Split (1)

train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M	input stringlengths 73 2.09M
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	simp only [ModelWithCorners.prod, ModelWithCorners.source_eq, mem_univ, and_self, setOf_true, PartialEquiv.prod_target, ModelWithCorners.target_eq, ModelWithCorners.mk_coe, mem_inter_iff, mem_preimage, mem_prod] at m ⊢	case refine_1.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J)) ⊢ (fun x => x.2) x ∈ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J)	case refine_1.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : (↑I.symm x.1 ∈ f.source ∧ ↑J.symm x.2 ∈ g.source) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2)) ⊢ ↑J.symm x.2 ∈ g.source ∧ x.2 ∈ interior (range ↑J)	Please generate a tactic in lean4 to solve the state. STATE: case refine_1.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J)) ⊢ (fun x => x.2) x ∈ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	exact ⟨m.1.2, (subset_of_eq ei m.2).2⟩	case refine_1.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : (↑I.symm x.1 ∈ f.source ∧ ↑J.symm x.2 ∈ g.source) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2)) ⊢ ↑J.symm x.2 ∈ g.source ∧ x.2 ∈ interior (range ↑J)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case refine_1.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : (↑I.symm x.1 ∈ f.source ∧ ↑J.symm x.2 ∈ g.source) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2)) ⊢ ↑J.symm x.2 ∈ g.source ∧ x.2 ∈ interior (range ↑J) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	simp only [ModelWithCorners.prod, ModelWithCorners.source_eq, mem_univ, and_self, setOf_true, PartialEquiv.prod_target, ModelWithCorners.target_eq, ModelWithCorners.mk_symm, PartialEquiv.coe_symm_mk, PartialHomeomorph.prod_apply, ModelWithCorners.mk_coe, PartialHomeomorph.prod_toPartialEquiv, PartialEquiv.prod_source, mk_preimage_prod, image_subset_iff]	case refine_2 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ ↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J)) ⊆ (fun x => ↑(I.prod J) (↑(f.prod g) (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J))	case refine_2 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ (fun x => ↑I.symm x.1) ⁻¹' f.source ∩ (fun x => ↑J.symm x.2) ⁻¹' g.source ∩ interior (range fun x => (↑I x.1, ↑J x.2)) ⊆ (fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2))	Please generate a tactic in lean4 to solve the state. STATE: case refine_2 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ ↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J)) ⊆ (fun x => ↑(I.prod J) (↑(f.prod g) (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	intro x ⟨⟨m0,m1⟩,m2⟩	case refine_2 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ (fun x => ↑I.symm x.1) ⁻¹' f.source ∩ (fun x => ↑J.symm x.2) ⁻¹' g.source ∩ interior (range fun x => (↑I x.1, ↑J x.2)) ⊆ (fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2))	case refine_2 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.source m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.source m2 : x ∈ interior (range fun x => (↑I x.1, ↑J x.2)) ⊢ x ∈ (fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2))	Please generate a tactic in lean4 to solve the state. STATE: case refine_2 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ (fun x => ↑I.symm x.1) ⁻¹' f.source ∩ (fun x => ↑J.symm x.2) ⁻¹' g.source ∩ interior (range fun x => (↑I x.1, ↑J x.2)) ⊆ (fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	replace m2 := subset_of_eq ei m2	case refine_2 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.source m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.source m2 : x ∈ interior (range fun x => (↑I x.1, ↑J x.2)) ⊢ x ∈ (fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2))	case refine_2 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.source m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.source m2 : x ∈ interior (range ↑I) ×ˢ interior (range ↑J) ⊢ x ∈ (fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2))	Please generate a tactic in lean4 to solve the state. STATE: case refine_2 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.source m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.source m2 : x ∈ interior (range fun x => (↑I x.1, ↑J x.2)) ⊢ x ∈ (fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	exact subset_of_eq ei.symm ⟨fa.2.1.2 ⟨m0,m2.1⟩, ga.2.1.2 ⟨m1,m2.2⟩⟩	case refine_2 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.source m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.source m2 : x ∈ interior (range ↑I) ×ˢ interior (range ↑J) ⊢ x ∈ (fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case refine_2 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.source m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.source m2 : x ∈ interior (range ↑I) ×ˢ interior (range ↑J) ⊢ x ∈ (fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	apply AnalyticOn.prod	case refine_3 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ AnalyticOn 𝕜 (fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x))) (↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)))	case refine_3.hf 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ AnalyticOn 𝕜 (fun x => ↑I (↑(f.prod g).symm (↑(I.prod J).symm x)).1) (↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J))) case refine_3.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ AnalyticOn 𝕜 (fun x => ↑J (↑(f.prod g).symm (↑(I.prod J).symm x)).2) (↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)))	Please generate a tactic in lean4 to solve the state. STATE: case refine_3 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ AnalyticOn 𝕜 (fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x))) (↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J))) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	simp only [ModelWithCorners.symm, modelWithCorners_prod_toPartialEquiv, PartialEquiv.prod_symm, PartialEquiv.prod_coe, ModelWithCorners.toPartialEquiv_coe_symm, PartialHomeomorph.prod_apply, PartialHomeomorph.prod_toPartialEquiv, PartialEquiv.prod_source]	case refine_3.hf 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ AnalyticOn 𝕜 (fun x => ↑I (↑(f.prod g).symm (↑(I.prod J).symm x)).1) (↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)))	case refine_3.hf 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ AnalyticOn 𝕜 (fun x => ↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)))	Please generate a tactic in lean4 to solve the state. STATE: case refine_3.hf 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ AnalyticOn 𝕜 (fun x => ↑I (↑(f.prod g).symm (↑(I.prod J).symm x)).1) (↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J))) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	refine fa.2.2.1.comp (analyticOn_fst _) ?_	case refine_3.hf 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ AnalyticOn 𝕜 (fun x => ↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)))	case refine_3.hf 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ MapsTo (fun x => x.1) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I))	Please generate a tactic in lean4 to solve the state. STATE: case refine_3.hf 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ AnalyticOn 𝕜 (fun x => ↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	intro x m	case refine_3.hf 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ MapsTo (fun x => x.1) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I))	case refine_3.hf 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)) ⊢ (fun x => x.1) x ∈ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I)	Please generate a tactic in lean4 to solve the state. STATE: case refine_3.hf 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ MapsTo (fun x => x.1) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	simp only [ModelWithCorners.prod, ModelWithCorners.source_eq, mem_univ, and_self, setOf_true, PartialEquiv.prod_target, ModelWithCorners.target_eq, ModelWithCorners.mk_coe, mem_inter_iff, mem_preimage, mem_prod] at m ⊢	case refine_3.hf 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)) ⊢ (fun x => x.1) x ∈ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I)	case refine_3.hf 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : (↑I.symm x.1 ∈ f.target ∧ ↑J.symm x.2 ∈ g.target) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2)) ⊢ ↑I.symm x.1 ∈ f.target ∧ x.1 ∈ interior (range ↑I)	Please generate a tactic in lean4 to solve the state. STATE: case refine_3.hf 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)) ⊢ (fun x => x.1) x ∈ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	exact ⟨m.1.1, (subset_of_eq ei m.2).1⟩	case refine_3.hf 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : (↑I.symm x.1 ∈ f.target ∧ ↑J.symm x.2 ∈ g.target) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2)) ⊢ ↑I.symm x.1 ∈ f.target ∧ x.1 ∈ interior (range ↑I)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case refine_3.hf 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : (↑I.symm x.1 ∈ f.target ∧ ↑J.symm x.2 ∈ g.target) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2)) ⊢ ↑I.symm x.1 ∈ f.target ∧ x.1 ∈ interior (range ↑I) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	simp only [ModelWithCorners.symm, modelWithCorners_prod_toPartialEquiv, PartialEquiv.prod_symm, PartialEquiv.prod_coe, ModelWithCorners.toPartialEquiv_coe_symm, PartialHomeomorph.prod_apply, PartialHomeomorph.prod_toPartialEquiv, PartialEquiv.prod_source]	case refine_3.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ AnalyticOn 𝕜 (fun x => ↑J (↑(f.prod g).symm (↑(I.prod J).symm x)).2) (↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)))	case refine_3.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ AnalyticOn 𝕜 (fun x => ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)))	Please generate a tactic in lean4 to solve the state. STATE: case refine_3.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ AnalyticOn 𝕜 (fun x => ↑J (↑(f.prod g).symm (↑(I.prod J).symm x)).2) (↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J))) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	refine ga.2.2.1.comp (analyticOn_snd _) ?_	case refine_3.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ AnalyticOn 𝕜 (fun x => ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)))	case refine_3.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ MapsTo (fun x => x.2) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J))	Please generate a tactic in lean4 to solve the state. STATE: case refine_3.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ AnalyticOn 𝕜 (fun x => ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	intro x m	case refine_3.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ MapsTo (fun x => x.2) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J))	case refine_3.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)) ⊢ (fun x => x.2) x ∈ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J)	Please generate a tactic in lean4 to solve the state. STATE: case refine_3.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ MapsTo (fun x => x.2) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	simp only [ModelWithCorners.prod, ModelWithCorners.source_eq, mem_univ, and_self, setOf_true, PartialEquiv.prod_target, ModelWithCorners.target_eq, ModelWithCorners.mk_coe, mem_inter_iff, mem_preimage, mem_prod] at m ⊢	case refine_3.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)) ⊢ (fun x => x.2) x ∈ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J)	case refine_3.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : (↑I.symm x.1 ∈ f.target ∧ ↑J.symm x.2 ∈ g.target) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2)) ⊢ ↑J.symm x.2 ∈ g.target ∧ x.2 ∈ interior (range ↑J)	Please generate a tactic in lean4 to solve the state. STATE: case refine_3.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)) ⊢ (fun x => x.2) x ∈ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	exact ⟨m.1.2, (subset_of_eq ei m.2).2⟩	case refine_3.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : (↑I.symm x.1 ∈ f.target ∧ ↑J.symm x.2 ∈ g.target) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2)) ⊢ ↑J.symm x.2 ∈ g.target ∧ x.2 ∈ interior (range ↑J)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case refine_3.hg 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m : (↑I.symm x.1 ∈ f.target ∧ ↑J.symm x.2 ∈ g.target) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2)) ⊢ ↑J.symm x.2 ∈ g.target ∧ x.2 ∈ interior (range ↑J) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	simp only [ModelWithCorners.prod, ModelWithCorners.source_eq, mem_univ, and_self, setOf_true, PartialEquiv.prod_target, ModelWithCorners.target_eq, ModelWithCorners.mk_symm, PartialEquiv.coe_symm_mk, PartialHomeomorph.prod_apply, ModelWithCorners.mk_coe, PartialHomeomorph.prod_toPartialEquiv, PartialEquiv.prod_source, mk_preimage_prod, image_subset_iff]	case refine_4 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ ↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)) ⊆ (fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J))	case refine_4 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ (fun x => ↑I.symm x.1) ⁻¹' f.target ∩ (fun x => ↑J.symm x.2) ⁻¹' g.target ∩ interior (range fun x => (↑I x.1, ↑J x.2)) ⊆ (fun x => (↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2))	Please generate a tactic in lean4 to solve the state. STATE: case refine_4 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ ↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)) ⊆ (fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	intro x ⟨⟨m0,m1⟩,m2⟩	case refine_4 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ (fun x => ↑I.symm x.1) ⁻¹' f.target ∩ (fun x => ↑J.symm x.2) ⁻¹' g.target ∩ interior (range fun x => (↑I x.1, ↑J x.2)) ⊆ (fun x => (↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2))	case refine_4 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.target m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.target m2 : x ∈ interior (range fun x => (↑I x.1, ↑J x.2)) ⊢ x ∈ (fun x => (↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2))	Please generate a tactic in lean4 to solve the state. STATE: case refine_4 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) ⊢ (fun x => ↑I.symm x.1) ⁻¹' f.target ∩ (fun x => ↑J.symm x.2) ⁻¹' g.target ∩ interior (range fun x => (↑I x.1, ↑J x.2)) ⊆ (fun x => (↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	replace m2 := subset_of_eq ei m2	case refine_4 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.target m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.target m2 : x ∈ interior (range fun x => (↑I x.1, ↑J x.2)) ⊢ x ∈ (fun x => (↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2))	case refine_4 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.target m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.target m2 : x ∈ interior (range ↑I) ×ˢ interior (range ↑J) ⊢ x ∈ (fun x => (↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2))	Please generate a tactic in lean4 to solve the state. STATE: case refine_4 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.target m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.target m2 : x ∈ interior (range fun x => (↑I x.1, ↑J x.2)) ⊢ x ∈ (fun x => (↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticGroupoid_prod	[79, 1]	[150, 72]	exact subset_of_eq ei.symm ⟨fa.2.2.2 ⟨m0,m2.1⟩, ga.2.2.2 ⟨m1,m2.2⟩⟩	case refine_4 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.target m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.target m2 : x ∈ interior (range ↑I) ×ˢ interior (range ↑J) ⊢ x ∈ (fun x => (↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case refine_4 𝕜 : Type inst✝⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : TopologicalSpace A F B : Type inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace 𝕜 F inst✝ : TopologicalSpace B I : ModelWithCorners 𝕜 E A J : ModelWithCorners 𝕜 F B f : PartialHomeomorph A A g : PartialHomeomorph B B er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) fa : f ∈ contDiffGroupoid ⊤ I ∧ (AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧ AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I) ga : g ∈ contDiffGroupoid ⊤ J ∧ (AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧ AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J) x : E × F m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.target m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.target m2 : x ∈ interior (range ↑I) ×ˢ interior (range ↑J) ⊢ x ∈ (fun x => (↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	extChartAt_self_analytic	[208, 1]	[223, 25]	apply AnalyticOn.congr (f := fun z ↦ z)	𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E ⊢ AnalyticOn 𝕜 (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) (↑𝓘(𝕜, E) '' (f.symm.trans f).source)	case hs 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E ⊢ IsOpen (↑𝓘(𝕜, E) '' (f.symm.trans f).source) case hf 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E ⊢ AnalyticOn 𝕜 (fun z => z) (↑𝓘(𝕜, E) '' (f.symm.trans f).source) case hg 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E ⊢ EqOn (fun z => z) (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) (↑𝓘(𝕜, E) '' (f.symm.trans f).source)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E ⊢ AnalyticOn 𝕜 (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) (↑𝓘(𝕜, E) '' (f.symm.trans f).source) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	extChartAt_self_analytic	[208, 1]	[223, 25]	simp only [modelWithCornersSelf_coe, id_eq, image_id', PartialHomeomorph.trans_toPartialEquiv, PartialHomeomorph.symm_toPartialEquiv, PartialEquiv.trans_source, PartialEquiv.symm_source, PartialHomeomorph.coe_coe_symm]	case hs 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E ⊢ IsOpen (↑𝓘(𝕜, E) '' (f.symm.trans f).source)	case hs 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E ⊢ IsOpen (f.target ∩ ↑f.symm ⁻¹' f.source)	Please generate a tactic in lean4 to solve the state. STATE: case hs 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E ⊢ IsOpen (↑𝓘(𝕜, E) '' (f.symm.trans f).source) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	extChartAt_self_analytic	[208, 1]	[223, 25]	exact f.isOpen_inter_preimage_symm f.open_source	case hs 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E ⊢ IsOpen (f.target ∩ ↑f.symm ⁻¹' f.source)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hs 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E ⊢ IsOpen (f.target ∩ ↑f.symm ⁻¹' f.source) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	extChartAt_self_analytic	[208, 1]	[223, 25]	exact analyticOn_id _	case hf 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E ⊢ AnalyticOn 𝕜 (fun z => z) (↑𝓘(𝕜, E) '' (f.symm.trans f).source)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hf 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E ⊢ AnalyticOn 𝕜 (fun z => z) (↑𝓘(𝕜, E) '' (f.symm.trans f).source) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	extChartAt_self_analytic	[208, 1]	[223, 25]	intro x m	case hg 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E ⊢ EqOn (fun z => z) (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) (↑𝓘(𝕜, E) '' (f.symm.trans f).source)	case hg 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E x : E m : x ∈ ↑𝓘(𝕜, E) '' (f.symm.trans f).source ⊢ (fun z => z) x = (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) x	Please generate a tactic in lean4 to solve the state. STATE: case hg 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E ⊢ EqOn (fun z => z) (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) (↑𝓘(𝕜, E) '' (f.symm.trans f).source) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	extChartAt_self_analytic	[208, 1]	[223, 25]	simp only [modelWithCornersSelf_coe, id, image_id', PartialHomeomorph.trans_toPartialEquiv, PartialHomeomorph.symm_toPartialEquiv, PartialEquiv.trans_source, PartialEquiv.symm_source, PartialHomeomorph.coe_coe_symm, mem_inter_iff, mem_preimage, Function.comp, modelWithCornersSelf_coe_symm, PartialHomeomorph.coe_trans] at m ⊢	case hg 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E x : E m : x ∈ ↑𝓘(𝕜, E) '' (f.symm.trans f).source ⊢ (fun z => z) x = (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) x	case hg 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E x : E m : x ∈ f.target ∧ ↑f.symm x ∈ f.source ⊢ x = ↑f (↑f.symm x)	Please generate a tactic in lean4 to solve the state. STATE: case hg 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E x : E m : x ∈ ↑𝓘(𝕜, E) '' (f.symm.trans f).source ⊢ (fun z => z) x = (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) x TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	extChartAt_self_analytic	[208, 1]	[223, 25]	rw [f.right_inv m.1]	case hg 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E x : E m : x ∈ f.target ∧ ↑f.symm x ∈ f.source ⊢ x = ↑f (↑f.symm x)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hg 𝕜 : Type inst✝³ : NontriviallyNormedField 𝕜 E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace 𝕜 E M : Type inst✝ : TopologicalSpace M f : PartialHomeomorph M E x : E m : x ∈ f.target ∧ ↑f.symm x ∈ f.source ⊢ x = ↑f (↑f.symm x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_iff	[263, 1]	[267, 54]	simp only [HolomorphicAt, ChartedSpace.liftPropAt_iff, extChartAt, PartialHomeomorph.extend_coe, PartialHomeomorph.extend_coe_symm, Function.comp]	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N x : M ⊢ HolomorphicAt I J f x ↔ ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	no goals	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N x : M ⊢ HolomorphicAt I J f x ↔ ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphic_iff	[270, 1]	[274, 92]	simp only [Holomorphic, holomorphicAt_iff, continuous_iff_continuousAt]	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N ⊢ Holomorphic I J f ↔ Continuous f ∧ ∀ (x : M), AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N ⊢ (∀ (x : M), ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)) ↔ (∀ (x : M), ContinuousAt f x) ∧ ∀ (x : M), AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N ⊢ Holomorphic I J f ↔ Continuous f ∧ ∀ (x : M), AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphic_iff	[270, 1]	[274, 92]	exact forall_and	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N ⊢ (∀ (x : M), ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)) ↔ (∀ (x : M), ContinuousAt f x) ∧ ∀ (x : M), AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	no goals	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N ⊢ (∀ (x : M), ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)) ↔ (∀ (x : M), ContinuousAt f x) ∧ ∀ (x : M), AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticAt_iff_holomorphicAt	[295, 1]	[300, 32]	simp only [holomorphicAt_iff, extChartAt_eq_refl, PartialEquiv.refl_coe, PartialEquiv.refl_symm, Function.id_comp, Function.comp_id, id_eq, iff_and_self]	𝕜 : Type inst✝³² : NontriviallyNormedField 𝕜 E A : Type inst✝³¹ : NormedAddCommGroup E inst✝³⁰ : NormedSpace 𝕜 E inst✝²⁹ : CompleteSpace E inst✝²⁸ : TopologicalSpace A F B : Type inst✝²⁷ : NormedAddCommGroup F inst✝²⁶ : NormedSpace 𝕜 F inst✝²⁵ : CompleteSpace F inst✝²⁴ : TopologicalSpace B G C : Type inst✝²³ : NormedAddCommGroup G inst✝²² : NormedSpace 𝕜 G inst✝²¹ : TopologicalSpace C H D : Type inst✝²⁰ : NormedAddCommGroup H inst✝¹⁹ : NormedSpace 𝕜 H inst✝¹⁸ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹⁷ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁶ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹⁵ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁴ : TopologicalSpace P inst✝¹³ : I.Boundaryless inst✝¹² : ChartedSpace A M cm : AnalyticManifold I M inst✝¹¹ : J.Boundaryless inst✝¹⁰ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁹ : K.Boundaryless inst✝⁸ : ChartedSpace C O co : AnalyticManifold K O inst✝⁷ : L.Boundaryless inst✝⁶ : ChartedSpace D P cp : AnalyticManifold L P inst✝⁵ : ChartedSpace A E inst✝⁴ : AnalyticManifold I E inst✝³ : ChartedSpace B F inst✝² : AnalyticManifold J F inst✝¹ : ExtChartEqRefl I inst✝ : ExtChartEqRefl J f : E → F x : E ⊢ AnalyticAt 𝕜 f x ↔ HolomorphicAt I J f x	𝕜 : Type inst✝³² : NontriviallyNormedField 𝕜 E A : Type inst✝³¹ : NormedAddCommGroup E inst✝³⁰ : NormedSpace 𝕜 E inst✝²⁹ : CompleteSpace E inst✝²⁸ : TopologicalSpace A F B : Type inst✝²⁷ : NormedAddCommGroup F inst✝²⁶ : NormedSpace 𝕜 F inst✝²⁵ : CompleteSpace F inst✝²⁴ : TopologicalSpace B G C : Type inst✝²³ : NormedAddCommGroup G inst✝²² : NormedSpace 𝕜 G inst✝²¹ : TopologicalSpace C H D : Type inst✝²⁰ : NormedAddCommGroup H inst✝¹⁹ : NormedSpace 𝕜 H inst✝¹⁸ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹⁷ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁶ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹⁵ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁴ : TopologicalSpace P inst✝¹³ : I.Boundaryless inst✝¹² : ChartedSpace A M cm : AnalyticManifold I M inst✝¹¹ : J.Boundaryless inst✝¹⁰ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁹ : K.Boundaryless inst✝⁸ : ChartedSpace C O co : AnalyticManifold K O inst✝⁷ : L.Boundaryless inst✝⁶ : ChartedSpace D P cp : AnalyticManifold L P inst✝⁵ : ChartedSpace A E inst✝⁴ : AnalyticManifold I E inst✝³ : ChartedSpace B F inst✝² : AnalyticManifold J F inst✝¹ : ExtChartEqRefl I inst✝ : ExtChartEqRefl J f : E → F x : E ⊢ AnalyticAt 𝕜 f x → ContinuousAt f x	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝³² : NontriviallyNormedField 𝕜 E A : Type inst✝³¹ : NormedAddCommGroup E inst✝³⁰ : NormedSpace 𝕜 E inst✝²⁹ : CompleteSpace E inst✝²⁸ : TopologicalSpace A F B : Type inst✝²⁷ : NormedAddCommGroup F inst✝²⁶ : NormedSpace 𝕜 F inst✝²⁵ : CompleteSpace F inst✝²⁴ : TopologicalSpace B G C : Type inst✝²³ : NormedAddCommGroup G inst✝²² : NormedSpace 𝕜 G inst✝²¹ : TopologicalSpace C H D : Type inst✝²⁰ : NormedAddCommGroup H inst✝¹⁹ : NormedSpace 𝕜 H inst✝¹⁸ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹⁷ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁶ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹⁵ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁴ : TopologicalSpace P inst✝¹³ : I.Boundaryless inst✝¹² : ChartedSpace A M cm : AnalyticManifold I M inst✝¹¹ : J.Boundaryless inst✝¹⁰ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁹ : K.Boundaryless inst✝⁸ : ChartedSpace C O co : AnalyticManifold K O inst✝⁷ : L.Boundaryless inst✝⁶ : ChartedSpace D P cp : AnalyticManifold L P inst✝⁵ : ChartedSpace A E inst✝⁴ : AnalyticManifold I E inst✝³ : ChartedSpace B F inst✝² : AnalyticManifold J F inst✝¹ : ExtChartEqRefl I inst✝ : ExtChartEqRefl J f : E → F x : E ⊢ AnalyticAt 𝕜 f x ↔ HolomorphicAt I J f x TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	analyticAt_iff_holomorphicAt	[295, 1]	[300, 32]	exact AnalyticAt.continuousAt	𝕜 : Type inst✝³² : NontriviallyNormedField 𝕜 E A : Type inst✝³¹ : NormedAddCommGroup E inst✝³⁰ : NormedSpace 𝕜 E inst✝²⁹ : CompleteSpace E inst✝²⁸ : TopologicalSpace A F B : Type inst✝²⁷ : NormedAddCommGroup F inst✝²⁶ : NormedSpace 𝕜 F inst✝²⁵ : CompleteSpace F inst✝²⁴ : TopologicalSpace B G C : Type inst✝²³ : NormedAddCommGroup G inst✝²² : NormedSpace 𝕜 G inst✝²¹ : TopologicalSpace C H D : Type inst✝²⁰ : NormedAddCommGroup H inst✝¹⁹ : NormedSpace 𝕜 H inst✝¹⁸ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹⁷ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁶ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹⁵ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁴ : TopologicalSpace P inst✝¹³ : I.Boundaryless inst✝¹² : ChartedSpace A M cm : AnalyticManifold I M inst✝¹¹ : J.Boundaryless inst✝¹⁰ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁹ : K.Boundaryless inst✝⁸ : ChartedSpace C O co : AnalyticManifold K O inst✝⁷ : L.Boundaryless inst✝⁶ : ChartedSpace D P cp : AnalyticManifold L P inst✝⁵ : ChartedSpace A E inst✝⁴ : AnalyticManifold I E inst✝³ : ChartedSpace B F inst✝² : AnalyticManifold J F inst✝¹ : ExtChartEqRefl I inst✝ : ExtChartEqRefl J f : E → F x : E ⊢ AnalyticAt 𝕜 f x → ContinuousAt f x	no goals	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝³² : NontriviallyNormedField 𝕜 E A : Type inst✝³¹ : NormedAddCommGroup E inst✝³⁰ : NormedSpace 𝕜 E inst✝²⁹ : CompleteSpace E inst✝²⁸ : TopologicalSpace A F B : Type inst✝²⁷ : NormedAddCommGroup F inst✝²⁶ : NormedSpace 𝕜 F inst✝²⁵ : CompleteSpace F inst✝²⁴ : TopologicalSpace B G C : Type inst✝²³ : NormedAddCommGroup G inst✝²² : NormedSpace 𝕜 G inst✝²¹ : TopologicalSpace C H D : Type inst✝²⁰ : NormedAddCommGroup H inst✝¹⁹ : NormedSpace 𝕜 H inst✝¹⁸ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹⁷ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁶ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹⁵ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁴ : TopologicalSpace P inst✝¹³ : I.Boundaryless inst✝¹² : ChartedSpace A M cm : AnalyticManifold I M inst✝¹¹ : J.Boundaryless inst✝¹⁰ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁹ : K.Boundaryless inst✝⁸ : ChartedSpace C O co : AnalyticManifold K O inst✝⁷ : L.Boundaryless inst✝⁶ : ChartedSpace D P cp : AnalyticManifold L P inst✝⁵ : ChartedSpace A E inst✝⁴ : AnalyticManifold I E inst✝³ : ChartedSpace B F inst✝² : AnalyticManifold J F inst✝¹ : ExtChartEqRefl I inst✝ : ExtChartEqRefl J f : E → F x : E ⊢ AnalyticAt 𝕜 f x → ContinuousAt f x TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	rw [holomorphicAt_iff] at fh gh ⊢	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : HolomorphicAt J K f (g x) gh : HolomorphicAt I J g x ⊢ HolomorphicAt I K (fun x => f (g x)) x	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt (fun x => f (g x)) x ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : HolomorphicAt J K f (g x) gh : HolomorphicAt I J g x ⊢ HolomorphicAt I K (fun x => f (g x)) x TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	use fh.1.comp gh.1	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt (fun x => f (g x)) x ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt (fun x => f (g x)) x ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	have e : extChartAt J (g x) (g x) = (extChartAt J (g x) ∘ g ∘ (extChartAt I x).symm) (extChartAt I x x) := by simp only [Function.comp_apply, PartialEquiv.left_inv _ (mem_extChartAt_source I x)]	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	rw [e] at fh	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ((↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	apply (fh.2.comp gh.2).congr	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ((↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ((↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq ((↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ∘ ↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm)	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ((↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	clear e fh	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ((↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq ((↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ∘ ↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq ((↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ∘ ↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm)	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ((↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq ((↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ∘ ↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	simp only [Function.comp]	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq ((↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ∘ ↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq (fun x_1 => ↑(extChartAt K (f (g x))) (f (↑(extChartAt J (g x)).symm (↑(extChartAt J (g x)) (g (↑(extChartAt I x).symm x_1)))))) fun x_1 => ↑(extChartAt K (f (g x))) (f (g (↑(extChartAt I x).symm x_1)))	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq ((↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ∘ ↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	refine m.mp (eventually_of_forall fun y m ↦ ?_)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) m : ∀ᶠ (y : E) in 𝓝 (↑(extChartAt I x) x), g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source ⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq (fun x_1 => ↑(extChartAt K (f (g x))) (f (↑(extChartAt J (g x)).symm (↑(extChartAt J (g x)) (g (↑(extChartAt I x).symm x_1)))))) fun x_1 => ↑(extChartAt K (f (g x))) (f (g (↑(extChartAt I x).symm x_1)))	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) m✝ : ∀ᶠ (y : E) in 𝓝 (↑(extChartAt I x) x), g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source y : E m : g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source ⊢ (fun x_1 => ↑(extChartAt K (f (g x))) (f (↑(extChartAt J (g x)).symm (↑(extChartAt J (g x)) (g (↑(extChartAt I x).symm x_1)))))) y = (fun x_1 => ↑(extChartAt K (f (g x))) (f (g (↑(extChartAt I x).symm x_1)))) y	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) m : ∀ᶠ (y : E) in 𝓝 (↑(extChartAt I x) x), g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source ⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq (fun x_1 => ↑(extChartAt K (f (g x))) (f (↑(extChartAt J (g x)).symm (↑(extChartAt J (g x)) (g (↑(extChartAt I x).symm x_1)))))) fun x_1 => ↑(extChartAt K (f (g x))) (f (g (↑(extChartAt I x).symm x_1))) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	simp_rw [PartialEquiv.left_inv _ m]	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) m✝ : ∀ᶠ (y : E) in 𝓝 (↑(extChartAt I x) x), g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source y : E m : g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source ⊢ (fun x_1 => ↑(extChartAt K (f (g x))) (f (↑(extChartAt J (g x)).symm (↑(extChartAt J (g x)) (g (↑(extChartAt I x).symm x_1)))))) y = (fun x_1 => ↑(extChartAt K (f (g x))) (f (g (↑(extChartAt I x).symm x_1)))) y	no goals	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) m✝ : ∀ᶠ (y : E) in 𝓝 (↑(extChartAt I x) x), g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source y : E m : g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source ⊢ (fun x_1 => ↑(extChartAt K (f (g x))) (f (↑(extChartAt J (g x)).symm (↑(extChartAt J (g x)) (g (↑(extChartAt I x).symm x_1)))))) y = (fun x_1 => ↑(extChartAt K (f (g x))) (f (g (↑(extChartAt I x).symm x_1)))) y TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	simp only [Function.comp_apply, PartialEquiv.left_inv _ (mem_extChartAt_source I x)]	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	no goals	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M fh : ContinuousAt f (g x) ∧ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x)) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	apply ContinuousAt.eventually_mem	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ∀ᶠ (y : E) in 𝓝 (↑(extChartAt I x) x), g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source	case hf 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt (fun y => g (↑(extChartAt I x).symm y)) (↑(extChartAt I x) x) case hs 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ (extChartAt J (g x)).source ∈ 𝓝 (g (↑(extChartAt I x).symm (↑(extChartAt I x) x)))	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ∀ᶠ (y : E) in 𝓝 (↑(extChartAt I x) x), g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	apply ContinuousAt.comp	case hf 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt (fun y => g (↑(extChartAt I x).symm y)) (↑(extChartAt I x) x)	case hf.hg 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt g (↑(extChartAt I x).symm (↑(extChartAt I x) x)) case hf.hf 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt (fun y => ↑(extChartAt I x).symm y) (↑(extChartAt I x) x)	Please generate a tactic in lean4 to solve the state. STATE: case hf 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt (fun y => g (↑(extChartAt I x).symm y)) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	rw [PartialEquiv.left_inv _ (mem_extChartAt_source _ _)]	case hf.hg 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt g (↑(extChartAt I x).symm (↑(extChartAt I x) x))	case hf.hg 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt g x	Please generate a tactic in lean4 to solve the state. STATE: case hf.hg 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt g (↑(extChartAt I x).symm (↑(extChartAt I x) x)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	exact gh.1	case hf.hg 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt g x	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hf.hg 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt g x TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	exact continuousAt_extChartAt_symm I x	case hf.hf 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt (fun y => ↑(extChartAt I x).symm y) (↑(extChartAt I x) x)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hf.hf 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt (fun y => ↑(extChartAt I x).symm y) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	rw [PartialEquiv.left_inv _ (mem_extChartAt_source _ _)]	case hs 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ (extChartAt J (g x)).source ∈ 𝓝 (g (↑(extChartAt I x).symm (↑(extChartAt I x) x)))	case hs 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ (extChartAt J (g x)).source ∈ 𝓝 (g x)	Please generate a tactic in lean4 to solve the state. STATE: case hs 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ (extChartAt J (g x)).source ∈ 𝓝 (g (↑(extChartAt I x).symm (↑(extChartAt I x) x))) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp	[321, 1]	[338, 38]	exact extChartAt_source_mem_nhds _ _	case hs 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ (extChartAt J (g x)).source ∈ 𝓝 (g x)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hs 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ (extChartAt J (g x)).source ∈ 𝓝 (g x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp_of_eq	[345, 1]	[348, 35]	rw [← e] at fh	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M y : N fh : HolomorphicAt J K f y gh : HolomorphicAt I J g x e : g x = y ⊢ HolomorphicAt I K (fun x => f (g x)) x	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M y : N fh : HolomorphicAt J K f (g x) gh : HolomorphicAt I J g x e : g x = y ⊢ HolomorphicAt I K (fun x => f (g x)) x	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M y : N fh : HolomorphicAt J K f y gh : HolomorphicAt I J g x e : g x = y ⊢ HolomorphicAt I K (fun x => f (g x)) x TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp_of_eq	[345, 1]	[348, 35]	exact fh.comp gh	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M y : N fh : HolomorphicAt J K f (g x) gh : HolomorphicAt I J g x e : g x = y ⊢ HolomorphicAt I K (fun x => f (g x)) x	no goals	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : N → O g : M → N x : M y : N fh : HolomorphicAt J K f (g x) gh : HolomorphicAt I J g x e : g x = y ⊢ HolomorphicAt I K (fun x => f (g x)) x TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.prod	[351, 1]	[355, 76]	rw [holomorphicAt_iff] at fh gh ⊢	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N g : M → O x : M fh : HolomorphicAt I J f x gh : HolomorphicAt I K g x ⊢ HolomorphicAt I (J.prod K) (fun x => (f x, g x)) x	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N g : M → O x : M fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt (fun x => (f x, g x)) x ∧ AnalyticAt 𝕜 (↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N g : M → O x : M fh : HolomorphicAt I J f x gh : HolomorphicAt I K g x ⊢ HolomorphicAt I (J.prod K) (fun x => (f x, g x)) x TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.prod	[351, 1]	[355, 76]	use fh.1.prod gh.1	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N g : M → O x : M fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt (fun x => (f x, g x)) x ∧ AnalyticAt 𝕜 (↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N g : M → O x : M fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ AnalyticAt 𝕜 (↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N g : M → O x : M fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ ContinuousAt (fun x => (f x, g x)) x ∧ AnalyticAt 𝕜 (↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.prod	[351, 1]	[355, 76]	refine (fh.2.prod gh.2).congr (eventually_of_forall fun y ↦ ?_)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N g : M → O x : M fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ AnalyticAt 𝕜 (↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N g : M → O x : M fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) y : E ⊢ (fun x_1 => ((↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) x_1, (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) x_1)) y = (↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm) y	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N g : M → O x : M fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) ⊢ AnalyticAt 𝕜 (↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.prod	[351, 1]	[355, 76]	simp only [extChartAt_prod, Function.comp, PartialEquiv.prod_coe]	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N g : M → O x : M fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) y : E ⊢ (fun x_1 => ((↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) x_1, (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) x_1)) y = (↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm) y	no goals	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P f : M → N g : M → O x : M fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) y : E ⊢ (fun x_1 => ((↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) x_1, (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) x_1)) y = (↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm) y TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp₂_of_eq	[368, 1]	[371, 91]	rw [← e] at ha	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P h : N → O → P f : M → N g : M → O x : M y : N × O ha : HolomorphicAt (J.prod K) L (uncurry h) y fa : HolomorphicAt I J f x ga : HolomorphicAt I K g x e : (f x, g x) = y ⊢ HolomorphicAt I L (fun x => h (f x) (g x)) x	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P h : N → O → P f : M → N g : M → O x : M y : N × O ha : HolomorphicAt (J.prod K) L (uncurry h) (f x, g x) fa : HolomorphicAt I J f x ga : HolomorphicAt I K g x e : (f x, g x) = y ⊢ HolomorphicAt I L (fun x => h (f x) (g x)) x	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P h : N → O → P f : M → N g : M → O x : M y : N × O ha : HolomorphicAt (J.prod K) L (uncurry h) y fa : HolomorphicAt I J f x ga : HolomorphicAt I K g x e : (f x, g x) = y ⊢ HolomorphicAt I L (fun x => h (f x) (g x)) x TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.comp₂_of_eq	[368, 1]	[371, 91]	exact ha.comp₂ fa ga	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P h : N → O → P f : M → N g : M → O x : M y : N × O ha : HolomorphicAt (J.prod K) L (uncurry h) (f x, g x) fa : HolomorphicAt I J f x ga : HolomorphicAt I K g x e : (f x, g x) = y ⊢ HolomorphicAt I L (fun x => h (f x) (g x)) x	no goals	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P h : N → O → P f : M → N g : M → O x : M y : N × O ha : HolomorphicAt (J.prod K) L (uncurry h) (f x, g x) fa : HolomorphicAt I J f x ga : HolomorphicAt I K g x e : (f x, g x) = y ⊢ HolomorphicAt I L (fun x => h (f x) (g x)) x TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_id	[374, 1]	[378, 60]	rw [holomorphicAt_iff]	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M ⊢ HolomorphicAt I I (fun x => x) x	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M ⊢ ContinuousAt (fun x => x) x ∧ AnalyticAt 𝕜 (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M ⊢ HolomorphicAt I I (fun x => x) x TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_id	[374, 1]	[378, 60]	use continuousAt_id	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M ⊢ ContinuousAt (fun x => x) x ∧ AnalyticAt 𝕜 (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M ⊢ AnalyticAt 𝕜 (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M ⊢ ContinuousAt (fun x => x) x ∧ AnalyticAt 𝕜 (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_id	[374, 1]	[378, 60]	apply (analyticAt_id _ _).congr	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M ⊢ AnalyticAt 𝕜 (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M ⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq id (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm)	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M ⊢ AnalyticAt 𝕜 (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_id	[374, 1]	[378, 60]	filter_upwards [((isOpen_extChartAt_target I x).eventually_mem (mem_extChartAt_target I x))]	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M ⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq id (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm)	case h 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M ⊢ ∀ a ∈ (extChartAt I x).target, id a = (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) a	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M ⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq id (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_id	[374, 1]	[378, 60]	intro y m	case h 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M ⊢ ∀ a ∈ (extChartAt I x).target, id a = (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) a	case h 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M y : E m : y ∈ (extChartAt I x).target ⊢ id y = (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) y	Please generate a tactic in lean4 to solve the state. STATE: case h 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M ⊢ ∀ a ∈ (extChartAt I x).target, id a = (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) a TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_id	[374, 1]	[378, 60]	simp only [Function.comp, PartialEquiv.right_inv _ m, id]	case h 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M y : E m : y ∈ (extChartAt I x).target ⊢ id y = (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) y	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M y : E m : y ∈ (extChartAt I x).target ⊢ id y = (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) y TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_const	[384, 1]	[386, 70]	rw [holomorphicAt_iff]	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M c : N ⊢ HolomorphicAt I J (fun x => c) x	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M c : N ⊢ ContinuousAt (fun x => c) x ∧ AnalyticAt 𝕜 (↑(extChartAt J c) ∘ (fun x => c) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M c : N ⊢ HolomorphicAt I J (fun x => c) x TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_const	[384, 1]	[386, 70]	use continuousAt_const	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M c : N ⊢ ContinuousAt (fun x => c) x ∧ AnalyticAt 𝕜 (↑(extChartAt J c) ∘ (fun x => c) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M c : N ⊢ AnalyticAt 𝕜 (↑(extChartAt J c) ∘ (fun x => c) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M c : N ⊢ ContinuousAt (fun x => c) x ∧ AnalyticAt 𝕜 (↑(extChartAt J c) ∘ (fun x => c) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_const	[384, 1]	[386, 70]	simp only [Prod.fst_comp_mk, Function.comp]	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M c : N ⊢ AnalyticAt 𝕜 (↑(extChartAt J c) ∘ (fun x => c) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M c : N ⊢ AnalyticAt 𝕜 (fun x => ↑(extChartAt J c) c) (↑(extChartAt I x) x)	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M c : N ⊢ AnalyticAt 𝕜 (↑(extChartAt J c) ∘ (fun x => c) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_const	[384, 1]	[386, 70]	exact analyticAt_const	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M c : N ⊢ AnalyticAt 𝕜 (fun x => ↑(extChartAt J c) c) (↑(extChartAt I x) x)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M c : N ⊢ AnalyticAt 𝕜 (fun x => ↑(extChartAt J c) c) (↑(extChartAt I x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_fst	[411, 1]	[419, 48]	rw [holomorphicAt_iff]	𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ HolomorphicAt (I.prod J) I (fun p => p.1) x	𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ ContinuousAt (fun p => p.1) x ∧ AnalyticAt 𝕜 (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) (↑(extChartAt (I.prod J) x) x)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ HolomorphicAt (I.prod J) I (fun p => p.1) x TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_fst	[411, 1]	[419, 48]	use continuousAt_fst	𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ ContinuousAt (fun p => p.1) x ∧ AnalyticAt 𝕜 (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) (↑(extChartAt (I.prod J) x) x)	case right 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ AnalyticAt 𝕜 (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) (↑(extChartAt (I.prod J) x) x)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ ContinuousAt (fun p => p.1) x ∧ AnalyticAt 𝕜 (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) (↑(extChartAt (I.prod J) x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_fst	[411, 1]	[419, 48]	refine (analyticAt_fst _).congr ?_	case right 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ AnalyticAt 𝕜 (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) (↑(extChartAt (I.prod J) x) x)	case right 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ (𝓝 (↑(extChartAt (I.prod J) x) x)).EventuallyEq (fun p => p.1) (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm)	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ AnalyticAt 𝕜 (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) (↑(extChartAt (I.prod J) x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_fst	[411, 1]	[419, 48]	filter_upwards [((isOpen_extChartAt_target _ x).eventually_mem (mem_extChartAt_target _ _))]	case right 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ (𝓝 (↑(extChartAt (I.prod J) x) x)).EventuallyEq (fun p => p.1) (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm)	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ ∀ a ∈ (extChartAt (I.prod J) x).target, a.1 = (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) a	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ (𝓝 (↑(extChartAt (I.prod J) x) x)).EventuallyEq (fun p => p.1) (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_fst	[411, 1]	[419, 48]	intro y m	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ ∀ a ∈ (extChartAt (I.prod J) x).target, a.1 = (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) a	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y ∈ (extChartAt (I.prod J) x).target ⊢ y.1 = (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) y	Please generate a tactic in lean4 to solve the state. STATE: case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ ∀ a ∈ (extChartAt (I.prod J) x).target, a.1 = (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) a TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_fst	[411, 1]	[419, 48]	rw [extChartAt_prod] at m	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y ∈ (extChartAt (I.prod J) x).target ⊢ y.1 = (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) y	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y ∈ ((extChartAt I x.1).prod (extChartAt J x.2)).target ⊢ y.1 = (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) y	Please generate a tactic in lean4 to solve the state. STATE: case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y ∈ (extChartAt (I.prod J) x).target ⊢ y.1 = (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) y TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_fst	[411, 1]	[419, 48]	simp only [PartialHomeomorph.prod_toPartialEquiv, PartialEquiv.prod_target, mem_prod] at m	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y ∈ ((extChartAt I x.1).prod (extChartAt J x.2)).target ⊢ y.1 = (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) y	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y.1 ∈ (extChartAt I x.1).target ∧ y.2 ∈ (extChartAt J x.2).target ⊢ y.1 = (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) y	Please generate a tactic in lean4 to solve the state. STATE: case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y ∈ ((extChartAt I x.1).prod (extChartAt J x.2)).target ⊢ y.1 = (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) y TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_fst	[411, 1]	[419, 48]	simp only [extChartAt_prod, Function.comp, PartialEquiv.prod_coe_symm]	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y.1 ∈ (extChartAt I x.1).target ∧ y.2 ∈ (extChartAt J x.2).target ⊢ y.1 = (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) y	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y.1 ∈ (extChartAt I x.1).target ∧ y.2 ∈ (extChartAt J x.2).target ⊢ y.1 = ↑(extChartAt I x.1) (↑(extChartAt I x.1).symm y.1)	Please generate a tactic in lean4 to solve the state. STATE: case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y.1 ∈ (extChartAt I x.1).target ∧ y.2 ∈ (extChartAt J x.2).target ⊢ y.1 = (↑(extChartAt I x.1) ∘ (fun p => p.1) ∘ ↑(extChartAt (I.prod J) x).symm) y TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_fst	[411, 1]	[419, 48]	exact ((extChartAt I x.1).right_inv m.1).symm	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y.1 ∈ (extChartAt I x.1).target ∧ y.2 ∈ (extChartAt J x.2).target ⊢ y.1 = ↑(extChartAt I x.1) (↑(extChartAt I x.1).symm y.1)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y.1 ∈ (extChartAt I x.1).target ∧ y.2 ∈ (extChartAt J x.2).target ⊢ y.1 = ↑(extChartAt I x.1) (↑(extChartAt I x.1).symm y.1) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_snd	[422, 1]	[430, 48]	rw [holomorphicAt_iff]	𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ HolomorphicAt (I.prod J) J (fun p => p.2) x	𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ ContinuousAt (fun p => p.2) x ∧ AnalyticAt 𝕜 (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) (↑(extChartAt (I.prod J) x) x)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ HolomorphicAt (I.prod J) J (fun p => p.2) x TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_snd	[422, 1]	[430, 48]	use continuousAt_snd	𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ ContinuousAt (fun p => p.2) x ∧ AnalyticAt 𝕜 (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) (↑(extChartAt (I.prod J) x) x)	case right 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ AnalyticAt 𝕜 (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) (↑(extChartAt (I.prod J) x) x)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ ContinuousAt (fun p => p.2) x ∧ AnalyticAt 𝕜 (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) (↑(extChartAt (I.prod J) x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_snd	[422, 1]	[430, 48]	refine (analyticAt_snd _).congr ?_	case right 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ AnalyticAt 𝕜 (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) (↑(extChartAt (I.prod J) x) x)	case right 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ (𝓝 (↑(extChartAt (I.prod J) x) x)).EventuallyEq (fun p => p.2) (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm)	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ AnalyticAt 𝕜 (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) (↑(extChartAt (I.prod J) x) x) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_snd	[422, 1]	[430, 48]	filter_upwards [((isOpen_extChartAt_target _ x).eventually_mem (mem_extChartAt_target _ _))]	case right 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ (𝓝 (↑(extChartAt (I.prod J) x) x)).EventuallyEq (fun p => p.2) (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm)	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ ∀ a ∈ (extChartAt (I.prod J) x).target, a.2 = (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) a	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ (𝓝 (↑(extChartAt (I.prod J) x) x)).EventuallyEq (fun p => p.2) (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_snd	[422, 1]	[430, 48]	intro y m	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ ∀ a ∈ (extChartAt (I.prod J) x).target, a.2 = (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) a	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y ∈ (extChartAt (I.prod J) x).target ⊢ y.2 = (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) y	Please generate a tactic in lean4 to solve the state. STATE: case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N ⊢ ∀ a ∈ (extChartAt (I.prod J) x).target, a.2 = (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) a TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_snd	[422, 1]	[430, 48]	rw [extChartAt_prod] at m	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y ∈ (extChartAt (I.prod J) x).target ⊢ y.2 = (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) y	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y ∈ ((extChartAt I x.1).prod (extChartAt J x.2)).target ⊢ y.2 = (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) y	Please generate a tactic in lean4 to solve the state. STATE: case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y ∈ (extChartAt (I.prod J) x).target ⊢ y.2 = (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) y TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_snd	[422, 1]	[430, 48]	simp only [PartialHomeomorph.prod_toPartialEquiv, PartialEquiv.prod_target, mem_prod] at m	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y ∈ ((extChartAt I x.1).prod (extChartAt J x.2)).target ⊢ y.2 = (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) y	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y.1 ∈ (extChartAt I x.1).target ∧ y.2 ∈ (extChartAt J x.2).target ⊢ y.2 = (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) y	Please generate a tactic in lean4 to solve the state. STATE: case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y ∈ ((extChartAt I x.1).prod (extChartAt J x.2)).target ⊢ y.2 = (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) y TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_snd	[422, 1]	[430, 48]	simp only [extChartAt_prod, Function.comp, PartialEquiv.prod_coe_symm]	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y.1 ∈ (extChartAt I x.1).target ∧ y.2 ∈ (extChartAt J x.2).target ⊢ y.2 = (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) y	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y.1 ∈ (extChartAt I x.1).target ∧ y.2 ∈ (extChartAt J x.2).target ⊢ y.2 = ↑(extChartAt J x.2) (↑(extChartAt J x.2).symm y.2)	Please generate a tactic in lean4 to solve the state. STATE: case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y.1 ∈ (extChartAt I x.1).target ∧ y.2 ∈ (extChartAt J x.2).target ⊢ y.2 = (↑(extChartAt J x.2) ∘ (fun p => p.2) ∘ ↑(extChartAt (I.prod J) x).symm) y TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	holomorphicAt_snd	[422, 1]	[430, 48]	exact ((extChartAt J x.2).right_inv m.2).symm	case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y.1 ∈ (extChartAt I x.1).target ∧ y.2 ∈ (extChartAt J x.2).target ⊢ y.2 = ↑(extChartAt J x.2) (↑(extChartAt J x.2).symm y.2)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h 𝕜 : Type inst✝²⁸ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁷ : NormedAddCommGroup E inst✝²⁶ : NormedSpace 𝕜 E inst✝²⁵ : CompleteSpace E inst✝²⁴ : TopologicalSpace A F B : Type inst✝²³ : NormedAddCommGroup F inst✝²² : NormedSpace 𝕜 F inst✝²¹ : CompleteSpace F inst✝²⁰ : TopologicalSpace B G C : Type inst✝¹⁹ : NormedAddCommGroup G inst✝¹⁸ : NormedSpace 𝕜 G inst✝¹⁷ : TopologicalSpace C H D : Type inst✝¹⁶ : NormedAddCommGroup H inst✝¹⁵ : NormedSpace 𝕜 H inst✝¹⁴ : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹³ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹² : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝¹¹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝¹⁰ : TopologicalSpace P inst✝⁹ : I.Boundaryless inst✝⁸ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁷ : J.Boundaryless inst✝⁶ : ChartedSpace B N cn : AnalyticManifold J N inst✝⁵ : K.Boundaryless inst✝⁴ : ChartedSpace C O co : AnalyticManifold K O inst✝³ : L.Boundaryless inst✝² : ChartedSpace D P cp : AnalyticManifold L P inst✝¹ : I.Boundaryless inst✝ : J.Boundaryless x : M × N y : E × F m : y.1 ∈ (extChartAt I x.1).target ∧ y.2 ∈ (extChartAt J x.2).target ⊢ y.2 = ↑(extChartAt J x.2) (↑(extChartAt J x.2).symm y.2) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.extChartAt	[448, 1]	[466, 47]	rw [holomorphicAt_iff]	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ HolomorphicAt I 𝓘(𝕜, E) (↑(_root_.extChartAt I x)) y	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ ContinuousAt (↑(_root_.extChartAt I x)) y ∧ AnalyticAt 𝕜 (↑(_root_.extChartAt 𝓘(𝕜, E) (↑(_root_.extChartAt I x) y)) ∘ ↑(_root_.extChartAt I x) ∘ ↑(_root_.extChartAt I y).symm) (↑(_root_.extChartAt I y) y)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ HolomorphicAt I 𝓘(𝕜, E) (↑(_root_.extChartAt I x)) y TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.extChartAt	[448, 1]	[466, 47]	use continuousAt_extChartAt' I ys	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ ContinuousAt (↑(_root_.extChartAt I x)) y ∧ AnalyticAt 𝕜 (↑(_root_.extChartAt 𝓘(𝕜, E) (↑(_root_.extChartAt I x) y)) ∘ ↑(_root_.extChartAt I x) ∘ ↑(_root_.extChartAt I y).symm) (↑(_root_.extChartAt I y) y)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ AnalyticAt 𝕜 (↑(_root_.extChartAt 𝓘(𝕜, E) (↑(_root_.extChartAt I x) y)) ∘ ↑(_root_.extChartAt I x) ∘ ↑(_root_.extChartAt I y).symm) (↑(_root_.extChartAt I y) y)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ ContinuousAt (↑(_root_.extChartAt I x)) y ∧ AnalyticAt 𝕜 (↑(_root_.extChartAt 𝓘(𝕜, E) (↑(_root_.extChartAt I x) y)) ∘ ↑(_root_.extChartAt I x) ∘ ↑(_root_.extChartAt I y).symm) (↑(_root_.extChartAt I y) y) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.extChartAt	[448, 1]	[466, 47]	simp only [Function.comp, extChartAt, PartialHomeomorph.extend, PartialEquiv.coe_trans, PartialHomeomorph.toFun_eq_coe, ModelWithCorners.toPartialEquiv_coe, PartialHomeomorph.refl_partialEquiv, PartialEquiv.refl_source, PartialHomeomorph.singletonChartedSpace_chartAt_eq, modelWithCornersSelf_partialEquiv, PartialEquiv.trans_refl, PartialEquiv.trans_symm_eq_symm_trans_symm, ModelWithCorners.toPartialEquiv_coe_symm, PartialHomeomorph.coe_coe_symm, PartialEquiv.refl_coe, id, _root_.extChartAt]	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ AnalyticAt 𝕜 (↑(_root_.extChartAt 𝓘(𝕜, E) (↑(_root_.extChartAt I x) y)) ∘ ↑(_root_.extChartAt I x) ∘ ↑(_root_.extChartAt I y).symm) (↑(_root_.extChartAt I y) y)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ AnalyticAt 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A x) (↑(_root_.chartAt A y).symm (↑I.symm x_1)))) (↑I (↑(_root_.chartAt A y) y))	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ AnalyticAt 𝕜 (↑(_root_.extChartAt 𝓘(𝕜, E) (↑(_root_.extChartAt I x) y)) ∘ ↑(_root_.extChartAt I x) ∘ ↑(_root_.extChartAt I y).symm) (↑(_root_.extChartAt I y) y) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.extChartAt	[448, 1]	[466, 47]	have a : (chartAt A x).symm ≫ₕ chartAt A y ∈ analyticGroupoid I := by apply StructureGroupoid.compatible_of_mem_maximalAtlas exact (@StructureGroupoid.chart_mem_maximalAtlas _ _ _ _ _ (analyticGroupoid I) cm.toHasGroupoid x) exact (@StructureGroupoid.chart_mem_maximalAtlas _ _ _ _ _ (analyticGroupoid I) cm.toHasGroupoid y)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ AnalyticAt 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A x) (↑(_root_.chartAt A y).symm (↑I.symm x_1)))) (↑I (↑(_root_.chartAt A y) y))	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source a : (_root_.chartAt A x).symm.trans (_root_.chartAt A y) ∈ analyticGroupoid I ⊢ AnalyticAt 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A x) (↑(_root_.chartAt A y).symm (↑I.symm x_1)))) (↑I (↑(_root_.chartAt A y) y))	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ AnalyticAt 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A x) (↑(_root_.chartAt A y).symm (↑I.symm x_1)))) (↑I (↑(_root_.chartAt A y) y)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.extChartAt	[448, 1]	[466, 47]	simp only [mem_analyticGroupoid_of_boundaryless, PartialHomeomorph.trans_symm_eq_symm_trans_symm, Function.comp, PartialHomeomorph.trans_apply] at a	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source a : (_root_.chartAt A x).symm.trans (_root_.chartAt A y) ∈ analyticGroupoid I ⊢ AnalyticAt 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A x) (↑(_root_.chartAt A y).symm (↑I.symm x_1)))) (↑I (↑(_root_.chartAt A y) y))	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source a : AnalyticOn 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A y) (↑(_root_.chartAt A x).symm (↑I.symm x_1)))) (↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).source) ∧ AnalyticOn 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A x).symm.symm (↑(_root_.chartAt A y).symm (↑I.symm x_1)))) (↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).target) ⊢ AnalyticAt 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A x) (↑(_root_.chartAt A y).symm (↑I.symm x_1)))) (↑I (↑(_root_.chartAt A y) y))	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source a : (_root_.chartAt A x).symm.trans (_root_.chartAt A y) ∈ analyticGroupoid I ⊢ AnalyticAt 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A x) (↑(_root_.chartAt A y).symm (↑I.symm x_1)))) (↑I (↑(_root_.chartAt A y) y)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.extChartAt	[448, 1]	[466, 47]	apply a.2	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source a : AnalyticOn 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A y) (↑(_root_.chartAt A x).symm (↑I.symm x_1)))) (↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).source) ∧ AnalyticOn 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A x).symm.symm (↑(_root_.chartAt A y).symm (↑I.symm x_1)))) (↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).target) ⊢ AnalyticAt 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A x) (↑(_root_.chartAt A y).symm (↑I.symm x_1)))) (↑I (↑(_root_.chartAt A y) y))	case right.a 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source a : AnalyticOn 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A y) (↑(_root_.chartAt A x).symm (↑I.symm x_1)))) (↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).source) ∧ AnalyticOn 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A x).symm.symm (↑(_root_.chartAt A y).symm (↑I.symm x_1)))) (↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).target) ⊢ ↑I (↑(_root_.chartAt A y) y) ∈ ↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).target	Please generate a tactic in lean4 to solve the state. STATE: case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source a : AnalyticOn 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A y) (↑(_root_.chartAt A x).symm (↑I.symm x_1)))) (↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).source) ∧ AnalyticOn 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A x).symm.symm (↑(_root_.chartAt A y).symm (↑I.symm x_1)))) (↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).target) ⊢ AnalyticAt 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A x) (↑(_root_.chartAt A y).symm (↑I.symm x_1)))) (↑I (↑(_root_.chartAt A y) y)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.extChartAt	[448, 1]	[466, 47]	clear a	case right.a 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source a : AnalyticOn 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A y) (↑(_root_.chartAt A x).symm (↑I.symm x_1)))) (↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).source) ∧ AnalyticOn 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A x).symm.symm (↑(_root_.chartAt A y).symm (↑I.symm x_1)))) (↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).target) ⊢ ↑I (↑(_root_.chartAt A y) y) ∈ ↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).target	case right.a 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ ↑I (↑(_root_.chartAt A y) y) ∈ ↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).target	Please generate a tactic in lean4 to solve the state. STATE: case right.a 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source a : AnalyticOn 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A y) (↑(_root_.chartAt A x).symm (↑I.symm x_1)))) (↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).source) ∧ AnalyticOn 𝕜 (fun x_1 => ↑I (↑(_root_.chartAt A x).symm.symm (↑(_root_.chartAt A y).symm (↑I.symm x_1)))) (↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).target) ⊢ ↑I (↑(_root_.chartAt A y) y) ∈ ↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).target TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.extChartAt	[448, 1]	[466, 47]	use chartAt A y y	case right.a 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ ↑I (↑(_root_.chartAt A y) y) ∈ ↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).target	case h 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ ↑(_root_.chartAt A y) y ∈ ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).target ∧ ↑I (↑(_root_.chartAt A y) y) = ↑I (↑(_root_.chartAt A y) y)	Please generate a tactic in lean4 to solve the state. STATE: case right.a 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ ↑I (↑(_root_.chartAt A y) y) ∈ ↑I '' ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).target TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.extChartAt	[448, 1]	[466, 47]	aesop	case h 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ ↑(_root_.chartAt A y) y ∈ ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).target ∧ ↑I (↑(_root_.chartAt A y) y) = ↑I (↑(_root_.chartAt A y) y)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ ↑(_root_.chartAt A y) y ∈ ((_root_.chartAt A x).symm.trans (_root_.chartAt A y)).target ∧ ↑I (↑(_root_.chartAt A y) y) = ↑I (↑(_root_.chartAt A y) y) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.extChartAt	[448, 1]	[466, 47]	apply StructureGroupoid.compatible_of_mem_maximalAtlas	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ (_root_.chartAt A x).symm.trans (_root_.chartAt A y) ∈ analyticGroupoid I	case he 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ _root_.chartAt A x ∈ StructureGroupoid.maximalAtlas M (analyticGroupoid I) case he' 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ _root_.chartAt A y ∈ StructureGroupoid.maximalAtlas M (analyticGroupoid I)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ (_root_.chartAt A x).symm.trans (_root_.chartAt A y) ∈ analyticGroupoid I TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.extChartAt	[448, 1]	[466, 47]	exact (@StructureGroupoid.chart_mem_maximalAtlas _ _ _ _ _ (analyticGroupoid I) cm.toHasGroupoid x)	case he 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ _root_.chartAt A x ∈ StructureGroupoid.maximalAtlas M (analyticGroupoid I) case he' 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ _root_.chartAt A y ∈ StructureGroupoid.maximalAtlas M (analyticGroupoid I)	case he' 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ _root_.chartAt A y ∈ StructureGroupoid.maximalAtlas M (analyticGroupoid I)	Please generate a tactic in lean4 to solve the state. STATE: case he 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ _root_.chartAt A x ∈ StructureGroupoid.maximalAtlas M (analyticGroupoid I) case he' 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ _root_.chartAt A y ∈ StructureGroupoid.maximalAtlas M (analyticGroupoid I) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.extChartAt	[448, 1]	[466, 47]	exact (@StructureGroupoid.chart_mem_maximalAtlas _ _ _ _ _ (analyticGroupoid I) cm.toHasGroupoid y)	case he' 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ _root_.chartAt A y ∈ StructureGroupoid.maximalAtlas M (analyticGroupoid I)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case he' 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x y : M ys : y ∈ (_root_.extChartAt I x).source ⊢ _root_.chartAt A y ∈ StructureGroupoid.maximalAtlas M (analyticGroupoid I) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.extChartAt_symm	[469, 1]	[487, 42]	rw [holomorphicAt_iff]	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M y : E ys : y ∈ (_root_.extChartAt I x).target ⊢ HolomorphicAt 𝓘(𝕜, E) I (↑(_root_.extChartAt I x).symm) y	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M y : E ys : y ∈ (_root_.extChartAt I x).target ⊢ ContinuousAt (↑(_root_.extChartAt I x).symm) y ∧ AnalyticAt 𝕜 (↑(_root_.extChartAt I (↑(_root_.extChartAt I x).symm y)) ∘ ↑(_root_.extChartAt I x).symm ∘ ↑(_root_.extChartAt 𝓘(𝕜, E) y).symm) (↑(_root_.extChartAt 𝓘(𝕜, E) y) y)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M y : E ys : y ∈ (_root_.extChartAt I x).target ⊢ HolomorphicAt 𝓘(𝕜, E) I (↑(_root_.extChartAt I x).symm) y TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/AnalyticManifold/AnalyticManifold.lean	HolomorphicAt.extChartAt_symm	[469, 1]	[487, 42]	use continuousAt_extChartAt_symm'' I ys	𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M y : E ys : y ∈ (_root_.extChartAt I x).target ⊢ ContinuousAt (↑(_root_.extChartAt I x).symm) y ∧ AnalyticAt 𝕜 (↑(_root_.extChartAt I (↑(_root_.extChartAt I x).symm y)) ∘ ↑(_root_.extChartAt I x).symm ∘ ↑(_root_.extChartAt 𝓘(𝕜, E) y).symm) (↑(_root_.extChartAt 𝓘(𝕜, E) y) y)	case right 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M y : E ys : y ∈ (_root_.extChartAt I x).target ⊢ AnalyticAt 𝕜 (↑(_root_.extChartAt I (↑(_root_.extChartAt I x).symm y)) ∘ ↑(_root_.extChartAt I x).symm ∘ ↑(_root_.extChartAt 𝓘(𝕜, E) y).symm) (↑(_root_.extChartAt 𝓘(𝕜, E) y) y)	Please generate a tactic in lean4 to solve the state. STATE: 𝕜 : Type inst✝²⁶ : NontriviallyNormedField 𝕜 E A : Type inst✝²⁵ : NormedAddCommGroup E inst✝²⁴ : NormedSpace 𝕜 E inst✝²³ : CompleteSpace E inst✝²² : TopologicalSpace A F B : Type inst✝²¹ : NormedAddCommGroup F inst✝²⁰ : NormedSpace 𝕜 F inst✝¹⁹ : CompleteSpace F inst✝¹⁸ : TopologicalSpace B G C : Type inst✝¹⁷ : NormedAddCommGroup G inst✝¹⁶ : NormedSpace 𝕜 G inst✝¹⁵ : TopologicalSpace C H D : Type inst✝¹⁴ : NormedAddCommGroup H inst✝¹³ : NormedSpace 𝕜 H inst✝¹² : TopologicalSpace D M : Type I : ModelWithCorners 𝕜 E A inst✝¹¹ : TopologicalSpace M N : Type J : ModelWithCorners 𝕜 F B inst✝¹⁰ : TopologicalSpace N O : Type K : ModelWithCorners 𝕜 G C inst✝⁹ : TopologicalSpace O P : Type L : ModelWithCorners 𝕜 H D inst✝⁸ : TopologicalSpace P inst✝⁷ : I.Boundaryless inst✝⁶ : ChartedSpace A M cm : AnalyticManifold I M inst✝⁵ : J.Boundaryless inst✝⁴ : ChartedSpace B N cn : AnalyticManifold J N inst✝³ : K.Boundaryless inst✝² : ChartedSpace C O co : AnalyticManifold K O inst✝¹ : L.Boundaryless inst✝ : ChartedSpace D P cp : AnalyticManifold L P x : M y : E ys : y ∈ (_root_.extChartAt I x).target ⊢ ContinuousAt (↑(_root_.extChartAt I x).symm) y ∧ AnalyticAt 𝕜 (↑(_root_.extChartAt I (↑(_root_.extChartAt I x).symm y)) ∘ ↑(_root_.extChartAt I x).symm ∘ ↑(_root_.extChartAt 𝓘(𝕜, E) y).symm) (↑(_root_.extChartAt 𝓘(𝕜, E) y) y) TACTIC:

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