restrict (↑(toEndomorphism R L M) x) h) x✝) = ↑(↑(↑(toEndomorphism R L { x // x ∈ ↑N }) x) x✝)",
"tactic": "simp [LinearMap.restrict_apply]"
}
] |
[
282,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Analysis/NormedSpace/Banach.lean
|
ContinuousLinearMap.isOpenMap
|
[
{
"state_after": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\n⊢ IsOpen (↑f '' s)",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\n⊢ IsOpenMap ↑f",
"tactic": "intro s hs"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\n⊢ IsOpen (↑f '' s)",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\n⊢ IsOpen (↑f '' s)",
"tactic": "rcases exists_preimage_norm_le f surj with ⟨C, Cpos, hC⟩"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\n⊢ ∃ ε, ε > 0 ∧ ball y ε ⊆ ↑f '' s",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\n⊢ IsOpen (↑f '' s)",
"tactic": "refine' isOpen_iff.2 fun y yfs => _"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\n⊢ ∃ ε, ε > 0 ∧ ball y ε ⊆ ↑f '' s",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\n⊢ ∃ ε, ε > 0 ∧ ball y ε ⊆ ↑f '' s",
"tactic": "rcases mem_image_iff_bex.1 yfs with ⟨x, xs, fxy⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\n⊢ ∃ ε, ε > 0 ∧ ball y ε ⊆ ↑f '' s",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\n⊢ ∃ ε, ε > 0 ∧ ball y ε ⊆ ↑f '' s",
"tactic": "rcases isOpen_iff.1 hs x xs with ⟨ε, εpos, hε⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\n⊢ z ∈ ↑f '' s",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\n⊢ ∃ ε, ε > 0 ∧ ball y ε ⊆ ↑f '' s",
"tactic": "refine' ⟨ε / C, div_pos εpos Cpos, fun z hz => _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\nw : E\nwim : ↑f w = z - y\nwnorm : ‖w‖ ≤ C * ‖z - y‖\n⊢ z ∈ ↑f '' s",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\n⊢ z ∈ ↑f '' s",
"tactic": "rcases hC (z - y) with ⟨w, wim, wnorm⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\nw : E\nwim : ↑f w = z - y\nwnorm : ‖w‖ ≤ C * ‖z - y‖\nthis : ↑f (x + w) = z\n⊢ z ∈ ↑f '' s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\nw : E\nwim : ↑f w = z - y\nwnorm : ‖w‖ ≤ C * ‖z - y‖\n⊢ z ∈ ↑f '' s",
"tactic": "have : f (x + w) = z := by rw [f.map_add, wim, fxy, add_sub_cancel'_right]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\nw : E\nwim : ↑f w = z - y\nwnorm : ‖w‖ ≤ C * ‖z - y‖\nthis : ↑f (x + w) = z\n⊢ ↑f (x + w) ∈ ↑f '' s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\nw : E\nwim : ↑f w = z - y\nwnorm : ‖w‖ ≤ C * ‖z - y‖\nthis : ↑f (x + w) = z\n⊢ z ∈ ↑f '' s",
"tactic": "rw [← this]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\nw : E\nwim : ↑f w = z - y\nwnorm : ‖w‖ ≤ C * ‖z - y‖\nthis✝ : ↑f (x + w) = z\nthis : x + w ∈ ball x ε\n⊢ ↑f (x + w) ∈ ↑f '' s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\nw : E\nwim : ↑f w = z - y\nwnorm : ‖w‖ ≤ C * ‖z - y‖\nthis : ↑f (x + w) = z\n⊢ ↑f (x + w) ∈ ↑f '' s",
"tactic": "have : x + w ∈ ball x ε :=\n calc\n dist (x + w) x = ‖w‖ := by\n rw [dist_eq_norm]\n simp\n _ ≤ C * ‖z - y‖ := wnorm\n _ < C * (ε / C) := by\n apply mul_lt_mul_of_pos_left _ Cpos\n rwa [mem_ball, dist_eq_norm] at hz\n _ = ε := mul_div_cancel' _ (ne_of_gt Cpos)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\nw : E\nwim : ↑f w = z - y\nwnorm : ‖w‖ ≤ C * ‖z - y‖\nthis✝ : ↑f (x + w) = z\nthis : x + w ∈ ball x ε\n⊢ ↑f (x + w) ∈ ↑f '' s",
"tactic": "exact Set.mem_image_of_mem _ (hε this)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\nw : E\nwim : ↑f w = z - y\nwnorm : ‖w‖ ≤ C * ‖z - y‖\n⊢ ↑f (x + w) = z",
"tactic": "rw [f.map_add, wim, fxy, add_sub_cancel'_right]"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\nw : E\nwim : ↑f w = z - y\nwnorm : ‖w‖ ≤ C * ‖z - y‖\nthis : ↑f (x + w) = z\n⊢ ‖x + w - x‖ = ‖w‖",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\nw : E\nwim : ↑f w = z - y\nwnorm : ‖w‖ ≤ C * ‖z - y‖\nthis : ↑f (x + w) = z\n⊢ dist (x + w) x = ‖w‖",
"tactic": "rw [dist_eq_norm]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\nw : E\nwim : ↑f w = z - y\nwnorm : ‖w‖ ≤ C * ‖z - y‖\nthis : ↑f (x + w) = z\n⊢ ‖x + w - x‖ = ‖w‖",
"tactic": "simp"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\nw : E\nwim : ↑f w = z - y\nwnorm : ‖w‖ ≤ C * ‖z - y‖\nthis : ↑f (x + w) = z\n⊢ ‖z - y‖ < ε / C",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\nw : E\nwim : ↑f w = z - y\nwnorm : ‖w‖ ≤ C * ‖z - y‖\nthis : ↑f (x + w) = z\n⊢ C * ‖z - y‖ < C * (ε / C)",
"tactic": "apply mul_lt_mul_of_pos_left _ Cpos"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\ns : Set E\nhs : IsOpen s\nC : ℝ\nCpos : C > 0\nhC : ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖\ny : F\nyfs : y ∈ ↑f '' s\nx : E\nxs : x ∈ s\nfxy : ↑f x = y\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ s\nz : F\nhz : z ∈ ball y (ε / C)\nw : E\nwim : ↑f w = z - y\nwnorm : ‖w‖ ≤ C * ‖z - y‖\nthis : ↑f (x + w) = z\n⊢ ‖z - y‖ < ε / C",
"tactic": "rwa [mem_ball, dist_eq_norm] at hz"
}
] |
[
262,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
11
] |
Mathlib/Order/UpperLower/Basic.lean
|
UpperSet.Ioi_top
|
[] |
[
1123,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1122,
8
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.distortion_disjUnion
|
[] |
[
694,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
692,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.