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 |
|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.inCharts | [183, 1] | [196, 77] | rw [PartialEquiv.left_inv _ m, PartialEquiv.left_inv _ (mem_extChartAt_source I z)] at fn | case nonconst
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : Analyti... | case nonconst
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : Analyti... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.inCharts | [183, 1] | [196, 77] | exact ((PartialEquiv.injOn _).eq_iff fm (mem_extChartAt_source _ _)).mp fn | case nonconst
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : Analyti... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | refine le_antisymm ?_ n.holomorphicAt.continuousAt | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | generalize hg : (fun x ↦ extChartAt I (f z) (f ((extChartAt I z).symm x))) = g | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | have ga : AnalyticAt ℂ g (extChartAt I z z) := by rw [← hg]; exact n.holomorphicAt.2 | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | cases' ga.eventually_constant_or_nhds_le_map_nhds with h h | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | rw [← hg] | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | exact n.holomorphicAt.2 | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | contrapose h | case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... | case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | clear h | case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... | case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | simp only [Filter.not_eventually] | case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... | case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | apply n.inCharts.nonconst.mp | case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... | case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | simp only [← hg, Ne, imp_self, Filter.eventually_true] | case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | simp only [← extChartAt_map_nhds' I z, Filter.map_map] at h | case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... | case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | replace h := @Filter.map_mono _ _ (extChartAt I (f z)).symm _ _ h | case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... | case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | simp only [← hg] at h | case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... | case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | rw [PartialEquiv.left_inv _ (mem_extChartAt_source I z)] at h | case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... | case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | simp only [extChartAt_symm_map_nhds' I (f z), Filter.map_map, Function.comp] at h | case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... | case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | have e : (fun w ↦ (extChartAt I (f z)).symm
(extChartAt I (f z) (f ((extChartAt I z).symm (extChartAt I z w))))) =ᶠ[𝓝 z] f := by
apply ((isOpen_extChartAt_source I z).eventually_mem (mem_extChartAt_source I z)).mp
apply (n.holomorphicAt.continuousAt.eventually_mem (extChartAt_source_mem_nhds I (f z))).mp
ref... | case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... | case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | rw [Filter.map_congr e] at h | case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... | case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | exact h | case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMani... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | apply ((isOpen_extChartAt_source I z).eventually_mem (mem_extChartAt_source I z)).mp | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | apply (n.holomorphicAt.continuousAt.eventually_mem (extChartAt_source_mem_nhds I (f z))).mp | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | refine eventually_of_forall fun w fm m ↦ ?_ | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | simp only [PartialEquiv.left_inv _ m, PartialEquiv.left_inv _ fm] | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | refine le_antisymm ?_ (continuousAt_fst.prod fa.continuousAt) | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | generalize hg : (fun e x ↦ extChartAt I (f c z) (f e ((extChartAt I z).symm x))) = g | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | have ga : AnalyticAt ℂ (uncurry g) (c, extChartAt I z z) := by
rw [← hg]; exact (holomorphicAt_iff.mp fa).2 | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | have gn : NontrivialHolomorphicAt (g c) (extChartAt I z z) := by rw [← hg]; exact n.inCharts | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | have h := gn.nhds_le_map_nhds_param' ga | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | simp only [nhds_prod_eq, ← extChartAt_map_nhds' I z, Filter.map_map, Filter.prod_map_id_map_eq,
Function.comp] at h | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | replace h := @Filter.map_mono _ _ (fun p : ℂ × ℂ ↦ (p.1, (extChartAt I (f c z)).symm p.2)) _ _ h | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | simp only [← hg] at h | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | rw [PartialEquiv.left_inv _ (mem_extChartAt_source I z)] at h | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | have pe := Filter.prod_map_id_map_eq (f := 𝓝 c) (g := 𝓝 (extChartAt I (f c z) (f c z)))
(m := fun x ↦ (extChartAt I (f c z)).symm x) | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | rw [extChartAt_symm_map_nhds', ←nhds_prod_eq] at pe | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | refine _root_.trans (le_of_eq pe) (_root_.trans h (le_of_eq ?_)) | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | clear h pe | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | rw [←nhds_prod_eq, Filter.map_map] | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | apply Filter.map_congr | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | case h
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | apply ((isOpen_extChartAt_source II (c, z)).eventually_mem (mem_extChartAt_source II (c, z))).mp | case h
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifo... | case h
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | apply (fa.continuousAt.eventually_mem (extChartAt_source_mem_nhds I (f c z))).mp | case h
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifo... | case h
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | apply eventually_of_forall | case h
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifo... | case h.hp
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMan... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | intro ⟨e, w⟩ fm m | case h.hp
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMan... | case h.hp
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMan... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | simp only [Function.comp, uncurry, extChartAt_prod, PartialEquiv.prod_source, mem_prod_eq] at fm m | case h.hp
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMan... | case h.hp
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMan... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | simp only [Function.comp, PartialEquiv.left_inv _ m.2, PartialEquiv.left_inv _ fm] | case h.hp
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticMan... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | rw [← hg] | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | exact (holomorphicAt_iff.mp fa).2 | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | rw [← hg] | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | exact n.inCharts | X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | set fl := fun n z ↦ log (f n z) | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have near1 : ∀ n z, z ∈ s → abs (f n z - 1) ≤ 1 / 2 := by
intro n z zs
calc abs (f n z - 1)
_ ≤ c * a ^ n := hf n z zs
_ ≤ (1 / 2 : ℝ) * (1:ℝ) ^ n := by bound
_ = 1 / 2 := by norm_num | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have near1' : ∀ n z, z ∈ s → abs (f n z - 1) < 1 := fun n z zs ↦
lt_of_le_of_lt (near1 n z zs) (by linarith) | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g... | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have expfl : ∀ n z, z ∈ s → exp (fl n z) = f n z := by
intro n z zs; refine Complex.exp_log ?_
exact near_one_avoids_zero (near1' n z zs) | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have hl : ∀ n, AnalyticOn ℂ (fl n) s := fun n ↦
(h n).log (fun z m ↦ mem_slitPlane_of_near_one (near1' n z m)) | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | set c2 := 2 * c | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have hfl : ∀ n z, z ∈ s → abs (fl n z) ≤ c2 * a ^ n := by
intro n z zs
calc abs (fl n z)
_ = abs (log (f n z)) := rfl
_ ≤ 2 * abs (f n z - 1) := (log_small (near1 n z zs))
_ ≤ 2 * (c * a ^ n) := by linarith [hf n z zs]
_ = 2 * c * a ^ n := by ring
_ = c2 * a ^ n := rfl | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rcases fast_series_converge o a0 a1 hl hfl with ⟨gl, gla, us⟩ | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... | case intro.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | generalize hg : (fun z ↦ exp (gl z)) = g | case intro.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ ... | case intro.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | use g | case intro.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ ... | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Com... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | refine ⟨?_, ?_, ?_⟩ | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Com... | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro n z zs | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
⊢ ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2 | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
n : ℕ
z : ℂ
zs : z ∈ s
⊢ Complex.abs (f n z - 1) ≤ 1 / 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | calc abs (f n z - 1)
_ ≤ c * a ^ n := hf n z zs
_ ≤ (1 / 2 : ℝ) * (1:ℝ) ^ n := by bound
_ = 1 / 2 := by norm_num | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
n : ℕ
z : ℂ
zs : z ∈ s
⊢ Complex.abs (f n z - 1) ≤ 1 / 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | bound | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
n : ℕ
z : ℂ
zs : z ∈ s
⊢ c * a ^ n ≤ 1 / 2 * 1 ^ n | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | norm_num | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
n : ℕ
z : ℂ
zs : z ∈ s
⊢ 1 / 2 * 1 ^ n = 1 / 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | linarith | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
n : ℕ
z : ℂ
zs : z ∈ s
⊢ 1 / 2 < 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro n z zs | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | refine Complex.exp_log ?_ | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | exact near_one_avoids_zero (near1' n z zs) | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro n z zs | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | calc abs (fl n z)
_ = abs (log (f n z)) := rfl
_ ≤ 2 * abs (f n z - 1) := (log_small (near1 n z zs))
_ ≤ 2 * (c * a ^ n) := by linarith [hf n z zs]
_ = 2 * c * a ^ n := by ring
_ = c2 * a ^ n := rfl | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | linarith [hf n z zs] | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | ring | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro z zs | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | specialize us z zs | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | simp at us | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have comp :
Filter.Tendsto (exp ∘ fun N : Finset ℕ ↦ N.sum fun n ↦ fl n z) atTop (𝓝 (exp (gl z))) :=
Filter.Tendsto.comp (Continuous.tendsto Complex.continuous_exp _) us | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have expsum0 : (exp ∘ fun N : Finset ℕ ↦ N.sum fun n ↦ fl n z) = fun N : Finset ℕ ↦
N.prod fun n ↦ f n z := by
apply funext; intro N; simp; rw [Complex.exp_sum]; simp_rw [expfl _ z zs] | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rw [expsum0] at comp | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rw [← hg] | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | assumption | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | apply funext | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.ab... | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Com... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro N | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Com... | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Com... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | simp | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Com... | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Com... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rw [Complex.exp_sum] | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Com... | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Com... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | simp_rw [expfl _ z zs] | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Com... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rw [← hg] | case h.refine_2
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... | case h.refine_2
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | exact fun z zs ↦ AnalyticAt.exp.comp (gla z zs) | case h.refine_2
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | simp only [Complex.exp_ne_zero, Ne, not_false_iff, imp_true_iff, ← hg] | case h.refine_3
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | rcases fast_products_converge o c12 a0 a1 h hf with ⟨g, gp, ga, g0⟩ | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
⊢ ProdExistsOn f s ∧ AnalyticOn ℂ (tprodOn f) s ∧ ∀ z ∈ s, tprodOn f z ≠ 0 | case intro.intro.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
⊢ ProdExistsOn f s ∧ Analytic... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | refine ⟨?_, ?_, ?_⟩ | case intro.intro.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
⊢ ProdExistsOn f s ∧ Analytic... | case intro.intro.intro.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
⊢ ProdExistsOn f s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | exact fun z zs ↦ ⟨g z, gp z zs⟩ | case intro.intro.intro.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
⊢ ProdExistsOn f s | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | rwa [← analyticOn_congr o fun z zs ↦ (gp.tprodOn_eq z zs).symm] | case intro.intro.intro.refine_2
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
⊢ AnalyticOn ℂ (tpro... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | intro z zs | case intro.intro.intro.refine_3
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
⊢ ∀ z ∈ s, tprodOn f... | case intro.intro.intro.refine_3
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
z : ℂ
zs : z ∈ s
⊢ t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | rw [gp.tprodOn_eq z zs] | case intro.intro.intro.refine_3
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
z : ℂ
zs : z ∈ s
⊢ t... | case intro.intro.intro.refine_3
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
z : ℂ
zs : z ∈ s
⊢ g... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | exact g0 z zs | case intro.intro.intro.refine_3
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
z : ℂ
zs : z ∈ s
⊢ g... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow | [112, 1] | [115, 72] | rw [HasProd] | f : ℕ → ℂ
g : ℂ
p : ℕ
h : HasProd f g
⊢ HasProd (fun n => f n ^ p) (g ^ p) | f : ℕ → ℂ
g : ℂ
p : ℕ
h : HasProd f g
⊢ Filter.Tendsto (fun s => s.prod fun b => f b ^ p) atTop (𝓝 (g ^ p)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow | [112, 1] | [115, 72] | simp_rw [Finset.prod_pow] | f : ℕ → ℂ
g : ℂ
p : ℕ
h : HasProd f g
⊢ Filter.Tendsto (fun s => s.prod fun b => f b ^ p) atTop (𝓝 (g ^ p)) | f : ℕ → ℂ
g : ℂ
p : ℕ
h : HasProd f g
⊢ Filter.Tendsto (fun s => (s.prod fun x => f x) ^ p) atTop (𝓝 (g ^ p)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow | [112, 1] | [115, 72] | exact Filter.Tendsto.comp (Continuous.tendsto (continuous_pow p) g) h | f : ℕ → ℂ
g : ℂ
p : ℕ
h : HasProd f g
⊢ Filter.Tendsto (fun s => (s.prod fun x => f x) ^ p) atTop (𝓝 (g ^ p)) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow' | [118, 1] | [120, 96] | rcases h with ⟨g, h⟩ | f : ℕ → ℂ
p : ℕ
h : ProdExists f
⊢ tprod f ^ p = ∏' (n : ℕ), f n ^ p | case intro
f : ℕ → ℂ
p : ℕ
g : ℂ
h : HasProd f g
⊢ tprod f ^ p = ∏' (n : ℕ), f n ^ p |
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