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/PNat/Basic.lean
|
PNat.mul_div_exact
|
[
{
"state_after": "case a\nm k : ℕ+\nh : k ∣ m\n⊢ ↑(k * divExact m k) = ↑m",
"state_before": "m k : ℕ+\nh : k ∣ m\n⊢ k * divExact m k = m",
"tactic": "apply PNat.eq"
},
{
"state_after": "case a\nm k : ℕ+\nh : k ∣ m\n⊢ ↑k * ↑(divExact m k) = ↑m",
"state_before": "case a\nm k : ℕ+\nh : k ∣ m\n⊢ ↑(k * divExact m k) = ↑m",
"tactic": "rw [mul_coe]"
},
{
"state_after": "case a\nm k : ℕ+\nh : k ∣ m\n⊢ ↑k * succ (div m k) = ↑m",
"state_before": "case a\nm k : ℕ+\nh : k ∣ m\n⊢ ↑k * ↑(divExact m k) = ↑m",
"tactic": "change (k : ℕ) * (div m k).succ = m"
},
{
"state_after": "no goals",
"state_before": "case a\nm k : ℕ+\nh : k ∣ m\n⊢ ↑k * succ (div m k) = ↑m",
"tactic": "rw [← div_add_mod m k, dvd_iff'.mp h, Nat.mul_succ]"
}
] |
[
435,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/Topology/Category/TopCat/Limits/Konig.lean
|
TopCat.partialSections.directed
|
[
{
"state_after": "J✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\n⊢ ∃ z,\n (fun G => partialSections F G.snd) A ⊇ (fun G => partialSections F G.snd) z ∧\n (fun G => partialSections F G.snd) B ⊇ (fun G => partialSections F G.snd) z",
"state_before": "J✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\n⊢ Directed Superset fun G => partialSections F G.snd",
"tactic": "intro A B"
},
{
"state_after": "J✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\n⊢ ∃ z,\n (fun G => partialSections F G.snd) A ⊇ (fun G => partialSections F G.snd) z ∧\n (fun G => partialSections F G.snd) B ⊇ (fun G => partialSections F G.snd) z",
"state_before": "J✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\n⊢ ∃ z,\n (fun G => partialSections F G.snd) A ⊇ (fun G => partialSections F G.snd) z ∧\n (fun G => partialSections F G.snd) B ⊇ (fun G => partialSections F G.snd) z",
"tactic": "let ιA : FiniteDiagramArrow A.1 → FiniteDiagramArrow (A.1 ⊔ B.1) := fun f =>\n ⟨f.1, f.2.1, Finset.mem_union_left _ f.2.2.1, Finset.mem_union_left _ f.2.2.2.1, f.2.2.2.2⟩"
},
{
"state_after": "J✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\n⊢ ∃ z,\n (fun G => partialSections F G.snd) A ⊇ (fun G => partialSections F G.snd) z ∧\n (fun G => partialSections F G.snd) B ⊇ (fun G => partialSections F G.snd) z",
"state_before": "J✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\n⊢ ∃ z,\n (fun G => partialSections F G.snd) A ⊇ (fun G => partialSections F G.snd) z ∧\n (fun G => partialSections F G.snd) B ⊇ (fun G => partialSections F G.snd) z",
"tactic": "let ιB : FiniteDiagramArrow B.1 → FiniteDiagramArrow (A.1 ⊔ B.1) := fun f =>\n ⟨f.1, f.2.1, Finset.mem_union_right _ f.2.2.1, Finset.mem_union_right _ f.2.2.2.1, f.2.2.2.2⟩"
},
{
"state_after": "case refine'_1\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\n⊢ (fun G => partialSections F G.snd) A ⊇\n (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\n\ncase refine'_2\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\n⊢ (fun G => partialSections F G.snd) B ⊇\n (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }",
"state_before": "J✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\n⊢ ∃ z,\n (fun G => partialSections F G.snd) A ⊇ (fun G => partialSections F G.snd) z ∧\n (fun G => partialSections F G.snd) B ⊇ (fun G => partialSections F G.snd) z",
"tactic": "refine' ⟨⟨A.1 ⊔ B.1, A.2.image ιA ⊔ B.2.image ιB⟩, _, _⟩"
},
{
"state_after": "case refine'_1\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow A.fst\nhf : f ∈ A.snd\n⊢ (forget TopCat).map (F.map f.snd.snd.snd.snd) (u f.fst) = u f.snd.fst",
"state_before": "case refine'_1\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\n⊢ (fun G => partialSections F G.snd) A ⊇\n (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }",
"tactic": "rintro u hu f hf"
},
{
"state_after": "case refine'_1\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow A.fst\nhf : f ∈ A.snd\nthis : ιA f ∈ Finset.image ιA A.snd ⊔ Finset.image ιB B.snd\n⊢ (forget TopCat).map (F.map f.snd.snd.snd.snd) (u f.fst) = u f.snd.fst",
"state_before": "case refine'_1\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow A.fst\nhf : f ∈ A.snd\n⊢ (forget TopCat).map (F.map f.snd.snd.snd.snd) (u f.fst) = u f.snd.fst",
"tactic": "have : ιA f ∈ A.2.image ιA ⊔ B.2.image ιB := by\n apply Finset.mem_union_left\n rw [Finset.mem_image]\n refine' ⟨f, hf, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow A.fst\nhf : f ∈ A.snd\nthis : ιA f ∈ Finset.image ιA A.snd ⊔ Finset.image ιB B.snd\n⊢ (forget TopCat).map (F.map f.snd.snd.snd.snd) (u f.fst) = u f.snd.fst",
"tactic": "exact hu this"
},
{
"state_after": "case h\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow A.fst\nhf : f ∈ A.snd\n⊢ ιA f ∈ Finset.image ιA A.snd",
"state_before": "J✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow A.fst\nhf : f ∈ A.snd\n⊢ ιA f ∈ Finset.image ιA A.snd ⊔ Finset.image ιB B.snd",
"tactic": "apply Finset.mem_union_left"
},
{
"state_after": "case h\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow A.fst\nhf : f ∈ A.snd\n⊢ ∃ a, a ∈ A.snd ∧ ιA a = ιA f",
"state_before": "case h\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow A.fst\nhf : f ∈ A.snd\n⊢ ιA f ∈ Finset.image ιA A.snd",
"tactic": "rw [Finset.mem_image]"
},
{
"state_after": "no goals",
"state_before": "case h\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow A.fst\nhf : f ∈ A.snd\n⊢ ∃ a, a ∈ A.snd ∧ ιA a = ιA f",
"tactic": "refine' ⟨f, hf, rfl⟩"
},
{
"state_after": "case refine'_2\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow B.fst\nhf : f ∈ B.snd\n⊢ (forget TopCat).map (F.map f.snd.snd.snd.snd) (u f.fst) = u f.snd.fst",
"state_before": "case refine'_2\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\n⊢ (fun G => partialSections F G.snd) B ⊇\n (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }",
"tactic": "rintro u hu f hf"
},
{
"state_after": "case refine'_2\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow B.fst\nhf : f ∈ B.snd\nthis : ιB f ∈ Finset.image ιA A.snd ⊔ Finset.image ιB B.snd\n⊢ (forget TopCat).map (F.map f.snd.snd.snd.snd) (u f.fst) = u f.snd.fst",
"state_before": "case refine'_2\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow B.fst\nhf : f ∈ B.snd\n⊢ (forget TopCat).map (F.map f.snd.snd.snd.snd) (u f.fst) = u f.snd.fst",
"tactic": "have : ιB f ∈ A.2.image ιA ⊔ B.2.image ιB := by\n apply Finset.mem_union_right\n rw [Finset.mem_image]\n refine' ⟨f, hf, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow B.fst\nhf : f ∈ B.snd\nthis : ιB f ∈ Finset.image ιA A.snd ⊔ Finset.image ιB B.snd\n⊢ (forget TopCat).map (F.map f.snd.snd.snd.snd) (u f.fst) = u f.snd.fst",
"tactic": "exact hu this"
},
{
"state_after": "case h\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow B.fst\nhf : f ∈ B.snd\n⊢ ιB f ∈ Finset.image ιB B.snd",
"state_before": "J✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow B.fst\nhf : f ∈ B.snd\n⊢ ιB f ∈ Finset.image ιA A.snd ⊔ Finset.image ιB B.snd",
"tactic": "apply Finset.mem_union_right"
},
{
"state_after": "case h\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow B.fst\nhf : f ∈ B.snd\n⊢ ∃ a, a ∈ B.snd ∧ ιB a = ιB f",
"state_before": "case h\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow B.fst\nhf : f ∈ B.snd\n⊢ ιB f ∈ Finset.image ιB B.snd",
"tactic": "rw [Finset.mem_image]"
},
{
"state_after": "no goals",
"state_before": "case h\nJ✝ : Type v\ninst✝¹ : SmallCategory J✝\nJ : Type u\ninst✝ : SmallCategory J\nF : J ⥤ TopCat\nA B : TopCat.FiniteDiagram J\nιA : TopCat.FiniteDiagramArrow A.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nιB : TopCat.FiniteDiagramArrow B.fst → TopCat.FiniteDiagramArrow (A.fst ⊔ B.fst) :=\n fun f =>\n { fst := f.fst,\n snd :=\n { fst := f.snd.fst,\n snd :=\n { fst := (_ : f.fst ∈ A.fst ∪ B.fst),\n snd := { fst := (_ : f.snd.fst ∈ A.fst ∪ B.fst), snd := f.snd.snd.snd.snd } } } }\nu : (j : J) → ↑(F.obj j)\nhu :\n u ∈ (fun G => partialSections F G.snd) { fst := A.fst ⊔ B.fst, snd := Finset.image ιA A.snd ⊔ Finset.image ιB B.snd }\nf : TopCat.FiniteDiagramArrow B.fst\nhf : f ∈ B.snd\n⊢ ∃ a, a ∈ B.snd ∧ ιB a = ιB f",
"tactic": "refine' ⟨f, hf, rfl⟩"
}
] |
[
131,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.coe_one
|
[] |
[
479,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
478,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_filter_ne_one
|
[] |
[
769,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
767,
1
] |
Mathlib/MeasureTheory/PiSystem.lean
|
isPiSystem_Ico_mem
|
[] |
[
200,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Topology/Order/Basic.lean
|
exists_Ico_subset_of_mem_nhds'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhs : s ∈ 𝓝 a\nu : α\nhu : a < u\n⊢ ∃ u', u' ∈ Ioc a u ∧ Ico a u' ⊆ s",
"tactic": "simpa only [OrderDual.exists, exists_prop, dual_Ico, dual_Ioc] using\n exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual"
}
] |
[
1238,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1235,
1
] |
Mathlib/Algebra/Hom/Centroid.lean
|
CentroidHom.id_comp
|
[] |
[
228,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.Tendsto.mono_left
|
[] |
[
2922,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2920,
1
] |
Mathlib/Analysis/NormedSpace/AddTorsor.lean
|
nndist_self_homothety
|
[] |
[
162,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.I_to_real
|
[] |
[
854,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
853,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
mem_nhdsWithin_subtype
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.302592\nγ : Type ?u.302595\nδ : Type ?u.302598\ninst✝ : TopologicalSpace α\ns : Set α\na : { x // x ∈ s }\nt u : Set { x // x ∈ s }\n⊢ t ∈ 𝓝[u] a ↔ t ∈ comap Subtype.val (𝓝[Subtype.val '' u] ↑a)",
"tactic": "rw [nhdsWithin, nhds_subtype, principal_subtype, ← comap_inf, ← nhdsWithin]"
}
] |
[
488,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/Topology/Inseparable.lean
|
Specializes.prod
|
[] |
[
212,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
contDiffWithinAt_zero
|
[
{
"state_after": "case mp\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\n⊢ ContDiffWithinAt 𝕜 0 f s x → ∃ u, u ∈ 𝓝[s] x ∧ ContinuousOn f (s ∩ u)\n\ncase mpr\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\n⊢ (∃ u, u ∈ 𝓝[s] x ∧ ContinuousOn f (s ∩ u)) → ContDiffWithinAt 𝕜 0 f s x",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\n⊢ ContDiffWithinAt 𝕜 0 f s x ↔ ∃ u, u ∈ 𝓝[s] x ∧ ContinuousOn f (s ∩ u)",
"tactic": "constructor"
},
{
"state_after": "case mp\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nh : ContDiffWithinAt 𝕜 0 f s x\n⊢ ∃ u, u ∈ 𝓝[s] x ∧ ContinuousOn f (s ∩ u)",
"state_before": "case mp\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\n⊢ ContDiffWithinAt 𝕜 0 f s x → ∃ u, u ∈ 𝓝[s] x ∧ ContinuousOn f (s ∩ u)",
"tactic": "intro h"
},
{
"state_after": "case mp.intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np✝ : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nh : ContDiffWithinAt 𝕜 0 f s x\nu : Set E\nH : u ∈ 𝓝[insert x s] x\np : E → FormalMultilinearSeries 𝕜 E F\nhp : HasFTaylorSeriesUpToOn (↑0) f p u\n⊢ ∃ u, u ∈ 𝓝[s] x ∧ ContinuousOn f (s ∩ u)",
"state_before": "case mp\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nh : ContDiffWithinAt 𝕜 0 f s x\n⊢ ∃ u, u ∈ 𝓝[s] x ∧ ContinuousOn f (s ∩ u)",
"tactic": "obtain ⟨u, H, p, hp⟩ := h 0 le_rfl"
},
{
"state_after": "case mp.intro.intro.intro.refine'_1\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np✝ : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nh : ContDiffWithinAt 𝕜 0 f s x\nu : Set E\nH : u ∈ 𝓝[insert x s] x\np : E → FormalMultilinearSeries 𝕜 E F\nhp : HasFTaylorSeriesUpToOn (↑0) f p u\n⊢ u ∈ 𝓝[s] x\n\ncase mp.intro.intro.intro.refine'_2\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np✝ : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nh : ContDiffWithinAt 𝕜 0 f s x\nu : Set E\nH : u ∈ 𝓝[insert x s] x\np : E → FormalMultilinearSeries 𝕜 E F\nhp : HasFTaylorSeriesUpToOn (↑0) f p u\n⊢ ContinuousOn f (s ∩ u)",
"state_before": "case mp.intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np✝ : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nh : ContDiffWithinAt 𝕜 0 f s x\nu : Set E\nH : u ∈ 𝓝[insert x s] x\np : E → FormalMultilinearSeries 𝕜 E F\nhp : HasFTaylorSeriesUpToOn (↑0) f p u\n⊢ ∃ u, u ∈ 𝓝[s] x ∧ ContinuousOn f (s ∩ u)",
"tactic": "refine' ⟨u, _, _⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.refine'_1\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np✝ : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nh : ContDiffWithinAt 𝕜 0 f s x\nu : Set E\nH : u ∈ 𝓝[insert x s] x\np : E → FormalMultilinearSeries 𝕜 E F\nhp : HasFTaylorSeriesUpToOn (↑0) f p u\n⊢ u ∈ 𝓝[s] x",
"tactic": "simpa [hx] using H"
},
{
"state_after": "case mp.intro.intro.intro.refine'_2\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np✝ : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nh : ContDiffWithinAt 𝕜 0 f s x\nu : Set E\nH : u ∈ 𝓝[insert x s] x\np : E → FormalMultilinearSeries 𝕜 E F\nhp : ContinuousOn f u ∧ ∀ (x : E), x ∈ u → ContinuousMultilinearMap.uncurry0 (p x 0) = f x\n⊢ ContinuousOn f (s ∩ u)",
"state_before": "case mp.intro.intro.intro.refine'_2\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np✝ : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nh : ContDiffWithinAt 𝕜 0 f s x\nu : Set E\nH : u ∈ 𝓝[insert x s] x\np : E → FormalMultilinearSeries 𝕜 E F\nhp : HasFTaylorSeriesUpToOn (↑0) f p u\n⊢ ContinuousOn f (s ∩ u)",
"tactic": "simp only [Nat.cast_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.refine'_2\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np✝ : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nh : ContDiffWithinAt 𝕜 0 f s x\nu : Set E\nH : u ∈ 𝓝[insert x s] x\np : E → FormalMultilinearSeries 𝕜 E F\nhp : ContinuousOn f u ∧ ∀ (x : E), x ∈ u → ContinuousMultilinearMap.uncurry0 (p x 0) = f x\n⊢ ContinuousOn f (s ∩ u)",
"tactic": "exact hp.1.mono (inter_subset_right s u)"
},
{
"state_after": "case mpr.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nu : Set E\nH : u ∈ 𝓝[s] x\nhu : ContinuousOn f (s ∩ u)\n⊢ ContDiffWithinAt 𝕜 0 f s x",
"state_before": "case mpr\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\n⊢ (∃ u, u ∈ 𝓝[s] x ∧ ContinuousOn f (s ∩ u)) → ContDiffWithinAt 𝕜 0 f s x",
"tactic": "rintro ⟨u, H, hu⟩"
},
{
"state_after": "case mpr.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nu : Set E\nH : u ∈ 𝓝[s] x\nhu : ContinuousOn f (s ∩ u)\n⊢ ContDiffWithinAt 𝕜 0 f (s ∩ u) x",
"state_before": "case mpr.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nu : Set E\nH : u ∈ 𝓝[s] x\nhu : ContinuousOn f (s ∩ u)\n⊢ ContDiffWithinAt 𝕜 0 f s x",
"tactic": "rw [← contDiffWithinAt_inter' H]"
},
{
"state_after": "case mpr.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nu : Set E\nH : u ∈ 𝓝[s] x\nhu : ContinuousOn f (s ∩ u)\nh' : x ∈ s ∩ u\n⊢ ContDiffWithinAt 𝕜 0 f (s ∩ u) x",
"state_before": "case mpr.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nu : Set E\nH : u ∈ 𝓝[s] x\nhu : ContinuousOn f (s ∩ u)\n⊢ ContDiffWithinAt 𝕜 0 f (s ∩ u) x",
"tactic": "have h' : x ∈ s ∩ u := ⟨hx, mem_of_mem_nhdsWithin hx H⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhx : x ∈ s\nu : Set E\nH : u ∈ 𝓝[s] x\nhu : ContinuousOn f (s ∩ u)\nh' : x ∈ s ∩ u\n⊢ ContDiffWithinAt 𝕜 0 f (s ∩ u) x",
"tactic": "exact (contDiffOn_zero.mpr hu).contDiffWithinAt h'"
}
] |
[
981,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
969,
1
] |
Mathlib/Analysis/Complex/Basic.lean
|
IsROrC.normSq_to_complex
|
[] |
[
477,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
476,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
|
MeasureTheory.Measure.measurePreserving_swap
|
[] |
[
518,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
517,
1
] |
Mathlib/Data/List/MinMax.lean
|
List.argmax_concat
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder β\ninst✝ : DecidableRel fun x x_1 => x < x_1\nf✝ : α → β\nl✝ : List α\no : Option α\na✝ m : α\nf : α → β\na : α\nl : List α\n⊢ foldl (argAux fun b c => f c < f b) none (l ++ [a]) =\n Option.casesOn (foldl (argAux fun b c => f c < f b) none l) (some a) fun c => if f c < f a then some a else some c",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder β\ninst✝ : DecidableRel fun x x_1 => x < x_1\nf✝ : α → β\nl✝ : List α\no : Option α\na✝ m : α\nf : α → β\na : α\nl : List α\n⊢ argmax f (l ++ [a]) = Option.casesOn (argmax f l) (some a) fun c => if f c < f a then some a else some c",
"tactic": "rw [argmax, argmax]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder β\ninst✝ : DecidableRel fun x x_1 => x < x_1\nf✝ : α → β\nl✝ : List α\no : Option α\na✝ m : α\nf : α → β\na : α\nl : List α\n⊢ foldl (argAux fun b c => f c < f b) none (l ++ [a]) =\n Option.casesOn (foldl (argAux fun b c => f c < f b) none l) (some a) fun c => if f c < f a then some a else some c",
"tactic": "simp [argAux]"
}
] |
[
143,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/LinearAlgebra/Ray.lean
|
SameRay.trans
|
[
{
"state_after": "case inl\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\ny z : M\nhyz : SameRay R y z\nhxy : SameRay R 0 y\nhy : y = 0 → 0 = 0 ∨ z = 0\n⊢ SameRay R 0 z\n\ncase inr\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx y z : M\nhxy : SameRay R x y\nhyz : SameRay R y z\nhy : y = 0 → x = 0 ∨ z = 0\nhx : x ≠ 0\n⊢ SameRay R x z",
"state_before": "R : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx y z : M\nhxy : SameRay R x y\nhyz : SameRay R y z\nhy : y = 0 → x = 0 ∨ z = 0\n⊢ SameRay R x z",
"tactic": "rcases eq_or_ne x 0 with (rfl | hx)"
},
{
"state_after": "case inr.inl\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx y : M\nhxy : SameRay R x y\nhx : x ≠ 0\nhyz : SameRay R y 0\nhy : y = 0 → x = 0 ∨ 0 = 0\n⊢ SameRay R x 0\n\ncase inr.inr\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx y z : M\nhxy : SameRay R x y\nhyz : SameRay R y z\nhy : y = 0 → x = 0 ∨ z = 0\nhx : x ≠ 0\nhz : z ≠ 0\n⊢ SameRay R x z",
"state_before": "case inr\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx y z : M\nhxy : SameRay R x y\nhyz : SameRay R y z\nhy : y = 0 → x = 0 ∨ z = 0\nhx : x ≠ 0\n⊢ SameRay R x z",
"tactic": "rcases eq_or_ne z 0 with (rfl | hz)"
},
{
"state_after": "case inr.inr.inl\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx z : M\nhx : x ≠ 0\nhz : z ≠ 0\nhxy : SameRay R x 0\nhyz : SameRay R 0 z\nhy : 0 = 0 → x = 0 ∨ z = 0\n⊢ SameRay R x z\n\ncase inr.inr.inr\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx y z : M\nhxy : SameRay R x y\nhyz : SameRay R y z\nhy✝ : y = 0 → x = 0 ∨ z = 0\nhx : x ≠ 0\nhz : z ≠ 0\nhy : y ≠ 0\n⊢ SameRay R x z",
"state_before": "case inr.inr\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx y z : M\nhxy : SameRay R x y\nhyz : SameRay R y z\nhy : y = 0 → x = 0 ∨ z = 0\nhx : x ≠ 0\nhz : z ≠ 0\n⊢ SameRay R x z",
"tactic": "rcases eq_or_ne y 0 with (rfl | hy)"
},
{
"state_after": "case inr.inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx y z : M\nhxy : SameRay R x y\nhyz : SameRay R y z\nhy✝ : y = 0 → x = 0 ∨ z = 0\nhx : x ≠ 0\nhz : z ≠ 0\nhy : y ≠ 0\nr₁ r₂ : R\nhr₁ : 0 < r₁\nhr₂ : 0 < r₂\nh₁ : r₁ • x = r₂ • y\n⊢ SameRay R x z",
"state_before": "case inr.inr.inr\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx y z : M\nhxy : SameRay R x y\nhyz : SameRay R y z\nhy✝ : y = 0 → x = 0 ∨ z = 0\nhx : x ≠ 0\nhz : z ≠ 0\nhy : y ≠ 0\n⊢ SameRay R x z",
"tactic": "rcases hxy.exists_pos hx hy with ⟨r₁, r₂, hr₁, hr₂, h₁⟩"
},
{
"state_after": "case inr.inr.inr.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx y z : M\nhxy : SameRay R x y\nhyz : SameRay R y z\nhy✝ : y = 0 → x = 0 ∨ z = 0\nhx : x ≠ 0\nhz : z ≠ 0\nhy : y ≠ 0\nr₁ r₂ : R\nhr₁ : 0 < r₁\nhr₂ : 0 < r₂\nh₁ : r₁ • x = r₂ • y\nr₃ r₄ : R\nhr₃ : 0 < r₃\nhr₄ : 0 < r₄\nh₂ : r₃ • y = r₄ • z\n⊢ SameRay R x z",
"state_before": "case inr.inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx y z : M\nhxy : SameRay R x y\nhyz : SameRay R y z\nhy✝ : y = 0 → x = 0 ∨ z = 0\nhx : x ≠ 0\nhz : z ≠ 0\nhy : y ≠ 0\nr₁ r₂ : R\nhr₁ : 0 < r₁\nhr₂ : 0 < r₂\nh₁ : r₁ • x = r₂ • y\n⊢ SameRay R x z",
"tactic": "rcases hyz.exists_pos hy hz with ⟨r₃, r₄, hr₃, hr₄, h₂⟩"
},
{
"state_after": "case inr.inr.inr.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx y z : M\nhxy : SameRay R x y\nhyz : SameRay R y z\nhy✝ : y = 0 → x = 0 ∨ z = 0\nhx : x ≠ 0\nhz : z ≠ 0\nhy : y ≠ 0\nr₁ r₂ : R\nhr₁ : 0 < r₁\nhr₂ : 0 < r₂\nh₁ : r₁ • x = r₂ • y\nr₃ r₄ : R\nhr₃ : 0 < r₃\nhr₄ : 0 < r₄\nh₂ : r₃ • y = r₄ • z\n⊢ (r₃ * r₁) • x = (r₂ * r₄) • z",
"state_before": "case inr.inr.inr.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx y z : M\nhxy : SameRay R x y\nhyz : SameRay R y z\nhy✝ : y = 0 → x = 0 ∨ z = 0\nhx : x ≠ 0\nhz : z ≠ 0\nhy : y ≠ 0\nr₁ r₂ : R\nhr₁ : 0 < r₁\nhr₂ : 0 < r₂\nh₁ : r₁ • x = r₂ • y\nr₃ r₄ : R\nhr₃ : 0 < r₃\nhr₄ : 0 < r₄\nh₂ : r₃ • y = r₄ • z\n⊢ SameRay R x z",
"tactic": "refine' Or.inr (Or.inr <| ⟨r₃ * r₁, r₂ * r₄, mul_pos hr₃ hr₁, mul_pos hr₂ hr₄, _⟩)"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx y z : M\nhxy : SameRay R x y\nhyz : SameRay R y z\nhy✝ : y = 0 → x = 0 ∨ z = 0\nhx : x ≠ 0\nhz : z ≠ 0\nhy : y ≠ 0\nr₁ r₂ : R\nhr₁ : 0 < r₁\nhr₂ : 0 < r₂\nh₁ : r₁ • x = r₂ • y\nr₃ r₄ : R\nhr₃ : 0 < r₃\nhr₄ : 0 < r₄\nh₂ : r₃ • y = r₄ • z\n⊢ (r₃ * r₁) • x = (r₂ * r₄) • z",
"tactic": "rw [mul_smul, mul_smul, h₁, ← h₂, smul_comm]"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\ny z : M\nhyz : SameRay R y z\nhxy : SameRay R 0 y\nhy : y = 0 → 0 = 0 ∨ z = 0\n⊢ SameRay R 0 z",
"tactic": "exact zero_left z"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx y : M\nhxy : SameRay R x y\nhx : x ≠ 0\nhyz : SameRay R y 0\nhy : y = 0 → x = 0 ∨ 0 = 0\n⊢ SameRay R x 0",
"tactic": "exact zero_right x"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inl\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type ?u.13611\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type ?u.13693\ninst✝ : DecidableEq ι\nx z : M\nhx : x ≠ 0\nhz : z ≠ 0\nhxy : SameRay R x 0\nhyz : SameRay R 0 z\nhy : 0 = 0 → x = 0 ∨ z = 0\n⊢ SameRay R x z",
"tactic": "exact (hy rfl).elim (fun h => (hx h).elim) fun h => (hz h).elim"
}
] |
[
117,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.Nonempty.of_mul_left
|
[] |
[
386,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
Mathlib/Analysis/NormedSpace/AddTorsor.lean
|
nndist_right_midpoint
|
[] |
[
212,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/Data/Polynomial/Reverse.lean
|
Polynomial.reflect_eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nf✝ : R[X]\nN : ℕ\nf : R[X]\n⊢ reflect N f = 0 ↔ f = 0",
"tactic": "rw [ofFinsupp_eq_zero, reflect, embDomain_eq_zero, ofFinsupp_eq_zero]"
}
] |
[
131,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.unit_left
|
[] |
[
201,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.ofReal_mul'
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.814833\nβ : Type ?u.814836\na b c d : ℝ≥0∞\nr p✝ q✝ : ℝ≥0\np q : ℝ\nhq : 0 ≤ q\n⊢ ENNReal.ofReal (p * q) = ENNReal.ofReal p * ENNReal.ofReal q",
"tactic": "rw [mul_comm, ofReal_mul hq, mul_comm]"
}
] |
[
2174,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2172,
1
] |
Mathlib/Data/Real/Irrational.lean
|
irrational_int_mul_iff
|
[
{
"state_after": "no goals",
"state_before": "q : ℚ\nm : ℤ\nn : ℕ\nx : ℝ\n⊢ Irrational (↑m * x) ↔ m ≠ 0 ∧ Irrational x",
"tactic": "rw [← cast_coe_int, irrational_rat_mul_iff, Int.cast_ne_zero]"
}
] |
[
614,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
613,
1
] |
src/lean/Init/Prelude.lean
|
Eq.symm
|
[] |
[
311,
10
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
310,
1
] |
Mathlib/Data/Nat/Digits.lean
|
Nat.digits_add_two_add_one
|
[
{
"state_after": "no goals",
"state_before": "n✝ b n : ℕ\n⊢ digits (b + 2) (n + 1) = (n + 1) % (b + 2) :: digits (b + 2) ((n + 1) / (b + 2))",
"tactic": "simp [digits, digitsAux_def]"
}
] |
[
121,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Analysis/SpecialFunctions/Exponential.lean
|
hasStrictFDerivAt_exp_smul_const'
|
[] |
[
405,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
401,
1
] |
Mathlib/Data/List/Intervals.lean
|
List.Ico.filter_le_of_top_le
|
[
{
"state_after": "n m l : ℕ\nhml : m ≤ l\nk : ℕ\nhk : k ∈ Ico n m\n⊢ ¬l ≤ k",
"state_before": "n m l : ℕ\nhml : m ≤ l\nk : ℕ\nhk : k ∈ Ico n m\n⊢ ¬decide (l ≤ k) = true",
"tactic": "rw [decide_eq_true_eq]"
},
{
"state_after": "no goals",
"state_before": "n m l : ℕ\nhml : m ≤ l\nk : ℕ\nhk : k ∈ Ico n m\n⊢ ¬l ≤ k",
"tactic": "exact not_le_of_gt (lt_of_lt_of_le (mem.1 hk).2 hml)"
}
] |
[
197,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Data/List/Basic.lean
|
List.getLastI_eq_getLast?
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.38004\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : Inhabited α\n⊢ getLastI [] = Option.iget (getLast? [])",
"tactic": "simp [getLastI, Inhabited.default]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.38004\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : Inhabited α\nhead✝¹ head✝ c : α\nl : List α\n⊢ getLastI (head✝¹ :: head✝ :: c :: l) = Option.iget (getLast? (head✝¹ :: head✝ :: c :: l))",
"tactic": "simp [getLastI, getLastI_eq_getLast? (c :: l)]"
}
] |
[
830,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
825,
1
] |
Mathlib/CategoryTheory/Sites/CompatiblePlus.lean
|
CategoryTheory.GrothendieckTopology.plusCompIso_whiskerLeft
|
[
{
"state_after": "case w.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\n⊢ (whiskerLeft (plusObj J P) η ≫ (plusCompIso J G P).hom).app X =\n ((plusCompIso J F P).hom ≫ plusMap J (whiskerLeft P η)).app X",
"state_before": "C : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\n⊢ whiskerLeft (plusObj J P) η ≫ (plusCompIso J G P).hom = (plusCompIso J F P).hom ≫ plusMap J (whiskerLeft P η)",
"tactic": "ext X"
},
{
"state_after": "case w.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\n⊢ ∀ (j : (Cover J X.unop)ᵒᵖ),\n (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app j ≫\n (whiskerLeft (plusObj J P) η ≫ (plusCompIso J G P).hom).app X =\n (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app j ≫\n ((plusCompIso J F P).hom ≫ plusMap J (whiskerLeft P η)).app X",
"state_before": "case w.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\n⊢ (whiskerLeft (plusObj J P) η ≫ (plusCompIso J G P).hom).app X =\n ((plusCompIso J F P).hom ≫ plusMap J (whiskerLeft P η)).app X",
"tactic": "apply (isColimitOfPreserves F (colimit.isColimit (J.diagram P X.unop))).hom_ext"
},
{
"state_after": "case w.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app W ≫\n (whiskerLeft (plusObj J P) η ≫ (plusCompIso J G P).hom).app X =\n (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app W ≫\n ((plusCompIso J F P).hom ≫ plusMap J (whiskerLeft P η)).app X",
"state_before": "case w.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\n⊢ ∀ (j : (Cover J X.unop)ᵒᵖ),\n (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app j ≫\n (whiskerLeft (plusObj J P) η ≫ (plusCompIso J G P).hom).app X =\n (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app j ≫\n ((plusCompIso J F P).hom ≫ plusMap J (whiskerLeft P η)).app X",
"tactic": "intro W"
},
{
"state_after": "case w.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map (colimit.ι (diagram J P X.unop) W) ≫ η.app (colimit (diagram J P X.unop)) ≫ (plusCompIso J G P).hom.app X =\n F.map (colimit.ι (diagram J P X.unop) W) ≫\n (plusCompIso J F P).hom.app X ≫ colimMap (diagramNatTrans J (whiskerLeft P η) X.unop)",
"state_before": "case w.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app W ≫\n (whiskerLeft (plusObj J P) η ≫ (plusCompIso J G P).hom).app X =\n (F.mapCocone (colimit.cocone (diagram J P X.unop))).ι.app W ≫\n ((plusCompIso J F P).hom ≫ plusMap J (whiskerLeft P η)).app X",
"tactic": "dsimp [plusObj, plusMap]"
},
{
"state_after": "case w.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ η.app (multiequalizer (Cover.index W.unop P)) ≫\n (diagramCompIso J G P X.unop).hom.app W ≫ colimit.ι (diagram J (P ⋙ G) X.unop) W =\n (diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (P ⋙ G)) ((diagram J (P ⋙ F) X.unop).obj W)\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (_ :\n ∀ (b : (Cover.index W.unop (P ⋙ G)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.fstTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.fst (Cover.index W.unop (P ⋙ G)) b =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.sndTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.snd (Cover.index W.unop (P ⋙ G)) b) ≫\n colimit.ι (diagram J (P ⋙ G) X.unop) W",
"state_before": "case w.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ F.map (colimit.ι (diagram J P X.unop) W) ≫ η.app (colimit (diagram J P X.unop)) ≫ (plusCompIso J G P).hom.app X =\n F.map (colimit.ι (diagram J P X.unop) W) ≫\n (plusCompIso J F P).hom.app X ≫ colimMap (diagramNatTrans J (whiskerLeft P η) X.unop)",
"tactic": "simp only [ι_plusCompIso_hom, ι_colimMap, whiskerLeft_app, ι_plusCompIso_hom_assoc,\n NatTrans.naturality_assoc, GrothendieckTopology.diagramNatTrans_app]"
},
{
"state_after": "case w.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ (η.app (multiequalizer (Cover.index W.unop P)) ≫ (diagramCompIso J G P X.unop).hom.app W) ≫\n colimit.ι (diagram J (P ⋙ G) X.unop) W =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (P ⋙ G)) ((diagram J (P ⋙ F) X.unop).obj W)\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (_ :\n ∀ (b : (Cover.index W.unop (P ⋙ G)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.fstTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.fst (Cover.index W.unop (P ⋙ G)) b =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.sndTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.snd (Cover.index W.unop (P ⋙ G)) b)) ≫\n colimit.ι (diagram J (P ⋙ G) X.unop) W",
"state_before": "case w.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ η.app (multiequalizer (Cover.index W.unop P)) ≫\n (diagramCompIso J G P X.unop).hom.app W ≫ colimit.ι (diagram J (P ⋙ G) X.unop) W =\n (diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (P ⋙ G)) ((diagram J (P ⋙ F) X.unop).obj W)\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (_ :\n ∀ (b : (Cover.index W.unop (P ⋙ G)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.fstTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.fst (Cover.index W.unop (P ⋙ G)) b =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.sndTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.snd (Cover.index W.unop (P ⋙ G)) b) ≫\n colimit.ι (diagram J (P ⋙ G) X.unop) W",
"tactic": "simp only [← Category.assoc]"
},
{
"state_after": "case w.h.e_a\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ η.app (multiequalizer (Cover.index W.unop P)) ≫ (diagramCompIso J G P X.unop).hom.app W =\n (diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (P ⋙ G)) ((diagram J (P ⋙ F) X.unop).obj W)\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (_ :\n ∀ (b : (Cover.index W.unop (P ⋙ G)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.fstTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.fst (Cover.index W.unop (P ⋙ G)) b =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.sndTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.snd (Cover.index W.unop (P ⋙ G)) b)",
"state_before": "case w.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ (η.app (multiequalizer (Cover.index W.unop P)) ≫ (diagramCompIso J G P X.unop).hom.app W) ≫\n colimit.ι (diagram J (P ⋙ G) X.unop) W =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (P ⋙ G)) ((diagram J (P ⋙ F) X.unop).obj W)\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (_ :\n ∀ (b : (Cover.index W.unop (P ⋙ G)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.fstTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.fst (Cover.index W.unop (P ⋙ G)) b =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.sndTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.snd (Cover.index W.unop (P ⋙ G)) b)) ≫\n colimit.ι (diagram J (P ⋙ G) X.unop) W",
"tactic": "congr 1"
},
{
"state_after": "case w.h.e_a.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ ∀ (a : (Cover.index W.unop (P ⋙ G)).L),\n (η.app (multiequalizer (Cover.index W.unop P)) ≫ (diagramCompIso J G P X.unop).hom.app W) ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ G)) a =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (P ⋙ G)) ((diagram J (P ⋙ F) X.unop).obj W)\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (_ :\n ∀ (b : (Cover.index W.unop (P ⋙ G)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.fstTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.fst (Cover.index W.unop (P ⋙ G)) b =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.sndTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.snd (Cover.index W.unop (P ⋙ G)) b)) ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ G)) a",
"state_before": "case w.h.e_a\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ η.app (multiequalizer (Cover.index W.unop P)) ≫ (diagramCompIso J G P X.unop).hom.app W =\n (diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (P ⋙ G)) ((diagram J (P ⋙ F) X.unop).obj W)\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (_ :\n ∀ (b : (Cover.index W.unop (P ⋙ G)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.fstTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.fst (Cover.index W.unop (P ⋙ G)) b =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.sndTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.snd (Cover.index W.unop (P ⋙ G)) b)",
"tactic": "apply Multiequalizer.hom_ext"
},
{
"state_after": "case w.h.e_a.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\na : (Cover.index W.unop (P ⋙ G)).L\n⊢ (η.app (multiequalizer (Cover.index W.unop P)) ≫ (diagramCompIso J G P X.unop).hom.app W) ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ G)) a =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (P ⋙ G)) ((diagram J (P ⋙ F) X.unop).obj W)\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (_ :\n ∀ (b : (Cover.index W.unop (P ⋙ G)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.fstTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.fst (Cover.index W.unop (P ⋙ G)) b =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.sndTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.snd (Cover.index W.unop (P ⋙ G)) b)) ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ G)) a",
"state_before": "case w.h.e_a.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\n⊢ ∀ (a : (Cover.index W.unop (P ⋙ G)).L),\n (η.app (multiequalizer (Cover.index W.unop P)) ≫ (diagramCompIso J G P X.unop).hom.app W) ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ G)) a =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (P ⋙ G)) ((diagram J (P ⋙ F) X.unop).obj W)\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (_ :\n ∀ (b : (Cover.index W.unop (P ⋙ G)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.fstTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.fst (Cover.index W.unop (P ⋙ G)) b =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.sndTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.snd (Cover.index W.unop (P ⋙ G)) b)) ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ G)) a",
"tactic": "intro a"
},
{
"state_after": "case w.h.e_a.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\na : (Cover.index W.unop (P ⋙ G)).L\n⊢ (η.app (multiequalizer (Cover.index W.unop P)) ≫ (diagramCompIso J G P X.unop).hom.app W) ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ G)) a =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (P ⋙ G)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (P ⋙ G)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerLeft P η).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (P ⋙ G)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (P ⋙ G)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerLeft P η).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (P ⋙ G)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (P ⋙ G)) i)) ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ G)) a",
"state_before": "case w.h.e_a.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\na : (Cover.index W.unop (P ⋙ G)).L\n⊢ (η.app (multiequalizer (Cover.index W.unop P)) ≫ (diagramCompIso J G P X.unop).hom.app W) ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ G)) a =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (P ⋙ G)) ((diagram J (P ⋙ F) X.unop).obj W)\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (_ :\n ∀ (b : (Cover.index W.unop (P ⋙ G)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.fstTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.fst (Cover.index W.unop (P ⋙ G)) b =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (MulticospanIndex.sndTo (Cover.index W.unop (P ⋙ G)) b) ≫\n MulticospanIndex.snd (Cover.index W.unop (P ⋙ G)) b)) ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ G)) a",
"tactic": "dsimp"
},
{
"state_after": "case w.h.e_a.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\na : (Cover.index W.unop (P ⋙ G)).L\n⊢ η.app (multiequalizer (Cover.index W.unop P)) ≫ G.map (Multiequalizer.ι (Cover.index W.unop P) a) =\n F.map (Multiequalizer.ι (Cover.index W.unop P) a) ≫ η.app (P.obj a.Y.op)",
"state_before": "case w.h.e_a.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\na : (Cover.index W.unop (P ⋙ G)).L\n⊢ (η.app (multiequalizer (Cover.index W.unop P)) ≫ (diagramCompIso J G P X.unop).hom.app W) ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ G)) a =\n ((diagramCompIso J F P X.unop).hom.app W ≫\n Multiequalizer.lift (Cover.index W.unop (P ⋙ G)) (multiequalizer (Cover.index W.unop (P ⋙ F)))\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ η.app (P.obj i.Y.op))\n (_ :\n ∀ (i : (Cover.index W.unop (P ⋙ G)).R),\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerLeft P η).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index W.unop (P ⋙ G)) i) ≫\n MulticospanIndex.fst (Cover.index W.unop (P ⋙ G)) i =\n (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerLeft P η).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index W.unop (P ⋙ G)) i) ≫\n MulticospanIndex.snd (Cover.index W.unop (P ⋙ G)) i)) ≫\n Multiequalizer.ι (Cover.index W.unop (P ⋙ G)) a",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case w.h.e_a.h\nC : Type u\ninst✝¹² : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝¹¹ : Category D\nE : Type w₂\ninst✝¹⁰ : Category E\nF✝ : D ⥤ E\ninst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁷ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ : Cᵒᵖ ⥤ D\ninst✝⁶ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝⁴ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\nF G : D ⥤ E\nη : F ⟶ G\nP : Cᵒᵖ ⥤ D\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ G\ninst✝ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : Cᵒᵖ\nW : (Cover J X.unop)ᵒᵖ\na : (Cover.index W.unop (P ⋙ G)).L\n⊢ η.app (multiequalizer (Cover.index W.unop P)) ≫ G.map (Multiequalizer.ι (Cover.index W.unop P) a) =\n F.map (Multiequalizer.ι (Cover.index W.unop P) a) ≫ η.app (P.obj a.Y.op)",
"tactic": "erw [η.naturality]"
}
] |
[
156,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Algebra/Associated.lean
|
Associates.mk_dvd_mk
|
[] |
[
948,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
947,
1
] |
Mathlib/CategoryTheory/Filtered.lean
|
CategoryTheory.IsCofiltered.inf_objs_exists
|
[
{
"state_after": "case empty\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\n⊢ ∃ S, ∀ {X : C}, X ∈ ∅ → _root_.Nonempty (S ⟶ X)\n\ncase insert\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nX : C\nO' : Finset C\nnm : ¬X ∈ O'\nh : ∃ S, ∀ {X : C}, X ∈ O' → _root_.Nonempty (S ⟶ X)\n⊢ ∃ S, ∀ {X_1 : C}, X_1 ∈ insert X O' → _root_.Nonempty (S ⟶ X_1)",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\n⊢ ∃ S, ∀ {X : C}, X ∈ O → _root_.Nonempty (S ⟶ X)",
"tactic": "induction' O using Finset.induction with X O' nm h"
},
{
"state_after": "case insert\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nX : C\nO' : Finset C\nnm : ¬X ∈ O'\nh : ∃ S, ∀ {X : C}, X ∈ O' → _root_.Nonempty (S ⟶ X)\n⊢ ∃ S, ∀ {X_1 : C}, X_1 ∈ insert X O' → _root_.Nonempty (S ⟶ X_1)",
"state_before": "case empty\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\n⊢ ∃ S, ∀ {X : C}, X ∈ ∅ → _root_.Nonempty (S ⟶ X)\n\ncase insert\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nX : C\nO' : Finset C\nnm : ¬X ∈ O'\nh : ∃ S, ∀ {X : C}, X ∈ O' → _root_.Nonempty (S ⟶ X)\n⊢ ∃ S, ∀ {X_1 : C}, X_1 ∈ insert X O' → _root_.Nonempty (S ⟶ X_1)",
"tactic": ". exact ⟨Classical.choice IsCofiltered.Nonempty, by intro; simp⟩"
},
{
"state_after": "no goals",
"state_before": "case empty\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\n⊢ ∃ S, ∀ {X : C}, X ∈ ∅ → _root_.Nonempty (S ⟶ X)",
"tactic": "exact ⟨Classical.choice IsCofiltered.Nonempty, by intro; simp⟩"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nX✝ : C\n⊢ X✝ ∈ ∅ → _root_.Nonempty (Classical.choice (_ : _root_.Nonempty C) ⟶ X✝)",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\n⊢ ∀ {X : C}, X ∈ ∅ → _root_.Nonempty (Classical.choice (_ : _root_.Nonempty C) ⟶ X)",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nX✝ : C\n⊢ X✝ ∈ ∅ → _root_.Nonempty (Classical.choice (_ : _root_.Nonempty C) ⟶ X✝)",
"tactic": "simp"
},
{
"state_after": "case insert.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nX : C\nO' : Finset C\nnm : ¬X ∈ O'\nS' : C\nw' : ∀ {X : C}, X ∈ O' → _root_.Nonempty (S' ⟶ X)\n⊢ ∃ S, ∀ {X_1 : C}, X_1 ∈ insert X O' → _root_.Nonempty (S ⟶ X_1)",
"state_before": "case insert\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nX : C\nO' : Finset C\nnm : ¬X ∈ O'\nh : ∃ S, ∀ {X : C}, X ∈ O' → _root_.Nonempty (S ⟶ X)\n⊢ ∃ S, ∀ {X_1 : C}, X_1 ∈ insert X O' → _root_.Nonempty (S ⟶ X_1)",
"tactic": "obtain ⟨S', w'⟩ := h"
},
{
"state_after": "case insert.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nX : C\nO' : Finset C\nnm : ¬X ∈ O'\nS' : C\nw' : ∀ {X : C}, X ∈ O' → _root_.Nonempty (S' ⟶ X)\n⊢ ∀ {X_1 : C}, X_1 ∈ insert X O' → _root_.Nonempty (min X S' ⟶ X_1)",
"state_before": "case insert.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nX : C\nO' : Finset C\nnm : ¬X ∈ O'\nS' : C\nw' : ∀ {X : C}, X ∈ O' → _root_.Nonempty (S' ⟶ X)\n⊢ ∃ S, ∀ {X_1 : C}, X_1 ∈ insert X O' → _root_.Nonempty (S ⟶ X_1)",
"tactic": "use min X S'"
},
{
"state_after": "case insert.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nX : C\nO' : Finset C\nnm : ¬X ∈ O'\nS' : C\nw' : ∀ {X : C}, X ∈ O' → _root_.Nonempty (S' ⟶ X)\nY : C\nmY : Y ∈ insert X O'\n⊢ _root_.Nonempty (min X S' ⟶ Y)",
"state_before": "case insert.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nX : C\nO' : Finset C\nnm : ¬X ∈ O'\nS' : C\nw' : ∀ {X : C}, X ∈ O' → _root_.Nonempty (S' ⟶ X)\n⊢ ∀ {X_1 : C}, X_1 ∈ insert X O' → _root_.Nonempty (min X S' ⟶ X_1)",
"tactic": "rintro Y mY"
},
{
"state_after": "case insert.intro.inl\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO' : Finset C\nS' : C\nw' : ∀ {X : C}, X ∈ O' → _root_.Nonempty (S' ⟶ X)\nY : C\nnm : ¬Y ∈ O'\nmY : Y ∈ insert Y O'\n⊢ _root_.Nonempty (min Y S' ⟶ Y)\n\ncase insert.intro.inr\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nX : C\nO' : Finset C\nnm : ¬X ∈ O'\nS' : C\nw' : ∀ {X : C}, X ∈ O' → _root_.Nonempty (S' ⟶ X)\nY : C\nmY : Y ∈ insert X O'\nh : Y ≠ X\n⊢ _root_.Nonempty (min X S' ⟶ Y)",
"state_before": "case insert.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nX : C\nO' : Finset C\nnm : ¬X ∈ O'\nS' : C\nw' : ∀ {X : C}, X ∈ O' → _root_.Nonempty (S' ⟶ X)\nY : C\nmY : Y ∈ insert X O'\n⊢ _root_.Nonempty (min X S' ⟶ Y)",
"tactic": "obtain rfl | h := eq_or_ne Y X"
},
{
"state_after": "no goals",
"state_before": "case insert.intro.inl\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO' : Finset C\nS' : C\nw' : ∀ {X : C}, X ∈ O' → _root_.Nonempty (S' ⟶ X)\nY : C\nnm : ¬Y ∈ O'\nmY : Y ∈ insert Y O'\n⊢ _root_.Nonempty (min Y S' ⟶ Y)",
"tactic": "exact ⟨minToLeft _ _⟩"
},
{
"state_after": "no goals",
"state_before": "case insert.intro.inr\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nX : C\nO' : Finset C\nnm : ¬X ∈ O'\nS' : C\nw' : ∀ {X : C}, X ∈ O' → _root_.Nonempty (S' ⟶ X)\nY : C\nmY : Y ∈ insert X O'\nh : Y ≠ X\n⊢ _root_.Nonempty (min X S' ⟶ Y)",
"tactic": "exact ⟨minToRight _ _ ≫ (w' (Finset.mem_of_mem_insert_of_ne mY h)).some⟩"
}
] |
[
631,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
622,
1
] |
Mathlib/Order/Filter/Prod.lean
|
Filter.prod_mono_left
|
[] |
[
237,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Set.IsWf.min_le
|
[] |
[
643,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
642,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
Algebra.ofId_apply
|
[] |
[
562,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
561,
1
] |
Mathlib/Analysis/Quaternion.lean
|
Quaternion.coeComplex_one
|
[] |
[
150,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.nth_zip
|
[] |
[
209,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Std/Data/List/Lemmas.lean
|
List.length_tail
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List α\n⊢ length (tail l) = length l - 1",
"tactic": "cases l <;> rfl"
}
] |
[
865,
96
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
865,
9
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.prod_pprod
|
[] |
[
898,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
896,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
subset_upperBounds_lowerBounds
|
[] |
[
262,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
|
BoxIntegral.TaggedPrepartition.mem_mk
|
[] |
[
61,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.isNormal_iff_strictMono_limit
|
[] |
[
428,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean
|
DoubleCentralizer.sub_toProd
|
[] |
[
233,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/Analysis/InnerProductSpace/PiL2.lean
|
Complex.toBasis_orthonormalBasisOneI
|
[] |
[
644,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
642,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.set_eventuallyLE_iff_inf_principal_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.209114\nι : Sort x\ns t : Set α\nl : Filter α\n⊢ t ∈ l ⊓ 𝓟 s ↔ l ⊓ 𝓟 s ≤ l ⊓ 𝓟 t",
"tactic": "simp only [le_inf_iff, inf_le_left, true_and_iff, le_principal_iff]"
}
] |
[
1744,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1741,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.degreeOf_def
|
[
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.278467\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : DecidableEq σ\nn : σ\np : MvPolynomial σ R\n⊢ Multiset.count n (degrees p) = Multiset.count n (degrees p)",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.278467\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : DecidableEq σ\nn : σ\np : MvPolynomial σ R\n⊢ degreeOf n p = Multiset.count n (degrees p)",
"tactic": "rw [degreeOf]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.278467\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q : MvPolynomial σ R\ninst✝ : DecidableEq σ\nn : σ\np : MvPolynomial σ R\n⊢ Multiset.count n (degrees p) = Multiset.count n (degrees p)",
"tactic": "convert rfl"
}
] |
[
493,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
492,
1
] |
Mathlib/CategoryTheory/Abelian/Basic.lean
|
CategoryTheory.Abelian.monoLift_comp
|
[] |
[
502,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
499,
1
] |
Mathlib/Data/Real/Irrational.lean
|
Irrational.add_nat
|
[] |
[
259,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.integral_Ici_eq_integral_Ioi'
|
[] |
[
642,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
640,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ici_bot
|
[] |
[
1023,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1022,
1
] |
Mathlib/Data/Sum/Basic.lean
|
Sum.Lex.mono_right
|
[] |
[
524,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
523,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
strictAnti_comp_ofDual_iff
|
[] |
[
187,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.last_sub
|
[
{
"state_after": "no goals",
"state_before": "n m : ℕ\ni : Fin (n + 1)\n⊢ ↑(last n - i) = ↑(↑rev i)",
"tactic": "rw [coe_sub_iff_le.2 i.le_last, val_last, val_rev, Nat.succ_sub_succ_eq_sub]"
}
] |
[
2009,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2008,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
IsCompact.compl_mem_cocompact
|
[] |
[
538,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
537,
1
] |
Mathlib/Computability/Ackermann.lean
|
add_lt_ack
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ 0 + n < ack 0 n",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "m : ℕ\n⊢ m + 1 + 0 < ack (m + 1) 0",
"tactic": "simpa using add_lt_ack m 1"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ m + 1 + n + 1 ≤ m + (m + n + 2)",
"tactic": "linarith"
},
{
"state_after": "m n : ℕ\n⊢ m + n + 2 = m + 1 + n + 1",
"state_before": "m n : ℕ\n⊢ m + n + 2 = succ (m + 1 + n)",
"tactic": "rw [succ_eq_add_one]"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ m + n + 2 = m + 1 + n + 1",
"tactic": "ring_nf"
}
] |
[
188,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.mul_add
|
[] |
[
193,
25
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
192,
11
] |
Mathlib/GroupTheory/Coset.lean
|
Subgroup.card_eq_card_quotient_mul_card_subgroup
|
[
{
"state_after": "α : Type u_1\ninst✝³ : Group α\ns✝ t : Subgroup α\ninst✝² : Fintype α\ns : Subgroup α\ninst✝¹ : Fintype { x // x ∈ s }\ninst✝ : DecidablePred fun a => a ∈ s\n⊢ Fintype.card α = Fintype.card ((α ⧸ s) × { x // x ∈ s })",
"state_before": "α : Type u_1\ninst✝³ : Group α\ns✝ t : Subgroup α\ninst✝² : Fintype α\ns : Subgroup α\ninst✝¹ : Fintype { x // x ∈ s }\ninst✝ : DecidablePred fun a => a ∈ s\n⊢ Fintype.card α = Fintype.card (α ⧸ s) * Fintype.card { x // x ∈ s }",
"tactic": "rw [← Fintype.card_prod]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : Group α\ns✝ t : Subgroup α\ninst✝² : Fintype α\ns : Subgroup α\ninst✝¹ : Fintype { x // x ∈ s }\ninst✝ : DecidablePred fun a => a ∈ s\n⊢ Fintype.card α = Fintype.card ((α ⧸ s) × { x // x ∈ s })",
"tactic": "exact Fintype.card_congr Subgroup.groupEquivQuotientProdSubgroup"
}
] |
[
785,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
783,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.card_zero
|
[] |
[
626,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
625,
1
] |
Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean
|
Euclidean.isOpen_ball
|
[] |
[
78,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Localization.one_rel
|
[] |
[
420,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.biUnion_subset_biUnion_of_subset_left
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.518185\ninst✝ : DecidableEq β\ns s₁ s₂ : Finset α\nt✝ t₁ t₂ t : α → Finset β\nh : s₁ ⊆ s₂\nx : β\n⊢ x ∈ Finset.biUnion s₁ t → x ∈ Finset.biUnion s₂ t",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.518185\ninst✝ : DecidableEq β\ns s₁ s₂ : Finset α\nt✝ t₁ t₂ t : α → Finset β\nh : s₁ ⊆ s₂\n⊢ Finset.biUnion s₁ t ⊆ Finset.biUnion s₂ t",
"tactic": "intro x"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.518185\ninst✝ : DecidableEq β\ns s₁ s₂ : Finset α\nt✝ t₁ t₂ t : α → Finset β\nh : s₁ ⊆ s₂\nx : β\n⊢ (∃ a, a ∈ s₁ ∧ x ∈ t a) → ∃ a, a ∈ s₂ ∧ x ∈ t a",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.518185\ninst✝ : DecidableEq β\ns s₁ s₂ : Finset α\nt✝ t₁ t₂ t : α → Finset β\nh : s₁ ⊆ s₂\nx : β\n⊢ x ∈ Finset.biUnion s₁ t → x ∈ Finset.biUnion s₂ t",
"tactic": "simp only [and_imp, mem_biUnion, exists_prop]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.518185\ninst✝ : DecidableEq β\ns s₁ s₂ : Finset α\nt✝ t₁ t₂ t : α → Finset β\nh : s₁ ⊆ s₂\nx : β\n⊢ (∃ a, a ∈ s₁ ∧ x ∈ t a) → ∃ a, a ∈ s₂ ∧ x ∈ t a",
"tactic": "exact Exists.imp fun a ha => ⟨h ha.1, ha.2⟩"
}
] |
[
3616,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3612,
1
] |
src/lean/Init/Data/Array/Basic.lean
|
Array.size_push
|
[] |
[
65,
24
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
64,
9
] |
Mathlib/Analysis/Calculus/LHopital.lean
|
HasDerivAt.lhopital_zero_atBot
|
[
{
"state_after": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∃ v, v ∈ atBot ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt f (f' y) y\nhgg' : ∃ v, v ∈ atBot ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt g (g' y) y\nhg' : ∃ v, v ∈ atBot ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\n⊢ Tendsto (fun x => f x / g x) atBot l",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in atBot, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in atBot, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in atBot, g' x ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\n⊢ Tendsto (fun x => f x / g x) atBot l",
"tactic": "rw [eventually_iff_exists_mem] at *"
},
{
"state_after": "case intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhgg' : ∃ v, v ∈ atBot ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt g (g' y) y\nhg' : ∃ v, v ∈ atBot ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\n⊢ Tendsto (fun x => f x / g x) atBot l",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∃ v, v ∈ atBot ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt f (f' y) y\nhgg' : ∃ v, v ∈ atBot ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt g (g' y) y\nhg' : ∃ v, v ∈ atBot ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\n⊢ Tendsto (fun x => f x / g x) atBot l",
"tactic": "rcases hff' with ⟨s₁, hs₁, hff'⟩"
},
{
"state_after": "case intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhg' : ∃ v, v ∈ atBot ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\n⊢ Tendsto (fun x => f x / g x) atBot l",
"state_before": "case intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhgg' : ∃ v, v ∈ atBot ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt g (g' y) y\nhg' : ∃ v, v ∈ atBot ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\n⊢ Tendsto (fun x => f x / g x) atBot l",
"tactic": "rcases hgg' with ⟨s₂, hs₂, hgg'⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atBot\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\n⊢ Tendsto (fun x => f x / g x) atBot l",
"state_before": "case intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhg' : ∃ v, v ∈ atBot ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\n⊢ Tendsto (fun x => f x / g x) atBot l",
"tactic": "rcases hg' with ⟨s₃, hs₃, hg'⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atBot\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\n⊢ Tendsto (fun x => f x / g x) atBot l",
"state_before": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atBot\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\n⊢ Tendsto (fun x => f x / g x) atBot l",
"tactic": "let s := s₁ ∩ s₂ ∩ s₃"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atBot\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nhs : s ∈ atBot\n⊢ Tendsto (fun x => f x / g x) atBot l",
"state_before": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atBot\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\n⊢ Tendsto (fun x => f x / g x) atBot l",
"tactic": "have hs : s ∈ atBot := inter_mem (inter_mem hs₁ hs₂) hs₃"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atBot\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nhs : ∃ a, ∀ (b : ℝ), b ≤ a → b ∈ s\n⊢ Tendsto (fun x => f x / g x) atBot l",
"state_before": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atBot\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nhs : s ∈ atBot\n⊢ Tendsto (fun x => f x / g x) atBot l",
"tactic": "rw [mem_atBot_sets] at hs"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl✝ : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l✝\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atBot\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nl : ℝ\nhl : ∀ (b : ℝ), b ≤ l → b ∈ s\n⊢ Tendsto (fun x => f x / g x) atBot l✝",
"state_before": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atBot\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nhs : ∃ a, ∀ (b : ℝ), b ≤ a → b ∈ s\n⊢ Tendsto (fun x => f x / g x) atBot l",
"tactic": "rcases hs with ⟨l, hl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl✝ : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l✝\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atBot\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nl : ℝ\nhl : ∀ (b : ℝ), b ≤ l → b ∈ s\nhl' : Iio l ⊆ s\n⊢ Tendsto (fun x => f x / g x) atBot l✝",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl✝ : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l✝\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atBot\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nl : ℝ\nhl : ∀ (b : ℝ), b ≤ l → b ∈ s\n⊢ Tendsto (fun x => f x / g x) atBot l✝",
"tactic": "have hl' : Iio l ⊆ s := fun x hx => hl x (le_of_lt hx)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl✝ : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l✝\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atBot\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nl : ℝ\nhl : ∀ (b : ℝ), b ≤ l → b ∈ s\nhl' : Iio l ⊆ s\n⊢ Tendsto (fun x => f x / g x) atBot l✝",
"tactic": "refine' lhopital_zero_atBot_on_Iio _ _ (fun x hx => hg' x <| (hl' hx).2) hfbot hgbot hdiv <;>\n intro x hx <;> apply_assumption <;> first | exact (hl' hx).1.1| exact (hl' hx).1.2"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : ℝ\nhab : a < b\nl✝ : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l✝\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atBot\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nl : ℝ\nhl : ∀ (b : ℝ), b ≤ l → b ∈ s\nhl' : Iio l ⊆ s\nx : ℝ\nhx : x ∈ Iio l\n⊢ x ∈ s₂",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : ℝ\nhab : a < b\nl✝ : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l✝\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atBot\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nl : ℝ\nhl : ∀ (b : ℝ), b ≤ l → b ∈ s\nhl' : Iio l ⊆ s\nx : ℝ\nhx : x ∈ Iio l\n⊢ x ∈ s₂",
"tactic": "exact (hl' hx).1.1"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : ℝ\nhab : a < b\nl✝ : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l✝\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atBot\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nl : ℝ\nhl : ∀ (b : ℝ), b ≤ l → b ∈ s\nhl' : Iio l ⊆ s\nx : ℝ\nhx : x ∈ Iio l\n⊢ x ∈ s₂",
"tactic": "exact (hl' hx).1.2"
}
] |
[
375,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.map_nsmul
|
[] |
[
1224,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1223,
1
] |
Mathlib/Topology/MetricSpace/Polish.lean
|
PolishSpace.IsClopenable.compl
|
[
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.67736\ninst✝ : TopologicalSpace α\ns : Set α\nt : TopologicalSpace α\nt_le : t ≤ inst✝\nt_polish : PolishSpace α\nh : IsClosed s\nh' : IsOpen s\n⊢ IsClopenable (sᶜ)",
"state_before": "α : Type u_1\nβ : Type ?u.67736\ninst✝ : TopologicalSpace α\ns : Set α\nhs : IsClopenable s\n⊢ IsClopenable (sᶜ)",
"tactic": "rcases hs with ⟨t, t_le, t_polish, h, h'⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.67736\ninst✝ : TopologicalSpace α\ns : Set α\nt : TopologicalSpace α\nt_le : t ≤ inst✝\nt_polish : PolishSpace α\nh : IsClosed s\nh' : IsOpen s\n⊢ IsClopenable (sᶜ)",
"tactic": "exact ⟨t, t_le, t_polish, @IsOpen.isClosed_compl α t s h', @IsClosed.isOpen_compl α t s h⟩"
}
] |
[
403,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion_congr_of_surjective
|
[] |
[
444,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
IsCompact.isSeparable
|
[] |
[
2247,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2245,
1
] |
Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
|
CategoryTheory.tensorLeftHomEquiv_id_tensor_comp_evaluation
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nY Z : C\ninst✝ : HasLeftDual Z\nf : Y ⟶ ᘁZ\n⊢ (λ_ Y).inv ≫ (η_ (ᘁZ) Z ⊗ 𝟙 Y) ≫ (α_ (ᘁZ) Z Y).hom ≫ (𝟙 ᘁZ ⊗ (𝟙 Z ⊗ f) ≫ ε_ (ᘁZ) Z) = f ≫ (ρ_ ᘁZ).inv",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nY Z : C\ninst✝ : HasLeftDual Z\nf : Y ⟶ ᘁZ\n⊢ ↑(tensorLeftHomEquiv Y (ᘁZ) Z (𝟙_ C)) ((𝟙 Z ⊗ f) ≫ ε_ (ᘁZ) Z) = f ≫ (ρ_ ᘁZ).inv",
"tactic": "dsimp [tensorLeftHomEquiv]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nY Z : C\ninst✝ : HasLeftDual Z\nf : Y ⟶ ᘁZ\n⊢ (λ_ Y).inv ≫ (η_ (ᘁZ) Z ⊗ 𝟙 Y) ≫ (α_ (ᘁZ) Z Y).hom ≫ (𝟙 ᘁZ ⊗ 𝟙 Z ⊗ f) ≫ (𝟙 ᘁZ ⊗ ε_ (ᘁZ) Z) = f ≫ (ρ_ ᘁZ).inv",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nY Z : C\ninst✝ : HasLeftDual Z\nf : Y ⟶ ᘁZ\n⊢ (λ_ Y).inv ≫ (η_ (ᘁZ) Z ⊗ 𝟙 Y) ≫ (α_ (ᘁZ) Z Y).hom ≫ (𝟙 ᘁZ ⊗ (𝟙 Z ⊗ f) ≫ ε_ (ᘁZ) Z) = f ≫ (ρ_ ᘁZ).inv",
"tactic": "rw [id_tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nY Z : C\ninst✝ : HasLeftDual Z\nf : Y ⟶ ᘁZ\n⊢ (λ_ Y).inv ≫ (η_ (ᘁZ) Z ⊗ 𝟙 Y) ≫ (((𝟙 ᘁZ ⊗ 𝟙 Z) ⊗ f) ≫ (α_ (ᘁZ) Z ᘁZ).hom) ≫ (𝟙 ᘁZ ⊗ ε_ (ᘁZ) Z) = f ≫ (ρ_ ᘁZ).inv",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nY Z : C\ninst✝ : HasLeftDual Z\nf : Y ⟶ ᘁZ\n⊢ (λ_ Y).inv ≫ (η_ (ᘁZ) Z ⊗ 𝟙 Y) ≫ (α_ (ᘁZ) Z Y).hom ≫ (𝟙 ᘁZ ⊗ 𝟙 Z ⊗ f) ≫ (𝟙 ᘁZ ⊗ ε_ (ᘁZ) Z) = f ≫ (ρ_ ᘁZ).inv",
"tactic": "slice_lhs 3 4 => rw [← associator_naturality]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nY Z : C\ninst✝ : HasLeftDual Z\nf : Y ⟶ ᘁZ\n⊢ (λ_ Y).inv ≫ (((𝟙 tensorUnit' ⊗ f) ≫ (η_ (ᘁZ) Z ⊗ 𝟙 ᘁZ)) ≫ (α_ (ᘁZ) Z ᘁZ).hom) ≫ (𝟙 ᘁZ ⊗ ε_ (ᘁZ) Z) = f ≫ (ρ_ ᘁZ).inv",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nY Z : C\ninst✝ : HasLeftDual Z\nf : Y ⟶ ᘁZ\n⊢ (λ_ Y).inv ≫ (η_ (ᘁZ) Z ⊗ 𝟙 Y) ≫ (((𝟙 ᘁZ ⊗ 𝟙 Z) ⊗ f) ≫ (α_ (ᘁZ) Z ᘁZ).hom) ≫ (𝟙 ᘁZ ⊗ ε_ (ᘁZ) Z) = f ≫ (ρ_ ᘁZ).inv",
"tactic": "slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor, ← id_tensor_comp_tensor_id]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nY Z : C\ninst✝ : HasLeftDual Z\nf : Y ⟶ ᘁZ\n⊢ (λ_ Y).inv ≫ (𝟙 tensorUnit' ⊗ f) ≫ (λ_ ᘁZ).hom ≫ (ρ_ ᘁZ).inv = f ≫ (ρ_ ᘁZ).inv",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nY Z : C\ninst✝ : HasLeftDual Z\nf : Y ⟶ ᘁZ\n⊢ (λ_ Y).inv ≫ (((𝟙 tensorUnit' ⊗ f) ≫ (η_ (ᘁZ) Z ⊗ 𝟙 ᘁZ)) ≫ (α_ (ᘁZ) Z ᘁZ).hom) ≫ (𝟙 ᘁZ ⊗ ε_ (ᘁZ) Z) = f ≫ (ρ_ ᘁZ).inv",
"tactic": "slice_lhs 3 5 => rw [evaluation_coevaluation]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nY Z : C\ninst✝ : HasLeftDual Z\nf : Y ⟶ ᘁZ\n⊢ (λ_ Y).inv ≫ (𝟙 tensorUnit' ⊗ f) ≫ (λ_ ᘁZ).hom ≫ (ρ_ ᘁZ).inv = f ≫ (ρ_ ᘁZ).inv",
"tactic": "simp"
}
] |
[
516,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
509,
1
] |
Mathlib/Geometry/Euclidean/Sphere/Basic.lean
|
EuclideanGeometry.Sphere.coe_mk
|
[] |
[
86,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Order/RelClasses.lean
|
transitive_lt
|
[] |
[
905,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
904,
1
] |
Mathlib/Data/Sigma/Basic.lean
|
Sigma.forall
|
[] |
[
92,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Algebra/Order/Interval.lean
|
NonemptyInterval.mul_eq_one_iff
|
[
{
"state_after": "case refine'_1\nι : Type ?u.338842\nα : Type u_1\ninst✝ : OrderedCommGroup α\ns t : NonemptyInterval α\nh : s * t = 1\n⊢ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1\n\ncase refine'_2\nι : Type ?u.338842\nα : Type u_1\ninst✝ : OrderedCommGroup α\ns t : NonemptyInterval α\n⊢ (∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1) → s * t = 1",
"state_before": "ι : Type ?u.338842\nα : Type u_1\ninst✝ : OrderedCommGroup α\ns t : NonemptyInterval α\n⊢ s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1",
"tactic": "refine' ⟨fun h => _, _⟩"
},
{
"state_after": "case refine'_1\nι : Type ?u.338842\nα : Type u_1\ninst✝ : OrderedCommGroup α\ns t : NonemptyInterval α\nh : (s * t).fst = 1.fst ∧ (s * t).snd = 1.snd\n⊢ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1",
"state_before": "case refine'_1\nι : Type ?u.338842\nα : Type u_1\ninst✝ : OrderedCommGroup α\ns t : NonemptyInterval α\nh : s * t = 1\n⊢ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1",
"tactic": "rw [ext_iff, Prod.ext_iff] at h"
},
{
"state_after": "case refine'_1\nι : Type ?u.338842\nα : Type u_1\ninst✝ : OrderedCommGroup α\ns t : NonemptyInterval α\nh : (s * t).fst = 1.fst ∧ (s * t).snd = 1.snd\nthis : s.fst = s.snd ∧ t.fst = t.snd\n⊢ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1",
"state_before": "case refine'_1\nι : Type ?u.338842\nα : Type u_1\ninst✝ : OrderedCommGroup α\ns t : NonemptyInterval α\nh : (s * t).fst = 1.fst ∧ (s * t).snd = 1.snd\n⊢ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1",
"tactic": "have := (mul_le_mul_iff_of_ge s.fst_le_snd t.fst_le_snd).1 (h.2.trans h.1.symm).le"
},
{
"state_after": "case refine'_1.refine'_1.h\nι : Type ?u.338842\nα : Type u_1\ninst✝ : OrderedCommGroup α\ns t : NonemptyInterval α\nh : (s * t).fst = 1.fst ∧ (s * t).snd = 1.snd\nthis : s.fst = s.snd ∧ t.fst = t.snd\n⊢ s.toProd = (s.fst, s.fst)\n\ncase refine'_1.refine'_2.h\nι : Type ?u.338842\nα : Type u_1\ninst✝ : OrderedCommGroup α\ns t : NonemptyInterval α\nh : (s * t).fst = 1.fst ∧ (s * t).snd = 1.snd\nthis : s.fst = s.snd ∧ t.fst = t.snd\n⊢ t.toProd = (t.fst, t.fst)",
"state_before": "case refine'_1\nι : Type ?u.338842\nα : Type u_1\ninst✝ : OrderedCommGroup α\ns t : NonemptyInterval α\nh : (s * t).fst = 1.fst ∧ (s * t).snd = 1.snd\nthis : s.fst = s.snd ∧ t.fst = t.snd\n⊢ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1",
"tactic": "refine' ⟨s.fst, t.fst, _, _, h.1⟩ <;> apply NonemptyInterval.ext <;> dsimp [pure]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.refine'_1.h\nι : Type ?u.338842\nα : Type u_1\ninst✝ : OrderedCommGroup α\ns t : NonemptyInterval α\nh : (s * t).fst = 1.fst ∧ (s * t).snd = 1.snd\nthis : s.fst = s.snd ∧ t.fst = t.snd\n⊢ s.toProd = (s.fst, s.fst)",
"tactic": "nth_rw 2 [this.1]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.refine'_2.h\nι : Type ?u.338842\nα : Type u_1\ninst✝ : OrderedCommGroup α\ns t : NonemptyInterval α\nh : (s * t).fst = 1.fst ∧ (s * t).snd = 1.snd\nthis : s.fst = s.snd ∧ t.fst = t.snd\n⊢ t.toProd = (t.fst, t.fst)",
"tactic": "nth_rw 2 [this.2]"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro\nι : Type ?u.338842\nα : Type u_1\ninst✝ : OrderedCommGroup α\nb c : α\nh : b * c = 1\n⊢ pure b * pure c = 1",
"state_before": "case refine'_2\nι : Type ?u.338842\nα : Type u_1\ninst✝ : OrderedCommGroup α\ns t : NonemptyInterval α\n⊢ (∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1) → s * t = 1",
"tactic": "rintro ⟨b, c, rfl, rfl, h⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.intro.intro\nι : Type ?u.338842\nα : Type u_1\ninst✝ : OrderedCommGroup α\nb c : α\nh : b * c = 1\n⊢ pure b * pure c = 1",
"tactic": "rw [pure_mul_pure, h, pure_one]"
}
] |
[
540,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
532,
11
] |
Mathlib/Data/Int/GCD.lean
|
Int.gcd_div_gcd_div_gcd
|
[
{
"state_after": "no goals",
"state_before": "i j : ℤ\nH : 0 < gcd i j\n⊢ gcd (i / ↑(gcd i j)) (j / ↑(gcd i j)) = 1",
"tactic": "rw [gcd_div (gcd_dvd_left i j) (gcd_dvd_right i j), natAbs_ofNat, Nat.div_self H]"
}
] |
[
334,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
1
] |
Mathlib/Algebra/GradedMonoid.lean
|
List.dProdIndex_cons
|
[] |
[
373,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/Topology/UnitInterval.lean
|
unitInterval.two_mul_sub_one_mem_iff
|
[
{
"state_after": "no goals",
"state_before": "t : ℝ\n⊢ 2 * t - 1 ∈ I ↔ t ∈ Icc (1 / 2) 1",
"tactic": "constructor <;> rintro ⟨h₁, h₂⟩ <;> constructor <;> linarith"
}
] |
[
178,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Std/Logic.lean
|
exists_eq_left'
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u_1\np q : α → Prop\nb : Prop\na' : α\n⊢ (∃ a, a' = a ∧ p a) ↔ p a'",
"tactic": "simp [@eq_comm _ a']"
}
] |
[
467,
88
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
467,
9
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Function.Injective.map_atTop_finset_prod_eq
|
[
{
"state_after": "ι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\n⊢ Filter.map (fun s => ∏ i in s, f (g i)) atTop = Filter.map (fun s => ∏ i in s, f i) atTop",
"state_before": "ι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\n⊢ Filter.map (fun s => ∏ i in s, f (g i)) atTop = Filter.map (fun s => ∏ i in s, f i) atTop",
"tactic": "haveI := Classical.decEq β"
},
{
"state_after": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\n⊢ ∃ v, ∀ (v' : Finset γ), v ⊆ v' → ∃ u', s ⊆ u' ∧ ∏ x in u', f x = ∏ b in v', f (g b)\n\ncase a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset γ\n⊢ ∃ v, ∀ (v' : Finset β), v ⊆ v' → ∃ u', s ⊆ u' ∧ ∏ x in u', f (g x) = ∏ b in v', f b",
"state_before": "ι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\n⊢ Filter.map (fun s => ∏ i in s, f (g i)) atTop = Filter.map (fun s => ∏ i in s, f i) atTop",
"tactic": "apply le_antisymm <;> refine' map_atTop_finset_prod_le_of_prod_eq fun s => _"
},
{
"state_after": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\nt : Finset γ\nht : Finset.preimage s g (_ : InjOn g (g ⁻¹' ↑s)) ⊆ t\n⊢ ∃ u', s ⊆ u' ∧ ∏ x in u', f x = ∏ b in t, f (g b)",
"state_before": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\n⊢ ∃ v, ∀ (v' : Finset γ), v ⊆ v' → ∃ u', s ⊆ u' ∧ ∏ x in u', f x = ∏ b in v', f (g b)",
"tactic": "refine' ⟨s.preimage g (hg.injOn _), fun t ht => _⟩"
},
{
"state_after": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\nt : Finset γ\nht : Finset.preimage s g (_ : InjOn g (g ⁻¹' ↑s)) ⊆ t\n⊢ ∏ x in Finset.image g t ∪ s, f x = ∏ b in t, f (g b)",
"state_before": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\nt : Finset γ\nht : Finset.preimage s g (_ : InjOn g (g ⁻¹' ↑s)) ⊆ t\n⊢ ∃ u', s ⊆ u' ∧ ∏ x in u', f x = ∏ b in t, f (g b)",
"tactic": "refine' ⟨t.image g ∪ s, Finset.subset_union_right _ _, _⟩"
},
{
"state_after": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\nt : Finset γ\nht : Finset.preimage s g (_ : InjOn g (g ⁻¹' ↑s)) ⊆ t\n⊢ ∏ x in Finset.image g t ∪ s, f x = ∏ x in Finset.image (fun x => g x) t, f x",
"state_before": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\nt : Finset γ\nht : Finset.preimage s g (_ : InjOn g (g ⁻¹' ↑s)) ⊆ t\n⊢ ∏ x in Finset.image g t ∪ s, f x = ∏ b in t, f (g b)",
"tactic": "rw [← Finset.prod_image (hg.injOn _)]"
},
{
"state_after": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\nt : Finset γ\nht : Finset.preimage s g (_ : InjOn g (g ⁻¹' ↑s)) ⊆ t\n⊢ ∀ (x : β), x ∈ Finset.image g t ∪ s → ¬x ∈ Finset.image g t → f x = 1",
"state_before": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\nt : Finset γ\nht : Finset.preimage s g (_ : InjOn g (g ⁻¹' ↑s)) ⊆ t\n⊢ ∏ x in Finset.image g t ∪ s, f x = ∏ x in Finset.image (fun x => g x) t, f x",
"tactic": "refine' (prod_subset (subset_union_left _ _) _).symm"
},
{
"state_after": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\nt : Finset γ\nht : Finset.preimage s g (_ : InjOn g (g ⁻¹' ↑s)) ⊆ t\n⊢ ∀ (x : β), (∃ a, a ∈ t ∧ g a = x) ∨ x ∈ s → (¬∃ a, a ∈ t ∧ g a = x) → f x = 1",
"state_before": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\nt : Finset γ\nht : Finset.preimage s g (_ : InjOn g (g ⁻¹' ↑s)) ⊆ t\n⊢ ∀ (x : β), x ∈ Finset.image g t ∪ s → ¬x ∈ Finset.image g t → f x = 1",
"tactic": "simp only [Finset.mem_union, Finset.mem_image]"
},
{
"state_after": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\nt : Finset γ\nht : Finset.preimage s g (_ : InjOn g (g ⁻¹' ↑s)) ⊆ t\ny : β\nhy : (∃ a, a ∈ t ∧ g a = y) ∨ y ∈ s\nhyt : ¬∃ a, a ∈ t ∧ g a = y\n⊢ y ∈ Set.range g → ∃ a, a ∈ t ∧ g a = y",
"state_before": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\nt : Finset γ\nht : Finset.preimage s g (_ : InjOn g (g ⁻¹' ↑s)) ⊆ t\n⊢ ∀ (x : β), (∃ a, a ∈ t ∧ g a = x) ∨ x ∈ s → (¬∃ a, a ∈ t ∧ g a = x) → f x = 1",
"tactic": "refine' fun y hy hyt => hf y (mt _ hyt)"
},
{
"state_after": "case a.intro\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\nt : Finset γ\nht : Finset.preimage s g (_ : InjOn g (g ⁻¹' ↑s)) ⊆ t\nx : γ\nhy : (∃ a, a ∈ t ∧ g a = g x) ∨ g x ∈ s\nhyt : ¬∃ a, a ∈ t ∧ g a = g x\n⊢ ∃ a, a ∈ t ∧ g a = g x",
"state_before": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\nt : Finset γ\nht : Finset.preimage s g (_ : InjOn g (g ⁻¹' ↑s)) ⊆ t\ny : β\nhy : (∃ a, a ∈ t ∧ g a = y) ∨ y ∈ s\nhyt : ¬∃ a, a ∈ t ∧ g a = y\n⊢ y ∈ Set.range g → ∃ a, a ∈ t ∧ g a = y",
"tactic": "rintro ⟨x, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case a.intro\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset β\nt : Finset γ\nht : Finset.preimage s g (_ : InjOn g (g ⁻¹' ↑s)) ⊆ t\nx : γ\nhy : (∃ a, a ∈ t ∧ g a = g x) ∨ g x ∈ s\nhyt : ¬∃ a, a ∈ t ∧ g a = g x\n⊢ ∃ a, a ∈ t ∧ g a = g x",
"tactic": "exact ⟨x, ht (Finset.mem_preimage.2 <| hy.resolve_left hyt), rfl⟩"
},
{
"state_after": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset γ\nt : Finset β\nht : Finset.image g s ⊆ t\n⊢ ∃ u', s ⊆ u' ∧ ∏ x in u', f (g x) = ∏ b in t, f b",
"state_before": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset γ\n⊢ ∃ v, ∀ (v' : Finset β), v ⊆ v' → ∃ u', s ⊆ u' ∧ ∏ x in u', f (g x) = ∏ b in v', f b",
"tactic": "refine' ⟨s.image g, fun t ht => _⟩"
},
{
"state_after": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset γ\nt : Finset β\nht : Finset.image g s ⊆ t\n⊢ ∃ u', s ⊆ u' ∧ ∏ x in u', f (g x) = ∏ x in Finset.preimage t g (_ : InjOn g (g ⁻¹' ↑t)), f (g x)",
"state_before": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset γ\nt : Finset β\nht : Finset.image g s ⊆ t\n⊢ ∃ u', s ⊆ u' ∧ ∏ x in u', f (g x) = ∏ b in t, f b",
"tactic": "simp only [← prod_preimage _ _ (hg.injOn _) _ fun x _ => hf x]"
},
{
"state_after": "no goals",
"state_before": "case a\nι : Type ?u.378253\nι' : Type ?u.378256\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\ninst✝ : CommMonoid α\ng : γ → β\nhg : Injective g\nf : β → α\nhf : ∀ (x : β), ¬x ∈ Set.range g → f x = 1\nthis : DecidableEq β\ns : Finset γ\nt : Finset β\nht : Finset.image g s ⊆ t\n⊢ ∃ u', s ⊆ u' ∧ ∏ x in u', f (g x) = ∏ x in Finset.preimage t g (_ : InjOn g (g ⁻¹' ↑t)), f (g x)",
"tactic": "exact ⟨_, (image_subset_iff_subset_preimage _).1 ht, rfl⟩"
}
] |
[
1972,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1957,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.toDual_inf
|
[] |
[
491,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/Data/QPF/Multivariate/Basic.lean
|
MvQPF.suppPreservation_iff_isUniform
|
[
{
"state_after": "case mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\n⊢ SuppPreservation → IsUniform\n\ncase mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\n⊢ IsUniform → SuppPreservation",
"state_before": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\n⊢ SuppPreservation ↔ IsUniform",
"tactic": "constructor"
},
{
"state_after": "case mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\nα : TypeVec n\na a' : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nf' : MvPFunctor.B (P F) a' ⟹ α\nh' : abs { fst := a, snd := f } = abs { fst := a', snd := f' }\ni : Fin2 n\n⊢ f i '' univ = f' i '' univ",
"state_before": "case mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\n⊢ SuppPreservation → IsUniform",
"tactic": "intro h α a a' f f' h' i"
},
{
"state_after": "no goals",
"state_before": "case mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\nα : TypeVec n\na a' : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nf' : MvPFunctor.B (P F) a' ⟹ α\nh' : abs { fst := a, snd := f } = abs { fst := a', snd := f' }\ni : Fin2 n\n⊢ f i '' univ = f' i '' univ",
"tactic": "rw [← MvPFunctor.supp_eq, ← MvPFunctor.supp_eq, ← h, h', h]"
},
{
"state_after": "case mpr.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\n⊢ supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }",
"state_before": "case mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\n⊢ IsUniform → SuppPreservation",
"tactic": "rintro h α ⟨a, f⟩"
},
{
"state_after": "case mpr.mk.h.h\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nx✝¹ : Fin2 n\nx✝ : α x✝¹\n⊢ x✝ ∈ supp (abs { fst := a, snd := f }) x✝¹ ↔ x✝ ∈ supp { fst := a, snd := f } x✝¹",
"state_before": "case mpr.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\n⊢ supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case mpr.mk.h.h\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : IsUniform\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nx✝¹ : Fin2 n\nx✝ : α x✝¹\n⊢ x✝ ∈ supp (abs { fst := a, snd := f }) x✝¹ ↔ x✝ ∈ supp { fst := a, snd := f } x✝¹",
"tactic": "rwa [supp_eq_of_isUniform, MvPFunctor.supp_eq]"
}
] |
[
267,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.add_toZNum
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.456739\nm n : Num\n⊢ toZNum (m + n) = toZNum m + toZNum n",
"tactic": "cases m <;> cases n <;> rfl"
}
] |
[
800,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
799,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieHom.mk_coe
|
[
{
"state_after": "case h\nR : Type u\nL₁ : Type v\nL₂ : Type w\nL₃ : Type w₁\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L₁\ninst✝⁴ : LieAlgebra R L₁\ninst✝³ : LieRing L₂\ninst✝² : LieAlgebra R L₂\ninst✝¹ : LieRing L₃\ninst✝ : LieAlgebra R L₃\nf : L₁ →ₗ⁅R⁆ L₂\nh₁ : ∀ (x y : L₁), ↑f (x + y) = ↑f x + ↑f y\nh₂ :\n ∀ (r : R) (x : L₁),\n AddHom.toFun { toFun := ↑f, map_add' := h₁ } (r • x) =\n ↑(RingHom.id R) r • AddHom.toFun { toFun := ↑f, map_add' := h₁ } x\nh₃ :\n ∀ {x y : L₁},\n AddHom.toFun { toAddHom := { toFun := ↑f, map_add' := h₁ }, map_smul' := h₂ }.toAddHom ⁅x, y⁆ =\n ⁅AddHom.toFun { toAddHom := { toFun := ↑f, map_add' := h₁ }, map_smul' := h₂ }.toAddHom x,\n AddHom.toFun { toAddHom := { toFun := ↑f, map_add' := h₁ }, map_smul' := h₂ }.toAddHom y⁆\nx✝ : L₁\n⊢ ↑{ toLinearMap := { toAddHom := { toFun := ↑f, map_add' := h₁ }, map_smul' := h₂ }, map_lie' := h₃ } x✝ = ↑f x✝",
"state_before": "R : Type u\nL₁ : Type v\nL₂ : Type w\nL₃ : Type w₁\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L₁\ninst✝⁴ : LieAlgebra R L₁\ninst✝³ : LieRing L₂\ninst✝² : LieAlgebra R L₂\ninst✝¹ : LieRing L₃\ninst✝ : LieAlgebra R L₃\nf : L₁ →ₗ⁅R⁆ L₂\nh₁ : ∀ (x y : L₁), ↑f (x + y) = ↑f x + ↑f y\nh₂ :\n ∀ (r : R) (x : L₁),\n AddHom.toFun { toFun := ↑f, map_add' := h₁ } (r • x) =\n ↑(RingHom.id R) r • AddHom.toFun { toFun := ↑f, map_add' := h₁ } x\nh₃ :\n ∀ {x y : L₁},\n AddHom.toFun { toAddHom := { toFun := ↑f, map_add' := h₁ }, map_smul' := h₂ }.toAddHom ⁅x, y⁆ =\n ⁅AddHom.toFun { toAddHom := { toFun := ↑f, map_add' := h₁ }, map_smul' := h₂ }.toAddHom x,\n AddHom.toFun { toAddHom := { toFun := ↑f, map_add' := h₁ }, map_smul' := h₂ }.toAddHom y⁆\n⊢ { toLinearMap := { toAddHom := { toFun := ↑f, map_add' := h₁ }, map_smul' := h₂ }, map_lie' := h₃ } = f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nL₁ : Type v\nL₂ : Type w\nL₃ : Type w₁\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L₁\ninst✝⁴ : LieAlgebra R L₁\ninst✝³ : LieRing L₂\ninst✝² : LieAlgebra R L₂\ninst✝¹ : LieRing L₃\ninst✝ : LieAlgebra R L₃\nf : L₁ →ₗ⁅R⁆ L₂\nh₁ : ∀ (x y : L₁), ↑f (x + y) = ↑f x + ↑f y\nh₂ :\n ∀ (r : R) (x : L₁),\n AddHom.toFun { toFun := ↑f, map_add' := h₁ } (r • x) =\n ↑(RingHom.id R) r • AddHom.toFun { toFun := ↑f, map_add' := h₁ } x\nh₃ :\n ∀ {x y : L₁},\n AddHom.toFun { toAddHom := { toFun := ↑f, map_add' := h₁ }, map_smul' := h₂ }.toAddHom ⁅x, y⁆ =\n ⁅AddHom.toFun { toAddHom := { toFun := ↑f, map_add' := h₁ }, map_smul' := h₂ }.toAddHom x,\n AddHom.toFun { toAddHom := { toFun := ↑f, map_add' := h₁ }, map_smul' := h₂ }.toAddHom y⁆\nx✝ : L₁\n⊢ ↑{ toLinearMap := { toAddHom := { toFun := ↑f, map_add' := h₁ }, map_smul' := h₂ }, map_lie' := h₃ } x✝ = ↑f x✝",
"tactic": "rfl"
}
] |
[
401,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
399,
1
] |
Mathlib/Algebra/Ring/Equiv.lean
|
RingEquiv.map_pow
|
[] |
[
823,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
822,
11
] |
Mathlib/LinearAlgebra/Matrix/IsDiag.lean
|
Matrix.isDiag_neg_iff
|
[] |
[
92,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Algebra/GroupPower/Order.lean
|
MonoidHom.map_neg
|
[
{
"state_after": "no goals",
"state_before": "β : Type ?u.296855\nA : Type ?u.296858\nG : Type ?u.296861\nM : Type u_1\nR : Type u_2\ninst✝³ : Ring R\ninst✝² : Monoid M\ninst✝¹ : LinearOrder M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nf : R →* M\nx : R\n⊢ ↑f (-x) = ↑f x",
"tactic": "rw [← neg_one_mul, map_mul, map_neg_one, one_mul]"
}
] |
[
805,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
805,
1
] |
Mathlib/MeasureTheory/Measure/OpenPos.lean
|
EMetric.measure_closedBall_pos
|
[] |
[
207,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Order/SymmDiff.lean
|
symmDiff_right_injective
|
[] |
[
522,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
521,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.Measure.toSignedMeasure_eq_toSignedMeasure_iff
|
[
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.135386\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : toSignedMeasure μ = toSignedMeasure ν\n⊢ μ = ν\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.135386\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : μ = ν\n⊢ toSignedMeasure μ = toSignedMeasure ν",
"state_before": "α : Type u_1\nβ : Type ?u.135386\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\n⊢ toSignedMeasure μ = toSignedMeasure ν ↔ μ = ν",
"tactic": "refine' ⟨fun h => _, fun h => _⟩"
},
{
"state_after": "case refine'_1.h\nα : Type u_1\nβ : Type ?u.135386\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : toSignedMeasure μ = toSignedMeasure ν\ni : Set α\nhi : MeasurableSet i\n⊢ ↑↑μ i = ↑↑ν i",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.135386\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : toSignedMeasure μ = toSignedMeasure ν\n⊢ μ = ν",
"tactic": "ext1 i hi"
},
{
"state_after": "case refine'_1.h\nα : Type u_1\nβ : Type ?u.135386\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : toSignedMeasure μ = toSignedMeasure ν\ni : Set α\nhi : MeasurableSet i\nthis : ↑(toSignedMeasure μ) i = ↑(toSignedMeasure ν) i\n⊢ ↑↑μ i = ↑↑ν i",
"state_before": "case refine'_1.h\nα : Type u_1\nβ : Type ?u.135386\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : toSignedMeasure μ = toSignedMeasure ν\ni : Set α\nhi : MeasurableSet i\n⊢ ↑↑μ i = ↑↑ν i",
"tactic": "have : μ.toSignedMeasure i = ν.toSignedMeasure i := by rw [h]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.h\nα : Type u_1\nβ : Type ?u.135386\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : toSignedMeasure μ = toSignedMeasure ν\ni : Set α\nhi : MeasurableSet i\nthis : ↑(toSignedMeasure μ) i = ↑(toSignedMeasure ν) i\n⊢ ↑↑μ i = ↑↑ν i",
"tactic": "rwa [toSignedMeasure_apply_measurable hi, toSignedMeasure_apply_measurable hi,\n ENNReal.toReal_eq_toReal] at this\n <;> exact measure_ne_top _ _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.135386\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : toSignedMeasure μ = toSignedMeasure ν\ni : Set α\nhi : MeasurableSet i\n⊢ ↑(toSignedMeasure μ) i = ↑(toSignedMeasure ν) i",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.135386\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : μ = ν\n⊢ toSignedMeasure μ = toSignedMeasure ν",
"tactic": "congr"
}
] |
[
458,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
450,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Group.conjugates_subset_normal
|
[
{
"state_after": "G : Type u_1\nG' : Type ?u.435622\ninst✝² : Group G\ninst✝¹ : Group G'\nA : Type ?u.435631\ninst✝ : AddGroup A\ns : Set G\nN : Subgroup G\ntn : Subgroup.Normal N\na✝ : G\nh : a✝ ∈ N\na : G\nhc : a ∈ conjugatesOf a✝\n⊢ a ∈ ↑N",
"state_before": "G : Type u_1\nG' : Type ?u.435622\ninst✝² : Group G\ninst✝¹ : Group G'\nA : Type ?u.435631\ninst✝ : AddGroup A\ns : Set G\nN : Subgroup G\ntn : Subgroup.Normal N\na : G\nh : a ∈ N\n⊢ conjugatesOf a ⊆ ↑N",
"tactic": "rintro a hc"
},
{
"state_after": "case intro\nG : Type u_1\nG' : Type ?u.435622\ninst✝² : Group G\ninst✝¹ : Group G'\nA : Type ?u.435631\ninst✝ : AddGroup A\ns : Set G\nN : Subgroup G\ntn : Subgroup.Normal N\na : G\nh : a ∈ N\nc : G\nhc : c * a * c⁻¹ ∈ conjugatesOf a\n⊢ c * a * c⁻¹ ∈ ↑N",
"state_before": "G : Type u_1\nG' : Type ?u.435622\ninst✝² : Group G\ninst✝¹ : Group G'\nA : Type ?u.435631\ninst✝ : AddGroup A\ns : Set G\nN : Subgroup G\ntn : Subgroup.Normal N\na✝ : G\nh : a✝ ∈ N\na : G\nhc : a ∈ conjugatesOf a✝\n⊢ a ∈ ↑N",
"tactic": "obtain ⟨c, rfl⟩ := isConj_iff.1 hc"
},
{
"state_after": "no goals",
"state_before": "case intro\nG : Type u_1\nG' : Type ?u.435622\ninst✝² : Group G\ninst✝¹ : Group G'\nA : Type ?u.435631\ninst✝ : AddGroup A\ns : Set G\nN : Subgroup G\ntn : Subgroup.Normal N\na : G\nh : a ∈ N\nc : G\nhc : c * a * c⁻¹ ∈ conjugatesOf a\n⊢ c * a * c⁻¹ ∈ ↑N",
"tactic": "exact tn.conj_mem a h c"
}
] |
[
2430,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2426,
1
] |
Mathlib/CategoryTheory/Sums/Basic.lean
|
CategoryTheory.Functor.sum_map_inr
|
[] |
[
214,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.option_some
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.21098\nσ : Type ?u.21101\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\nn : ℕ\n⊢ (Nat.casesOn (encode (decode n)) 0 fun n => Nat.succ (Nat.succ n)) = encode (Option.map some (decode n))",
"tactic": "cases @decode α _ n <;> simp"
}
] |
[
246,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/GroupTheory/Perm/Option.lean
|
Equiv.Perm.decomposeOption_symm_sign
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ne : Perm α\n⊢ ↑sign (↑decomposeOption.symm (none, e)) = ↑sign e",
"tactic": "simp"
}
] |
[
81,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Tactic/NormNum/Core.lean
|
Mathlib.Meta.NormNum.IsNat.to_raw_eq
|
[] |
[
52,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
TopologicalSpace.Opens.localHomeomorphSubtypeCoe_coe
|
[] |
[
1337,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1336,
1
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.prod_same_eq
|
[] |
[
433,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/Order/Lattice.lean
|
le_sup_left
|
[] |
[
132,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Combinatorics/Pigeonhole.lean
|
Finset.exists_le_card_fiber_of_mul_le_card_of_maps_to
|
[] |
[
289,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
sup_himp_self_right
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.72997\nα : Type u_1\nβ : Type ?u.73003\ninst✝ : GeneralizedHeytingAlgebra α\na✝ b✝ c d a b : α\n⊢ a ⊔ b ⇨ b = a ⇨ b",
"tactic": "rw [sup_himp_distrib, himp_self, inf_top_eq]"
}
] |
[
433,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.range_snd
|
[] |
[
1285,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1284,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ring.not_isField_iff_exists_prime
|
[] |
[
774,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
768,
1
] |
Mathlib/Logic/Equiv/TransferInstance.lean
|
Equiv.mulEquiv_symm_apply
|
[
{
"state_after": "α : Type u\nβ : Type v\ne✝ e : α ≃ β\ninst✝ : Mul β\nb : β\n⊢ ↑(MulEquiv.symm (mulEquiv e)) b = ↑e.symm b",
"state_before": "α : Type u\nβ : Type v\ne✝ e : α ≃ β\ninst✝ : Mul β\nb : β\n⊢ ↑(MulEquiv.symm (mulEquiv e)) b = ↑e.symm b",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ne✝ e : α ≃ β\ninst✝ : Mul β\nb : β\n⊢ ↑(MulEquiv.symm (mulEquiv e)) b = ↑e.symm b",
"tactic": "rfl"
}
] |
[
158,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Topology/Instances/Real.lean
|
Real.uniformContinuous_const_mul
|
[] |
[
126,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
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