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leandojo-lean4-formal-informal-strings-split

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Metric.ediam_cthickening_le ** ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε✝ : ℝ s t : Set α x : α ε : ℝ≥0 ⊢ EMetric.diam (cthickening (↑ε) s) ≤ EMetric.diam s + 2 * ↑ε refine' diam_le fun x hx y hy => ENNReal.le_of_forall_pos_le_add fun δ hδ _ => _ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s t : Set α x✝¹ : α ε : ℝ≥0 x : α hx : x ∈ cthickening (↑ε) s y : α hy : y ∈ cthickening (↑ε) s δ : ℝ≥0 hδ : 0 < δ x✝ : EMetric.diam s + 2 * ↑ε < ⊤ ⊢ edist x y ≤ EMetric.diam s + 2 * ↑ε + ↑δ rw [mem_cthickening_iff, ENNReal.ofReal_coe_nnreal] at hx hy ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s t : Set α x✝¹ : α ε : ℝ≥0 x : α hx : infEdist x s ≤ ↑ε y : α hy : infEdist y s ≤ ↑ε δ : ℝ≥0 hδ : 0 < δ x✝ : EMetric.diam s + 2 * ↑ε < ⊤ ⊢ edist x y ≤ EMetric.diam s + 2 * ↑ε + ↑δ have hε : (ε : ℝ≥0∞) < ε + δ := ENNReal.coe_lt_coe.2 (lt_add_of_pos_right _ hδ) ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s t : Set α x✝¹ : α ε : ℝ≥0 x : α hx : infEdist x s ≤ ↑ε y : α hy : infEdist y s ≤ ↑ε δ : ℝ≥0 hδ : 0 < δ x✝ : EMetric.diam s + 2 * ↑ε < ⊤ hε : ↑ε < ↑ε + ↑δ ⊢ edist x y ≤ EMetric.diam s + 2 * ↑ε + ↑δ replace hx := hx.trans_lt hε ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s t : Set α x✝¹ : α ε : ℝ≥0 x y : α hy : infEdist y s ≤ ↑ε δ : ℝ≥0 hδ : 0 < δ x✝ : EMetric.diam s + 2 * ↑ε < ⊤ hε : ↑ε < ↑ε + ↑δ hx : infEdist x s < ↑ε + ↑δ ⊢ edist x y ≤ EMetric.diam s + 2 * ↑ε + ↑δ obtain ⟨x', hx', hxx'⟩ := infEdist_lt_iff.mp hx case intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s t : Set α x✝¹ : α ε : ℝ≥0 x y : α hy : infEdist y s ≤ ↑ε δ : ℝ≥0 hδ : 0 < δ x✝ : EMetric.diam s + 2 * ↑ε < ⊤ hε : ↑ε < ↑ε + ↑δ hx : infEdist x s < ↑ε + ↑δ x' : α hx' : x' ∈ s hxx' : edist x x' < ↑ε + ↑δ ⊢ edist x y ≤ EMetric.diam s + 2 * ↑ε + ↑δ calc edist x y ≤ edist x x' + edist y x' := edist_triangle_right _ _ _ _ ≤ ε + δ + (infEdist y s + EMetric.diam s) := add_le_add hxx'.le (edist_le_infEdist_add_ediam hx') _ ≤ ε + δ + (ε + EMetric.diam s) := add_le_add_left (add_le_add_right hy _) _ _ = _ := by rw [two_mul]; ac_rfl ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s t : Set α x✝¹ : α ε : ℝ≥0 x y : α hy : infEdist y s ≤ ↑ε δ : ℝ≥0 hδ : 0 < δ x✝ : EMetric.diam s + 2 * ↑ε < ⊤ hε : ↑ε < ↑ε + ↑δ hx : infEdist x s < ↑ε + ↑δ x' : α hx' : x' ∈ s hxx' : edist x x' < ↑ε + ↑δ ⊢ ↑ε + ↑δ + (↑ε + EMetric.diam s) = EMetric.diam s + 2 * ↑ε + ↑δ rw [two_mul] ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s t : Set α x✝¹ : α ε : ℝ≥0 x y : α hy : infEdist y s ≤ ↑ε δ : ℝ≥0 hδ : 0 < δ x✝ : EMetric.diam s + 2 * ↑ε < ⊤ hε : ↑ε < ↑ε + ↑δ hx : infEdist x s < ↑ε + ↑δ x' : α hx' : x' ∈ s hxx' : edist x x' < ↑ε + ↑δ ⊢ ↑ε + ↑δ + (↑ε + EMetric.diam s) = EMetric.diam s + (↑ε + ↑ε) + ↑δ ac_rfl Qed
Metric.diam_cthickening_le ** ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ ε : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α hε : 0 ≤ ε ⊢ diam (cthickening ε s) ≤ diam s + 2 * ε lift ε to ℝ≥0 using hε case intro ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α ε : ℝ≥0 ⊢ diam (cthickening (↑ε) s) ≤ diam s + 2 * ↑ε refine (toReal_le_add' (ediam_cthickening_le _) ?_ ?_).trans_eq ?_ case intro.refine_1 ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α ε : ℝ≥0 ⊢ EMetric.diam s = ⊤ → EMetric.diam (cthickening (↑ε) s) = ⊤ exact fun h ↦ top_unique <\| h ▸ EMetric.diam_mono (self_subset_cthickening _) case intro.refine_2 ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α ε : ℝ≥0 ⊢ 2 * ↑ε = ⊤ → EMetric.diam (cthickening (↑ε) s) = ⊤ simp [mul_eq_top] case intro.refine_3 ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α ε : ℝ≥0 ⊢ ENNReal.toReal (EMetric.diam s) + ENNReal.toReal (2 * ↑ε) = diam s + 2 * ↑ε simp [diam] Qed
Metric.diam_thickening_le ** ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ ε : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α hε : 0 ≤ ε ⊢ diam (thickening ε s) ≤ diam s + 2 * ε by_cases hs : IsBounded s case neg ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ ε : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α hε : 0 ≤ ε hs : ¬Bornology.IsBounded s ⊢ diam (thickening ε s) ≤ diam s + 2 * ε obtain rfl \| hε := hε.eq_or_lt case pos ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ ε : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α hε : 0 ≤ ε hs : Bornology.IsBounded s ⊢ diam (thickening ε s) ≤ diam s + 2 * ε exact (diam_mono (thickening_subset_cthickening _ _) hs.cthickening).trans (diam_cthickening_le _ hε) case neg.inl ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α hs : ¬Bornology.IsBounded s hε : 0 ≤ 0 ⊢ diam (thickening 0 s) ≤ diam s + 2 * 0 simp [thickening_of_nonpos, diam_nonneg] case neg.inr ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ ε : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α hε✝ : 0 ≤ ε hs : ¬Bornology.IsBounded s hε : 0 < ε ⊢ 0 ≤ diam s + 2 * ε positivity Qed
Metric.thickening_closure ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α ⊢ thickening δ (closure s) = thickening δ s simp_rw [thickening, infEdist_closure] ** Qed
Metric.cthickening_closure ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α ⊢ cthickening δ (closure s) = cthickening δ s simp_rw [cthickening, infEdist_closure] ** Qed
Disjoint.exists_thickenings ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t ⊢ ∃ δ, 0 < δ ∧ Disjoint (thickening δ s) (thickening δ t) obtain ⟨r, hr, h⟩ := exists_pos_forall_lt_edist hs ht hst case intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y ⊢ ∃ δ, 0 < δ ∧ Disjoint (thickening δ s) (thickening δ t) refine' ⟨r / 2, half_pos (NNReal.coe_pos.2 hr), _⟩ case intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y ⊢ Disjoint (thickening (↑r / 2) s) (thickening (↑r / 2) t) rw [disjoint_iff_inf_le] case intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y ⊢ thickening (↑r / 2) s ⊓ thickening (↑r / 2) t ≤ ⊥ rintro z ⟨hzs, hzt⟩ case intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y z : α hzs : z ∈ thickening (↑r / 2) s hzt : z ∈ thickening (↑r / 2) t ⊢ z ∈ ⊥ rw [mem_thickening_iff_exists_edist_lt] at hzs hzt case intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y z : α hzs : ∃ z_1, z_1 ∈ s ∧ edist z z_1 < ENNReal.ofReal (↑r / 2) hzt : ∃ z_1, z_1 ∈ t ∧ edist z z_1 < ENNReal.ofReal (↑r / 2) ⊢ z ∈ ⊥ rw [← NNReal.coe_two, ← NNReal.coe_div, ENNReal.ofReal_coe_nnreal] at hzs hzt case intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y z : α hzs : ∃ z_1, z_1 ∈ s ∧ edist z z_1 < ↑(r / 2) hzt : ∃ z_1, z_1 ∈ t ∧ edist z z_1 < ↑(r / 2) ⊢ z ∈ ⊥ obtain ⟨x, hx, hzx⟩ := hzs case intro.intro.intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x✝ : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y z : α hzt : ∃ z_1, z_1 ∈ t ∧ edist z z_1 < ↑(r / 2) x : α hx : x ∈ s hzx : edist z x < ↑(r / 2) ⊢ z ∈ ⊥ obtain ⟨y, hy, hzy⟩ := hzt case intro.intro.intro.intro.intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x✝ : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y z x : α hx : x ∈ s hzx : edist z x < ↑(r / 2) y : α hy : y ∈ t hzy : edist z y < ↑(r / 2) ⊢ z ∈ ⊥ refine' (h x hx y hy).not_le _ case intro.intro.intro.intro.intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x✝ : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y z x : α hx : x ∈ s hzx : edist z x < ↑(r / 2) y : α hy : y ∈ t hzy : edist z y < ↑(r / 2) ⊢ edist x y ≤ ↑r calc edist x y ≤ edist z x + edist z y := edist_triangle_left _ _ _ _ ≤ ↑(r / 2) + ↑(r / 2) := (add_le_add hzx.le hzy.le) _ = r := by rw [← ENNReal.coe_add, add_halves] ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x✝ : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y z x : α hx : x ∈ s hzx : edist z x < ↑(r / 2) y : α hy : y ∈ t hzy : edist z y < ↑(r / 2) ⊢ ↑(r / 2) + ↑(r / 2) = ↑r rw [← ENNReal.coe_add, add_halves] ** Qed
Disjoint.exists_cthickenings ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t ⊢ ∃ δ, 0 < δ ∧ Disjoint (cthickening δ s) (cthickening δ t) obtain ⟨δ, hδ, h⟩ := hst.exists_thickenings hs ht case intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t δ : ℝ hδ : 0 < δ h : Disjoint (thickening δ s) (thickening δ t) ⊢ ∃ δ, 0 < δ ∧ Disjoint (cthickening δ s) (cthickening δ t) refine' ⟨δ / 2, half_pos hδ, h.mono _ _⟩ <;> exact cthickening_subset_thickening' hδ (half_lt_self hδ) _ ** Qed
IsCompact.exists_isCompact_cthickening ι : Sort u_1 α : Type u β : Type v inst✝¹ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α inst✝ : LocallyCompactSpace α hs : IsCompact s ⊢ ∃ δ, 0 < δ ∧ IsCompact (cthickening δ s) rcases exists_compact_superset hs with ⟨K, K_compact, hK⟩ case intro.intro ι : Sort u_1 α : Type u β : Type v inst✝¹ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α inst✝ : LocallyCompactSpace α hs : IsCompact s K : Set α K_compact : IsCompact K hK : s ⊆ interior K ⊢ ∃ δ, 0 < δ ∧ IsCompact (cthickening δ s) rcases hs.exists_cthickening_subset_open isOpen_interior hK with ⟨δ, δpos, hδ⟩ case intro.intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝¹ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α inst✝ : LocallyCompactSpace α hs : IsCompact s K : Set α K_compact : IsCompact K hK : s ⊆ interior K δ : ℝ δpos : 0 < δ hδ : cthickening δ s ⊆ interior K ⊢ ∃ δ, 0 < δ ∧ IsCompact (cthickening δ s) refine ⟨δ, δpos, ?_⟩ case intro.intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝¹ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α inst✝ : LocallyCompactSpace α hs : IsCompact s K : Set α K_compact : IsCompact K hK : s ⊆ interior K δ : ℝ δpos : 0 < δ hδ : cthickening δ s ⊆ interior K ⊢ IsCompact (cthickening δ s) exact K_compact.of_isClosed_subset isClosed_cthickening (hδ.trans interior_subset) ** Qed
Metric.cthickening_eq_iInter_cthickening' ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ⊢ cthickening δ E = ⋂ ε ∈ s, cthickening ε E apply Subset.antisymm case h₁ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ⊢ cthickening δ E ⊆ ⋂ ε ∈ s, cthickening ε E exact subset_iInter₂ fun _ hε => cthickening_mono (le_of_lt (hsδ hε)) E case h₂ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ⊢ ⋂ ε ∈ s, cthickening ε E ⊆ cthickening δ E unfold cthickening case h₂ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ⊢ ⋂ ε ∈ s, {x \| infEdist x E ≤ ENNReal.ofReal ε} ⊆ {x \| infEdist x E ≤ ENNReal.ofReal δ} intro x hx case h₂ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x✝ : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α x : α hx : x ∈ ⋂ ε ∈ s, {x \| infEdist x E ≤ ENNReal.ofReal ε} ⊢ x ∈ {x \| infEdist x E ≤ ENNReal.ofReal δ} simp only [mem_iInter, mem_setOf_eq] at * case h₂ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x✝ : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α x : α hx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i ⊢ infEdist x E ≤ ENNReal.ofReal δ apply ENNReal.le_of_forall_pos_le_add case h₂.h ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x✝ : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α x : α hx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i ⊢ ∀ (ε : ℝ≥0), 0 < ε → ENNReal.ofReal δ < ⊤ → infEdist x E ≤ ENNReal.ofReal δ + ↑ε intro η η_pos _ case h₂.h ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x✝ : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α x : α hx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i η : ℝ≥0 η_pos : 0 < η a✝ : ENNReal.ofReal δ < ⊤ ⊢ infEdist x E ≤ ENNReal.ofReal δ + ↑η rcases hs (δ + η) (lt_add_of_pos_right _ (NNReal.coe_pos.mpr η_pos)) with ⟨ε, ⟨hsε, hε⟩⟩ case h₂.h.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x✝ : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α x : α hx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i η : ℝ≥0 η_pos : 0 < η a✝ : ENNReal.ofReal δ < ⊤ ε : ℝ hsε : ε ∈ s hε : ε ∈ Ioc δ (δ + ↑η) ⊢ infEdist x E ≤ ENNReal.ofReal δ + ↑η apply ((hx ε hsε).trans (ENNReal.ofReal_le_ofReal hε.2)).trans case h₂.h.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x✝ : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α x : α hx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i η : ℝ≥0 η_pos : 0 < η a✝ : ENNReal.ofReal δ < ⊤ ε : ℝ hsε : ε ∈ s hε : ε ∈ Ioc δ (δ + ↑η) ⊢ ENNReal.ofReal (δ + ↑η) ≤ ENNReal.ofReal δ + ↑η rw [ENNReal.coe_nnreal_eq η] case h₂.h.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x✝ : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α x : α hx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i η : ℝ≥0 η_pos : 0 < η a✝ : ENNReal.ofReal δ < ⊤ ε : ℝ hsε : ε ∈ s hε : ε ∈ Ioc δ (δ + ↑η) ⊢ ENNReal.ofReal (δ + ↑η) ≤ ENNReal.ofReal δ + ENNReal.ofReal ↑η exact ENNReal.ofReal_add_le ** Qed
Metric.cthickening_eq_iInter_cthickening ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ E : Set α ⊢ cthickening δ E = ⋂ ε, ⋂ (_ : δ < ε), cthickening ε E apply cthickening_eq_iInter_cthickening' (Ioi δ) rfl.subset case hs ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ E : Set α ⊢ ∀ (ε : ℝ), δ < ε → Set.Nonempty (Ioi δ ∩ Ioc δ ε) simp_rw [inter_eq_right.mpr Ioc_subset_Ioi_self] case hs ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ E : Set α ⊢ ∀ (ε : ℝ), δ < ε → Set.Nonempty (Ioc δ ε) exact fun _ hε => nonempty_Ioc.mpr hε ** Qed
Metric.cthickening_eq_iInter_thickening' ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ⊢ cthickening δ E = ⋂ ε ∈ s, thickening ε E refine' (subset_iInter₂ fun ε hε => _).antisymm _ case refine'_1 ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ε : ℝ hε : ε ∈ s ⊢ cthickening δ E ⊆ thickening ε E obtain ⟨ε', -, hε'⟩ := hs ε (hsδ hε) case refine'_1.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ε : ℝ hε : ε ∈ s ε' : ℝ hε' : ε' ∈ Ioc δ ε ⊢ cthickening δ E ⊆ thickening ε E have ss := cthickening_subset_thickening' (lt_of_le_of_lt δ_nn hε'.1) hε'.1 E case refine'_1.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ε : ℝ hε : ε ∈ s ε' : ℝ hε' : ε' ∈ Ioc δ ε ss : cthickening δ E ⊆ thickening ε' E ⊢ cthickening δ E ⊆ thickening ε E exact ss.trans (thickening_mono hε'.2 E) case refine'_2 ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ⊢ ⋂ i ∈ s, thickening i E ⊆ cthickening δ E rw [cthickening_eq_iInter_cthickening' s hsδ hs E] case refine'_2 ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ⊢ ⋂ i ∈ s, thickening i E ⊆ ⋂ ε ∈ s, cthickening ε E exact iInter₂_mono fun ε _ => thickening_subset_cthickening ε E ** Qed
Metric.cthickening_eq_iInter_thickening ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ E : Set α ⊢ cthickening δ E = ⋂ ε, ⋂ (_ : δ < ε), thickening ε E apply cthickening_eq_iInter_thickening' δ_nn (Ioi δ) rfl.subset case hs ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ E : Set α ⊢ ∀ (ε : ℝ), δ < ε → Set.Nonempty (Ioi δ ∩ Ioc δ ε) simp_rw [inter_eq_right.mpr Ioc_subset_Ioi_self] case hs ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ E : Set α ⊢ ∀ (ε : ℝ), δ < ε → Set.Nonempty (Ioc δ ε) exact fun _ hε => nonempty_Ioc.mpr hε ** Qed
Metric.cthickening_eq_iInter_thickening'' ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ E : Set α ⊢ cthickening δ E = ⋂ ε, ⋂ (_ : max 0 δ < ε), thickening ε E rw [← cthickening_max_zero, cthickening_eq_iInter_thickening] case δ_nn ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ E : Set α ⊢ 0 ≤ max 0 δ exact le_max_left _ _ ** Qed
Metric.closure_eq_iInter_cthickening' ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) ⊢ closure E = ⋂ δ ∈ s, cthickening δ E by_cases hs₀ : s ⊆ Ioi 0 case neg ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : ¬s ⊆ Ioi 0 ⊢ closure E = ⋂ δ ∈ s, cthickening δ E obtain ⟨δ, hδs, δ_nonpos⟩ := not_subset.mp hs₀ case neg.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : ¬s ⊆ Ioi 0 δ : ℝ hδs : δ ∈ s δ_nonpos : ¬δ ∈ Ioi 0 ⊢ closure E = ⋂ δ ∈ s, cthickening δ E rw [Set.mem_Ioi, not_lt] at δ_nonpos case neg.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : ¬s ⊆ Ioi 0 δ : ℝ hδs : δ ∈ s δ_nonpos : δ ≤ 0 ⊢ closure E = ⋂ δ ∈ s, cthickening δ E apply Subset.antisymm case pos ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : s ⊆ Ioi 0 ⊢ closure E = ⋂ δ ∈ s, cthickening δ E rw [← cthickening_zero] case pos ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : s ⊆ Ioi 0 ⊢ cthickening 0 E = ⋂ δ ∈ s, cthickening δ E apply cthickening_eq_iInter_cthickening' _ hs₀ hs case neg.intro.intro.h₁ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : ¬s ⊆ Ioi 0 δ : ℝ hδs : δ ∈ s δ_nonpos : δ ≤ 0 ⊢ closure E ⊆ ⋂ δ ∈ s, cthickening δ E exact subset_iInter₂ fun ε _ => closure_subset_cthickening ε E case neg.intro.intro.h₂ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : ¬s ⊆ Ioi 0 δ : ℝ hδs : δ ∈ s δ_nonpos : δ ≤ 0 ⊢ ⋂ δ ∈ s, cthickening δ E ⊆ closure E rw [← cthickening_of_nonpos δ_nonpos E] case neg.intro.intro.h₂ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : ¬s ⊆ Ioi 0 δ : ℝ hδs : δ ∈ s δ_nonpos : δ ≤ 0 ⊢ ⋂ δ ∈ s, cthickening δ E ⊆ cthickening δ E exact biInter_subset_of_mem hδs ** Qed
Metric.closure_eq_iInter_cthickening ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α E : Set α ⊢ closure E = ⋂ δ, ⋂ (_ : 0 < δ), cthickening δ E rw [← cthickening_zero] ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α E : Set α ⊢ cthickening 0 E = ⋂ δ, ⋂ (_ : 0 < δ), cthickening δ E exact cthickening_eq_iInter_cthickening E ** Qed
Metric.closure_eq_iInter_thickening' ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs₀ : s ⊆ Ioi 0 hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) ⊢ closure E = ⋂ δ ∈ s, thickening δ E rw [← cthickening_zero] ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs₀ : s ⊆ Ioi 0 hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) ⊢ cthickening 0 E = ⋂ δ ∈ s, thickening δ E apply cthickening_eq_iInter_thickening' le_rfl _ hs₀ hs ** Qed
Metric.closure_eq_iInter_thickening ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α E : Set α ⊢ closure E = ⋂ δ, ⋂ (_ : 0 < δ), thickening δ E rw [← cthickening_zero] ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α E : Set α ⊢ cthickening 0 E = ⋂ δ, ⋂ (_ : 0 < δ), thickening δ E exact cthickening_eq_iInter_thickening rfl.ge E ** Qed
Metric.closedBall_subset_cthickening ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α x : α E : Set α hx : x ∈ E δ : ℝ ⊢ closedBall x δ ⊆ cthickening δ E refine' (closedBall_subset_cthickening_singleton _ _).trans (cthickening_subset_of_subset _ _) ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α x : α E : Set α hx : x ∈ E δ : ℝ ⊢ {x} ⊆ E simpa using hx ** Qed
Metric.cthickening_subset_iUnion_closedBall_of_lt ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α E : Set α δ δ' : ℝ hδ₀ : 0 < δ' hδδ' : δ < δ' ⊢ cthickening δ E ⊆ ⋃ x ∈ E, closedBall x δ' refine' (cthickening_subset_thickening' hδ₀ hδδ' E).trans fun x hx => _ ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α E : Set α δ δ' : ℝ hδ₀ : 0 < δ' hδδ' : δ < δ' x : α hx : x ∈ thickening δ' E ⊢ x ∈ ⋃ x ∈ E, closedBall x δ' obtain ⟨y, hy₁, hy₂⟩ := mem_thickening_iff.mp hx case intro.intro ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α E : Set α δ δ' : ℝ hδ₀ : 0 < δ' hδδ' : δ < δ' x : α hx : x ∈ thickening δ' E y : α hy₁ : y ∈ E hy₂ : dist x y < δ' ⊢ x ∈ ⋃ x ∈ E, closedBall x δ' exact mem_iUnion₂.mpr ⟨y, hy₁, hy₂.le⟩ ** Qed
IsCompact.cthickening_eq_biUnion_closedBall ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ ⊢ cthickening δ E = ⋃ x ∈ E, closedBall x δ rcases eq_empty_or_nonempty E with (rfl \| hne) case inr ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ hne : Set.Nonempty E ⊢ cthickening δ E = ⋃ x ∈ E, closedBall x δ refine Subset.antisymm (fun x hx ↦ ?_) (iUnion₂_subset fun x hx ↦ closedBall_subset_cthickening hx _) case inr ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ E ⊢ x ∈ ⋃ x ∈ E, closedBall x δ obtain ⟨y, yE, hy⟩ : ∃ y ∈ E, infEdist x E = edist x y := hE.exists_infEdist_eq_edist hne _ case inr.intro.intro ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ E y : α yE : y ∈ E hy : infEdist x E = edist x y ⊢ x ∈ ⋃ x ∈ E, closedBall x δ have D1 : edist x y ≤ ENNReal.ofReal δ := (le_of_eq hy.symm).trans hx case inr.intro.intro ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ E y : α yE : y ∈ E hy : infEdist x E = edist x y D1 : edist x y ≤ ENNReal.ofReal δ ⊢ x ∈ ⋃ x ∈ E, closedBall x δ have D2 : dist x y ≤ δ := by rw [edist_dist] at D1 exact (ENNReal.ofReal_le_ofReal_iff hδ).1 D1 case inr.intro.intro ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ E y : α yE : y ∈ E hy : infEdist x E = edist x y D1 : edist x y ≤ ENNReal.ofReal δ D2 : dist x y ≤ δ ⊢ x ∈ ⋃ x ∈ E, closedBall x δ exact mem_biUnion yE D2 case inl ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ hδ : 0 ≤ δ hE : IsCompact ∅ ⊢ cthickening δ ∅ = ⋃ x ∈ ∅, closedBall x δ simp only [cthickening_empty, biUnion_empty] ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ E y : α yE : y ∈ E hy : infEdist x E = edist x y D1 : edist x y ≤ ENNReal.ofReal δ ⊢ dist x y ≤ δ rw [edist_dist] at D1 ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ E y : α yE : y ∈ E hy : infEdist x E = edist x y D1 : ENNReal.ofReal (dist x y) ≤ ENNReal.ofReal δ ⊢ dist x y ≤ δ exact (ENNReal.ofReal_le_ofReal_iff hδ).1 D1 ** Qed
Metric.cthickening_eq_biUnion_closedBall ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α E : Set α hδ : 0 ≤ δ ⊢ cthickening δ E = ⋃ x ∈ closure E, closedBall x δ rcases eq_empty_or_nonempty E with (rfl \| hne) case inr ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α E : Set α hδ : 0 ≤ δ hne : Set.Nonempty E ⊢ cthickening δ E = ⋃ x ∈ closure E, closedBall x δ rw [← cthickening_closure] case inr ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α E : Set α hδ : 0 ≤ δ hne : Set.Nonempty E ⊢ cthickening δ (closure E) = ⋃ x ∈ closure E, closedBall x δ refine Subset.antisymm (fun x hx ↦ ?_) (iUnion₂_subset fun x hx ↦ closedBall_subset_cthickening hx _) case inr ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α E : Set α hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ (closure E) ⊢ x ∈ ⋃ x ∈ closure E, closedBall x δ obtain ⟨y, yE, hy⟩ : ∃ y ∈ closure E, infDist x (closure E) = dist x y := isClosed_closure.exists_infDist_eq_dist (closure_nonempty_iff.mpr hne) x case inr.intro.intro ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α E : Set α hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ (closure E) y : α yE : y ∈ closure E hy : infDist x (closure E) = dist x y ⊢ x ∈ ⋃ x ∈ closure E, closedBall x δ replace hy : dist x y ≤ δ := (ENNReal.ofReal_le_ofReal_iff hδ).mp (((congr_arg ENNReal.ofReal hy.symm).le.trans ENNReal.ofReal_toReal_le).trans hx) case inr.intro.intro ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α E : Set α hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ (closure E) y : α yE : y ∈ closure E hy : dist x y ≤ δ ⊢ x ∈ ⋃ x ∈ closure E, closedBall x δ exact mem_biUnion yE hy case inl ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α hδ : 0 ≤ δ ⊢ cthickening δ ∅ = ⋃ x ∈ closure ∅, closedBall x δ simp only [cthickening_empty, biUnion_empty, closure_empty] ** Qed
IsClosed.cthickening_eq_biUnion_closedBall ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α E : Set α hE : IsClosed E hδ : 0 ≤ δ ⊢ cthickening δ E = ⋃ x ∈ E, closedBall x δ rw [cthickening_eq_biUnion_closedBall E hδ, hE.closure_eq] ** Qed
Metric.infEdist_le_infEdist_cthickening_add ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α ⊢ infEdist x s ≤ infEdist x (cthickening δ s) + ENNReal.ofReal δ refine' le_of_forall_lt' fun r h => _ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α r : ℝ≥0∞ h : infEdist x (cthickening δ s) + ENNReal.ofReal δ < r ⊢ infEdist x s < r simp_rw [← lt_tsub_iff_right, infEdist_lt_iff, mem_cthickening_iff] at h ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α r : ℝ≥0∞ h : ∃ y, infEdist y s ≤ ENNReal.ofReal δ ∧ edist x y < r - ENNReal.ofReal δ ⊢ infEdist x s < r obtain ⟨y, hy, hxy⟩ := h case intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α r : ℝ≥0∞ y : α hy : infEdist y s ≤ ENNReal.ofReal δ hxy : edist x y < r - ENNReal.ofReal δ ⊢ infEdist x s < r exact infEdist_le_edist_add_infEdist.trans_lt ((ENNReal.add_lt_add_of_lt_of_le (hy.trans_lt ENNReal.ofReal_lt_top).ne hxy hy).trans_eq (tsub_add_cancel_of_le <\| le_self_add.trans (lt_tsub_iff_left.1 hxy).le)) ** Qed
Metric.thickening_thickening_subset ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α ε δ : ℝ s : Set α ⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s obtain hε \| hε := le_total ε 0 case inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α ε δ : ℝ s : Set α hε : 0 ≤ ε ⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s obtain hδ \| hδ := le_total δ 0 case inr.inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α ε δ : ℝ s : Set α hε : 0 ≤ ε hδ : 0 ≤ δ ⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s intro x case inr.inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x✝ : α ε δ : ℝ s : Set α hε : 0 ≤ ε hδ : 0 ≤ δ x : α ⊢ x ∈ thickening ε (thickening δ s) → x ∈ thickening (ε + δ) s simp_rw [mem_thickening_iff_exists_edist_lt, ENNReal.ofReal_add hε hδ] case inr.inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x✝ : α ε δ : ℝ s : Set α hε : 0 ≤ ε hδ : 0 ≤ δ x : α ⊢ (∃ z, (∃ z_1, z_1 ∈ s ∧ edist z z_1 < ENNReal.ofReal δ) ∧ edist x z < ENNReal.ofReal ε) → ∃ z, z ∈ s ∧ edist x z < ENNReal.ofReal ε + ENNReal.ofReal δ exact fun ⟨y, ⟨z, hz, hy⟩, hx⟩ => ⟨z, hz, (edist_triangle _ _ _).trans_lt <\| ENNReal.add_lt_add hx hy⟩ case inl ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α ε δ : ℝ s : Set α hε : ε ≤ 0 ⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s simp only [thickening_of_nonpos hε, empty_subset] case inr.inl ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α ε δ : ℝ s : Set α hε : 0 ≤ ε hδ : δ ≤ 0 ⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s simp only [thickening_of_nonpos hδ, thickening_empty, empty_subset] ** Qed
Metric.thickening_cthickening_subset ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε✝ : ℝ s✝ t : Set α x : α ε : ℝ hδ : 0 ≤ δ s : Set α ⊢ thickening ε (cthickening δ s) ⊆ thickening (ε + δ) s obtain hε \| hε := le_total ε 0 case inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε✝ : ℝ s✝ t : Set α x : α ε : ℝ hδ : 0 ≤ δ s : Set α hε : 0 ≤ ε ⊢ thickening ε (cthickening δ s) ⊆ thickening (ε + δ) s intro x case inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε✝ : ℝ s✝ t : Set α x✝ : α ε : ℝ hδ : 0 ≤ δ s : Set α hε : 0 ≤ ε x : α ⊢ x ∈ thickening ε (cthickening δ s) → x ∈ thickening (ε + δ) s simp_rw [mem_thickening_iff_exists_edist_lt, mem_cthickening_iff, ← infEdist_lt_iff, ENNReal.ofReal_add hε hδ] case inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε✝ : ℝ s✝ t : Set α x✝ : α ε : ℝ hδ : 0 ≤ δ s : Set α hε : 0 ≤ ε x : α ⊢ (∃ z, infEdist z s ≤ ENNReal.ofReal δ ∧ edist x z < ENNReal.ofReal ε) → infEdist x s < ENNReal.ofReal ε + ENNReal.ofReal δ rintro ⟨y, hy, hxy⟩ case inr.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε✝ : ℝ s✝ t : Set α x✝ : α ε : ℝ hδ : 0 ≤ δ s : Set α hε : 0 ≤ ε x y : α hy : infEdist y s ≤ ENNReal.ofReal δ hxy : edist x y < ENNReal.ofReal ε ⊢ infEdist x s < ENNReal.ofReal ε + ENNReal.ofReal δ exact infEdist_le_edist_add_infEdist.trans_lt (ENNReal.add_lt_add_of_lt_of_le (hy.trans_lt ENNReal.ofReal_lt_top).ne hxy hy) case inl ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε✝ : ℝ s✝ t : Set α x : α ε : ℝ hδ : 0 ≤ δ s : Set α hε : ε ≤ 0 ⊢ thickening ε (cthickening δ s) ⊆ thickening (ε + δ) s simp only [thickening_of_nonpos hε, empty_subset] ** Qed
Metric.cthickening_thickening_subset ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α hε : 0 ≤ ε δ : ℝ s : Set α ⊢ cthickening ε (thickening δ s) ⊆ cthickening (ε + δ) s obtain hδ \| hδ := le_total δ 0 case inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α hε : 0 ≤ ε δ : ℝ s : Set α hδ : 0 ≤ δ ⊢ cthickening ε (thickening δ s) ⊆ cthickening (ε + δ) s intro x case inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x✝ : α hε : 0 ≤ ε δ : ℝ s : Set α hδ : 0 ≤ δ x : α ⊢ x ∈ cthickening ε (thickening δ s) → x ∈ cthickening (ε + δ) s simp_rw [mem_cthickening_iff, ENNReal.ofReal_add hε hδ] case inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x✝ : α hε : 0 ≤ ε δ : ℝ s : Set α hδ : 0 ≤ δ x : α ⊢ infEdist x (thickening δ s) ≤ ENNReal.ofReal ε → infEdist x s ≤ ENNReal.ofReal ε + ENNReal.ofReal δ exact fun hx => infEdist_le_infEdist_thickening_add.trans (add_le_add_right hx _) case inl ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α hε : 0 ≤ ε δ : ℝ s : Set α hδ : δ ≤ 0 ⊢ cthickening ε (thickening δ s) ⊆ cthickening (ε + δ) s simp only [thickening_of_nonpos hδ, cthickening_empty, empty_subset] ** Qed
Metric.cthickening_cthickening_subset ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x : α hε : 0 ≤ ε hδ : 0 ≤ δ s : Set α ⊢ cthickening ε (cthickening δ s) ⊆ cthickening (ε + δ) s intro x ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x✝ : α hε : 0 ≤ ε hδ : 0 ≤ δ s : Set α x : α ⊢ x ∈ cthickening ε (cthickening δ s) → x ∈ cthickening (ε + δ) s simp_rw [mem_cthickening_iff, ENNReal.ofReal_add hε hδ] ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x✝ : α hε : 0 ≤ ε hδ : 0 ≤ δ s : Set α x : α ⊢ infEdist x (cthickening δ s) ≤ ENNReal.ofReal ε → infEdist x s ≤ ENNReal.ofReal ε + ENNReal.ofReal δ exact fun hx => infEdist_le_infEdist_cthickening_add.trans (add_le_add_right hx _) ** Qed
Metric.frontier_cthickening_disjoint ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α A : Set α r₁ r₂ : ℝ≥0 hr : r₁ ≠ r₂ ⊢ ENNReal.ofReal ↑r₁ ≠ ENNReal.ofReal ↑r₂ simpa ** Qed
TopCat.Presheaf.germ_res_apply C : Type u inst✝² : Category.{v, u} C inst✝¹ : HasColimits C X Y Z : TopCat F : Presheaf C X U V : Opens ↑X i : U ⟶ V x : { x // x ∈ U } inst✝ : ConcreteCategory C s : (forget C).obj (F.obj (op V)) ⊢ ↑(germ F x) (↑(F.map i.op) s) = ↑(germ F ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑V) }) x)) s rw [←comp_apply, germ_res] ** Qed
TopCat.Presheaf.stalk_hom_ext C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y✝ Z : TopCat F : Presheaf C X x : ↑X Y : C f₁ f₂ : stalk F x ⟶ Y ih : ∀ (U : Opens ↑X) (hxU : x ∈ U), germ F { val := x, property := hxU } ≫ f₁ = germ F { val := x, property := hxU } ≫ f₂ U : (OpenNhds x)ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U ≫ f₁ = colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U ≫ f₂ induction' U using Opposite.rec with U case mk C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y✝ Z : TopCat F : Presheaf C X x : ↑X Y : C f₁ f₂ : stalk F x ⟶ Y ih : ∀ (U : Opens ↑X) (hxU : x ∈ U), germ F { val := x, property := hxU } ≫ f₁ = germ F { val := x, property := hxU } ≫ f₂ U : OpenNhds x ⊢ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) { unop := U } ≫ f₁ = colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) { unop := U } ≫ f₂ cases' U with U hxU case mk.mk C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y✝ Z : TopCat F : Presheaf C X x : ↑X Y : C f₁ f₂ : stalk F x ⟶ Y ih : ∀ (U : Opens ↑X) (hxU : x ∈ U), germ F { val := x, property := hxU } ≫ f₁ = germ F { val := x, property := hxU } ≫ f₂ U : Opens ↑X hxU : x ∈ U ⊢ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) { unop := { obj := U, property := hxU } } ≫ f₁ = colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) { unop := { obj := U, property := hxU } } ≫ f₂ exact ih U hxU ** Qed
TopCat.Presheaf.stalkPushforward_germ ** C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y F : Presheaf C X U : Opens ↑Y x : { x // x ∈ (Opens.map f).obj U } ⊢ germ (f _* F) { val := ↑f ↑x, property := (_ : ↑x ∈ (Opens.map f).obj U) } ≫ stalkPushforward C f F ↑x = germ F x rw [stalkPushforward, germ, colimit.ι_map_assoc, colimit.ι_pre, whiskerRight_app] C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y F : Presheaf C X U : Opens ↑Y x : { x // x ∈ (Opens.map f).obj U } ⊢ F.map ((NatTrans.op (OpenNhds.inclusionMapIso f ↑x).inv).app (op { obj := U, property := (_ : ↑{ val := ↑f ↑x, property := (_ : ↑x ∈ (Opens.map f).obj U) } ∈ U) })) ≫ colimit.ι (((whiskeringLeft (OpenNhds ↑x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ↑x).op).obj F) ((OpenNhds.map f ↑x).op.obj (op { obj := U, property := (_ : ↑{ val := ↑f ↑x, property := (_ : ↑x ∈ (Opens.map f).obj U) } ∈ U) })) = germ F x erw [CategoryTheory.Functor.map_id, Category.id_comp] C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y F : Presheaf C X U : Opens ↑Y x : { x // x ∈ (Opens.map f).obj U } ⊢ colimit.ι (((whiskeringLeft (OpenNhds ↑x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ↑x).op).obj F) ((OpenNhds.map f ↑x).op.obj (op { obj := U, property := (_ : ↑{ val := ↑f ↑x, property := (_ : ↑x ∈ (Opens.map f).obj U) } ∈ U) })) = germ F x rfl Qed
TopCat.Presheaf.stalkPushforward.id C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X x : ↑X ⊢ stalkPushforward C (𝟙 X) ℱ x = (stalkFunctor C x).map (Pushforward.id ℱ).hom ext1 j ** case w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X x : ↑X j : (OpenNhds (↑(𝟙 X) x))ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑(𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion (↑(𝟙 X) x)).op).obj (𝟙 X _* ℱ)) j ≫ stalkPushforward C (𝟙 X) ℱ x = colimit.ι (((whiskeringLeft (OpenNhds (↑(𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion (↑(𝟙 X) x)).op).obj (𝟙 X _* ℱ)) j ≫ (stalkFunctor C x).map (Pushforward.id ℱ).hom induction' j with j case w.h C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X x : ↑X j : OpenNhds (↑(𝟙 X) x) ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑(𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion (↑(𝟙 X) x)).op).obj (𝟙 X _* ℱ)) (op j) ≫ stalkPushforward C (𝟙 X) ℱ x = colimit.ι (((whiskeringLeft (OpenNhds (↑(𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion (↑(𝟙 X) x)).op).obj (𝟙 X _* ℱ)) (op j) ≫ (stalkFunctor C x).map (Pushforward.id ℱ).hom rcases j with ⟨⟨_, _⟩, _⟩ case w.h.mk.mk C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X x : ↑X carrier✝ : Set ↑X is_open'✝ : IsOpen carrier✝ property✝ : ↑(𝟙 X) x ∈ { carrier := carrier✝, is_open' := is_open'✝ } ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑(𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion (↑(𝟙 X) x)).op).obj (𝟙 X _* ℱ)) (op { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }) ≫ stalkPushforward C (𝟙 X) ℱ x = colimit.ι (((whiskeringLeft (OpenNhds (↑(𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion (↑(𝟙 X) x)).op).obj (𝟙 X _* ℱ)) (op { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }) ≫ (stalkFunctor C x).map (Pushforward.id ℱ).hom erw [colimit.ι_map_assoc] case w.h.mk.mk C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X x : ↑X carrier✝ : Set ↑X is_open'✝ : IsOpen carrier✝ property✝ : ↑(𝟙 X) x ∈ { carrier := carrier✝, is_open' := is_open'✝ } ⊢ (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso (𝟙 X) x).inv) ℱ).app (op { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }) ≫ colimit.ι ((OpenNhds.map (𝟙 X) x).op ⋙ ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj ℱ) (op { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }) ≫ colimit.pre (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj ℱ) (OpenNhds.map (𝟙 X) x).op = colimit.ι (((whiskeringLeft (OpenNhds (↑(𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion (↑(𝟙 X) x)).op).obj (𝟙 X _* ℱ)) (op { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }) ≫ (stalkFunctor C x).map (Pushforward.id ℱ).hom simp [stalkFunctor, stalkPushforward] Qed
TopCat.Presheaf.stalkPushforward.comp ** C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X f : X ⟶ Y g : Y ⟶ Z x : ↑X ⊢ stalkPushforward C (f ≫ g) ℱ x = stalkPushforward C g (f _* ℱ) (↑f x) ≫ stalkPushforward C f ℱ x ext U case w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X f : X ⟶ Y g : Y ⟶ Z x : ↑X U : (OpenNhds (↑(f ≫ g) x))ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑(f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj (OpenNhds.inclusion (↑(f ≫ g) x)).op).obj ((f ≫ g) _* ℱ)) U ≫ stalkPushforward C (f ≫ g) ℱ x = colimit.ι (((whiskeringLeft (OpenNhds (↑(f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj (OpenNhds.inclusion (↑(f ≫ g) x)).op).obj ((f ≫ g) _* ℱ)) U ≫ stalkPushforward C g (f _* ℱ) (↑f x) ≫ stalkPushforward C f ℱ x rcases U with ⟨⟨_, _⟩, _⟩ case w.mk.mk.mk C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X f : X ⟶ Y g : Y ⟶ Z x : ↑X carrier✝ : Set ↑Z is_open'✝ : IsOpen carrier✝ property✝ : ↑(f ≫ g) x ∈ { carrier := carrier✝, is_open' := is_open'✝ } ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑(f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj (OpenNhds.inclusion (↑(f ≫ g) x)).op).obj ((f ≫ g) _* ℱ)) { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫ stalkPushforward C (f ≫ g) ℱ x = colimit.ι (((whiskeringLeft (OpenNhds (↑(f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj (OpenNhds.inclusion (↑(f ≫ g) x)).op).obj ((f ≫ g) _* ℱ)) { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫ stalkPushforward C g (f _* ℱ) (↑f x) ≫ stalkPushforward C f ℱ x simp [stalkFunctor, stalkPushforward] Qed
TopCat.Presheaf.stalkPushforward.stalkPushforward_iso_of_openEmbedding C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X ⊢ IsIso (stalkPushforward C f F x) haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x) C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) ⊢ IsIso (stalkPushforward C f F x) convert IsIso.of_iso ((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op ((OpenNhds.inclusion (f x)).op ⋙ f _* F) : _).symm ≪≫ colim.mapIso _) ** case h.e'_5.h C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) ⊢ stalkPushforward C f F x = ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F)).symm ≪≫ colim.mapIso ?convert_2).hom case convert_2 C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) ⊢ (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ⋙ (OpenNhds.inclusion (↑f x)).op ⋙ f _* F ≅ ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F swap case convert_2 C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) ⊢ (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ⋙ (OpenNhds.inclusion (↑f x)).op ⋙ f _* F ≅ ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F fapply NatIso.ofComponents case convert_2.app C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) ⊢ (X_1 : (OpenNhds x)ᵒᵖ) → ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ⋙ (OpenNhds.inclusion (↑f x)).op ⋙ f _* F).obj X_1 ≅ (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F).obj X_1 intro U case convert_2.app C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) U : (OpenNhds x)ᵒᵖ ⊢ ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ⋙ (OpenNhds.inclusion (↑f x)).op ⋙ f _* F).obj U ≅ (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F).obj U refine' F.mapIso (eqToIso _) case convert_2.app C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) U : (OpenNhds x)ᵒᵖ ⊢ (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U dsimp only [Functor.op] case convert_2.app C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) U : (OpenNhds x)ᵒᵖ ⊢ op ((Opens.map f).obj (op ((OpenNhds.inclusion (↑f x)).obj (op ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).obj U.unop)).unop)).unop) = op ((OpenNhds.inclusion x).obj U.unop) exact congr_arg op (Opens.ext <\| Set.preimage_image_eq (unop U).1.1 hf.inj) case convert_2.naturality C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) ⊢ autoParam (∀ {X_1 Y_1 : (OpenNhds x)ᵒᵖ} (f_1 : X_1 ⟶ Y_1), ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ⋙ (OpenNhds.inclusion (↑f x)).op ⋙ f _* F).map f_1 ≫ (F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj Y_1)) = (OpenNhds.inclusion x).op.obj Y_1))).hom = (F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj X_1)) = (OpenNhds.inclusion x).op.obj X_1))).hom ≫ (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F).map f_1) _auto✝ intro U V i case convert_2.naturality C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) U V : (OpenNhds x)ᵒᵖ i : U ⟶ V ⊢ ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ⋙ (OpenNhds.inclusion (↑f x)).op ⋙ f _* F).map i ≫ (F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj V)) = (OpenNhds.inclusion x).op.obj V))).hom = (F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U))).hom ≫ (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F).map i erw [← F.map_comp, ← F.map_comp] case convert_2.naturality C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) U V : (OpenNhds x)ᵒᵖ i : U ⟶ V ⊢ F.map ((Opens.map f).op.map ((OpenNhds.inclusion (↑f x)).op.map ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.map i)) ≫ (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj V)) = (OpenNhds.inclusion x).op.obj V)).hom) = F.map ((eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U)).hom ≫ (OpenNhds.inclusion x).op.map i) congr 1 case h.e'_5.h C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) ⊢ stalkPushforward C f F x = ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F)).symm ≪≫ colim.mapIso (NatIso.ofComponents fun U => F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U)))).hom ext U case h.e'_5.h.w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f x)).op).obj (f _* F)) U ≫ stalkPushforward C f F x = colimit.ι (((whiskeringLeft (OpenNhds (↑f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f x)).op).obj (f _* F)) U ≫ ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F)).symm ≪≫ colim.mapIso (NatIso.ofComponents fun U => F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U)))).hom rw [← Iso.comp_inv_eq] case h.e'_5.h.w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ (colimit.ι (((whiskeringLeft (OpenNhds (↑f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f x)).op).obj (f _* F)) U ≫ stalkPushforward C f F x) ≫ ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F)).symm ≪≫ colim.mapIso (NatIso.ofComponents fun U => F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U)))).inv = colimit.ι (((whiskeringLeft (OpenNhds (↑f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f x)).op).obj (f _* F)) U erw [colimit.ι_map_assoc] case h.e'_5.h.w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ ((whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F).app U ≫ colimit.ι ((OpenNhds.map f x).op ⋙ ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U ≫ colimit.pre (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) (OpenNhds.map f x).op) ≫ ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F)).symm ≪≫ colim.mapIso (NatIso.ofComponents fun U => F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U)))).inv = colimit.ι (((whiskeringLeft (OpenNhds (↑f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f x)).op).obj (f _* F)) U rw [colimit.ι_pre, Category.assoc] case h.e'_5.h.w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F).app U ≫ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) ((OpenNhds.map f x).op.obj U) ≫ ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F)).symm ≪≫ colim.mapIso (NatIso.ofComponents fun U => F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U)))).inv = colimit.ι (((whiskeringLeft (OpenNhds (↑f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f x)).op).obj (f _* F)) U erw [colimit.ι_map_assoc, colimit.ι_pre, ← F.map_comp_assoc] case h.e'_5.h.w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ F.map ((NatTrans.op (OpenNhds.inclusionMapIso f x).inv).app U ≫ (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj ((OpenNhds.map f x).op.obj U))) = (OpenNhds.inclusion x).op.obj ((OpenNhds.map f x).op.obj U))).inv) ≫ colimit.ι ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj ((OpenNhds.map f x).op.obj U)) = colimit.ι (((whiskeringLeft (OpenNhds (↑f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f x)).op).obj (f _* F)) U ** apply colimit.w ((OpenNhds.inclusion (f x)).op ⋙ f _* F) _ ** C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ U ⟶ (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj ((OpenNhds.map f x).op.obj U) dsimp only [Functor.op] C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ U ⟶ op ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).obj (op ((OpenNhds.map f x).obj U.unop)).unop) refine' ((homOfLE _).op : op (unop U) ⟶ _) C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).obj (op ((OpenNhds.map f x).obj U.unop)).unop ≤ U.unop exact Set.image_preimage_subset _ _ Qed
TopCat.Presheaf.stalkSpecializes_refl C✝ : Type u inst✝³ : Category.{v, u} C✝ inst✝² : HasColimits C✝ X✝ Y Z : TopCat C : Type u_1 inst✝¹ : Category.{u_2, u_1} C inst✝ : HasColimits C X : TopCat F : Presheaf C X x : ↑X ⊢ stalkSpecializes F (_ : x ⤳ x) = 𝟙 (stalk F x) ext case ih C✝ : Type u inst✝³ : Category.{v, u} C✝ inst✝² : HasColimits C✝ X✝ Y Z : TopCat C : Type u_1 inst✝¹ : Category.{u_2, u_1} C inst✝ : HasColimits C X : TopCat F : Presheaf C X x : ↑X U✝ : Opens ↑X hxU✝ : x ∈ U✝ ⊢ germ F { val := x, property := hxU✝ } ≫ stalkSpecializes F (_ : x ⤳ x) = germ F { val := x, property := hxU✝ } ≫ 𝟙 (stalk F x) simp ** Qed
TopCat.Presheaf.stalkSpecializes_comp C✝ : Type u inst✝³ : Category.{v, u} C✝ inst✝² : HasColimits C✝ X✝ Y Z : TopCat C : Type u_1 inst✝¹ : Category.{u_2, u_1} C inst✝ : HasColimits C X : TopCat F : Presheaf C X x y z : ↑X h : x ⤳ y h' : y ⤳ z ⊢ stalkSpecializes F h' ≫ stalkSpecializes F h = stalkSpecializes F (_ : x ⤳ z) ext case ih C✝ : Type u inst✝³ : Category.{v, u} C✝ inst✝² : HasColimits C✝ X✝ Y Z : TopCat C : Type u_1 inst✝¹ : Category.{u_2, u_1} C inst✝ : HasColimits C X : TopCat F : Presheaf C X x y z : ↑X h : x ⤳ y h' : y ⤳ z U✝ : Opens ↑X hxU✝ : z ∈ U✝ ⊢ germ F { val := z, property := hxU✝ } ≫ stalkSpecializes F h' ≫ stalkSpecializes F h = germ F { val := z, property := hxU✝ } ≫ stalkSpecializes F (_ : x ⤳ z) simp ** Qed
TopCat.Presheaf.stalkSpecializes_stalkFunctor_map C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat F G : Presheaf C X f : F ⟶ G x y : ↑X h : x ⤳ y ⊢ stalkSpecializes F h ≫ (stalkFunctor C x).map f = (stalkFunctor C y).map f ≫ stalkSpecializes G h ext case w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat F G : Presheaf C X f : F ⟶ G x y : ↑X h : x ⤳ y j✝ : (OpenNhds y)ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F) j✝ ≫ stalkSpecializes F h ≫ (stalkFunctor C x).map f = colimit.ι (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F) j✝ ≫ (stalkFunctor C y).map f ≫ stalkSpecializes G h delta stalkFunctor case w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat F G : Presheaf C X f : F ⟶ G x y : ↑X h : x ⤳ y j✝ : (OpenNhds y)ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F) j✝ ≫ stalkSpecializes F h ≫ ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op ⋙ colim).map f = colimit.ι (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F) j✝ ≫ ((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op ⋙ colim).map f ≫ stalkSpecializes G h simpa [stalkSpecializes] using by rfl C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat F G : Presheaf C X f : F ⟶ G x y : ↑X h : x ⤳ y j✝ : (OpenNhds y)ᵒᵖ ⊢ f.app (op j✝.unop.obj) ≫ colimit.ι ((OpenNhds.inclusion x).op ⋙ G) (op { obj := j✝.unop.obj, property := (_ : x ∈ j✝.unop.obj.carrier) }) = f.app (op ((OpenNhds.inclusion y).obj j✝.unop)) ≫ colimit.ι ((OpenNhds.inclusion x).op ⋙ G) (op { obj := j✝.unop.obj, property := (_ : x ∈ j✝.unop.obj.carrier) }) rfl ** Qed
TopCat.Presheaf.stalkSpecializes_stalkPushforward ** C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y F : Presheaf C X x y : ↑X h : x ⤳ y ⊢ stalkSpecializes (f _* F) (_ : ↑f x ⤳ ↑f y) ≫ stalkPushforward C f F x = stalkPushforward C f F y ≫ stalkSpecializes F h ext case w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y F : Presheaf C X x y : ↑X h : x ⤳ y j✝ : (OpenNhds (↑f y))ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑f y))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f y)).op).obj (f _* F)) j✝ ≫ stalkSpecializes (f _* F) (_ : ↑f x ⤳ ↑f y) ≫ stalkPushforward C f F x = colimit.ι (((whiskeringLeft (OpenNhds (↑f y))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f y)).op).obj (f _* F)) j✝ ≫ stalkPushforward C f F y ≫ stalkSpecializes F h delta stalkPushforward case w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y F : Presheaf C X x y : ↑X h : x ⤳ y j✝ : (OpenNhds (↑f y))ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑f y))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f y)).op).obj (f _* F)) j✝ ≫ stalkSpecializes (f _* F) (_ : ↑f x ⤳ ↑f y) ≫ colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F) ≫ colimit.pre (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) (OpenNhds.map f x).op = colimit.ι (((whiskeringLeft (OpenNhds (↑f y))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f y)).op).obj (f _* F)) j✝ ≫ (colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f y).inv) F) ≫ colimit.pre (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F) (OpenNhds.map f y).op) ≫ stalkSpecializes F h simp only [stalkSpecializes, colimit.ι_desc_assoc, colimit.ι_map_assoc, colimit.ι_pre, Category.assoc, colimit.pre_desc, colimit.ι_desc] case w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y F : Presheaf C X x y : ↑X h : x ⤳ y j✝ : (OpenNhds (↑f y))ᵒᵖ ⊢ (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F).app (op { obj := j✝.unop.obj, property := (_ : ↑f x ∈ j✝.unop.obj.carrier) }) ≫ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) ((OpenNhds.map f x).op.obj (op { obj := j✝.unop.obj, property := (_ : ↑f x ∈ j✝.unop.obj.carrier) })) = (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f y).inv) F).app j✝ ≫ (Cocone.whisker (OpenNhds.map f y).op { pt := colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F), ι := NatTrans.mk fun U => colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }) }).ι.app j✝ rfl Qed
TopCat.Presheaf.germ_ext C : Type u inst✝² : Category.{v, u} C inst✝¹ : HasColimits C X Y Z : TopCat inst✝ : ConcreteCategory C F : Presheaf C X U V : Opens ↑X x : ↑X hxU : x ∈ U hxV : x ∈ V W : Opens ↑X hxW : x ∈ W iWU : W ⟶ U iWV : W ⟶ V sU : (forget C).obj (F.obj (op U)) sV : (forget C).obj (F.obj (op V)) ih : ↑(F.map iWU.op) sU = ↑(F.map iWV.op) sV ⊢ ↑(germ F { val := x, property := hxU }) sU = ↑(germ F { val := x, property := hxV }) sV erw [← F.germ_res iWU ⟨x, hxW⟩, ← F.germ_res iWV ⟨x, hxW⟩, comp_apply, comp_apply, ih] ** Qed
TopCat.Presheaf.germ_exist C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X x : ↑X t : (forget C).obj (stalk F x) ⊢ ∃ U m s, ↑(germ F { val := x, property := m }) s = t obtain ⟨U, s, e⟩ := Types.jointly_surjective.{v, v} _ (isColimitOfPreserves (forget C) (colimit.isColimit _)) t case intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X x : ↑X t : (forget C).obj (stalk F x) U : (OpenNhds x)ᵒᵖ s : (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙ forget C).obj U e : ((forget C).mapCocone (colimit.cocone (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F))).ι.app U s = t ⊢ ∃ U m s, ↑(germ F { val := x, property := m }) s = t revert s e case intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X x : ↑X t : (forget C).obj (stalk F x) U : (OpenNhds x)ᵒᵖ ⊢ ∀ (s : (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙ forget C).obj U), ((forget C).mapCocone (colimit.cocone (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F))).ι.app U s = t → ∃ U m s, ↑(germ F { val := x, property := m }) s = t induction U with \| h U => ?_ case intro.intro.h C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X x : ↑X t : (forget C).obj (stalk F x) U : OpenNhds x ⊢ ∀ (s : (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙ forget C).obj (op U)), ((forget C).mapCocone (colimit.cocone (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F))).ι.app (op U) s = t → ∃ U m s, ↑(germ F { val := x, property := m }) s = t cases' U with V m case intro.intro.h.mk C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X x : ↑X t : (forget C).obj (stalk F x) V : Opens ↑X m : x ∈ V ⊢ ∀ (s : (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙ forget C).obj (op { obj := V, property := m })), ((forget C).mapCocone (colimit.cocone (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F))).ι.app (op { obj := V, property := m }) s = t → ∃ U m s, ↑(germ F { val := x, property := m }) s = t intro s e case intro.intro.h.mk C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X x : ↑X t : (forget C).obj (stalk F x) V : Opens ↑X m : x ∈ V s : (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙ forget C).obj (op { obj := V, property := m }) e : ((forget C).mapCocone (colimit.cocone (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F))).ι.app (op { obj := V, property := m }) s = t ⊢ ∃ U m s, ↑(germ F { val := x, property := m }) s = t exact ⟨V, m, s, e⟩ ** Qed
TopCat.Presheaf.germ_eq C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X U V : Opens ↑X x : ↑X mU : x ∈ U mV : x ∈ V s : (forget C).obj (F.obj (op U)) t : (forget C).obj (F.obj (op V)) h : ↑(germ F { val := x, property := mU }) s = ↑(germ F { val := x, property := mV }) t ⊢ ∃ W _m iU iV, ↑(F.map iU.op) s = ↑(F.map iV.op) t obtain ⟨W, iU, iV, e⟩ := (Types.FilteredColimit.isColimit_eq_iff.{v, v} _ (isColimitOfPreserves _ (colimit.isColimit ((OpenNhds.inclusion x).op ⋙ F)))).mp h case intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X U V : Opens ↑X x : ↑X mU : x ∈ U mV : x ∈ V s : (forget C).obj (F.obj (op U)) t : (forget C).obj (F.obj (op V)) h : ↑(germ F { val := x, property := mU }) s = ↑(germ F { val := x, property := mV }) t W : (OpenNhds x)ᵒᵖ iU : op { obj := U, property := (_ : ↑{ val := x, property := mU } ∈ U) } ⟶ W iV : op { obj := V, property := (_ : ↑{ val := x, property := mV } ∈ V) } ⟶ W e : (((OpenNhds.inclusion x).op ⋙ F) ⋙ forget C).map iU s = (((OpenNhds.inclusion x).op ⋙ F) ⋙ forget C).map iV t ⊢ ∃ W _m iU iV, ↑(F.map iU.op) s = ↑(F.map iV.op) t exact ⟨(unop W).1, (unop W).2, iU.unop, iV.unop, e⟩ ** Qed
TopCat.Presheaf.stalkFunctor_map_injective_of_app_injective C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X s t : (forget C).obj ((stalkFunctor C x).obj F) hst : ↑((stalkFunctor C x).map f) s = ↑((stalkFunctor C x).map f) t ⊢ s = t rcases germ_exist F x s with ⟨U₁, hxU₁, s, rfl⟩ case intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X t : (forget C).obj ((stalkFunctor C x).obj F) U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) hst : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) t ⊢ ↑(germ F { val := x, property := hxU₁ }) s = t rcases germ_exist F x t with ⟨U₂, hxU₂, t, rfl⟩ case intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) ⊢ ↑(germ F { val := x, property := hxU₁ }) s = ↑(germ F { val := x, property := hxU₂ }) t erw [stalkFunctor_map_germ_apply _ ⟨x, _⟩] at hst case intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst✝ : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) ⊢ ↑(germ F { val := x, property := hxU₁ }) s = ↑(germ F { val := x, property := hxU₂ }) t erw [stalkFunctor_map_germ_apply _ ⟨x, _⟩] at hst case intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst✝¹ : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst✝ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₂ }).op ⋙ G) (op { obj := U₂, property := (_ : ↑{ val := x, property := hxU₂ } ∈ U₂) })) (↑(f.app (op U₂)) t) ⊢ ↑(germ F { val := x, property := hxU₁ }) s = ↑(germ F { val := x, property := hxU₂ }) t obtain ⟨W, hxW, iWU₁, iWU₂, heq⟩ := G.germ_eq x hxU₁ hxU₂ _ _ hst case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst✝¹ : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst✝ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₂ }).op ⋙ G) (op { obj := U₂, property := (_ : ↑{ val := x, property := hxU₂ } ∈ U₂) })) (↑(f.app (op U₂)) t) W : Opens ↑X hxW : x ∈ W iWU₁ : W ⟶ U₁ iWU₂ : W ⟶ U₂ heq : ↑(G.map iWU₁.op) (↑(f.app (op U₁)) s) = ↑(G.map iWU₂.op) (↑(f.app (op U₂)) t) ⊢ ↑(germ F { val := x, property := hxU₁ }) s = ↑(germ F { val := x, property := hxU₂ }) t rw [← comp_apply, ← comp_apply, ← f.naturality, ← f.naturality, comp_apply, comp_apply] at heq case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst✝¹ : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst✝ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₂ }).op ⋙ G) (op { obj := U₂, property := (_ : ↑{ val := x, property := hxU₂ } ∈ U₂) })) (↑(f.app (op U₂)) t) W : Opens ↑X hxW : x ∈ W iWU₁ : W ⟶ U₁ iWU₂ : W ⟶ U₂ heq : ↑(f.app (op W)) (↑(F.map iWU₁.op) s) = ↑(f.app (op W)) (↑(F.map iWU₂.op) t) ⊢ ↑(germ F { val := x, property := hxU₁ }) s = ↑(germ F { val := x, property := hxU₂ }) t replace heq := h W heq case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst✝¹ : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst✝ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₂ }).op ⋙ G) (op { obj := U₂, property := (_ : ↑{ val := x, property := hxU₂ } ∈ U₂) })) (↑(f.app (op U₂)) t) W : Opens ↑X hxW : x ∈ W iWU₁ : W ⟶ U₁ iWU₂ : W ⟶ U₂ heq : ↑(F.map iWU₁.op) s = ↑(F.map iWU₂.op) t ⊢ ↑(germ F { val := x, property := hxU₁ }) s = ↑(germ F { val := x, property := hxU₂ }) t convert congr_arg (F.germ ⟨x, hxW⟩) heq using 1 case h.e'_2.h C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst✝¹ : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst✝ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₂ }).op ⋙ G) (op { obj := U₂, property := (_ : ↑{ val := x, property := hxU₂ } ∈ U₂) })) (↑(f.app (op U₂)) t) W : Opens ↑X hxW : x ∈ W iWU₁ : W ⟶ U₁ iWU₂ : W ⟶ U₂ heq : ↑(F.map iWU₁.op) s = ↑(F.map iWU₂.op) t e_1✝ : (forget C).obj ((stalkFunctor C x).obj F) = (forget C).obj (stalk F ↑{ val := x, property := hxW }) ⊢ ↑(germ F { val := x, property := hxU₁ }) s = ↑(germ F { val := x, property := hxW }) (↑(F.map iWU₁.op) s) case h.e'_3.h C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst✝¹ : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst✝ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₂ }).op ⋙ G) (op { obj := U₂, property := (_ : ↑{ val := x, property := hxU₂ } ∈ U₂) })) (↑(f.app (op U₂)) t) W : Opens ↑X hxW : x ∈ W iWU₁ : W ⟶ U₁ iWU₂ : W ⟶ U₂ heq : ↑(F.map iWU₁.op) s = ↑(F.map iWU₂.op) t e_1✝ : (forget C).obj ((stalkFunctor C x).obj F) = (forget C).obj (stalk F ↑{ val := x, property := hxW }) ⊢ ↑(germ F { val := x, property := hxU₂ }) t = ↑(germ F { val := x, property := hxW }) (↑(F.map iWU₂.op) t) exacts [(F.germ_res_apply iWU₁ ⟨x, hxW⟩ s).symm, (F.germ_res_apply iWU₂ ⟨x, hxW⟩ t).symm] ** Qed
TopCat.Presheaf.section_ext C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X U : Opens ↑X s t : (forget C).obj (F.val.obj (op U)) h : ∀ (x : { x // x ∈ U }), ↑(germ (Sheaf.presheaf F) x) s = ↑(germ (Sheaf.presheaf F) x) t ⊢ s = t choose V m i₁ i₂ heq using fun x : U => F.presheaf.germ_eq x.1 x.2 x.2 s t (h x) C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X U : Opens ↑X s t : (forget C).obj (F.val.obj (op U)) h : ∀ (x : { x // x ∈ U }), ↑(germ (Sheaf.presheaf F) x) s = ↑(germ (Sheaf.presheaf F) x) t V : { x // x ∈ U } → Opens ↑X m : ∀ (x : { x // x ∈ U }), ↑x ∈ V x i₁ i₂ : (x : { x // x ∈ U }) → V x ⟶ U heq : ∀ (x : { x // x ∈ U }), ↑((Sheaf.presheaf F).map (i₁ x).op) s = ↑((Sheaf.presheaf F).map (i₂ x).op) t ⊢ s = t apply F.eq_of_locally_eq' V U i₁ case hcover C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X U : Opens ↑X s t : (forget C).obj (F.val.obj (op U)) h : ∀ (x : { x // x ∈ U }), ↑(germ (Sheaf.presheaf F) x) s = ↑(germ (Sheaf.presheaf F) x) t V : { x // x ∈ U } → Opens ↑X m : ∀ (x : { x // x ∈ U }), ↑x ∈ V x i₁ i₂ : (x : { x // x ∈ U }) → V x ⟶ U heq : ∀ (x : { x // x ∈ U }), ↑((Sheaf.presheaf F).map (i₁ x).op) s = ↑((Sheaf.presheaf F).map (i₂ x).op) t ⊢ U ≤ iSup V intro x hxU case hcover C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X U : Opens ↑X s t : (forget C).obj (F.val.obj (op U)) h : ∀ (x : { x // x ∈ U }), ↑(germ (Sheaf.presheaf F) x) s = ↑(germ (Sheaf.presheaf F) x) t V : { x // x ∈ U } → Opens ↑X m : ∀ (x : { x // x ∈ U }), ↑x ∈ V x i₁ i₂ : (x : { x // x ∈ U }) → V x ⟶ U heq : ∀ (x : { x // x ∈ U }), ↑((Sheaf.presheaf F).map (i₁ x).op) s = ↑((Sheaf.presheaf F).map (i₂ x).op) t x : ↑X hxU : x ∈ ↑U ⊢ x ∈ ↑(iSup V) simp only [Opens.coe_iSup, Set.mem_iUnion, SetLike.mem_coe] case hcover C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X U : Opens ↑X s t : (forget C).obj (F.val.obj (op U)) h : ∀ (x : { x // x ∈ U }), ↑(germ (Sheaf.presheaf F) x) s = ↑(germ (Sheaf.presheaf F) x) t V : { x // x ∈ U } → Opens ↑X m : ∀ (x : { x // x ∈ U }), ↑x ∈ V x i₁ i₂ : (x : { x // x ∈ U }) → V x ⟶ U heq : ∀ (x : { x // x ∈ U }), ↑((Sheaf.presheaf F).map (i₁ x).op) s = ↑((Sheaf.presheaf F).map (i₂ x).op) t x : ↑X hxU : x ∈ ↑U ⊢ ∃ i, x ∈ V i exact ⟨⟨x, hxU⟩, m ⟨x, hxU⟩⟩ case h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X U : Opens ↑X s t : (forget C).obj (F.val.obj (op U)) h : ∀ (x : { x // x ∈ U }), ↑(germ (Sheaf.presheaf F) x) s = ↑(germ (Sheaf.presheaf F) x) t V : { x // x ∈ U } → Opens ↑X m : ∀ (x : { x // x ∈ U }), ↑x ∈ V x i₁ i₂ : (x : { x // x ∈ U }) → V x ⟶ U heq : ∀ (x : { x // x ∈ U }), ↑((Sheaf.presheaf F).map (i₁ x).op) s = ↑((Sheaf.presheaf F).map (i₂ x).op) t ⊢ ∀ (i : { x // x ∈ U }), ↑(F.val.map (i₁ i).op) s = ↑(F.val.map (i₁ i).op) t intro x case h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X U : Opens ↑X s t : (forget C).obj (F.val.obj (op U)) h : ∀ (x : { x // x ∈ U }), ↑(germ (Sheaf.presheaf F) x) s = ↑(germ (Sheaf.presheaf F) x) t V : { x // x ∈ U } → Opens ↑X m : ∀ (x : { x // x ∈ U }), ↑x ∈ V x i₁ i₂ : (x : { x // x ∈ U }) → V x ⟶ U heq : ∀ (x : { x // x ∈ U }), ↑((Sheaf.presheaf F).map (i₁ x).op) s = ↑((Sheaf.presheaf F).map (i₂ x).op) t x : { x // x ∈ U } ⊢ ↑(F.val.map (i₁ x).op) s = ↑(F.val.map (i₁ x).op) t rw [heq, Subsingleton.elim (i₁ x) (i₂ x)] ** Qed
TopCat.Presheaf.app_injective_of_stalkFunctor_map_injective C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X G : Presheaf C X f : F.val ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f) s t : (forget C).obj (F.val.obj (op U)) hst : ↑(f.app (op U)) s = ↑(f.app (op U)) t x : { x // x ∈ U } ⊢ ↑((stalkFunctor C ↑x).map f) (↑(germ (Sheaf.presheaf F) x) s) = ↑((stalkFunctor C ↑x).map f) (↑(germ (Sheaf.presheaf F) x) t) erw [stalkFunctor_map_germ_apply, stalkFunctor_map_germ_apply, hst] ** Qed
TopCat.Presheaf.app_surjective_of_injective_of_locally_surjective C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t ⊢ Function.Surjective ↑(f.val.app (op U)) intro t C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) ⊢ ∃ a, ↑(f.val.app (op U)) a = t choose V mV iVU sf heq using hsurj t C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t ⊢ ∃ a, ↑(f.val.app (op U)) a = t have V_cover : U ≤ iSup V := by intro x hxU simp only [Opens.coe_iSup, Set.mem_iUnion, SetLike.mem_coe] exact ⟨⟨x, hxU⟩, mV ⟨x, hxU⟩⟩ C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V ⊢ IsCompatible F.val V sf intro x y C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V x y : { x // x ∈ U } ⊢ ↑(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x) = ↑(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y) apply section_ext case h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V x y : { x // x ∈ U } ⊢ ∀ (x_1 : { x_1 // x_1 ∈ V x ⊓ V y }), ↑(germ (Sheaf.presheaf F) x_1) (↑(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x)) = ↑(germ (Sheaf.presheaf F) x_1) (↑(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y)) intro z case h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V x y : { x // x ∈ U } z : { x_1 // x_1 ∈ V x ⊓ V y } ⊢ ↑(germ (Sheaf.presheaf F) z) (↑(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x)) = ↑(germ (Sheaf.presheaf F) z) (↑(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y)) apply hinj ⟨z, (iVU x).le ((inf_le_left : V x ⊓ V y ≤ V x) z.2)⟩ case h.a C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V x y : { x // x ∈ U } z : { x_1 // x_1 ∈ V x ⊓ V y } ⊢ ↑((stalkFunctor C ↑{ val := ↑z, property := (_ : ↑z ∈ ↑U) }).map f.val) (↑(germ (Sheaf.presheaf F) z) (↑(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x))) = ↑((stalkFunctor C ↑{ val := ↑z, property := (_ : ↑z ∈ ↑U) }).map f.val) (↑(germ (Sheaf.presheaf F) z) (↑(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y))) dsimp only case h.a C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V x y : { x // x ∈ U } z : { x_1 // x_1 ∈ V x ⊓ V y } ⊢ ↑((stalkFunctor C ↑z).map f.val) (↑(germ (Sheaf.presheaf F) z) (↑(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x))) = ↑((stalkFunctor C ↑z).map f.val) (↑(germ (Sheaf.presheaf F) z) (↑(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y))) erw [stalkFunctor_map_germ_apply, stalkFunctor_map_germ_apply] case h.a C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V x y : { x // x ∈ U } z : { x_1 // x_1 ∈ V x ⊓ V y } ⊢ ↑(colimit.ι ((OpenNhds.inclusion ↑z).op ⋙ G.val) (op { obj := V x ⊓ V y, property := (_ : ↑z ∈ V x ⊓ V y) })) (↑(f.val.app (op (V x ⊓ V y))) (↑(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x))) = ↑(colimit.ι ((OpenNhds.inclusion ↑z).op ⋙ G.val) (op { obj := V x ⊓ V y, property := (_ : ↑z ∈ V x ⊓ V y) })) (↑(f.val.app (op (V x ⊓ V y))) (↑(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y))) simp_rw [← comp_apply, f.1.naturality, comp_apply, heq, ← comp_apply, ← G.1.map_comp] case h.a C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V x y : { x // x ∈ U } z : { x_1 // x_1 ∈ V x ⊓ V y } ⊢ ↑(colimit.ι ((OpenNhds.inclusion ↑z).op ⋙ G.val) (op { obj := V x ⊓ V y, property := (_ : ↑z ∈ V x ⊓ V y) })) (↑(G.val.map ((iVU x).op ≫ (Opens.infLELeft (V x) (V y)).op)) t) = ↑(colimit.ι ((OpenNhds.inclusion ↑z).op ⋙ G.val) (op { obj := V x ⊓ V y, property := (_ : ↑z ∈ V x ⊓ V y) })) (↑(G.val.map ((iVU y).op ≫ (Opens.infLERight (V x) (V y)).op)) t) rfl C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t ⊢ U ≤ iSup V intro x hxU C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t x : ↑X hxU : x ∈ ↑U ⊢ x ∈ ↑(iSup V) simp only [Opens.coe_iSup, Set.mem_iUnion, SetLike.mem_coe] C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t x : ↑X hxU : x ∈ ↑U ⊢ ∃ i, x ∈ V i exact ⟨⟨x, hxU⟩, mV ⟨x, hxU⟩⟩ C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V this : IsCompatible F.val V sf ⊢ ∃ a, ↑(f.val.app (op U)) a = t obtain ⟨s, s_spec, -⟩ := F.existsUnique_gluing' V U iVU V_cover sf this case intro.intro C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V this : IsCompatible F.val V sf s : (forget C).obj (F.val.obj (op U)) s_spec : ∀ (i : { x // x ∈ U }), ↑(F.val.map (iVU i).op) s = sf i ⊢ ∃ a, ↑(f.val.app (op U)) a = t use s case h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V this : IsCompatible F.val V sf s : (forget C).obj (F.val.obj (op U)) s_spec : ∀ (i : { x // x ∈ U }), ↑(F.val.map (iVU i).op) s = sf i ⊢ ↑(f.val.app (op U)) s = t apply G.eq_of_locally_eq' V U iVU V_cover case h.h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V this : IsCompatible F.val V sf s : (forget C).obj (F.val.obj (op U)) s_spec : ∀ (i : { x // x ∈ U }), ↑(F.val.map (iVU i).op) s = sf i ⊢ ∀ (i : { x // x ∈ U }), ↑(G.val.map (iVU i).op) (↑(f.val.app (op U)) s) = ↑(G.val.map (iVU i).op) t intro x case h.h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V this : IsCompatible F.val V sf s : (forget C).obj (F.val.obj (op U)) s_spec : ∀ (i : { x // x ∈ U }), ↑(F.val.map (iVU i).op) s = sf i x : { x // x ∈ U } ⊢ ↑(G.val.map (iVU x).op) (↑(f.val.app (op U)) s) = ↑(G.val.map (iVU x).op) t rw [← comp_apply, ← f.1.naturality, comp_apply, s_spec, heq] ** Qed
TopCat.Presheaf.app_surjective_of_stalkFunctor_map_bijective C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) ⊢ Function.Surjective ↑(f.val.app (op U)) refine' app_surjective_of_injective_of_locally_surjective f U (fun x => (h x).1) fun t x => _ C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } ⊢ ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t obtain ⟨s₀, hs₀⟩ := (h x).2 (G.presheaf.germ x t) case intro C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } s₀ : (forget C).obj ((stalkFunctor C ↑x).obj F.val) hs₀ : ↑((stalkFunctor C ↑x).map f.val) s₀ = ↑(germ (Sheaf.presheaf G) x) t ⊢ ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t obtain ⟨V₁, hxV₁, s₁, hs₁⟩ := F.presheaf.germ_exist x.1 s₀ case intro.intro.intro.intro C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } s₀ : (forget C).obj ((stalkFunctor C ↑x).obj F.val) hs₀ : ↑((stalkFunctor C ↑x).map f.val) s₀ = ↑(germ (Sheaf.presheaf G) x) t V₁ : Opens ↑X hxV₁ : ↑x ∈ V₁ s₁ : (forget C).obj ((Sheaf.presheaf F).obj (op V₁)) hs₁ : ↑(germ (Sheaf.presheaf F) { val := ↑x, property := hxV₁ }) s₁ = s₀ ⊢ ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t subst hs₁ case intro.intro.intro.intro C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } V₁ : Opens ↑X hxV₁ : ↑x ∈ V₁ s₁ : (forget C).obj ((Sheaf.presheaf F).obj (op V₁)) hs₀ : ↑((stalkFunctor C ↑x).map f.val) (↑(germ (Sheaf.presheaf F) { val := ↑x, property := hxV₁ }) s₁) = ↑(germ (Sheaf.presheaf G) x) t ⊢ ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t rename' hs₀ => hs₁ case intro.intro.intro.intro C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } V₁ : Opens ↑X hxV₁ : ↑x ∈ V₁ s₁ : (forget C).obj ((Sheaf.presheaf F).obj (op V₁)) hs₁ : ↑((stalkFunctor C ↑x).map f.val) (↑(germ (Sheaf.presheaf F) { val := ↑x, property := hxV₁ }) s₁) = ↑(germ (Sheaf.presheaf G) x) t ⊢ ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t erw [stalkFunctor_map_germ_apply V₁ ⟨x.1, hxV₁⟩ f.1 s₁] at hs₁ case intro.intro.intro.intro C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } V₁ : Opens ↑X hxV₁ : ↑x ∈ V₁ s₁ : (forget C).obj ((Sheaf.presheaf F).obj (op V₁)) hs₁ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := ↑x, property := hxV₁ }).op ⋙ G.val) (op { obj := V₁, property := (_ : ↑{ val := ↑x, property := hxV₁ } ∈ V₁) })) (↑(f.val.app (op V₁)) s₁) = ↑(germ (Sheaf.presheaf G) x) t ⊢ ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t obtain ⟨V₂, hxV₂, iV₂V₁, iV₂U, heq⟩ := G.presheaf.germ_eq x.1 hxV₁ x.2 _ _ hs₁ case intro.intro.intro.intro.intro.intro.intro.intro C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } V₁ : Opens ↑X hxV₁ : ↑x ∈ V₁ s₁ : (forget C).obj ((Sheaf.presheaf F).obj (op V₁)) hs₁ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := ↑x, property := hxV₁ }).op ⋙ G.val) (op { obj := V₁, property := (_ : ↑{ val := ↑x, property := hxV₁ } ∈ V₁) })) (↑(f.val.app (op V₁)) s₁) = ↑(germ (Sheaf.presheaf G) x) t V₂ : Opens ↑X hxV₂ : ↑x ∈ V₂ iV₂V₁ : V₂ ⟶ V₁ iV₂U : V₂ ⟶ U heq : ↑((Sheaf.presheaf G).map iV₂V₁.op) (↑(f.val.app (op V₁)) s₁) = ↑((Sheaf.presheaf G).map iV₂U.op) t ⊢ ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t use V₂, hxV₂, iV₂U, F.1.map iV₂V₁.op s₁ case h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } V₁ : Opens ↑X hxV₁ : ↑x ∈ V₁ s₁ : (forget C).obj ((Sheaf.presheaf F).obj (op V₁)) hs₁ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := ↑x, property := hxV₁ }).op ⋙ G.val) (op { obj := V₁, property := (_ : ↑{ val := ↑x, property := hxV₁ } ∈ V₁) })) (↑(f.val.app (op V₁)) s₁) = ↑(germ (Sheaf.presheaf G) x) t V₂ : Opens ↑X hxV₂ : ↑x ∈ V₂ iV₂V₁ : V₂ ⟶ V₁ iV₂U : V₂ ⟶ U heq : ↑((Sheaf.presheaf G).map iV₂V₁.op) (↑(f.val.app (op V₁)) s₁) = ↑((Sheaf.presheaf G).map iV₂U.op) t ⊢ ↑(f.val.app (op V₂)) (↑(F.val.map iV₂V₁.op) s₁) = ↑(G.val.map iV₂U.op) t rw [← comp_apply, f.1.naturality, comp_apply, heq] ** Qed
TopCat.Presheaf.app_isIso_of_stalkFunctor_map_iso C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X inst✝ : ∀ (x : { x // x ∈ U }), IsIso ((stalkFunctor C ↑x).map f.val) ⊢ IsIso (f.val.app (op U)) suffices IsIso ((forget C).map (f.1.app (op U))) by exact isIso_of_reflects_iso (f.1.app (op U)) (forget C) C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X inst✝ : ∀ (x : { x // x ∈ U }), IsIso ((stalkFunctor C ↑x).map f.val) ⊢ IsIso ((forget C).map (f.val.app (op U))) rw [isIso_iff_bijective] C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X inst✝ : ∀ (x : { x // x ∈ U }), IsIso ((stalkFunctor C ↑x).map f.val) ⊢ Function.Bijective ((forget C).map (f.val.app (op U))) apply app_bijective_of_stalkFunctor_map_bijective case h C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X inst✝ : ∀ (x : { x // x ∈ U }), IsIso ((stalkFunctor C ↑x).map f.val) ⊢ ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) intro x case h C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X inst✝ : ∀ (x : { x // x ∈ U }), IsIso ((stalkFunctor C ↑x).map f.val) x : { x // x ∈ U } ⊢ Function.Bijective ↑((stalkFunctor C ↑x).map f.val) apply (isIso_iff_bijective _).mp case h C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X inst✝ : ∀ (x : { x // x ∈ U }), IsIso ((stalkFunctor C ↑x).map f.val) x : { x // x ∈ U } ⊢ IsIso ↑((stalkFunctor C ↑x).map f.val) exact Functor.map_isIso (forget C) ((stalkFunctor C x.1).map f.1) C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X inst✝ : ∀ (x : { x // x ∈ U }), IsIso ((stalkFunctor C ↑x).map f.val) this : IsIso ((forget C).map (f.val.app (op U))) ⊢ IsIso (f.val.app (op U)) exact isIso_of_reflects_iso (f.1.app (op U)) (forget C) ** Qed
TopCat.Presheaf.isIso_of_stalkFunctor_map_iso C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G inst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val) ⊢ IsIso f suffices IsIso ((Sheaf.forget C X).map f) by exact isIso_of_fully_faithful (Sheaf.forget C X) f C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G inst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val) ⊢ IsIso ((Sheaf.forget C X).map f) suffices ∀ U : (Opens X)ᵒᵖ, IsIso (f.1.app U) by exact @NatIso.isIso_of_isIso_app _ _ _ _ F.1 G.1 f.1 this C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G inst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val) ⊢ ∀ (U : (Opens ↑X)ᵒᵖ), IsIso (f.val.app U) intro U C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G inst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val) U : (Opens ↑X)ᵒᵖ ⊢ IsIso (f.val.app U) induction U case h C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G inst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val) X✝ : Opens ↑X ⊢ IsIso (f.val.app (op X✝)) apply app_isIso_of_stalkFunctor_map_iso C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G inst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val) this : IsIso ((Sheaf.forget C X).map f) ⊢ IsIso f exact isIso_of_fully_faithful (Sheaf.forget C X) f C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G inst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val) this : ∀ (U : (Opens ↑X)ᵒᵖ), IsIso (f.val.app U) ⊢ IsIso ((Sheaf.forget C X).map f) exact @NatIso.isIso_of_isIso_app _ _ _ _ F.1 G.1 f.1 this ** Qed
Ordinal.CNFRec_zero C : Ordinal.{u_2} → Sort u_1 b : Ordinal.{u_2} H0 : C 0 H : (o : Ordinal.{u_2}) → o ≠ 0 → C (o % b ^ log b o) → C o ⊢ CNFRec b H0 H 0 = H0 rw [CNFRec, dif_pos rfl] C : Ordinal.{u_2} → Sort u_1 b : Ordinal.{u_2} H0 : C 0 H : (o : Ordinal.{u_2}) → o ≠ 0 → C (o % b ^ log b o) → C o ⊢ Eq.mpr (_ : C 0 = C 0) H0 = H0 rfl ** Qed
Ordinal.CNFRec_pos b o : Ordinal.{u_2} C : Ordinal.{u_2} → Sort u_1 ho : o ≠ 0 H0 : C 0 H : (o : Ordinal.{u_2}) → o ≠ 0 → C (o % b ^ log b o) → C o ⊢ CNFRec b H0 H o = H o ho (CNFRec b H0 H (o % b ^ log b o)) rw [CNFRec, dif_neg ho] ** Qed
Ordinal.zero_CNF o : Ordinal.{u_1} ho : o ≠ 0 ⊢ CNF 0 o = [(0, o)] simp [CNF_ne_zero ho] ** Qed
Ordinal.one_CNF o : Ordinal.{u_1} ho : o ≠ 0 ⊢ CNF 1 o = [(0, o)] simp [CNF_ne_zero ho] ** Qed
Ordinal.CNF_of_le_one b o : Ordinal.{u_1} hb : b ≤ 1 ho : o ≠ 0 ⊢ CNF b o = [(0, o)] rcases le_one_iff.1 hb with (rfl \| rfl) case inl o : Ordinal.{u_1} ho : o ≠ 0 hb : 0 ≤ 1 ⊢ CNF 0 o = [(0, o)] exact zero_CNF ho case inr o : Ordinal.{u_1} ho : o ≠ 0 hb : 1 ≤ 1 ⊢ CNF 1 o = [(0, o)] exact one_CNF ho ** Qed
Ordinal.CNF_of_lt b o : Ordinal.{u_1} ho : o ≠ 0 hb : o < b ⊢ CNF b o = [(0, o)] simp only [CNF_ne_zero ho, log_eq_zero hb, opow_zero, div_one, mod_one, CNF_zero] ** Qed
Ordinal.CNF_foldr ** b o : Ordinal.{u_1} ⊢ foldr (fun p r => b ^ p.1 * p.2 + r) 0 (CNF b 0) = 0 rw [CNF_zero] b o : Ordinal.{u_1} ⊢ foldr (fun p r => b ^ p.1 * p.2 + r) 0 [] = 0 rfl b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : foldr (fun p r => b ^ p.1 * p.2 + r) 0 (CNF b (o % b ^ log b o)) = o % b ^ log b o ⊢ foldr (fun p r => b ^ p.1 * p.2 + r) 0 (CNF b o) = o rw [CNF_ne_zero ho, foldr_cons, IH, div_add_mod] Qed
Ordinal.CNF_fst_le_log b o : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} ⊢ x ∈ CNF b o → x.1 ≤ log b o refine' CNFRec b _ (fun o ho H ↦ _) o case refine'_1 b o : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} ⊢ x ∈ CNF b 0 → x.1 ≤ log b 0 rw [CNF_zero] case refine'_1 b o : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} ⊢ x ∈ [] → x.1 ≤ log b 0 intro contra case refine'_1 b o : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} contra : x ∈ [] ⊢ x.1 ≤ log b 0 contradiction case refine'_2 b o✝ : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o) ⊢ x ∈ CNF b o → x.1 ≤ log b o rw [CNF_ne_zero ho, mem_cons] case refine'_2 b o✝ : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o) ⊢ x = (log b o, o / b ^ log b o) ∨ x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b o rintro (rfl \| h) case refine'_2.inl b o✝ o : Ordinal.{u} ho : o ≠ 0 H : (log b o, o / b ^ log b o) ∈ CNF b (o % b ^ log b o) → (log b o, o / b ^ log b o).1 ≤ log b (o % b ^ log b o) ⊢ (log b o, o / b ^ log b o).1 ≤ log b o exact le_rfl case refine'_2.inr b o✝ : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o) h : x ∈ CNF b (o % b ^ log b o) ⊢ x.1 ≤ log b o exact (H h).trans (log_mono_right _ (mod_opow_log_lt_self b ho).le) ** Qed
Ordinal.CNF_lt_snd b o : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} ⊢ x ∈ CNF b o → 0 < x.2 refine' CNFRec b (by simp) (fun o ho IH ↦ _) o b o✝ : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → 0 < x.2 ⊢ x ∈ CNF b o → 0 < x.2 rw [CNF_ne_zero ho] b o✝ : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → 0 < x.2 ⊢ x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) → 0 < x.2 rintro (h \| ⟨_, h⟩) b o : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} ⊢ x ∈ CNF b 0 → 0 < x.2 simp case head b o✝ o : Ordinal.{u} ho : o ≠ 0 IH : (log b o, o / b ^ log b o) ∈ CNF b (o % b ^ log b o) → 0 < (log b o, o / b ^ log b o).2 ⊢ 0 < (log b o, o / b ^ log b o).2 exact div_opow_log_pos b ho case tail b o✝ : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → 0 < x.2 h : Mem x (CNF b (o % b ^ log b o)) ⊢ 0 < x.2 exact IH h ** Qed
Ordinal.CNF_snd_lt b o : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} ⊢ x ∈ CNF b o → x.2 < b refine' CNFRec b _ (fun o ho IH ↦ _) o case refine'_1 b o : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} ⊢ x ∈ CNF b 0 → x.2 < b simp only [CNF_zero, not_mem_nil, IsEmpty.forall_iff] case refine'_2 b o✝ : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b ⊢ x ∈ CNF b o → x.2 < b rw [CNF_ne_zero ho] case refine'_2 b o✝ : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b ⊢ x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) → x.2 < b intro h case refine'_2 b o✝ : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b h : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) ⊢ x.2 < b cases' (mem_cons.mp h) with h h case refine'_2.inl b o✝ : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) h : x = (log b o, o / b ^ log b o) ⊢ x.2 < b rw [h] case refine'_2.inl b o✝ : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) h : x = (log b o, o / b ^ log b o) ⊢ (log b o, o / b ^ log b o).2 < b simpa only using div_opow_log_lt o hb case refine'_2.inr b o✝ : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) h : x ∈ CNF b (o % b ^ log b o) ⊢ x.2 < b exact IH h ** Qed
Ordinal.CNF_sorted b o : Ordinal.{u_1} ⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) refine' CNFRec b _ (fun o ho IH ↦ _) o case refine'_1 b o : Ordinal.{u_1} ⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b 0)) simp only [CNF_zero] case refine'_2 b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) ⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) cases' le_or_lt b 1 with hb hb case refine'_2.inl b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : b ≤ 1 ⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) simp only [CNF_of_le_one hb ho, map] case refine'_2.inr b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : 1 < b ⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) cases' lt_or_le o b with hob hbo case refine'_2.inr.inl b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : 1 < b hob : o < b ⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) simp only [CNF_of_lt ho hob, map] case refine'_2.inr.inr b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : 1 < b hbo : b ≤ o ⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) rw [CNF_ne_zero ho, map_cons, sorted_cons] case refine'_2.inr.inr b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : 1 < b hbo : b ≤ o ⊢ (∀ (b_1 : Ordinal.{u_1}), b_1 ∈ map Prod.fst (CNF b (o % b ^ log b o)) → (log b o, o / b ^ log b o).1 > b_1) ∧ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) refine' ⟨fun a H ↦ _, IH⟩ case refine'_2.inr.inr b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : 1 < b hbo : b ≤ o a : Ordinal.{u_1} H : a ∈ map Prod.fst (CNF b (o % b ^ log b o)) ⊢ (log b o, o / b ^ log b o).1 > a rw [mem_map] at H case refine'_2.inr.inr b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : 1 < b hbo : b ≤ o a : Ordinal.{u_1} H : ∃ a_1, a_1 ∈ CNF b (o % b ^ log b o) ∧ a_1.1 = a ⊢ (log b o, o / b ^ log b o).1 > a rcases H with ⟨⟨a, a'⟩, H, rfl⟩ case refine'_2.inr.inr.intro.mk.intro b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : 1 < b hbo : b ≤ o a a' : Ordinal.{u_1} H : (a, a') ∈ CNF b (o % b ^ log b o) ⊢ (log b o, o / b ^ log b o).1 > (a, a').1 exact (CNF_fst_le_log H).trans_lt (log_mod_opow_log_lt_log_self hb ho hbo) ** Qed
Ordinal.nadd_def a✝ b✝ c a b : Ordinal.{u} ⊢ a ♯ b = max (blsub a fun a' x => a' ♯ b) (blsub b fun b' x => a ♯ b') rw [nadd] ** Qed
Ordinal.lt_nadd_iff a b c : Ordinal.{u} ⊢ a < b ♯ c ↔ (∃ b', b' < b ∧ a ≤ b' ♯ c) ∨ ∃ c', c' < c ∧ a ≤ b ♯ c' rw [nadd_def] a b c : Ordinal.{u} ⊢ a < max (blsub b fun a' x => a' ♯ c) (blsub c fun b' x => b ♯ b') ↔ (∃ b', b' < b ∧ a ≤ b' ♯ c) ∨ ∃ c', c' < c ∧ a ≤ b ♯ c' simp [lt_blsub_iff] ** Qed
Ordinal.nadd_le_iff a b c : Ordinal.{u} ⊢ b ♯ c ≤ a ↔ (∀ (b' : Ordinal.{u}), b' < b → b' ♯ c < a) ∧ ∀ (c' : Ordinal.{u}), c' < c → b ♯ c' < a rw [nadd_def] a b c : Ordinal.{u} ⊢ max (blsub b fun a' x => a' ♯ c) (blsub c fun b' x => b ♯ b') ≤ a ↔ (∀ (b' : Ordinal.{u}), b' < b → b' ♯ c < a) ∧ ∀ (c' : Ordinal.{u}), c' < c → b ♯ c' < a simp [blsub_le_iff] ** Qed
Ordinal.nadd_le_nadd_left a✝ b c : Ordinal.{u} h : b ≤ c a : Ordinal.{u} ⊢ a ♯ b ≤ a ♯ c rcases lt_or_eq_of_le h with (h \| rfl) case inl a✝ b c : Ordinal.{u} h✝ : b ≤ c a : Ordinal.{u} h : b < c ⊢ a ♯ b ≤ a ♯ c exact (nadd_lt_nadd_left h a).le case inr a✝ b a : Ordinal.{u} h : b ≤ b ⊢ a ♯ b ≤ a ♯ b exact le_rfl ** Qed
Ordinal.nadd_le_nadd_right a✝ b c : Ordinal.{u} h : b ≤ c a : Ordinal.{u} ⊢ b ♯ a ≤ c ♯ a rcases lt_or_eq_of_le h with (h \| rfl) case inl a✝ b c : Ordinal.{u} h✝ : b ≤ c a : Ordinal.{u} h : b < c ⊢ b ♯ a ≤ c ♯ a exact (nadd_lt_nadd_right h a).le case inr a✝ b a : Ordinal.{u} h : b ≤ b ⊢ b ♯ a ≤ b ♯ a exact le_rfl ** Qed
Ordinal.nadd_comm a✝ b✝ c : Ordinal.{u} a b : Ordinal.{u_1} ⊢ a ♯ b = b ♯ a rw [nadd_def, nadd_def, max_comm] a✝ b✝ c : Ordinal.{u} a b : Ordinal.{u_1} ⊢ max (blsub b fun b' x => a ♯ b') (blsub a fun a' x => a' ♯ b) = max (blsub b fun a' x => a' ♯ a) (blsub a fun b' x => b ♯ b') congr <;> ext <;> apply nadd_comm ** Qed
Ordinal.blsub_nadd_of_mono a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ blsub (a ♯ b) f = max (blsub a fun a' ha' => f (a' ♯ b) (_ : a' ♯ b < a ♯ b)) (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) apply (blsub_le_iff.2 fun i h => _).antisymm (max_le _ _) a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ ∀ (i : Ordinal.{u}) (h : i < a ♯ b), f i h < max (blsub a fun a' ha' => f (a' ♯ b) (_ : a' ♯ b < a ♯ b)) (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ (blsub a fun a' ha' => f (a' ♯ b) (_ : a' ♯ b < a ♯ b)) ≤ blsub (a ♯ b) f a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) ≤ blsub (a ♯ b) f intro i h a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ (blsub a fun a' ha' => f (a' ♯ b) (_ : a' ♯ b < a ♯ b)) ≤ blsub (a ♯ b) f a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) ≤ blsub (a ♯ b) f all_goals apply blsub_le_of_brange_subset.{u, u, v} rintro c ⟨d, hd, rfl⟩ apply mem_brange_self a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj i : Ordinal.{u} h : i < a ♯ b ⊢ f i h < max (blsub a fun a' ha' => f (a' ♯ b) (_ : a' ♯ b < a ♯ b)) (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) rcases lt_nadd_iff.1 h with (⟨a', ha', hi⟩ \| ⟨b', hb', hi⟩) case inl.intro.intro a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj i : Ordinal.{u} h : i < a ♯ b a' : Ordinal.{u} ha' : a' < a hi : i ≤ a' ♯ b ⊢ f i h < max (blsub a fun a' ha' => f (a' ♯ b) (_ : a' ♯ b < a ♯ b)) (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) exact lt_max_of_lt_left ((hf h (nadd_lt_nadd_right ha' b) hi).trans_lt (lt_blsub _ _ ha')) case inr.intro.intro a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj i : Ordinal.{u} h : i < a ♯ b b' : Ordinal.{u} hb' : b' < b hi : i ≤ a ♯ b' ⊢ f i h < max (blsub a fun a' ha' => f (a' ♯ b) (_ : a' ♯ b < a ♯ b)) (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) exact lt_max_of_lt_right ((hf h (nadd_lt_nadd_left hb' a) hi).trans_lt (lt_blsub _ _ hb')) a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) ≤ blsub (a ♯ b) f apply blsub_le_of_brange_subset.{u, u, v} a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ (brange b fun a_1 ha => (fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) a_1 ha) ⊆ brange (a ♯ b) fun a_1 ha => f a_1 ha rintro c ⟨d, hd, rfl⟩ case intro.intro a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj d : Ordinal.{u} hd : d < b ⊢ (fun a_1 ha => (fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) a_1 ha) d hd ∈ brange (a ♯ b) fun a_1 ha => f a_1 ha apply mem_brange_self ** Qed
Ordinal.nadd_assoc a✝ b✝ c✝ : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ a ♯ b ♯ c = a ♯ (b ♯ c) rw [nadd_def a (b ♯ c), nadd_def, blsub_nadd_of_mono, blsub_nadd_of_mono, max_assoc] a✝ b✝ c✝ : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ max (blsub a fun a' ha' => a' ♯ b ♯ c) (max (blsub b fun b' hb' => a ♯ b' ♯ c) (blsub c fun b' x => a ♯ b ♯ b')) = max (blsub a fun a' x => a' ♯ (b ♯ c)) (max (blsub b fun a' ha' => a ♯ (a' ♯ c)) (blsub c fun b' hb' => a ♯ (b ♯ b'))) congr <;> ext (d hd) <;> apply nadd_assoc case hf a✝ b✝ c✝ : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ ∀ {i j : Ordinal.{u_1}}, i < b ♯ c → j < b ♯ c → i ≤ j → a ♯ i ≤ a ♯ j exact fun _ _ h => nadd_le_nadd_left h a case hf a✝ b✝ c✝ : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ ∀ {i j : Ordinal.{u_1}}, i < a ♯ b → j < a ♯ b → i ≤ j → i ♯ c ≤ j ♯ c exact fun _ _ h => nadd_le_nadd_right h c ** Qed
Ordinal.nadd_zero a b c : Ordinal.{u} ⊢ a ♯ 0 = a induction' a using Ordinal.induction with a IH case h a✝ b c a : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < a → k ♯ 0 = k ⊢ a ♯ 0 = a rw [nadd_def, blsub_zero, max_zero_right] case h a✝ b c a : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < a → k ♯ 0 = k ⊢ (blsub a fun a' x => a' ♯ 0) = a convert blsub_id a case h.e'_2.h.e'_2.h.h a✝ b c a : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < a → k ♯ 0 = k x✝¹ : Ordinal.{u} x✝ : x✝¹ < a ⊢ x✝¹ ♯ 0 = x✝¹ rename_i hb case h.e'_2.h.e'_2.h.h a✝ b c a : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < a → k ♯ 0 = k x✝ : Ordinal.{u} hb : x✝ < a ⊢ x✝ ♯ 0 = x✝ exact IH _ hb ** Qed
Ordinal.zero_nadd a b c : Ordinal.{u} ⊢ 0 ♯ a = a rw [nadd_comm, nadd_zero] ** Qed
Ordinal.nadd_one a b c : Ordinal.{u} ⊢ a ♯ 1 = succ a induction' a using Ordinal.induction with a IH case h a✝ b c a : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < a → k ♯ 1 = succ k ⊢ a ♯ 1 = succ a rw [nadd_def, blsub_one, nadd_zero, max_eq_right_iff, blsub_le_iff] case h a✝ b c a : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < a → k ♯ 1 = succ k ⊢ ∀ (i : Ordinal.{u}), i < a → i ♯ 1 < succ a intro i hi case h a✝ b c a : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < a → k ♯ 1 = succ k i : Ordinal.{u} hi : i < a ⊢ i ♯ 1 < succ a rwa [IH i hi, succ_lt_succ_iff] ** Qed
Ordinal.one_nadd a b c : Ordinal.{u} ⊢ 1 ♯ a = succ a rw [nadd_comm, nadd_one] ** Qed
Ordinal.nadd_succ a b c : Ordinal.{u} ⊢ a ♯ succ b = succ (a ♯ b) rw [← nadd_one (a ♯ b), nadd_assoc, nadd_one] ** Qed
Ordinal.succ_nadd a b c : Ordinal.{u} ⊢ succ a ♯ b = succ (a ♯ b) rw [← one_nadd (a ♯ b), ← nadd_assoc, one_nadd] ** Qed
Ordinal.nadd_nat a b c : Ordinal.{u} n : ℕ ⊢ a ♯ ↑n = a + ↑n induction' n with n hn case zero a b c : Ordinal.{u} ⊢ a ♯ ↑Nat.zero = a + ↑Nat.zero simp case succ a b c : Ordinal.{u} n : ℕ hn : a ♯ ↑n = a + ↑n ⊢ a ♯ ↑(Nat.succ n) = a + ↑(Nat.succ n) rw [Nat.cast_succ, add_one_eq_succ, nadd_succ, add_succ, hn] ** Qed
Ordinal.nat_nadd a b c : Ordinal.{u} n : ℕ ⊢ ↑n ♯ a = a + ↑n rw [nadd_comm, nadd_nat] ** Qed
Ordinal.add_le_nadd a b c : Ordinal.{u} ⊢ a + b ≤ a ♯ b induction b using limitRecOn with \| H₁ => simp \| H₂ c h => rwa [add_succ, nadd_succ, succ_le_succ_iff] \| H₃ c hc H => simp_rw [← IsNormal.blsub_eq.{u, u} (add_isNormal a) hc, blsub_le_iff] exact fun i hi => (H i hi).trans_lt (nadd_lt_nadd_left hi a) case H₁ a b c : Ordinal.{u} ⊢ a + 0 ≤ a ♯ 0 simp case H₂ a b c✝ c : Ordinal.{u} h : a + c ≤ a ♯ c ⊢ a + succ c ≤ a ♯ succ c rwa [add_succ, nadd_succ, succ_le_succ_iff] case H₃ a b c✝ c : Ordinal.{u} hc : IsLimit c H : ∀ (o' : Ordinal.{u}), o' < c → a + o' ≤ a ♯ o' ⊢ a + c ≤ a ♯ c simp_rw [← IsNormal.blsub_eq.{u, u} (add_isNormal a) hc, blsub_le_iff] case H₃ a b c✝ c : Ordinal.{u} hc : IsLimit c H : ∀ (o' : Ordinal.{u}), o' < c → a + o' ≤ a ♯ o' ⊢ ∀ (i : Ordinal.{u}), i < c → a + i < a ♯ c exact fun i hi => (H i hi).trans_lt (nadd_lt_nadd_left hi a) ** Qed
NatOrdinal.toOrdinal_cast_nat n : ℕ ⊢ ↑toOrdinal ↑n = ↑n induction' n with n hn case zero ⊢ ↑toOrdinal ↑Nat.zero = ↑Nat.zero rfl case succ n : ℕ hn : ↑toOrdinal ↑n = ↑n ⊢ ↑toOrdinal ↑(Nat.succ n) = ↑(Nat.succ n) change (toOrdinal n) ♯ 1 = n + 1 case succ n : ℕ hn : ↑toOrdinal ↑n = ↑n ⊢ ↑toOrdinal ↑n ♯ 1 = ↑n + 1 rw [hn] case succ n : ℕ hn : ↑toOrdinal ↑n = ↑n ⊢ ↑n ♯ 1 = ↑n + 1 exact nadd_one n ** Qed
Ordinal.toNatOrdinal_cast_nat n : ℕ ⊢ ↑toNatOrdinal ↑n = ↑n rw [← toOrdinal_cast_nat n] n : ℕ ⊢ ↑toNatOrdinal (↑toOrdinal ↑n) = ↑n rfl ** Qed
Ordinal.le_nadd_self a b : Ordinal.{u_1} ⊢ a ≤ b ♯ a simpa using nadd_le_nadd_right (Ordinal.zero_le b) a ** Qed
Ordinal.le_self_nadd a b : Ordinal.{u_1} ⊢ a ≤ a ♯ b simpa using nadd_le_nadd_left (Ordinal.zero_le b) a ** Qed
Ordinal.nmul_def a✝ b✝ c d : Ordinal.{u} a b : Ordinal.{u_1} ⊢ a ⨳ b = sInf {c \| ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} rw [nmul] ** Qed
Ordinal.nmul_nadd_lt a b c d a' b' : Ordinal.{u} ha : a' < a hb : b' < b ⊢ a' ⨳ b ♯ a ⨳ b' < a ⨳ b ♯ a' ⨳ b' rw [nmul_def a b] a b c d a' b' : Ordinal.{u} ha : a' < a hb : b' < b ⊢ a' ⨳ b ♯ a ⨳ b' < sInf {c \| ∀ (a' : Ordinal.{u}), a' < a → ∀ (b' : Ordinal.{u}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} ♯ a' ⨳ b' exact csInf_mem (nmul_nonempty a b) a' ha b' hb ** Qed
Ordinal.nmul_nadd_le a b c d a' b' : Ordinal.{u} ha : a' ≤ a hb : b' ≤ b ⊢ a' ⨳ b ♯ a ⨳ b' ≤ a ⨳ b ♯ a' ⨳ b' rcases lt_or_eq_of_le ha with (ha \| rfl) case inl a b c d a' b' : Ordinal.{u} ha✝ : a' ≤ a hb : b' ≤ b ha : a' < a ⊢ a' ⨳ b ♯ a ⨳ b' ≤ a ⨳ b ♯ a' ⨳ b' rcases lt_or_eq_of_le hb with (hb \| rfl) case inl.inl a b c d a' b' : Ordinal.{u} ha✝ : a' ≤ a hb✝ : b' ≤ b ha : a' < a hb : b' < b ⊢ a' ⨳ b ♯ a ⨳ b' ≤ a ⨳ b ♯ a' ⨳ b' exact (nmul_nadd_lt ha hb).le case inl.inr a c d a' b' : Ordinal.{u} ha✝ : a' ≤ a ha : a' < a hb : b' ≤ b' ⊢ a' ⨳ b' ♯ a ⨳ b' ≤ a ⨳ b' ♯ a' ⨳ b' rw [nadd_comm] case inr b c d a' b' : Ordinal.{u} hb : b' ≤ b ha : a' ≤ a' ⊢ a' ⨳ b ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a' ⨳ b' exact le_rfl ** Qed
Ordinal.lt_nmul_iff a b c d : Ordinal.{u} ⊢ c < a ⨳ b ↔ ∃ a', a' < a ∧ ∃ b', b' < b ∧ c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' refine' ⟨fun h => _, _⟩ case refine'_1 a b c d : Ordinal.{u} h : c < a ⨳ b ⊢ ∃ a', a' < a ∧ ∃ b', b' < b ∧ c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' rw [nmul] at h case refine'_1 a b c d : Ordinal.{u} h : c < sInf {c \| ∀ (a' : Ordinal.{u}), a' < a → ∀ (b' : Ordinal.{u}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} ⊢ ∃ a', a' < a ∧ ∃ b', b' < b ∧ c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' simpa using not_mem_of_lt_csInf h ⟨0, fun _ _ => bot_le⟩ case refine'_2 a b c d : Ordinal.{u} ⊢ (∃ a', a' < a ∧ ∃ b', b' < b ∧ c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b') → c < a ⨳ b rintro ⟨a', ha, b', hb, h⟩ case refine'_2.intro.intro.intro.intro a b c d a' : Ordinal.{u} ha : a' < a b' : Ordinal.{u} hb : b' < b h : c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' ⊢ c < a ⨳ b have := h.trans_lt (nmul_nadd_lt ha hb) case refine'_2.intro.intro.intro.intro a b c d a' : Ordinal.{u} ha : a' < a b' : Ordinal.{u} hb : b' < b h : c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' this : c ♯ a' ⨳ b' < a ⨳ b ♯ a' ⨳ b' ⊢ c < a ⨳ b rwa [nadd_lt_nadd_iff_right] at this ** Qed
Ordinal.nmul_le_iff a b c d : Ordinal.{u} ⊢ a ⨳ b ≤ c ↔ ∀ (a' : Ordinal.{u}), a' < a → ∀ (b' : Ordinal.{u}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b' rw [← not_iff_not] a b c d : Ordinal.{u} ⊢ ¬a ⨳ b ≤ c ↔ ¬∀ (a' : Ordinal.{u}), a' < a → ∀ (b' : Ordinal.{u}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b' simp [lt_nmul_iff] ** Qed
Ordinal.nmul_comm a✝ b✝ c d : Ordinal.{u} a b : Ordinal.{u_1} ⊢ a ⨳ b = b ⨳ a rw [nmul, nmul] a✝ b✝ c d : Ordinal.{u} a b : Ordinal.{u_1} ⊢ sInf {c \| ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} = sInf {c \| ∀ (a' : Ordinal.{u_1}), a' < b → ∀ (b' : Ordinal.{u_1}), b' < a → a' ⨳ a ♯ b ⨳ b' < c ♯ a' ⨳ b'} congr case e_a a✝ b✝ c d : Ordinal.{u} a b : Ordinal.{u_1} ⊢ {c \| ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} = {c \| ∀ (a' : Ordinal.{u_1}), a' < b → ∀ (b' : Ordinal.{u_1}), b' < a → a' ⨳ a ♯ b ⨳ b' < c ♯ a' ⨳ b'} ext x case e_a.h a✝ b✝ c d : Ordinal.{u} a b x : Ordinal.{u_1} ⊢ x ∈ {c \| ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} ↔ x ∈ {c \| ∀ (a' : Ordinal.{u_1}), a' < b → ∀ (b' : Ordinal.{u_1}), b' < a → a' ⨳ a ♯ b ⨳ b' < c ♯ a' ⨳ b'} constructor <;> intro H c hc d hd case e_a.h.mp a✝ b✝ c✝ d✝ : Ordinal.{u} a b x : Ordinal.{u_1} H : x ∈ {c \| ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} c : Ordinal.{u_1} hc : c < b d : Ordinal.{u_1} hd : d < a ⊢ c ⨳ a ♯ b ⨳ d < x ♯ c ⨳ d rw [nadd_comm, ← nmul_comm d b, ← nmul_comm a c, ← nmul_comm d] case e_a.h.mp a✝ b✝ c✝ d✝ : Ordinal.{u} a b x : Ordinal.{u_1} H : x ∈ {c \| ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} c : Ordinal.{u_1} hc : c < b d : Ordinal.{u_1} hd : d < a ⊢ d ⨳ b ♯ a ⨳ c < x ♯ d ⨳ c exact H _ hd _ hc case e_a.h.mpr a✝ b✝ c✝ d✝ : Ordinal.{u} a b x : Ordinal.{u_1} H : x ∈ {c \| ∀ (a' : Ordinal.{u_1}), a' < b → ∀ (b' : Ordinal.{u_1}), b' < a → a' ⨳ a ♯ b ⨳ b' < c ♯ a' ⨳ b'} c : Ordinal.{u_1} hc : c < a d : Ordinal.{u_1} hd : d < b ⊢ c ⨳ b ♯ a ⨳ d < x ♯ c ⨳ d rw [nadd_comm, nmul_comm a d, nmul_comm c, nmul_comm c] case e_a.h.mpr a✝ b✝ c✝ d✝ : Ordinal.{u} a b x : Ordinal.{u_1} H : x ∈ {c \| ∀ (a' : Ordinal.{u_1}), a' < b → ∀ (b' : Ordinal.{u_1}), b' < a → a' ⨳ a ♯ b ⨳ b' < c ♯ a' ⨳ b'} c : Ordinal.{u_1} hc : c < a d : Ordinal.{u_1} hd : d < b ⊢ d ⨳ a ♯ b ⨳ c < x ♯ d ⨳ c exact H _ hd _ hc ** Qed
Ordinal.nmul_zero a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ a ⨳ 0 = 0 rw [← Ordinal.le_zero, nmul_le_iff] a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < 0 → a' ⨳ 0 ♯ a ⨳ b' < 0 ♯ a' ⨳ b' exact fun _ _ a ha => (Ordinal.not_lt_zero a ha).elim ** Qed
Ordinal.zero_nmul a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ 0 ⨳ a = 0 rw [nmul_comm, nmul_zero] ** Qed
Ordinal.nmul_one a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ a ⨳ 1 = a rw [nmul] a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ sInf {c \| ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < 1 → a' ⨳ 1 ♯ a ⨳ b' < c ♯ a' ⨳ b'} = a simp only [lt_one_iff_zero, forall_eq, nmul_zero, nadd_zero] a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ sInf {c \| ∀ (a' : Ordinal.{u_1}), a' < a → a' ⨳ 1 < c} = a convert csInf_Ici (α := Ordinal) case h.e'_2.h.e'_3 a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ {c \| ∀ (a' : Ordinal.{u_1}), a' < a → a' ⨳ 1 < c} = Set.Ici a ext b case h.e'_2.h.e'_3.h a✝ b✝ c d : Ordinal.{u} a b : Ordinal.{u_1} ⊢ b ∈ {c \| ∀ (a' : Ordinal.{u_1}), a' < a → a' ⨳ 1 < c} ↔ b ∈ Set.Ici a simp only [Set.mem_setOf_eq, Set.mem_Ici] case h.e'_2.h.e'_3.h a✝ b✝ c d : Ordinal.{u} a b : Ordinal.{u_1} ⊢ (∀ (a' : Ordinal.{u_1}), a' < a → a' ⨳ 1 < b) ↔ a ≤ b refine' ⟨fun H => le_of_forall_lt fun c hc => _, fun ha c hc => _⟩ case h.e'_2.h.e'_3.h.refine'_1 a✝ b✝ c✝ d : Ordinal.{u} a b : Ordinal.{u_1} H : ∀ (a' : Ordinal.{u_1}), a' < a → a' ⨳ 1 < b c : Ordinal.{u_1} hc : c < a ⊢ c < b simpa only [nmul_one c] using H c hc case h.e'_2.h.e'_3.h.refine'_2 a✝ b✝ c✝ d : Ordinal.{u} a b : Ordinal.{u_1} ha : a ≤ b c : Ordinal.{u_1} hc : c < a ⊢ c ⨳ 1 < b simpa only [nmul_one c] using hc.trans_le ha ** Qed
Ordinal.one_nmul a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ 1 ⨳ a = a rw [nmul_comm, nmul_one] ** Qed
Ordinal.nmul_lt_nmul_of_pos_left a b c d : Ordinal.{u} h₁ : a < b h₂ : 0 < c ⊢ c ⨳ a ♯ 0 ⨳ a ≤ 0 ⨳ b ♯ c ⨳ a simp ** Qed
Ordinal.nmul_lt_nmul_of_pos_right a b c d : Ordinal.{u} h₁ : a < b h₂ : 0 < c ⊢ a ⨳ c ♯ a ⨳ 0 ≤ a ⨳ c ♯ b ⨳ 0 simp ** Qed
Ordinal.nmul_le_nmul_of_nonneg_left a b c d : Ordinal.{u} h₁ : a ≤ b h₂ : 0 ≤ c ⊢ c ⨳ a ≤ c ⨳ b rcases lt_or_eq_of_le h₁ with (h₁ \| rfl) <;> rcases lt_or_eq_of_le h₂ with (h₂ \| rfl) case inl.inr a b d : Ordinal.{u} h₁✝ : a ≤ b h₁ : a < b h₂ : 0 ≤ 0 ⊢ 0 ⨳ a ≤ 0 ⨳ b case inr.inl a c d : Ordinal.{u} h₂✝ : 0 ≤ c h₁ : a ≤ a h₂ : 0 < c ⊢ c ⨳ a ≤ c ⨳ a case inr.inr a d : Ordinal.{u} h₁ : a ≤ a h₂ : 0 ≤ 0 ⊢ 0 ⨳ a ≤ 0 ⨳ a all_goals simp case inl.inl a b c d : Ordinal.{u} h₁✝ : a ≤ b h₂✝ : 0 ≤ c h₁ : a < b h₂ : 0 < c ⊢ c ⨳ a ≤ c ⨳ b exact (nmul_lt_nmul_of_pos_left h₁ h₂).le case inr.inr a d : Ordinal.{u} h₁ : a ≤ a h₂ : 0 ≤ 0 ⊢ 0 ⨳ a ≤ 0 ⨳ a simp ** Qed
Ordinal.nmul_le_nmul_of_nonneg_right a b c d : Ordinal.{u} h₁ : a ≤ b h₂ : 0 ≤ c ⊢ a ⨳ c ≤ b ⨳ c rw [nmul_comm, nmul_comm b] a b c d : Ordinal.{u} h₁ : a ≤ b h₂ : 0 ≤ c ⊢ c ⨳ a ≤ c ⨳ b exact nmul_le_nmul_of_nonneg_left h₁ h₂ ** Qed
Ordinal.nmul_nadd a✝ b✝ c✝ d : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ a ⨳ (b ♯ c) = a ⨳ b ♯ a ⨳ c refine le_antisymm (nmul_le_iff.2 fun a' ha d hd => ?_) (nadd_le_iff.2 ⟨fun d hd => ?_, fun d hd => ?_⟩) case refine_1 a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd : d < b ♯ c ⊢ a' ⨳ (b ♯ c) ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d rw [nmul_nadd a' b c] case refine_1 a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd : d < b ♯ c ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d rcases lt_nadd_iff.1 hd with (⟨b', hb, hd⟩ \| ⟨c', hc, hd⟩) case refine_1.inl.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c b' : Ordinal.{u_1} hb : b' < b hd : d ≤ b' ♯ c ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d have := nadd_lt_nadd_of_lt_of_le (nmul_nadd_lt ha hb) (nmul_nadd_le ha.le hd) case refine_1.inl.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c b' : Ordinal.{u_1} hb : b' < b hd : d ≤ b' ♯ c this : a' ⨳ b ♯ a ⨳ b' ♯ (a' ⨳ (b' ♯ c) ♯ a ⨳ d) < a ⨳ b ♯ a' ⨳ b' ♯ (a ⨳ (b' ♯ c) ♯ a' ⨳ d) ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d rw [nmul_nadd a' b' c, nmul_nadd a b' c] at this case refine_1.inl.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c b' : Ordinal.{u_1} hb : b' < b hd : d ≤ b' ♯ c this : a' ⨳ b ♯ a ⨳ b' ♯ (a' ⨳ b' ♯ a' ⨳ c ♯ a ⨳ d) < a ⨳ b ♯ a' ⨳ b' ♯ (a ⨳ b' ♯ a ⨳ c ♯ a' ⨳ d) ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d simp only [nadd_assoc] at this case refine_1.inl.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c b' : Ordinal.{u_1} hb : b' < b hd : d ≤ b' ♯ c this : a' ⨳ b ♯ (a ⨳ b' ♯ (a' ⨳ b' ♯ (a' ⨳ c ♯ a ⨳ d))) < a ⨳ b ♯ (a' ⨳ b' ♯ (a ⨳ b' ♯ (a ⨳ c ♯ a' ⨳ d))) ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d rwa [nadd_left_comm, nadd_left_comm _ (a ⨳ b'), nadd_left_comm (a ⨳ b), nadd_lt_nadd_iff_left, nadd_left_comm (a' ⨳ b), nadd_left_comm (a ⨳ b), nadd_lt_nadd_iff_left, ← nadd_assoc, ← nadd_assoc] at this case refine_1.inr.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c c' : Ordinal.{u_1} hc : c' < c hd : d ≤ b ♯ c' ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d have := nadd_lt_nadd_of_le_of_lt (nmul_nadd_le ha.le hd) (nmul_nadd_lt ha hc) case refine_1.inr.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c c' : Ordinal.{u_1} hc : c' < c hd : d ≤ b ♯ c' this : a' ⨳ (b ♯ c') ♯ a ⨳ d ♯ (a' ⨳ c ♯ a ⨳ c') < a ⨳ (b ♯ c') ♯ a' ⨳ d ♯ (a ⨳ c ♯ a' ⨳ c') ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d rw [nmul_nadd a' b c', nmul_nadd a b c'] at this case refine_1.inr.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c c' : Ordinal.{u_1} hc : c' < c hd : d ≤ b ♯ c' this : a' ⨳ b ♯ a' ⨳ c' ♯ a ⨳ d ♯ (a' ⨳ c ♯ a ⨳ c') < a ⨳ b ♯ a ⨳ c' ♯ a' ⨳ d ♯ (a ⨳ c ♯ a' ⨳ c') ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d simp only [nadd_assoc] at this case refine_1.inr.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c c' : Ordinal.{u_1} hc : c' < c hd : d ≤ b ♯ c' this : a' ⨳ b ♯ (a' ⨳ c' ♯ (a ⨳ d ♯ (a' ⨳ c ♯ a ⨳ c'))) < a ⨳ b ♯ (a ⨳ c' ♯ (a' ⨳ d ♯ (a ⨳ c ♯ a' ⨳ c'))) ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d rwa [nadd_left_comm, nadd_comm (a ⨳ c), nadd_left_comm (a' ⨳ d), nadd_left_comm (a ⨳ c'), nadd_left_comm (a ⨳ b), nadd_lt_nadd_iff_left, nadd_comm (a' ⨳ c), nadd_left_comm (a ⨳ d), nadd_left_comm (a' ⨳ b), nadd_left_comm (a ⨳ b), nadd_lt_nadd_iff_left, nadd_comm (a ⨳ d), nadd_comm (a' ⨳ d), ← nadd_assoc, ← nadd_assoc] at this case refine_2 a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd : d < a ⨳ b ⊢ d ♯ a ⨳ c < a ⨳ (b ♯ c) rcases lt_nmul_iff.1 hd with ⟨a', ha, b', hb, hd⟩ case refine_2.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ b a' : Ordinal.{u_1} ha : a' < a b' : Ordinal.{u_1} hb : b' < b hd : d ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' ⊢ d ♯ a ⨳ c < a ⨳ (b ♯ c) have := nadd_lt_nadd_of_le_of_lt hd (nmul_nadd_lt ha (nadd_lt_nadd_right hb c)) case refine_2.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ b a' : Ordinal.{u_1} ha : a' < a b' : Ordinal.{u_1} hb : b' < b hd : d ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' this : d ♯ a' ⨳ b' ♯ (a' ⨳ (b ♯ c) ♯ a ⨳ (b' ♯ c)) < a' ⨳ b ♯ a ⨳ b' ♯ (a ⨳ (b ♯ c) ♯ a' ⨳ (b' ♯ c)) ⊢ d ♯ a ⨳ c < a ⨳ (b ♯ c) rw [nmul_nadd a' b c, nmul_nadd a b' c, nmul_nadd a'] at this case refine_2.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ b a' : Ordinal.{u_1} ha : a' < a b' : Ordinal.{u_1} hb : b' < b hd : d ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' this : d ♯ a' ⨳ b' ♯ (a' ⨳ b ♯ a' ⨳ c ♯ (a ⨳ b' ♯ a ⨳ c)) < a' ⨳ b ♯ a ⨳ b' ♯ (a ⨳ (b ♯ c) ♯ (a' ⨳ b' ♯ a' ⨳ c)) ⊢ d ♯ a ⨳ c < a ⨳ (b ♯ c) simp only [nadd_assoc] at this case refine_2.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ b a' : Ordinal.{u_1} ha : a' < a b' : Ordinal.{u_1} hb : b' < b hd : d ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' this : d ♯ (a' ⨳ b' ♯ (a' ⨳ b ♯ (a' ⨳ c ♯ (a ⨳ b' ♯ a ⨳ c)))) < a' ⨳ b ♯ (a ⨳ b' ♯ (a ⨳ (b ♯ c) ♯ (a' ⨳ b' ♯ a' ⨳ c))) ⊢ d ♯ a ⨳ c < a ⨳ (b ♯ c) rwa [nadd_left_comm (a' ⨳ b'), nadd_left_comm, nadd_lt_nadd_iff_left, nadd_left_comm, nadd_left_comm _ (a' ⨳ b'), nadd_left_comm (a ⨳ b'), nadd_lt_nadd_iff_left, nadd_left_comm (a' ⨳ c), nadd_left_comm, nadd_lt_nadd_iff_left, nadd_left_comm, nadd_comm _ (a' ⨳ c), nadd_lt_nadd_iff_left] at this case refine_3 a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd : d < a ⨳ c ⊢ a ⨳ b ♯ d < a ⨳ (b ♯ c) rcases lt_nmul_iff.1 hd with ⟨a', ha, c', hc, hd⟩ case refine_3.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ c a' : Ordinal.{u_1} ha : a' < a c' : Ordinal.{u_1} hc : c' < c hd : d ♯ a' ⨳ c' ≤ a' ⨳ c ♯ a ⨳ c' ⊢ a ⨳ b ♯ d < a ⨳ (b ♯ c) have := nadd_lt_nadd_of_lt_of_le (nmul_nadd_lt ha (nadd_lt_nadd_left hc b)) hd case refine_3.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ c a' : Ordinal.{u_1} ha : a' < a c' : Ordinal.{u_1} hc : c' < c hd : d ♯ a' ⨳ c' ≤ a' ⨳ c ♯ a ⨳ c' this : a' ⨳ (b ♯ c) ♯ a ⨳ (b ♯ c') ♯ (d ♯ a' ⨳ c') < a ⨳ (b ♯ c) ♯ a' ⨳ (b ♯ c') ♯ (a' ⨳ c ♯ a ⨳ c') ⊢ a ⨳ b ♯ d < a ⨳ (b ♯ c) rw [nmul_nadd a' b c, nmul_nadd a b c', nmul_nadd a'] at this case refine_3.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ c a' : Ordinal.{u_1} ha : a' < a c' : Ordinal.{u_1} hc : c' < c hd : d ♯ a' ⨳ c' ≤ a' ⨳ c ♯ a ⨳ c' this : a' ⨳ b ♯ a' ⨳ c ♯ (a ⨳ b ♯ a ⨳ c') ♯ (d ♯ a' ⨳ c') < a ⨳ (b ♯ c) ♯ (a' ⨳ b ♯ a' ⨳ c') ♯ (a' ⨳ c ♯ a ⨳ c') ⊢ a ⨳ b ♯ d < a ⨳ (b ♯ c) simp only [nadd_assoc] at this case refine_3.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ c a' : Ordinal.{u_1} ha : a' < a c' : Ordinal.{u_1} hc : c' < c hd : d ♯ a' ⨳ c' ≤ a' ⨳ c ♯ a ⨳ c' this : a' ⨳ b ♯ (a' ⨳ c ♯ (a ⨳ b ♯ (a ⨳ c' ♯ (d ♯ a' ⨳ c')))) < a ⨳ (b ♯ c) ♯ (a' ⨳ b ♯ (a' ⨳ c' ♯ (a' ⨳ c ♯ a ⨳ c'))) ⊢ a ⨳ b ♯ d < a ⨳ (b ♯ c) rwa [nadd_left_comm _ (a' ⨳ b), nadd_lt_nadd_iff_left, nadd_left_comm (a' ⨳ c'), nadd_left_comm _ (a' ⨳ c), nadd_lt_nadd_iff_left, nadd_left_comm, nadd_comm (a' ⨳ c'), nadd_left_comm _ (a ⨳ c'), nadd_lt_nadd_iff_left, nadd_comm _ (a' ⨳ c'), nadd_comm _ (a' ⨳ c'), nadd_left_comm, nadd_lt_nadd_iff_left] at this ** Qed
Ordinal.nadd_nmul a✝ b✝ c✝ d : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ (a ♯ b) ⨳ c = a ⨳ c ♯ b ⨳ c rw [nmul_comm, nmul_nadd, nmul_comm, nmul_comm c] ** Qed
Ordinal.nmul_nadd_lt₃ a b c d a' b' c' : Ordinal.{u} ha : a' < a hb : b' < b hc : c' < c ⊢ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' < a ⨳ b ⨳ c ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' simpa only [nadd_nmul, ← nadd_assoc] using nmul_nadd_lt (nmul_nadd_lt ha hb) hc ** Qed
Ordinal.nmul_nadd_le₃ a b c d a' b' c' : Ordinal.{u} ha : a' ≤ a hb : b' ≤ b hc : c' ≤ c ⊢ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' ≤ a ⨳ b ⨳ c ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' simpa only [nadd_nmul, ← nadd_assoc] using nmul_nadd_le (nmul_nadd_le ha hb) hc ** Qed
Ordinal.nmul_nadd_lt₃' a b c d a' b' c' : Ordinal.{u} ha : a' < a hb : b' < b hc : c' < c ⊢ a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') < a ⨳ (b ⨳ c) ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') simp only [nmul_comm _ (_ ⨳ _)] case h.e'_4 a b c d a' b' c' : Ordinal.{u} ha : a' < a hb : b' < b hc : c' < c ⊢ b ⨳ c ⨳ a ♯ b' ⨳ c ⨳ a' ♯ b ⨳ c' ⨳ a' ♯ b' ⨳ c' ⨳ a = b ⨳ c ⨳ a ♯ b' ⨳ c' ⨳ a ♯ b' ⨳ c ⨳ a' ♯ b ⨳ c' ⨳ a' simp only [nadd_eq_add, NatOrdinal.toOrdinal_toNatOrdinal] case h.e'_4 a b c d a' b' c' : Ordinal.{u} ha : a' < a hb : b' < b hc : c' < c ⊢ ↑toOrdinal (↑toNatOrdinal (b ⨳ c ⨳ a) + ↑toNatOrdinal (b' ⨳ c ⨳ a') + ↑toNatOrdinal (b ⨳ c' ⨳ a') + ↑toNatOrdinal (b' ⨳ c' ⨳ a)) = ↑toOrdinal (↑toNatOrdinal (b ⨳ c ⨳ a) + ↑toNatOrdinal (b' ⨳ c' ⨳ a) + ↑toNatOrdinal (b' ⨳ c ⨳ a') + ↑toNatOrdinal (b ⨳ c' ⨳ a')) abel_nf ** Qed
Ordinal.nmul_nadd_le₃' a b c d a' b' c' : Ordinal.{u} ha : a' ≤ a hb : b' ≤ b hc : c' ≤ c ⊢ a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') ≤ a ⨳ (b ⨳ c) ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') simp only [nmul_comm _ (_ ⨳ _)] case h.e'_4 a b c d a' b' c' : Ordinal.{u} ha : a' ≤ a hb : b' ≤ b hc : c' ≤ c ⊢ b ⨳ c ⨳ a ♯ b' ⨳ c ⨳ a' ♯ b ⨳ c' ⨳ a' ♯ b' ⨳ c' ⨳ a = b ⨳ c ⨳ a ♯ b' ⨳ c' ⨳ a ♯ b' ⨳ c ⨳ a' ♯ b ⨳ c' ⨳ a' simp only [nadd_eq_add, NatOrdinal.toOrdinal_toNatOrdinal] case h.e'_4 a b c d a' b' c' : Ordinal.{u} ha : a' ≤ a hb : b' ≤ b hc : c' ≤ c ⊢ ↑toOrdinal (↑toNatOrdinal (b ⨳ c ⨳ a) + ↑toNatOrdinal (b' ⨳ c ⨳ a') + ↑toNatOrdinal (b ⨳ c' ⨳ a') + ↑toNatOrdinal (b' ⨳ c' ⨳ a)) = ↑toOrdinal (↑toNatOrdinal (b ⨳ c ⨳ a) + ↑toNatOrdinal (b' ⨳ c' ⨳ a) + ↑toNatOrdinal (b' ⨳ c ⨳ a') + ↑toNatOrdinal (b ⨳ c' ⨳ a')) abel_nf ** Qed

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formal stringlengths 41 427k	informal stringclasses 1 value
Metric.ediam_cthickening_le ** ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε✝ : ℝ s t : Set α x : α ε : ℝ≥0 ⊢ EMetric.diam (cthickening (↑ε) s) ≤ EMetric.diam s + 2 * ↑ε refine' diam_le fun x hx y hy => ENNReal.le_of_forall_pos_le_add fun δ hδ _ => _ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s t : Set α x✝¹ : α ε : ℝ≥0 x : α hx : x ∈ cthickening (↑ε) s y : α hy : y ∈ cthickening (↑ε) s δ : ℝ≥0 hδ : 0 < δ x✝ : EMetric.diam s + 2 * ↑ε < ⊤ ⊢ edist x y ≤ EMetric.diam s + 2 * ↑ε + ↑δ rw [mem_cthickening_iff, ENNReal.ofReal_coe_nnreal] at hx hy ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s t : Set α x✝¹ : α ε : ℝ≥0 x : α hx : infEdist x s ≤ ↑ε y : α hy : infEdist y s ≤ ↑ε δ : ℝ≥0 hδ : 0 < δ x✝ : EMetric.diam s + 2 * ↑ε < ⊤ ⊢ edist x y ≤ EMetric.diam s + 2 * ↑ε + ↑δ have hε : (ε : ℝ≥0∞) < ε + δ := ENNReal.coe_lt_coe.2 (lt_add_of_pos_right _ hδ) ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s t : Set α x✝¹ : α ε : ℝ≥0 x : α hx : infEdist x s ≤ ↑ε y : α hy : infEdist y s ≤ ↑ε δ : ℝ≥0 hδ : 0 < δ x✝ : EMetric.diam s + 2 * ↑ε < ⊤ hε : ↑ε < ↑ε + ↑δ ⊢ edist x y ≤ EMetric.diam s + 2 * ↑ε + ↑δ replace hx := hx.trans_lt hε ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s t : Set α x✝¹ : α ε : ℝ≥0 x y : α hy : infEdist y s ≤ ↑ε δ : ℝ≥0 hδ : 0 < δ x✝ : EMetric.diam s + 2 * ↑ε < ⊤ hε : ↑ε < ↑ε + ↑δ hx : infEdist x s < ↑ε + ↑δ ⊢ edist x y ≤ EMetric.diam s + 2 * ↑ε + ↑δ obtain ⟨x', hx', hxx'⟩ := infEdist_lt_iff.mp hx case intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s t : Set α x✝¹ : α ε : ℝ≥0 x y : α hy : infEdist y s ≤ ↑ε δ : ℝ≥0 hδ : 0 < δ x✝ : EMetric.diam s + 2 * ↑ε < ⊤ hε : ↑ε < ↑ε + ↑δ hx : infEdist x s < ↑ε + ↑δ x' : α hx' : x' ∈ s hxx' : edist x x' < ↑ε + ↑δ ⊢ edist x y ≤ EMetric.diam s + 2 * ↑ε + ↑δ calc edist x y ≤ edist x x' + edist y x' := edist_triangle_right _ _ _ _ ≤ ε + δ + (infEdist y s + EMetric.diam s) := add_le_add hxx'.le (edist_le_infEdist_add_ediam hx') _ ≤ ε + δ + (ε + EMetric.diam s) := add_le_add_left (add_le_add_right hy _) _ _ = _ := by rw [two_mul]; ac_rfl ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s t : Set α x✝¹ : α ε : ℝ≥0 x y : α hy : infEdist y s ≤ ↑ε δ : ℝ≥0 hδ : 0 < δ x✝ : EMetric.diam s + 2 * ↑ε < ⊤ hε : ↑ε < ↑ε + ↑δ hx : infEdist x s < ↑ε + ↑δ x' : α hx' : x' ∈ s hxx' : edist x x' < ↑ε + ↑δ ⊢ ↑ε + ↑δ + (↑ε + EMetric.diam s) = EMetric.diam s + 2 * ↑ε + ↑δ rw [two_mul] ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s t : Set α x✝¹ : α ε : ℝ≥0 x y : α hy : infEdist y s ≤ ↑ε δ : ℝ≥0 hδ : 0 < δ x✝ : EMetric.diam s + 2 * ↑ε < ⊤ hε : ↑ε < ↑ε + ↑δ hx : infEdist x s < ↑ε + ↑δ x' : α hx' : x' ∈ s hxx' : edist x x' < ↑ε + ↑δ ⊢ ↑ε + ↑δ + (↑ε + EMetric.diam s) = EMetric.diam s + (↑ε + ↑ε) + ↑δ ac_rfl Qed
Metric.diam_cthickening_le ** ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ ε : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α hε : 0 ≤ ε ⊢ diam (cthickening ε s) ≤ diam s + 2 * ε lift ε to ℝ≥0 using hε case intro ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α ε : ℝ≥0 ⊢ diam (cthickening (↑ε) s) ≤ diam s + 2 * ↑ε refine (toReal_le_add' (ediam_cthickening_le _) ?_ ?_).trans_eq ?_ case intro.refine_1 ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α ε : ℝ≥0 ⊢ EMetric.diam s = ⊤ → EMetric.diam (cthickening (↑ε) s) = ⊤ exact fun h ↦ top_unique <\| h ▸ EMetric.diam_mono (self_subset_cthickening _) case intro.refine_2 ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α ε : ℝ≥0 ⊢ 2 * ↑ε = ⊤ → EMetric.diam (cthickening (↑ε) s) = ⊤ simp [mul_eq_top] case intro.refine_3 ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α ε : ℝ≥0 ⊢ ENNReal.toReal (EMetric.diam s) + ENNReal.toReal (2 * ↑ε) = diam s + 2 * ↑ε simp [diam] Qed
Metric.diam_thickening_le ** ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ ε : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α hε : 0 ≤ ε ⊢ diam (thickening ε s) ≤ diam s + 2 * ε by_cases hs : IsBounded s case neg ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ ε : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α hε : 0 ≤ ε hs : ¬Bornology.IsBounded s ⊢ diam (thickening ε s) ≤ diam s + 2 * ε obtain rfl \| hε := hε.eq_or_lt case pos ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ ε : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α hε : 0 ≤ ε hs : Bornology.IsBounded s ⊢ diam (thickening ε s) ≤ diam s + 2 * ε exact (diam_mono (thickening_subset_cthickening _ _) hs.cthickening).trans (diam_cthickening_le _ hε) case neg.inl ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α hs : ¬Bornology.IsBounded s hε : 0 ≤ 0 ⊢ diam (thickening 0 s) ≤ diam s + 2 * 0 simp [thickening_of_nonpos, diam_nonneg] case neg.inr ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ ε : ℝ s✝ t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α s : Set α hε✝ : 0 ≤ ε hs : ¬Bornology.IsBounded s hε : 0 < ε ⊢ 0 ≤ diam s + 2 * ε positivity Qed
Metric.thickening_closure ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α ⊢ thickening δ (closure s) = thickening δ s simp_rw [thickening, infEdist_closure] ** Qed
Metric.cthickening_closure ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α ⊢ cthickening δ (closure s) = cthickening δ s simp_rw [cthickening, infEdist_closure] ** Qed
Disjoint.exists_thickenings ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t ⊢ ∃ δ, 0 < δ ∧ Disjoint (thickening δ s) (thickening δ t) obtain ⟨r, hr, h⟩ := exists_pos_forall_lt_edist hs ht hst case intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y ⊢ ∃ δ, 0 < δ ∧ Disjoint (thickening δ s) (thickening δ t) refine' ⟨r / 2, half_pos (NNReal.coe_pos.2 hr), _⟩ case intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y ⊢ Disjoint (thickening (↑r / 2) s) (thickening (↑r / 2) t) rw [disjoint_iff_inf_le] case intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y ⊢ thickening (↑r / 2) s ⊓ thickening (↑r / 2) t ≤ ⊥ rintro z ⟨hzs, hzt⟩ case intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y z : α hzs : z ∈ thickening (↑r / 2) s hzt : z ∈ thickening (↑r / 2) t ⊢ z ∈ ⊥ rw [mem_thickening_iff_exists_edist_lt] at hzs hzt case intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y z : α hzs : ∃ z_1, z_1 ∈ s ∧ edist z z_1 < ENNReal.ofReal (↑r / 2) hzt : ∃ z_1, z_1 ∈ t ∧ edist z z_1 < ENNReal.ofReal (↑r / 2) ⊢ z ∈ ⊥ rw [← NNReal.coe_two, ← NNReal.coe_div, ENNReal.ofReal_coe_nnreal] at hzs hzt case intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y z : α hzs : ∃ z_1, z_1 ∈ s ∧ edist z z_1 < ↑(r / 2) hzt : ∃ z_1, z_1 ∈ t ∧ edist z z_1 < ↑(r / 2) ⊢ z ∈ ⊥ obtain ⟨x, hx, hzx⟩ := hzs case intro.intro.intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x✝ : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y z : α hzt : ∃ z_1, z_1 ∈ t ∧ edist z z_1 < ↑(r / 2) x : α hx : x ∈ s hzx : edist z x < ↑(r / 2) ⊢ z ∈ ⊥ obtain ⟨y, hy, hzy⟩ := hzt case intro.intro.intro.intro.intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x✝ : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y z x : α hx : x ∈ s hzx : edist z x < ↑(r / 2) y : α hy : y ∈ t hzy : edist z y < ↑(r / 2) ⊢ z ∈ ⊥ refine' (h x hx y hy).not_le _ case intro.intro.intro.intro.intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x✝ : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y z x : α hx : x ∈ s hzx : edist z x < ↑(r / 2) y : α hy : y ∈ t hzy : edist z y < ↑(r / 2) ⊢ edist x y ≤ ↑r calc edist x y ≤ edist z x + edist z y := edist_triangle_left _ _ _ _ ≤ ↑(r / 2) + ↑(r / 2) := (add_le_add hzx.le hzy.le) _ = r := by rw [← ENNReal.coe_add, add_halves] ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x✝ : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t r : ℝ≥0 hr : 0 < r h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ t → ↑r < edist x y z x : α hx : x ∈ s hzx : edist z x < ↑(r / 2) y : α hy : y ∈ t hzy : edist z y < ↑(r / 2) ⊢ ↑(r / 2) + ↑(r / 2) = ↑r rw [← ENNReal.coe_add, add_halves] ** Qed
Disjoint.exists_cthickenings ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t ⊢ ∃ δ, 0 < δ ∧ Disjoint (cthickening δ s) (cthickening δ t) obtain ⟨δ, hδ, h⟩ := hst.exists_thickenings hs ht case intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α hst : Disjoint s t hs : IsCompact s ht : IsClosed t δ : ℝ hδ : 0 < δ h : Disjoint (thickening δ s) (thickening δ t) ⊢ ∃ δ, 0 < δ ∧ Disjoint (cthickening δ s) (cthickening δ t) refine' ⟨δ / 2, half_pos hδ, h.mono _ _⟩ <;> exact cthickening_subset_thickening' hδ (half_lt_self hδ) _ ** Qed
IsCompact.exists_isCompact_cthickening ι : Sort u_1 α : Type u β : Type v inst✝¹ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α inst✝ : LocallyCompactSpace α hs : IsCompact s ⊢ ∃ δ, 0 < δ ∧ IsCompact (cthickening δ s) rcases exists_compact_superset hs with ⟨K, K_compact, hK⟩ case intro.intro ι : Sort u_1 α : Type u β : Type v inst✝¹ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α inst✝ : LocallyCompactSpace α hs : IsCompact s K : Set α K_compact : IsCompact K hK : s ⊆ interior K ⊢ ∃ δ, 0 < δ ∧ IsCompact (cthickening δ s) rcases hs.exists_cthickening_subset_open isOpen_interior hK with ⟨δ, δpos, hδ⟩ case intro.intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝¹ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α inst✝ : LocallyCompactSpace α hs : IsCompact s K : Set α K_compact : IsCompact K hK : s ⊆ interior K δ : ℝ δpos : 0 < δ hδ : cthickening δ s ⊆ interior K ⊢ ∃ δ, 0 < δ ∧ IsCompact (cthickening δ s) refine ⟨δ, δpos, ?_⟩ case intro.intro.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝¹ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α inst✝ : LocallyCompactSpace α hs : IsCompact s K : Set α K_compact : IsCompact K hK : s ⊆ interior K δ : ℝ δpos : 0 < δ hδ : cthickening δ s ⊆ interior K ⊢ IsCompact (cthickening δ s) exact K_compact.of_isClosed_subset isClosed_cthickening (hδ.trans interior_subset) ** Qed
Metric.cthickening_eq_iInter_cthickening' ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ⊢ cthickening δ E = ⋂ ε ∈ s, cthickening ε E apply Subset.antisymm case h₁ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ⊢ cthickening δ E ⊆ ⋂ ε ∈ s, cthickening ε E exact subset_iInter₂ fun _ hε => cthickening_mono (le_of_lt (hsδ hε)) E case h₂ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ⊢ ⋂ ε ∈ s, cthickening ε E ⊆ cthickening δ E unfold cthickening case h₂ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ⊢ ⋂ ε ∈ s, {x \| infEdist x E ≤ ENNReal.ofReal ε} ⊆ {x \| infEdist x E ≤ ENNReal.ofReal δ} intro x hx case h₂ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x✝ : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α x : α hx : x ∈ ⋂ ε ∈ s, {x \| infEdist x E ≤ ENNReal.ofReal ε} ⊢ x ∈ {x \| infEdist x E ≤ ENNReal.ofReal δ} simp only [mem_iInter, mem_setOf_eq] at * case h₂ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x✝ : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α x : α hx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i ⊢ infEdist x E ≤ ENNReal.ofReal δ apply ENNReal.le_of_forall_pos_le_add case h₂.h ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x✝ : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α x : α hx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i ⊢ ∀ (ε : ℝ≥0), 0 < ε → ENNReal.ofReal δ < ⊤ → infEdist x E ≤ ENNReal.ofReal δ + ↑ε intro η η_pos _ case h₂.h ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x✝ : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α x : α hx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i η : ℝ≥0 η_pos : 0 < η a✝ : ENNReal.ofReal δ < ⊤ ⊢ infEdist x E ≤ ENNReal.ofReal δ + ↑η rcases hs (δ + η) (lt_add_of_pos_right _ (NNReal.coe_pos.mpr η_pos)) with ⟨ε, ⟨hsε, hε⟩⟩ case h₂.h.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x✝ : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α x : α hx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i η : ℝ≥0 η_pos : 0 < η a✝ : ENNReal.ofReal δ < ⊤ ε : ℝ hsε : ε ∈ s hε : ε ∈ Ioc δ (δ + ↑η) ⊢ infEdist x E ≤ ENNReal.ofReal δ + ↑η apply ((hx ε hsε).trans (ENNReal.ofReal_le_ofReal hε.2)).trans case h₂.h.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x✝ : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α x : α hx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i η : ℝ≥0 η_pos : 0 < η a✝ : ENNReal.ofReal δ < ⊤ ε : ℝ hsε : ε ∈ s hε : ε ∈ Ioc δ (δ + ↑η) ⊢ ENNReal.ofReal (δ + ↑η) ≤ ENNReal.ofReal δ + ↑η rw [ENNReal.coe_nnreal_eq η] case h₂.h.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x✝ : α δ : ℝ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α x : α hx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i η : ℝ≥0 η_pos : 0 < η a✝ : ENNReal.ofReal δ < ⊤ ε : ℝ hsε : ε ∈ s hε : ε ∈ Ioc δ (δ + ↑η) ⊢ ENNReal.ofReal (δ + ↑η) ≤ ENNReal.ofReal δ + ENNReal.ofReal ↑η exact ENNReal.ofReal_add_le ** Qed
Metric.cthickening_eq_iInter_cthickening ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ E : Set α ⊢ cthickening δ E = ⋂ ε, ⋂ (_ : δ < ε), cthickening ε E apply cthickening_eq_iInter_cthickening' (Ioi δ) rfl.subset case hs ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ E : Set α ⊢ ∀ (ε : ℝ), δ < ε → Set.Nonempty (Ioi δ ∩ Ioc δ ε) simp_rw [inter_eq_right.mpr Ioc_subset_Ioi_self] case hs ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ E : Set α ⊢ ∀ (ε : ℝ), δ < ε → Set.Nonempty (Ioc δ ε) exact fun _ hε => nonempty_Ioc.mpr hε ** Qed
Metric.cthickening_eq_iInter_thickening' ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ⊢ cthickening δ E = ⋂ ε ∈ s, thickening ε E refine' (subset_iInter₂ fun ε hε => _).antisymm _ case refine'_1 ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ε : ℝ hε : ε ∈ s ⊢ cthickening δ E ⊆ thickening ε E obtain ⟨ε', -, hε'⟩ := hs ε (hsδ hε) case refine'_1.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ε : ℝ hε : ε ∈ s ε' : ℝ hε' : ε' ∈ Ioc δ ε ⊢ cthickening δ E ⊆ thickening ε E have ss := cthickening_subset_thickening' (lt_of_le_of_lt δ_nn hε'.1) hε'.1 E case refine'_1.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ε : ℝ hε : ε ∈ s ε' : ℝ hε' : ε' ∈ Ioc δ ε ss : cthickening δ E ⊆ thickening ε' E ⊢ cthickening δ E ⊆ thickening ε E exact ss.trans (thickening_mono hε'.2 E) case refine'_2 ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ⊢ ⋂ i ∈ s, thickening i E ⊆ cthickening δ E rw [cthickening_eq_iInter_cthickening' s hsδ hs E] case refine'_2 ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ s : Set ℝ hsδ : s ⊆ Ioi δ hs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε) E : Set α ⊢ ⋂ i ∈ s, thickening i E ⊆ ⋂ ε ∈ s, cthickening ε E exact iInter₂_mono fun ε _ => thickening_subset_cthickening ε E ** Qed
Metric.cthickening_eq_iInter_thickening ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ E : Set α ⊢ cthickening δ E = ⋂ ε, ⋂ (_ : δ < ε), thickening ε E apply cthickening_eq_iInter_thickening' δ_nn (Ioi δ) rfl.subset case hs ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ E : Set α ⊢ ∀ (ε : ℝ), δ < ε → Set.Nonempty (Ioi δ ∩ Ioc δ ε) simp_rw [inter_eq_right.mpr Ioc_subset_Ioi_self] case hs ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ δ_nn : 0 ≤ δ E : Set α ⊢ ∀ (ε : ℝ), δ < ε → Set.Nonempty (Ioc δ ε) exact fun _ hε => nonempty_Ioc.mpr hε ** Qed
Metric.cthickening_eq_iInter_thickening'' ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ E : Set α ⊢ cthickening δ E = ⋂ ε, ⋂ (_ : max 0 δ < ε), thickening ε E rw [← cthickening_max_zero, cthickening_eq_iInter_thickening] case δ_nn ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s t : Set α x : α δ : ℝ E : Set α ⊢ 0 ≤ max 0 δ exact le_max_left _ _ ** Qed
Metric.closure_eq_iInter_cthickening' ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) ⊢ closure E = ⋂ δ ∈ s, cthickening δ E by_cases hs₀ : s ⊆ Ioi 0 case neg ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : ¬s ⊆ Ioi 0 ⊢ closure E = ⋂ δ ∈ s, cthickening δ E obtain ⟨δ, hδs, δ_nonpos⟩ := not_subset.mp hs₀ case neg.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : ¬s ⊆ Ioi 0 δ : ℝ hδs : δ ∈ s δ_nonpos : ¬δ ∈ Ioi 0 ⊢ closure E = ⋂ δ ∈ s, cthickening δ E rw [Set.mem_Ioi, not_lt] at δ_nonpos case neg.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : ¬s ⊆ Ioi 0 δ : ℝ hδs : δ ∈ s δ_nonpos : δ ≤ 0 ⊢ closure E = ⋂ δ ∈ s, cthickening δ E apply Subset.antisymm case pos ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : s ⊆ Ioi 0 ⊢ closure E = ⋂ δ ∈ s, cthickening δ E rw [← cthickening_zero] case pos ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : s ⊆ Ioi 0 ⊢ cthickening 0 E = ⋂ δ ∈ s, cthickening δ E apply cthickening_eq_iInter_cthickening' _ hs₀ hs case neg.intro.intro.h₁ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : ¬s ⊆ Ioi 0 δ : ℝ hδs : δ ∈ s δ_nonpos : δ ≤ 0 ⊢ closure E ⊆ ⋂ δ ∈ s, cthickening δ E exact subset_iInter₂ fun ε _ => closure_subset_cthickening ε E case neg.intro.intro.h₂ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : ¬s ⊆ Ioi 0 δ : ℝ hδs : δ ∈ s δ_nonpos : δ ≤ 0 ⊢ ⋂ δ ∈ s, cthickening δ E ⊆ closure E rw [← cthickening_of_nonpos δ_nonpos E] case neg.intro.intro.h₂ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) hs₀ : ¬s ⊆ Ioi 0 δ : ℝ hδs : δ ∈ s δ_nonpos : δ ≤ 0 ⊢ ⋂ δ ∈ s, cthickening δ E ⊆ cthickening δ E exact biInter_subset_of_mem hδs ** Qed
Metric.closure_eq_iInter_cthickening ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α E : Set α ⊢ closure E = ⋂ δ, ⋂ (_ : 0 < δ), cthickening δ E rw [← cthickening_zero] ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α E : Set α ⊢ cthickening 0 E = ⋂ δ, ⋂ (_ : 0 < δ), cthickening δ E exact cthickening_eq_iInter_cthickening E ** Qed
Metric.closure_eq_iInter_thickening' ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs₀ : s ⊆ Ioi 0 hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) ⊢ closure E = ⋂ δ ∈ s, thickening δ E rw [← cthickening_zero] ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x : α E : Set α s : Set ℝ hs₀ : s ⊆ Ioi 0 hs : ∀ (ε : ℝ), 0 < ε → Set.Nonempty (s ∩ Ioc 0 ε) ⊢ cthickening 0 E = ⋂ δ ∈ s, thickening δ E apply cthickening_eq_iInter_thickening' le_rfl _ hs₀ hs ** Qed
Metric.closure_eq_iInter_thickening ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α E : Set α ⊢ closure E = ⋂ δ, ⋂ (_ : 0 < δ), thickening δ E rw [← cthickening_zero] ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α E : Set α ⊢ cthickening 0 E = ⋂ δ, ⋂ (_ : 0 < δ), thickening δ E exact cthickening_eq_iInter_thickening rfl.ge E ** Qed
Metric.closedBall_subset_cthickening ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α x : α E : Set α hx : x ∈ E δ : ℝ ⊢ closedBall x δ ⊆ cthickening δ E refine' (closedBall_subset_cthickening_singleton _ _).trans (cthickening_subset_of_subset _ _) ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α x : α E : Set α hx : x ∈ E δ : ℝ ⊢ {x} ⊆ E simpa using hx ** Qed
Metric.cthickening_subset_iUnion_closedBall_of_lt ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α E : Set α δ δ' : ℝ hδ₀ : 0 < δ' hδδ' : δ < δ' ⊢ cthickening δ E ⊆ ⋃ x ∈ E, closedBall x δ' refine' (cthickening_subset_thickening' hδ₀ hδδ' E).trans fun x hx => _ ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α E : Set α δ δ' : ℝ hδ₀ : 0 < δ' hδδ' : δ < δ' x : α hx : x ∈ thickening δ' E ⊢ x ∈ ⋃ x ∈ E, closedBall x δ' obtain ⟨y, hy₁, hy₂⟩ := mem_thickening_iff.mp hx case intro.intro ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α E : Set α δ δ' : ℝ hδ₀ : 0 < δ' hδδ' : δ < δ' x : α hx : x ∈ thickening δ' E y : α hy₁ : y ∈ E hy₂ : dist x y < δ' ⊢ x ∈ ⋃ x ∈ E, closedBall x δ' exact mem_iUnion₂.mpr ⟨y, hy₁, hy₂.le⟩ ** Qed
IsCompact.cthickening_eq_biUnion_closedBall ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ ⊢ cthickening δ E = ⋃ x ∈ E, closedBall x δ rcases eq_empty_or_nonempty E with (rfl \| hne) case inr ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ hne : Set.Nonempty E ⊢ cthickening δ E = ⋃ x ∈ E, closedBall x δ refine Subset.antisymm (fun x hx ↦ ?_) (iUnion₂_subset fun x hx ↦ closedBall_subset_cthickening hx _) case inr ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ E ⊢ x ∈ ⋃ x ∈ E, closedBall x δ obtain ⟨y, yE, hy⟩ : ∃ y ∈ E, infEdist x E = edist x y := hE.exists_infEdist_eq_edist hne _ case inr.intro.intro ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ E y : α yE : y ∈ E hy : infEdist x E = edist x y ⊢ x ∈ ⋃ x ∈ E, closedBall x δ have D1 : edist x y ≤ ENNReal.ofReal δ := (le_of_eq hy.symm).trans hx case inr.intro.intro ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ E y : α yE : y ∈ E hy : infEdist x E = edist x y D1 : edist x y ≤ ENNReal.ofReal δ ⊢ x ∈ ⋃ x ∈ E, closedBall x δ have D2 : dist x y ≤ δ := by rw [edist_dist] at D1 exact (ENNReal.ofReal_le_ofReal_iff hδ).1 D1 case inr.intro.intro ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ E y : α yE : y ∈ E hy : infEdist x E = edist x y D1 : edist x y ≤ ENNReal.ofReal δ D2 : dist x y ≤ δ ⊢ x ∈ ⋃ x ∈ E, closedBall x δ exact mem_biUnion yE D2 case inl ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ hδ : 0 ≤ δ hE : IsCompact ∅ ⊢ cthickening δ ∅ = ⋃ x ∈ ∅, closedBall x δ simp only [cthickening_empty, biUnion_empty] ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ E y : α yE : y ∈ E hy : infEdist x E = edist x y D1 : edist x y ≤ ENNReal.ofReal δ ⊢ dist x y ≤ δ rw [edist_dist] at D1 ι : Sort u_1 α✝ : Type u β : Type v inst✝¹ : PseudoEMetricSpace α✝ δ✝ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝ : PseudoMetricSpace α δ : ℝ E : Set α hE : IsCompact E hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ E y : α yE : y ∈ E hy : infEdist x E = edist x y D1 : ENNReal.ofReal (dist x y) ≤ ENNReal.ofReal δ ⊢ dist x y ≤ δ exact (ENNReal.ofReal_le_ofReal_iff hδ).1 D1 ** Qed
Metric.cthickening_eq_biUnion_closedBall ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α E : Set α hδ : 0 ≤ δ ⊢ cthickening δ E = ⋃ x ∈ closure E, closedBall x δ rcases eq_empty_or_nonempty E with (rfl \| hne) case inr ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α E : Set α hδ : 0 ≤ δ hne : Set.Nonempty E ⊢ cthickening δ E = ⋃ x ∈ closure E, closedBall x δ rw [← cthickening_closure] case inr ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α E : Set α hδ : 0 ≤ δ hne : Set.Nonempty E ⊢ cthickening δ (closure E) = ⋃ x ∈ closure E, closedBall x δ refine Subset.antisymm (fun x hx ↦ ?_) (iUnion₂_subset fun x hx ↦ closedBall_subset_cthickening hx _) case inr ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α E : Set α hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ (closure E) ⊢ x ∈ ⋃ x ∈ closure E, closedBall x δ obtain ⟨y, yE, hy⟩ : ∃ y ∈ closure E, infDist x (closure E) = dist x y := isClosed_closure.exists_infDist_eq_dist (closure_nonempty_iff.mpr hne) x case inr.intro.intro ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α E : Set α hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ (closure E) y : α yE : y ∈ closure E hy : infDist x (closure E) = dist x y ⊢ x ∈ ⋃ x ∈ closure E, closedBall x δ replace hy : dist x y ≤ δ := (ENNReal.ofReal_le_ofReal_iff hδ).mp (((congr_arg ENNReal.ofReal hy.symm).le.trans ENNReal.ofReal_toReal_le).trans hx) case inr.intro.intro ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x✝ : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α E : Set α hδ : 0 ≤ δ hne : Set.Nonempty E x : α hx : x ∈ cthickening δ (closure E) y : α yE : y ∈ closure E hy : dist x y ≤ δ ⊢ x ∈ ⋃ x ∈ closure E, closedBall x δ exact mem_biUnion yE hy case inl ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α hδ : 0 ≤ δ ⊢ cthickening δ ∅ = ⋃ x ∈ closure ∅, closedBall x δ simp only [cthickening_empty, biUnion_empty, closure_empty] ** Qed
IsClosed.cthickening_eq_biUnion_closedBall ι : Sort u_1 α✝ : Type u β : Type v inst✝² : PseudoEMetricSpace α✝ δ ε : ℝ s t : Set α✝ x : α✝ α : Type u_2 inst✝¹ : PseudoMetricSpace α inst✝ : ProperSpace α E : Set α hE : IsClosed E hδ : 0 ≤ δ ⊢ cthickening δ E = ⋃ x ∈ E, closedBall x δ rw [cthickening_eq_biUnion_closedBall E hδ, hE.closure_eq] ** Qed
Metric.infEdist_le_infEdist_cthickening_add ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α ⊢ infEdist x s ≤ infEdist x (cthickening δ s) + ENNReal.ofReal δ refine' le_of_forall_lt' fun r h => _ ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α r : ℝ≥0∞ h : infEdist x (cthickening δ s) + ENNReal.ofReal δ < r ⊢ infEdist x s < r simp_rw [← lt_tsub_iff_right, infEdist_lt_iff, mem_cthickening_iff] at h ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α r : ℝ≥0∞ h : ∃ y, infEdist y s ≤ ENNReal.ofReal δ ∧ edist x y < r - ENNReal.ofReal δ ⊢ infEdist x s < r obtain ⟨y, hy, hxy⟩ := h case intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α r : ℝ≥0∞ y : α hy : infEdist y s ≤ ENNReal.ofReal δ hxy : edist x y < r - ENNReal.ofReal δ ⊢ infEdist x s < r exact infEdist_le_edist_add_infEdist.trans_lt ((ENNReal.add_lt_add_of_lt_of_le (hy.trans_lt ENNReal.ofReal_lt_top).ne hxy hy).trans_eq (tsub_add_cancel_of_le <\| le_self_add.trans (lt_tsub_iff_left.1 hxy).le)) ** Qed
Metric.thickening_thickening_subset ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α ε δ : ℝ s : Set α ⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s obtain hε \| hε := le_total ε 0 case inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α ε δ : ℝ s : Set α hε : 0 ≤ ε ⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s obtain hδ \| hδ := le_total δ 0 case inr.inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α ε δ : ℝ s : Set α hε : 0 ≤ ε hδ : 0 ≤ δ ⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s intro x case inr.inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x✝ : α ε δ : ℝ s : Set α hε : 0 ≤ ε hδ : 0 ≤ δ x : α ⊢ x ∈ thickening ε (thickening δ s) → x ∈ thickening (ε + δ) s simp_rw [mem_thickening_iff_exists_edist_lt, ENNReal.ofReal_add hε hδ] case inr.inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x✝ : α ε δ : ℝ s : Set α hε : 0 ≤ ε hδ : 0 ≤ δ x : α ⊢ (∃ z, (∃ z_1, z_1 ∈ s ∧ edist z z_1 < ENNReal.ofReal δ) ∧ edist x z < ENNReal.ofReal ε) → ∃ z, z ∈ s ∧ edist x z < ENNReal.ofReal ε + ENNReal.ofReal δ exact fun ⟨y, ⟨z, hz, hy⟩, hx⟩ => ⟨z, hz, (edist_triangle _ _ _).trans_lt <\| ENNReal.add_lt_add hx hy⟩ case inl ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α ε δ : ℝ s : Set α hε : ε ≤ 0 ⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s simp only [thickening_of_nonpos hε, empty_subset] case inr.inl ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε✝ : ℝ s✝ t : Set α x : α ε δ : ℝ s : Set α hε : 0 ≤ ε hδ : δ ≤ 0 ⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s simp only [thickening_of_nonpos hδ, thickening_empty, empty_subset] ** Qed
Metric.thickening_cthickening_subset ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε✝ : ℝ s✝ t : Set α x : α ε : ℝ hδ : 0 ≤ δ s : Set α ⊢ thickening ε (cthickening δ s) ⊆ thickening (ε + δ) s obtain hε \| hε := le_total ε 0 case inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε✝ : ℝ s✝ t : Set α x : α ε : ℝ hδ : 0 ≤ δ s : Set α hε : 0 ≤ ε ⊢ thickening ε (cthickening δ s) ⊆ thickening (ε + δ) s intro x case inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε✝ : ℝ s✝ t : Set α x✝ : α ε : ℝ hδ : 0 ≤ δ s : Set α hε : 0 ≤ ε x : α ⊢ x ∈ thickening ε (cthickening δ s) → x ∈ thickening (ε + δ) s simp_rw [mem_thickening_iff_exists_edist_lt, mem_cthickening_iff, ← infEdist_lt_iff, ENNReal.ofReal_add hε hδ] case inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε✝ : ℝ s✝ t : Set α x✝ : α ε : ℝ hδ : 0 ≤ δ s : Set α hε : 0 ≤ ε x : α ⊢ (∃ z, infEdist z s ≤ ENNReal.ofReal δ ∧ edist x z < ENNReal.ofReal ε) → infEdist x s < ENNReal.ofReal ε + ENNReal.ofReal δ rintro ⟨y, hy, hxy⟩ case inr.intro.intro ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε✝ : ℝ s✝ t : Set α x✝ : α ε : ℝ hδ : 0 ≤ δ s : Set α hε : 0 ≤ ε x y : α hy : infEdist y s ≤ ENNReal.ofReal δ hxy : edist x y < ENNReal.ofReal ε ⊢ infEdist x s < ENNReal.ofReal ε + ENNReal.ofReal δ exact infEdist_le_edist_add_infEdist.trans_lt (ENNReal.add_lt_add_of_lt_of_le (hy.trans_lt ENNReal.ofReal_lt_top).ne hxy hy) case inl ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε✝ : ℝ s✝ t : Set α x : α ε : ℝ hδ : 0 ≤ δ s : Set α hε : ε ≤ 0 ⊢ thickening ε (cthickening δ s) ⊆ thickening (ε + δ) s simp only [thickening_of_nonpos hε, empty_subset] ** Qed
Metric.cthickening_thickening_subset ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α hε : 0 ≤ ε δ : ℝ s : Set α ⊢ cthickening ε (thickening δ s) ⊆ cthickening (ε + δ) s obtain hδ \| hδ := le_total δ 0 case inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α hε : 0 ≤ ε δ : ℝ s : Set α hδ : 0 ≤ δ ⊢ cthickening ε (thickening δ s) ⊆ cthickening (ε + δ) s intro x case inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x✝ : α hε : 0 ≤ ε δ : ℝ s : Set α hδ : 0 ≤ δ x : α ⊢ x ∈ cthickening ε (thickening δ s) → x ∈ cthickening (ε + δ) s simp_rw [mem_cthickening_iff, ENNReal.ofReal_add hε hδ] case inr ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x✝ : α hε : 0 ≤ ε δ : ℝ s : Set α hδ : 0 ≤ δ x : α ⊢ infEdist x (thickening δ s) ≤ ENNReal.ofReal ε → infEdist x s ≤ ENNReal.ofReal ε + ENNReal.ofReal δ exact fun hx => infEdist_le_infEdist_thickening_add.trans (add_le_add_right hx _) case inl ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ✝ ε : ℝ s✝ t : Set α x : α hε : 0 ≤ ε δ : ℝ s : Set α hδ : δ ≤ 0 ⊢ cthickening ε (thickening δ s) ⊆ cthickening (ε + δ) s simp only [thickening_of_nonpos hδ, cthickening_empty, empty_subset] ** Qed
Metric.cthickening_cthickening_subset ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x : α hε : 0 ≤ ε hδ : 0 ≤ δ s : Set α ⊢ cthickening ε (cthickening δ s) ⊆ cthickening (ε + δ) s intro x ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x✝ : α hε : 0 ≤ ε hδ : 0 ≤ δ s : Set α x : α ⊢ x ∈ cthickening ε (cthickening δ s) → x ∈ cthickening (ε + δ) s simp_rw [mem_cthickening_iff, ENNReal.ofReal_add hε hδ] ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s✝ t : Set α x✝ : α hε : 0 ≤ ε hδ : 0 ≤ δ s : Set α x : α ⊢ infEdist x (cthickening δ s) ≤ ENNReal.ofReal ε → infEdist x s ≤ ENNReal.ofReal ε + ENNReal.ofReal δ exact fun hx => infEdist_le_infEdist_cthickening_add.trans (add_le_add_right hx _) ** Qed
Metric.frontier_cthickening_disjoint ι : Sort u_1 α : Type u β : Type v inst✝ : PseudoEMetricSpace α δ ε : ℝ s t : Set α x : α A : Set α r₁ r₂ : ℝ≥0 hr : r₁ ≠ r₂ ⊢ ENNReal.ofReal ↑r₁ ≠ ENNReal.ofReal ↑r₂ simpa ** Qed
TopCat.Presheaf.germ_res_apply C : Type u inst✝² : Category.{v, u} C inst✝¹ : HasColimits C X Y Z : TopCat F : Presheaf C X U V : Opens ↑X i : U ⟶ V x : { x // x ∈ U } inst✝ : ConcreteCategory C s : (forget C).obj (F.obj (op V)) ⊢ ↑(germ F x) (↑(F.map i.op) s) = ↑(germ F ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑V) }) x)) s rw [←comp_apply, germ_res] ** Qed
TopCat.Presheaf.stalk_hom_ext C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y✝ Z : TopCat F : Presheaf C X x : ↑X Y : C f₁ f₂ : stalk F x ⟶ Y ih : ∀ (U : Opens ↑X) (hxU : x ∈ U), germ F { val := x, property := hxU } ≫ f₁ = germ F { val := x, property := hxU } ≫ f₂ U : (OpenNhds x)ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U ≫ f₁ = colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U ≫ f₂ induction' U using Opposite.rec with U case mk C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y✝ Z : TopCat F : Presheaf C X x : ↑X Y : C f₁ f₂ : stalk F x ⟶ Y ih : ∀ (U : Opens ↑X) (hxU : x ∈ U), germ F { val := x, property := hxU } ≫ f₁ = germ F { val := x, property := hxU } ≫ f₂ U : OpenNhds x ⊢ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) { unop := U } ≫ f₁ = colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) { unop := U } ≫ f₂ cases' U with U hxU case mk.mk C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y✝ Z : TopCat F : Presheaf C X x : ↑X Y : C f₁ f₂ : stalk F x ⟶ Y ih : ∀ (U : Opens ↑X) (hxU : x ∈ U), germ F { val := x, property := hxU } ≫ f₁ = germ F { val := x, property := hxU } ≫ f₂ U : Opens ↑X hxU : x ∈ U ⊢ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) { unop := { obj := U, property := hxU } } ≫ f₁ = colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) { unop := { obj := U, property := hxU } } ≫ f₂ exact ih U hxU ** Qed
TopCat.Presheaf.stalkPushforward_germ ** C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y F : Presheaf C X U : Opens ↑Y x : { x // x ∈ (Opens.map f).obj U } ⊢ germ (f _* F) { val := ↑f ↑x, property := (_ : ↑x ∈ (Opens.map f).obj U) } ≫ stalkPushforward C f F ↑x = germ F x rw [stalkPushforward, germ, colimit.ι_map_assoc, colimit.ι_pre, whiskerRight_app] C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y F : Presheaf C X U : Opens ↑Y x : { x // x ∈ (Opens.map f).obj U } ⊢ F.map ((NatTrans.op (OpenNhds.inclusionMapIso f ↑x).inv).app (op { obj := U, property := (_ : ↑{ val := ↑f ↑x, property := (_ : ↑x ∈ (Opens.map f).obj U) } ∈ U) })) ≫ colimit.ι (((whiskeringLeft (OpenNhds ↑x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ↑x).op).obj F) ((OpenNhds.map f ↑x).op.obj (op { obj := U, property := (_ : ↑{ val := ↑f ↑x, property := (_ : ↑x ∈ (Opens.map f).obj U) } ∈ U) })) = germ F x erw [CategoryTheory.Functor.map_id, Category.id_comp] C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y F : Presheaf C X U : Opens ↑Y x : { x // x ∈ (Opens.map f).obj U } ⊢ colimit.ι (((whiskeringLeft (OpenNhds ↑x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ↑x).op).obj F) ((OpenNhds.map f ↑x).op.obj (op { obj := U, property := (_ : ↑{ val := ↑f ↑x, property := (_ : ↑x ∈ (Opens.map f).obj U) } ∈ U) })) = germ F x rfl Qed
TopCat.Presheaf.stalkPushforward.id C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X x : ↑X ⊢ stalkPushforward C (𝟙 X) ℱ x = (stalkFunctor C x).map (Pushforward.id ℱ).hom ext1 j ** case w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X x : ↑X j : (OpenNhds (↑(𝟙 X) x))ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑(𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion (↑(𝟙 X) x)).op).obj (𝟙 X _* ℱ)) j ≫ stalkPushforward C (𝟙 X) ℱ x = colimit.ι (((whiskeringLeft (OpenNhds (↑(𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion (↑(𝟙 X) x)).op).obj (𝟙 X _* ℱ)) j ≫ (stalkFunctor C x).map (Pushforward.id ℱ).hom induction' j with j case w.h C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X x : ↑X j : OpenNhds (↑(𝟙 X) x) ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑(𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion (↑(𝟙 X) x)).op).obj (𝟙 X _* ℱ)) (op j) ≫ stalkPushforward C (𝟙 X) ℱ x = colimit.ι (((whiskeringLeft (OpenNhds (↑(𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion (↑(𝟙 X) x)).op).obj (𝟙 X _* ℱ)) (op j) ≫ (stalkFunctor C x).map (Pushforward.id ℱ).hom rcases j with ⟨⟨_, _⟩, _⟩ case w.h.mk.mk C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X x : ↑X carrier✝ : Set ↑X is_open'✝ : IsOpen carrier✝ property✝ : ↑(𝟙 X) x ∈ { carrier := carrier✝, is_open' := is_open'✝ } ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑(𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion (↑(𝟙 X) x)).op).obj (𝟙 X _* ℱ)) (op { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }) ≫ stalkPushforward C (𝟙 X) ℱ x = colimit.ι (((whiskeringLeft (OpenNhds (↑(𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion (↑(𝟙 X) x)).op).obj (𝟙 X _* ℱ)) (op { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }) ≫ (stalkFunctor C x).map (Pushforward.id ℱ).hom erw [colimit.ι_map_assoc] case w.h.mk.mk C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X x : ↑X carrier✝ : Set ↑X is_open'✝ : IsOpen carrier✝ property✝ : ↑(𝟙 X) x ∈ { carrier := carrier✝, is_open' := is_open'✝ } ⊢ (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso (𝟙 X) x).inv) ℱ).app (op { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }) ≫ colimit.ι ((OpenNhds.map (𝟙 X) x).op ⋙ ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj ℱ) (op { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }) ≫ colimit.pre (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj ℱ) (OpenNhds.map (𝟙 X) x).op = colimit.ι (((whiskeringLeft (OpenNhds (↑(𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion (↑(𝟙 X) x)).op).obj (𝟙 X _* ℱ)) (op { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }) ≫ (stalkFunctor C x).map (Pushforward.id ℱ).hom simp [stalkFunctor, stalkPushforward] Qed
TopCat.Presheaf.stalkPushforward.comp ** C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X f : X ⟶ Y g : Y ⟶ Z x : ↑X ⊢ stalkPushforward C (f ≫ g) ℱ x = stalkPushforward C g (f _* ℱ) (↑f x) ≫ stalkPushforward C f ℱ x ext U case w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X f : X ⟶ Y g : Y ⟶ Z x : ↑X U : (OpenNhds (↑(f ≫ g) x))ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑(f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj (OpenNhds.inclusion (↑(f ≫ g) x)).op).obj ((f ≫ g) _* ℱ)) U ≫ stalkPushforward C (f ≫ g) ℱ x = colimit.ι (((whiskeringLeft (OpenNhds (↑(f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj (OpenNhds.inclusion (↑(f ≫ g) x)).op).obj ((f ≫ g) _* ℱ)) U ≫ stalkPushforward C g (f _* ℱ) (↑f x) ≫ stalkPushforward C f ℱ x rcases U with ⟨⟨_, _⟩, _⟩ case w.mk.mk.mk C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat ℱ : Presheaf C X f : X ⟶ Y g : Y ⟶ Z x : ↑X carrier✝ : Set ↑Z is_open'✝ : IsOpen carrier✝ property✝ : ↑(f ≫ g) x ∈ { carrier := carrier✝, is_open' := is_open'✝ } ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑(f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj (OpenNhds.inclusion (↑(f ≫ g) x)).op).obj ((f ≫ g) _* ℱ)) { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫ stalkPushforward C (f ≫ g) ℱ x = colimit.ι (((whiskeringLeft (OpenNhds (↑(f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj (OpenNhds.inclusion (↑(f ≫ g) x)).op).obj ((f ≫ g) _* ℱ)) { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫ stalkPushforward C g (f _* ℱ) (↑f x) ≫ stalkPushforward C f ℱ x simp [stalkFunctor, stalkPushforward] Qed
TopCat.Presheaf.stalkPushforward.stalkPushforward_iso_of_openEmbedding C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X ⊢ IsIso (stalkPushforward C f F x) haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x) C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) ⊢ IsIso (stalkPushforward C f F x) convert IsIso.of_iso ((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op ((OpenNhds.inclusion (f x)).op ⋙ f _* F) : _).symm ≪≫ colim.mapIso _) ** case h.e'_5.h C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) ⊢ stalkPushforward C f F x = ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F)).symm ≪≫ colim.mapIso ?convert_2).hom case convert_2 C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) ⊢ (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ⋙ (OpenNhds.inclusion (↑f x)).op ⋙ f _* F ≅ ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F swap case convert_2 C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) ⊢ (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ⋙ (OpenNhds.inclusion (↑f x)).op ⋙ f _* F ≅ ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F fapply NatIso.ofComponents case convert_2.app C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) ⊢ (X_1 : (OpenNhds x)ᵒᵖ) → ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ⋙ (OpenNhds.inclusion (↑f x)).op ⋙ f _* F).obj X_1 ≅ (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F).obj X_1 intro U case convert_2.app C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) U : (OpenNhds x)ᵒᵖ ⊢ ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ⋙ (OpenNhds.inclusion (↑f x)).op ⋙ f _* F).obj U ≅ (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F).obj U refine' F.mapIso (eqToIso _) case convert_2.app C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) U : (OpenNhds x)ᵒᵖ ⊢ (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U dsimp only [Functor.op] case convert_2.app C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) U : (OpenNhds x)ᵒᵖ ⊢ op ((Opens.map f).obj (op ((OpenNhds.inclusion (↑f x)).obj (op ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).obj U.unop)).unop)).unop) = op ((OpenNhds.inclusion x).obj U.unop) exact congr_arg op (Opens.ext <\| Set.preimage_image_eq (unop U).1.1 hf.inj) case convert_2.naturality C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) ⊢ autoParam (∀ {X_1 Y_1 : (OpenNhds x)ᵒᵖ} (f_1 : X_1 ⟶ Y_1), ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ⋙ (OpenNhds.inclusion (↑f x)).op ⋙ f _* F).map f_1 ≫ (F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj Y_1)) = (OpenNhds.inclusion x).op.obj Y_1))).hom = (F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj X_1)) = (OpenNhds.inclusion x).op.obj X_1))).hom ≫ (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F).map f_1) _auto✝ intro U V i case convert_2.naturality C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) U V : (OpenNhds x)ᵒᵖ i : U ⟶ V ⊢ ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ⋙ (OpenNhds.inclusion (↑f x)).op ⋙ f _* F).map i ≫ (F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj V)) = (OpenNhds.inclusion x).op.obj V))).hom = (F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U))).hom ≫ (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F).map i erw [← F.map_comp, ← F.map_comp] case convert_2.naturality C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) U V : (OpenNhds x)ᵒᵖ i : U ⟶ V ⊢ F.map ((Opens.map f).op.map ((OpenNhds.inclusion (↑f x)).op.map ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.map i)) ≫ (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj V)) = (OpenNhds.inclusion x).op.obj V)).hom) = F.map ((eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U)).hom ≫ (OpenNhds.inclusion x).op.map i) congr 1 case h.e'_5.h C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) ⊢ stalkPushforward C f F x = ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F)).symm ≪≫ colim.mapIso (NatIso.ofComponents fun U => F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U)))).hom ext U case h.e'_5.h.w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f x)).op).obj (f _* F)) U ≫ stalkPushforward C f F x = colimit.ι (((whiskeringLeft (OpenNhds (↑f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f x)).op).obj (f _* F)) U ≫ ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F)).symm ≪≫ colim.mapIso (NatIso.ofComponents fun U => F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U)))).hom rw [← Iso.comp_inv_eq] case h.e'_5.h.w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ (colimit.ι (((whiskeringLeft (OpenNhds (↑f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f x)).op).obj (f _* F)) U ≫ stalkPushforward C f F x) ≫ ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F)).symm ≪≫ colim.mapIso (NatIso.ofComponents fun U => F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U)))).inv = colimit.ι (((whiskeringLeft (OpenNhds (↑f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f x)).op).obj (f _* F)) U erw [colimit.ι_map_assoc] case h.e'_5.h.w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ ((whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F).app U ≫ colimit.ι ((OpenNhds.map f x).op ⋙ ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U ≫ colimit.pre (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) (OpenNhds.map f x).op) ≫ ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F)).symm ≪≫ colim.mapIso (NatIso.ofComponents fun U => F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U)))).inv = colimit.ι (((whiskeringLeft (OpenNhds (↑f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f x)).op).obj (f _* F)) U rw [colimit.ι_pre, Category.assoc] case h.e'_5.h.w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F).app U ≫ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) ((OpenNhds.map f x).op.obj U) ≫ ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F)).symm ≪≫ colim.mapIso (NatIso.ofComponents fun U => F.mapIso (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj U)) = (OpenNhds.inclusion x).op.obj U)))).inv = colimit.ι (((whiskeringLeft (OpenNhds (↑f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f x)).op).obj (f _* F)) U erw [colimit.ι_map_assoc, colimit.ι_pre, ← F.map_comp_assoc] case h.e'_5.h.w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ F.map ((NatTrans.op (OpenNhds.inclusionMapIso f x).inv).app U ≫ (eqToIso (_ : (Opens.map f).op.obj ((OpenNhds.inclusion (↑f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj ((OpenNhds.map f x).op.obj U))) = (OpenNhds.inclusion x).op.obj ((OpenNhds.map f x).op.obj U))).inv) ≫ colimit.ι ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj ((OpenNhds.map f x).op.obj U)) = colimit.ι (((whiskeringLeft (OpenNhds (↑f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f x)).op).obj (f _* F)) U ** apply colimit.w ((OpenNhds.inclusion (f x)).op ⋙ f _* F) _ ** C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ U ⟶ (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).op.obj ((OpenNhds.map f x).op.obj U) dsimp only [Functor.op] C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ U ⟶ op ((IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).obj (op ((OpenNhds.map f x).obj U.unop)).unop) refine' ((homOfLE _).op : op (unop U) ⟶ _) C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y hf : OpenEmbedding ↑f F : Presheaf C X x : ↑X this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x) e_3✝ : stalk (f _* F) (↑f x) = colimit ((OpenNhds.inclusion (↑f x)).op ⋙ f _* F) e_4✝ : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U : (OpenNhds (↑f x))ᵒᵖ ⊢ (IsOpenMap.functorNhds (_ : IsOpenMap ↑f) x).obj (op ((OpenNhds.map f x).obj U.unop)).unop ≤ U.unop exact Set.image_preimage_subset _ _ Qed
TopCat.Presheaf.stalkSpecializes_refl C✝ : Type u inst✝³ : Category.{v, u} C✝ inst✝² : HasColimits C✝ X✝ Y Z : TopCat C : Type u_1 inst✝¹ : Category.{u_2, u_1} C inst✝ : HasColimits C X : TopCat F : Presheaf C X x : ↑X ⊢ stalkSpecializes F (_ : x ⤳ x) = 𝟙 (stalk F x) ext case ih C✝ : Type u inst✝³ : Category.{v, u} C✝ inst✝² : HasColimits C✝ X✝ Y Z : TopCat C : Type u_1 inst✝¹ : Category.{u_2, u_1} C inst✝ : HasColimits C X : TopCat F : Presheaf C X x : ↑X U✝ : Opens ↑X hxU✝ : x ∈ U✝ ⊢ germ F { val := x, property := hxU✝ } ≫ stalkSpecializes F (_ : x ⤳ x) = germ F { val := x, property := hxU✝ } ≫ 𝟙 (stalk F x) simp ** Qed
TopCat.Presheaf.stalkSpecializes_comp C✝ : Type u inst✝³ : Category.{v, u} C✝ inst✝² : HasColimits C✝ X✝ Y Z : TopCat C : Type u_1 inst✝¹ : Category.{u_2, u_1} C inst✝ : HasColimits C X : TopCat F : Presheaf C X x y z : ↑X h : x ⤳ y h' : y ⤳ z ⊢ stalkSpecializes F h' ≫ stalkSpecializes F h = stalkSpecializes F (_ : x ⤳ z) ext case ih C✝ : Type u inst✝³ : Category.{v, u} C✝ inst✝² : HasColimits C✝ X✝ Y Z : TopCat C : Type u_1 inst✝¹ : Category.{u_2, u_1} C inst✝ : HasColimits C X : TopCat F : Presheaf C X x y z : ↑X h : x ⤳ y h' : y ⤳ z U✝ : Opens ↑X hxU✝ : z ∈ U✝ ⊢ germ F { val := z, property := hxU✝ } ≫ stalkSpecializes F h' ≫ stalkSpecializes F h = germ F { val := z, property := hxU✝ } ≫ stalkSpecializes F (_ : x ⤳ z) simp ** Qed
TopCat.Presheaf.stalkSpecializes_stalkFunctor_map C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat F G : Presheaf C X f : F ⟶ G x y : ↑X h : x ⤳ y ⊢ stalkSpecializes F h ≫ (stalkFunctor C x).map f = (stalkFunctor C y).map f ≫ stalkSpecializes G h ext case w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat F G : Presheaf C X f : F ⟶ G x y : ↑X h : x ⤳ y j✝ : (OpenNhds y)ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F) j✝ ≫ stalkSpecializes F h ≫ (stalkFunctor C x).map f = colimit.ι (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F) j✝ ≫ (stalkFunctor C y).map f ≫ stalkSpecializes G h delta stalkFunctor case w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat F G : Presheaf C X f : F ⟶ G x y : ↑X h : x ⤳ y j✝ : (OpenNhds y)ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F) j✝ ≫ stalkSpecializes F h ≫ ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op ⋙ colim).map f = colimit.ι (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F) j✝ ≫ ((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op ⋙ colim).map f ≫ stalkSpecializes G h simpa [stalkSpecializes] using by rfl C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat F G : Presheaf C X f : F ⟶ G x y : ↑X h : x ⤳ y j✝ : (OpenNhds y)ᵒᵖ ⊢ f.app (op j✝.unop.obj) ≫ colimit.ι ((OpenNhds.inclusion x).op ⋙ G) (op { obj := j✝.unop.obj, property := (_ : x ∈ j✝.unop.obj.carrier) }) = f.app (op ((OpenNhds.inclusion y).obj j✝.unop)) ≫ colimit.ι ((OpenNhds.inclusion x).op ⋙ G) (op { obj := j✝.unop.obj, property := (_ : x ∈ j✝.unop.obj.carrier) }) rfl ** Qed
TopCat.Presheaf.stalkSpecializes_stalkPushforward ** C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y F : Presheaf C X x y : ↑X h : x ⤳ y ⊢ stalkSpecializes (f _* F) (_ : ↑f x ⤳ ↑f y) ≫ stalkPushforward C f F x = stalkPushforward C f F y ≫ stalkSpecializes F h ext case w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y F : Presheaf C X x y : ↑X h : x ⤳ y j✝ : (OpenNhds (↑f y))ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑f y))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f y)).op).obj (f _* F)) j✝ ≫ stalkSpecializes (f _* F) (_ : ↑f x ⤳ ↑f y) ≫ stalkPushforward C f F x = colimit.ι (((whiskeringLeft (OpenNhds (↑f y))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f y)).op).obj (f _* F)) j✝ ≫ stalkPushforward C f F y ≫ stalkSpecializes F h delta stalkPushforward case w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y F : Presheaf C X x y : ↑X h : x ⤳ y j✝ : (OpenNhds (↑f y))ᵒᵖ ⊢ colimit.ι (((whiskeringLeft (OpenNhds (↑f y))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f y)).op).obj (f _* F)) j✝ ≫ stalkSpecializes (f _* F) (_ : ↑f x ⤳ ↑f y) ≫ colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F) ≫ colimit.pre (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) (OpenNhds.map f x).op = colimit.ι (((whiskeringLeft (OpenNhds (↑f y))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (↑f y)).op).obj (f _* F)) j✝ ≫ (colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f y).inv) F) ≫ colimit.pre (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F) (OpenNhds.map f y).op) ≫ stalkSpecializes F h simp only [stalkSpecializes, colimit.ι_desc_assoc, colimit.ι_map_assoc, colimit.ι_pre, Category.assoc, colimit.pre_desc, colimit.ι_desc] case w C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y Z : TopCat f : X ⟶ Y F : Presheaf C X x y : ↑X h : x ⤳ y j✝ : (OpenNhds (↑f y))ᵒᵖ ⊢ (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F).app (op { obj := j✝.unop.obj, property := (_ : ↑f x ∈ j✝.unop.obj.carrier) }) ≫ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) ((OpenNhds.map f x).op.obj (op { obj := j✝.unop.obj, property := (_ : ↑f x ∈ j✝.unop.obj.carrier) })) = (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f y).inv) F).app j✝ ≫ (Cocone.whisker (OpenNhds.map f y).op { pt := colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F), ι := NatTrans.mk fun U => colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }) }).ι.app j✝ rfl Qed
TopCat.Presheaf.germ_ext C : Type u inst✝² : Category.{v, u} C inst✝¹ : HasColimits C X Y Z : TopCat inst✝ : ConcreteCategory C F : Presheaf C X U V : Opens ↑X x : ↑X hxU : x ∈ U hxV : x ∈ V W : Opens ↑X hxW : x ∈ W iWU : W ⟶ U iWV : W ⟶ V sU : (forget C).obj (F.obj (op U)) sV : (forget C).obj (F.obj (op V)) ih : ↑(F.map iWU.op) sU = ↑(F.map iWV.op) sV ⊢ ↑(germ F { val := x, property := hxU }) sU = ↑(germ F { val := x, property := hxV }) sV erw [← F.germ_res iWU ⟨x, hxW⟩, ← F.germ_res iWV ⟨x, hxW⟩, comp_apply, comp_apply, ih] ** Qed
TopCat.Presheaf.germ_exist C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X x : ↑X t : (forget C).obj (stalk F x) ⊢ ∃ U m s, ↑(germ F { val := x, property := m }) s = t obtain ⟨U, s, e⟩ := Types.jointly_surjective.{v, v} _ (isColimitOfPreserves (forget C) (colimit.isColimit _)) t case intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X x : ↑X t : (forget C).obj (stalk F x) U : (OpenNhds x)ᵒᵖ s : (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙ forget C).obj U e : ((forget C).mapCocone (colimit.cocone (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F))).ι.app U s = t ⊢ ∃ U m s, ↑(germ F { val := x, property := m }) s = t revert s e case intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X x : ↑X t : (forget C).obj (stalk F x) U : (OpenNhds x)ᵒᵖ ⊢ ∀ (s : (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙ forget C).obj U), ((forget C).mapCocone (colimit.cocone (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F))).ι.app U s = t → ∃ U m s, ↑(germ F { val := x, property := m }) s = t induction U with \| h U => ?_ case intro.intro.h C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X x : ↑X t : (forget C).obj (stalk F x) U : OpenNhds x ⊢ ∀ (s : (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙ forget C).obj (op U)), ((forget C).mapCocone (colimit.cocone (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F))).ι.app (op U) s = t → ∃ U m s, ↑(germ F { val := x, property := m }) s = t cases' U with V m case intro.intro.h.mk C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X x : ↑X t : (forget C).obj (stalk F x) V : Opens ↑X m : x ∈ V ⊢ ∀ (s : (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙ forget C).obj (op { obj := V, property := m })), ((forget C).mapCocone (colimit.cocone (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F))).ι.app (op { obj := V, property := m }) s = t → ∃ U m s, ↑(germ F { val := x, property := m }) s = t intro s e case intro.intro.h.mk C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X x : ↑X t : (forget C).obj (stalk F x) V : Opens ↑X m : x ∈ V s : (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙ forget C).obj (op { obj := V, property := m }) e : ((forget C).mapCocone (colimit.cocone (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F))).ι.app (op { obj := V, property := m }) s = t ⊢ ∃ U m s, ↑(germ F { val := x, property := m }) s = t exact ⟨V, m, s, e⟩ ** Qed
TopCat.Presheaf.germ_eq C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X U V : Opens ↑X x : ↑X mU : x ∈ U mV : x ∈ V s : (forget C).obj (F.obj (op U)) t : (forget C).obj (F.obj (op V)) h : ↑(germ F { val := x, property := mU }) s = ↑(germ F { val := x, property := mV }) t ⊢ ∃ W _m iU iV, ↑(F.map iU.op) s = ↑(F.map iV.op) t obtain ⟨W, iU, iV, e⟩ := (Types.FilteredColimit.isColimit_eq_iff.{v, v} _ (isColimitOfPreserves _ (colimit.isColimit ((OpenNhds.inclusion x).op ⋙ F)))).mp h case intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F : Presheaf C X U V : Opens ↑X x : ↑X mU : x ∈ U mV : x ∈ V s : (forget C).obj (F.obj (op U)) t : (forget C).obj (F.obj (op V)) h : ↑(germ F { val := x, property := mU }) s = ↑(germ F { val := x, property := mV }) t W : (OpenNhds x)ᵒᵖ iU : op { obj := U, property := (_ : ↑{ val := x, property := mU } ∈ U) } ⟶ W iV : op { obj := V, property := (_ : ↑{ val := x, property := mV } ∈ V) } ⟶ W e : (((OpenNhds.inclusion x).op ⋙ F) ⋙ forget C).map iU s = (((OpenNhds.inclusion x).op ⋙ F) ⋙ forget C).map iV t ⊢ ∃ W _m iU iV, ↑(F.map iU.op) s = ↑(F.map iV.op) t exact ⟨(unop W).1, (unop W).2, iU.unop, iV.unop, e⟩ ** Qed
TopCat.Presheaf.stalkFunctor_map_injective_of_app_injective C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X s t : (forget C).obj ((stalkFunctor C x).obj F) hst : ↑((stalkFunctor C x).map f) s = ↑((stalkFunctor C x).map f) t ⊢ s = t rcases germ_exist F x s with ⟨U₁, hxU₁, s, rfl⟩ case intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X t : (forget C).obj ((stalkFunctor C x).obj F) U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) hst : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) t ⊢ ↑(germ F { val := x, property := hxU₁ }) s = t rcases germ_exist F x t with ⟨U₂, hxU₂, t, rfl⟩ case intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) ⊢ ↑(germ F { val := x, property := hxU₁ }) s = ↑(germ F { val := x, property := hxU₂ }) t erw [stalkFunctor_map_germ_apply _ ⟨x, _⟩] at hst case intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst✝ : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) ⊢ ↑(germ F { val := x, property := hxU₁ }) s = ↑(germ F { val := x, property := hxU₂ }) t erw [stalkFunctor_map_germ_apply _ ⟨x, _⟩] at hst case intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst✝¹ : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst✝ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₂ }).op ⋙ G) (op { obj := U₂, property := (_ : ↑{ val := x, property := hxU₂ } ∈ U₂) })) (↑(f.app (op U₂)) t) ⊢ ↑(germ F { val := x, property := hxU₁ }) s = ↑(germ F { val := x, property := hxU₂ }) t obtain ⟨W, hxW, iWU₁, iWU₂, heq⟩ := G.germ_eq x hxU₁ hxU₂ _ _ hst case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst✝¹ : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst✝ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₂ }).op ⋙ G) (op { obj := U₂, property := (_ : ↑{ val := x, property := hxU₂ } ∈ U₂) })) (↑(f.app (op U₂)) t) W : Opens ↑X hxW : x ∈ W iWU₁ : W ⟶ U₁ iWU₂ : W ⟶ U₂ heq : ↑(G.map iWU₁.op) (↑(f.app (op U₁)) s) = ↑(G.map iWU₂.op) (↑(f.app (op U₂)) t) ⊢ ↑(germ F { val := x, property := hxU₁ }) s = ↑(germ F { val := x, property := hxU₂ }) t rw [← comp_apply, ← comp_apply, ← f.naturality, ← f.naturality, comp_apply, comp_apply] at heq case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst✝¹ : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst✝ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₂ }).op ⋙ G) (op { obj := U₂, property := (_ : ↑{ val := x, property := hxU₂ } ∈ U₂) })) (↑(f.app (op U₂)) t) W : Opens ↑X hxW : x ∈ W iWU₁ : W ⟶ U₁ iWU₂ : W ⟶ U₂ heq : ↑(f.app (op W)) (↑(F.map iWU₁.op) s) = ↑(f.app (op W)) (↑(F.map iWU₂.op) t) ⊢ ↑(germ F { val := x, property := hxU₁ }) s = ↑(germ F { val := x, property := hxU₂ }) t replace heq := h W heq case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst✝¹ : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst✝ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₂ }).op ⋙ G) (op { obj := U₂, property := (_ : ↑{ val := x, property := hxU₂ } ∈ U₂) })) (↑(f.app (op U₂)) t) W : Opens ↑X hxW : x ∈ W iWU₁ : W ⟶ U₁ iWU₂ : W ⟶ U₂ heq : ↑(F.map iWU₁.op) s = ↑(F.map iWU₂.op) t ⊢ ↑(germ F { val := x, property := hxU₁ }) s = ↑(germ F { val := x, property := hxU₂ }) t convert congr_arg (F.germ ⟨x, hxW⟩) heq using 1 case h.e'_2.h C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst✝¹ : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst✝ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₂ }).op ⋙ G) (op { obj := U₂, property := (_ : ↑{ val := x, property := hxU₂ } ∈ U₂) })) (↑(f.app (op U₂)) t) W : Opens ↑X hxW : x ∈ W iWU₁ : W ⟶ U₁ iWU₂ : W ⟶ U₂ heq : ↑(F.map iWU₁.op) s = ↑(F.map iWU₂.op) t e_1✝ : (forget C).obj ((stalkFunctor C x).obj F) = (forget C).obj (stalk F ↑{ val := x, property := hxW }) ⊢ ↑(germ F { val := x, property := hxU₁ }) s = ↑(germ F { val := x, property := hxW }) (↑(F.map iWU₁.op) s) case h.e'_3.h C : Type u inst✝³ : Category.{v, u} C inst✝² : HasColimits C X Y Z : TopCat inst✝¹ : ConcreteCategory C inst✝ : PreservesFilteredColimits (forget C) F G : Presheaf C X f : F ⟶ G h : ∀ (U : Opens ↑X), Function.Injective ↑(f.app (op U)) x : ↑X U₁ : Opens ↑X hxU₁ : x ∈ U₁ s : (forget C).obj (F.obj (op U₁)) U₂ : Opens ↑X hxU₂ : x ∈ U₂ t : (forget C).obj (F.obj (op U₂)) hst✝¹ : ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₁ }) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst✝ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑((stalkFunctor C x).map f) (↑(germ F { val := x, property := hxU₂ }) t) hst : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₁ }).op ⋙ G) (op { obj := U₁, property := (_ : ↑{ val := x, property := hxU₁ } ∈ U₁) })) (↑(f.app (op U₁)) s) = ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := x, property := hxU₂ }).op ⋙ G) (op { obj := U₂, property := (_ : ↑{ val := x, property := hxU₂ } ∈ U₂) })) (↑(f.app (op U₂)) t) W : Opens ↑X hxW : x ∈ W iWU₁ : W ⟶ U₁ iWU₂ : W ⟶ U₂ heq : ↑(F.map iWU₁.op) s = ↑(F.map iWU₂.op) t e_1✝ : (forget C).obj ((stalkFunctor C x).obj F) = (forget C).obj (stalk F ↑{ val := x, property := hxW }) ⊢ ↑(germ F { val := x, property := hxU₂ }) t = ↑(germ F { val := x, property := hxW }) (↑(F.map iWU₂.op) t) exacts [(F.germ_res_apply iWU₁ ⟨x, hxW⟩ s).symm, (F.germ_res_apply iWU₂ ⟨x, hxW⟩ t).symm] ** Qed
TopCat.Presheaf.section_ext C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X U : Opens ↑X s t : (forget C).obj (F.val.obj (op U)) h : ∀ (x : { x // x ∈ U }), ↑(germ (Sheaf.presheaf F) x) s = ↑(germ (Sheaf.presheaf F) x) t ⊢ s = t choose V m i₁ i₂ heq using fun x : U => F.presheaf.germ_eq x.1 x.2 x.2 s t (h x) C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X U : Opens ↑X s t : (forget C).obj (F.val.obj (op U)) h : ∀ (x : { x // x ∈ U }), ↑(germ (Sheaf.presheaf F) x) s = ↑(germ (Sheaf.presheaf F) x) t V : { x // x ∈ U } → Opens ↑X m : ∀ (x : { x // x ∈ U }), ↑x ∈ V x i₁ i₂ : (x : { x // x ∈ U }) → V x ⟶ U heq : ∀ (x : { x // x ∈ U }), ↑((Sheaf.presheaf F).map (i₁ x).op) s = ↑((Sheaf.presheaf F).map (i₂ x).op) t ⊢ s = t apply F.eq_of_locally_eq' V U i₁ case hcover C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X U : Opens ↑X s t : (forget C).obj (F.val.obj (op U)) h : ∀ (x : { x // x ∈ U }), ↑(germ (Sheaf.presheaf F) x) s = ↑(germ (Sheaf.presheaf F) x) t V : { x // x ∈ U } → Opens ↑X m : ∀ (x : { x // x ∈ U }), ↑x ∈ V x i₁ i₂ : (x : { x // x ∈ U }) → V x ⟶ U heq : ∀ (x : { x // x ∈ U }), ↑((Sheaf.presheaf F).map (i₁ x).op) s = ↑((Sheaf.presheaf F).map (i₂ x).op) t ⊢ U ≤ iSup V intro x hxU case hcover C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X U : Opens ↑X s t : (forget C).obj (F.val.obj (op U)) h : ∀ (x : { x // x ∈ U }), ↑(germ (Sheaf.presheaf F) x) s = ↑(germ (Sheaf.presheaf F) x) t V : { x // x ∈ U } → Opens ↑X m : ∀ (x : { x // x ∈ U }), ↑x ∈ V x i₁ i₂ : (x : { x // x ∈ U }) → V x ⟶ U heq : ∀ (x : { x // x ∈ U }), ↑((Sheaf.presheaf F).map (i₁ x).op) s = ↑((Sheaf.presheaf F).map (i₂ x).op) t x : ↑X hxU : x ∈ ↑U ⊢ x ∈ ↑(iSup V) simp only [Opens.coe_iSup, Set.mem_iUnion, SetLike.mem_coe] case hcover C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X U : Opens ↑X s t : (forget C).obj (F.val.obj (op U)) h : ∀ (x : { x // x ∈ U }), ↑(germ (Sheaf.presheaf F) x) s = ↑(germ (Sheaf.presheaf F) x) t V : { x // x ∈ U } → Opens ↑X m : ∀ (x : { x // x ∈ U }), ↑x ∈ V x i₁ i₂ : (x : { x // x ∈ U }) → V x ⟶ U heq : ∀ (x : { x // x ∈ U }), ↑((Sheaf.presheaf F).map (i₁ x).op) s = ↑((Sheaf.presheaf F).map (i₂ x).op) t x : ↑X hxU : x ∈ ↑U ⊢ ∃ i, x ∈ V i exact ⟨⟨x, hxU⟩, m ⟨x, hxU⟩⟩ case h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X U : Opens ↑X s t : (forget C).obj (F.val.obj (op U)) h : ∀ (x : { x // x ∈ U }), ↑(germ (Sheaf.presheaf F) x) s = ↑(germ (Sheaf.presheaf F) x) t V : { x // x ∈ U } → Opens ↑X m : ∀ (x : { x // x ∈ U }), ↑x ∈ V x i₁ i₂ : (x : { x // x ∈ U }) → V x ⟶ U heq : ∀ (x : { x // x ∈ U }), ↑((Sheaf.presheaf F).map (i₁ x).op) s = ↑((Sheaf.presheaf F).map (i₂ x).op) t ⊢ ∀ (i : { x // x ∈ U }), ↑(F.val.map (i₁ i).op) s = ↑(F.val.map (i₁ i).op) t intro x case h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X U : Opens ↑X s t : (forget C).obj (F.val.obj (op U)) h : ∀ (x : { x // x ∈ U }), ↑(germ (Sheaf.presheaf F) x) s = ↑(germ (Sheaf.presheaf F) x) t V : { x // x ∈ U } → Opens ↑X m : ∀ (x : { x // x ∈ U }), ↑x ∈ V x i₁ i₂ : (x : { x // x ∈ U }) → V x ⟶ U heq : ∀ (x : { x // x ∈ U }), ↑((Sheaf.presheaf F).map (i₁ x).op) s = ↑((Sheaf.presheaf F).map (i₂ x).op) t x : { x // x ∈ U } ⊢ ↑(F.val.map (i₁ x).op) s = ↑(F.val.map (i₁ x).op) t rw [heq, Subsingleton.elim (i₁ x) (i₂ x)] ** Qed
TopCat.Presheaf.app_injective_of_stalkFunctor_map_injective C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F : Sheaf C X G : Presheaf C X f : F.val ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f) s t : (forget C).obj (F.val.obj (op U)) hst : ↑(f.app (op U)) s = ↑(f.app (op U)) t x : { x // x ∈ U } ⊢ ↑((stalkFunctor C ↑x).map f) (↑(germ (Sheaf.presheaf F) x) s) = ↑((stalkFunctor C ↑x).map f) (↑(germ (Sheaf.presheaf F) x) t) erw [stalkFunctor_map_germ_apply, stalkFunctor_map_germ_apply, hst] ** Qed
TopCat.Presheaf.app_surjective_of_injective_of_locally_surjective C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t ⊢ Function.Surjective ↑(f.val.app (op U)) intro t C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) ⊢ ∃ a, ↑(f.val.app (op U)) a = t choose V mV iVU sf heq using hsurj t C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t ⊢ ∃ a, ↑(f.val.app (op U)) a = t have V_cover : U ≤ iSup V := by intro x hxU simp only [Opens.coe_iSup, Set.mem_iUnion, SetLike.mem_coe] exact ⟨⟨x, hxU⟩, mV ⟨x, hxU⟩⟩ C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V ⊢ IsCompatible F.val V sf intro x y C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V x y : { x // x ∈ U } ⊢ ↑(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x) = ↑(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y) apply section_ext case h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V x y : { x // x ∈ U } ⊢ ∀ (x_1 : { x_1 // x_1 ∈ V x ⊓ V y }), ↑(germ (Sheaf.presheaf F) x_1) (↑(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x)) = ↑(germ (Sheaf.presheaf F) x_1) (↑(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y)) intro z case h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V x y : { x // x ∈ U } z : { x_1 // x_1 ∈ V x ⊓ V y } ⊢ ↑(germ (Sheaf.presheaf F) z) (↑(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x)) = ↑(germ (Sheaf.presheaf F) z) (↑(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y)) apply hinj ⟨z, (iVU x).le ((inf_le_left : V x ⊓ V y ≤ V x) z.2)⟩ case h.a C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V x y : { x // x ∈ U } z : { x_1 // x_1 ∈ V x ⊓ V y } ⊢ ↑((stalkFunctor C ↑{ val := ↑z, property := (_ : ↑z ∈ ↑U) }).map f.val) (↑(germ (Sheaf.presheaf F) z) (↑(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x))) = ↑((stalkFunctor C ↑{ val := ↑z, property := (_ : ↑z ∈ ↑U) }).map f.val) (↑(germ (Sheaf.presheaf F) z) (↑(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y))) dsimp only case h.a C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V x y : { x // x ∈ U } z : { x_1 // x_1 ∈ V x ⊓ V y } ⊢ ↑((stalkFunctor C ↑z).map f.val) (↑(germ (Sheaf.presheaf F) z) (↑(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x))) = ↑((stalkFunctor C ↑z).map f.val) (↑(germ (Sheaf.presheaf F) z) (↑(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y))) erw [stalkFunctor_map_germ_apply, stalkFunctor_map_germ_apply] case h.a C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V x y : { x // x ∈ U } z : { x_1 // x_1 ∈ V x ⊓ V y } ⊢ ↑(colimit.ι ((OpenNhds.inclusion ↑z).op ⋙ G.val) (op { obj := V x ⊓ V y, property := (_ : ↑z ∈ V x ⊓ V y) })) (↑(f.val.app (op (V x ⊓ V y))) (↑(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x))) = ↑(colimit.ι ((OpenNhds.inclusion ↑z).op ⋙ G.val) (op { obj := V x ⊓ V y, property := (_ : ↑z ∈ V x ⊓ V y) })) (↑(f.val.app (op (V x ⊓ V y))) (↑(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y))) simp_rw [← comp_apply, f.1.naturality, comp_apply, heq, ← comp_apply, ← G.1.map_comp] case h.a C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V x y : { x // x ∈ U } z : { x_1 // x_1 ∈ V x ⊓ V y } ⊢ ↑(colimit.ι ((OpenNhds.inclusion ↑z).op ⋙ G.val) (op { obj := V x ⊓ V y, property := (_ : ↑z ∈ V x ⊓ V y) })) (↑(G.val.map ((iVU x).op ≫ (Opens.infLELeft (V x) (V y)).op)) t) = ↑(colimit.ι ((OpenNhds.inclusion ↑z).op ⋙ G.val) (op { obj := V x ⊓ V y, property := (_ : ↑z ∈ V x ⊓ V y) })) (↑(G.val.map ((iVU y).op ≫ (Opens.infLERight (V x) (V y)).op)) t) rfl C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t ⊢ U ≤ iSup V intro x hxU C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t x : ↑X hxU : x ∈ ↑U ⊢ x ∈ ↑(iSup V) simp only [Opens.coe_iSup, Set.mem_iUnion, SetLike.mem_coe] C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t x : ↑X hxU : x ∈ ↑U ⊢ ∃ i, x ∈ V i exact ⟨⟨x, hxU⟩, mV ⟨x, hxU⟩⟩ C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V this : IsCompatible F.val V sf ⊢ ∃ a, ↑(f.val.app (op U)) a = t obtain ⟨s, s_spec, -⟩ := F.existsUnique_gluing' V U iVU V_cover sf this case intro.intro C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V this : IsCompatible F.val V sf s : (forget C).obj (F.val.obj (op U)) s_spec : ∀ (i : { x // x ∈ U }), ↑(F.val.map (iVU i).op) s = sf i ⊢ ∃ a, ↑(f.val.app (op U)) a = t use s case h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V this : IsCompatible F.val V sf s : (forget C).obj (F.val.obj (op U)) s_spec : ∀ (i : { x // x ∈ U }), ↑(F.val.map (iVU i).op) s = sf i ⊢ ↑(f.val.app (op U)) s = t apply G.eq_of_locally_eq' V U iVU V_cover case h.h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V this : IsCompatible F.val V sf s : (forget C).obj (F.val.obj (op U)) s_spec : ∀ (i : { x // x ∈ U }), ↑(F.val.map (iVU i).op) s = sf i ⊢ ∀ (i : { x // x ∈ U }), ↑(G.val.map (iVU i).op) (↑(f.val.app (op U)) s) = ↑(G.val.map (iVU i).op) t intro x case h.h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X hinj : ∀ (x : { x // x ∈ U }), Function.Injective ↑((stalkFunctor C ↑x).map f.val) hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : { x // x ∈ U }), ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t t : (forget C).obj (G.val.obj (op U)) V : { x // x ∈ U } → Opens ↑X mV : ∀ (x : { x // x ∈ U }), ↑x ∈ V x iVU : (x : { x // x ∈ U }) → V x ⟶ U sf : (x : { x // x ∈ U }) → (forget C).obj (F.val.obj (op (V x))) heq : ∀ (x : { x // x ∈ U }), ↑(f.val.app (op (V x))) (sf x) = ↑(G.val.map (iVU x).op) t V_cover : U ≤ iSup V this : IsCompatible F.val V sf s : (forget C).obj (F.val.obj (op U)) s_spec : ∀ (i : { x // x ∈ U }), ↑(F.val.map (iVU i).op) s = sf i x : { x // x ∈ U } ⊢ ↑(G.val.map (iVU x).op) (↑(f.val.app (op U)) s) = ↑(G.val.map (iVU x).op) t rw [← comp_apply, ← f.1.naturality, comp_apply, s_spec, heq] ** Qed
TopCat.Presheaf.app_surjective_of_stalkFunctor_map_bijective C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) ⊢ Function.Surjective ↑(f.val.app (op U)) refine' app_surjective_of_injective_of_locally_surjective f U (fun x => (h x).1) fun t x => _ C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } ⊢ ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t obtain ⟨s₀, hs₀⟩ := (h x).2 (G.presheaf.germ x t) case intro C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } s₀ : (forget C).obj ((stalkFunctor C ↑x).obj F.val) hs₀ : ↑((stalkFunctor C ↑x).map f.val) s₀ = ↑(germ (Sheaf.presheaf G) x) t ⊢ ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t obtain ⟨V₁, hxV₁, s₁, hs₁⟩ := F.presheaf.germ_exist x.1 s₀ case intro.intro.intro.intro C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } s₀ : (forget C).obj ((stalkFunctor C ↑x).obj F.val) hs₀ : ↑((stalkFunctor C ↑x).map f.val) s₀ = ↑(germ (Sheaf.presheaf G) x) t V₁ : Opens ↑X hxV₁ : ↑x ∈ V₁ s₁ : (forget C).obj ((Sheaf.presheaf F).obj (op V₁)) hs₁ : ↑(germ (Sheaf.presheaf F) { val := ↑x, property := hxV₁ }) s₁ = s₀ ⊢ ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t subst hs₁ case intro.intro.intro.intro C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } V₁ : Opens ↑X hxV₁ : ↑x ∈ V₁ s₁ : (forget C).obj ((Sheaf.presheaf F).obj (op V₁)) hs₀ : ↑((stalkFunctor C ↑x).map f.val) (↑(germ (Sheaf.presheaf F) { val := ↑x, property := hxV₁ }) s₁) = ↑(germ (Sheaf.presheaf G) x) t ⊢ ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t rename' hs₀ => hs₁ case intro.intro.intro.intro C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } V₁ : Opens ↑X hxV₁ : ↑x ∈ V₁ s₁ : (forget C).obj ((Sheaf.presheaf F).obj (op V₁)) hs₁ : ↑((stalkFunctor C ↑x).map f.val) (↑(germ (Sheaf.presheaf F) { val := ↑x, property := hxV₁ }) s₁) = ↑(germ (Sheaf.presheaf G) x) t ⊢ ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t erw [stalkFunctor_map_germ_apply V₁ ⟨x.1, hxV₁⟩ f.1 s₁] at hs₁ case intro.intro.intro.intro C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } V₁ : Opens ↑X hxV₁ : ↑x ∈ V₁ s₁ : (forget C).obj ((Sheaf.presheaf F).obj (op V₁)) hs₁ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := ↑x, property := hxV₁ }).op ⋙ G.val) (op { obj := V₁, property := (_ : ↑{ val := ↑x, property := hxV₁ } ∈ V₁) })) (↑(f.val.app (op V₁)) s₁) = ↑(germ (Sheaf.presheaf G) x) t ⊢ ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t obtain ⟨V₂, hxV₂, iV₂V₁, iV₂U, heq⟩ := G.presheaf.germ_eq x.1 hxV₁ x.2 _ _ hs₁ case intro.intro.intro.intro.intro.intro.intro.intro C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } V₁ : Opens ↑X hxV₁ : ↑x ∈ V₁ s₁ : (forget C).obj ((Sheaf.presheaf F).obj (op V₁)) hs₁ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := ↑x, property := hxV₁ }).op ⋙ G.val) (op { obj := V₁, property := (_ : ↑{ val := ↑x, property := hxV₁ } ∈ V₁) })) (↑(f.val.app (op V₁)) s₁) = ↑(germ (Sheaf.presheaf G) x) t V₂ : Opens ↑X hxV₂ : ↑x ∈ V₂ iV₂V₁ : V₂ ⟶ V₁ iV₂U : V₂ ⟶ U heq : ↑((Sheaf.presheaf G).map iV₂V₁.op) (↑(f.val.app (op V₁)) s₁) = ↑((Sheaf.presheaf G).map iV₂U.op) t ⊢ ∃ V x iVU s, ↑(f.val.app (op V)) s = ↑(G.val.map iVU.op) t use V₂, hxV₂, iV₂U, F.1.map iV₂V₁.op s₁ case h C : Type u inst✝⁶ : Category.{v, u} C inst✝⁵ : HasColimits C X Y Z : TopCat inst✝⁴ : ConcreteCategory C inst✝³ : PreservesFilteredColimits (forget C) inst✝² : HasLimits C inst✝¹ : PreservesLimits (forget C) inst✝ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X h : ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) t : (forget C).obj (G.val.obj (op U)) x : { x // x ∈ U } V₁ : Opens ↑X hxV₁ : ↑x ∈ V₁ s₁ : (forget C).obj ((Sheaf.presheaf F).obj (op V₁)) hs₁ : ↑(colimit.ι ((OpenNhds.inclusion ↑{ val := ↑x, property := hxV₁ }).op ⋙ G.val) (op { obj := V₁, property := (_ : ↑{ val := ↑x, property := hxV₁ } ∈ V₁) })) (↑(f.val.app (op V₁)) s₁) = ↑(germ (Sheaf.presheaf G) x) t V₂ : Opens ↑X hxV₂ : ↑x ∈ V₂ iV₂V₁ : V₂ ⟶ V₁ iV₂U : V₂ ⟶ U heq : ↑((Sheaf.presheaf G).map iV₂V₁.op) (↑(f.val.app (op V₁)) s₁) = ↑((Sheaf.presheaf G).map iV₂U.op) t ⊢ ↑(f.val.app (op V₂)) (↑(F.val.map iV₂V₁.op) s₁) = ↑(G.val.map iV₂U.op) t rw [← comp_apply, f.1.naturality, comp_apply, heq] ** Qed
TopCat.Presheaf.app_isIso_of_stalkFunctor_map_iso C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X inst✝ : ∀ (x : { x // x ∈ U }), IsIso ((stalkFunctor C ↑x).map f.val) ⊢ IsIso (f.val.app (op U)) suffices IsIso ((forget C).map (f.1.app (op U))) by exact isIso_of_reflects_iso (f.1.app (op U)) (forget C) C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X inst✝ : ∀ (x : { x // x ∈ U }), IsIso ((stalkFunctor C ↑x).map f.val) ⊢ IsIso ((forget C).map (f.val.app (op U))) rw [isIso_iff_bijective] C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X inst✝ : ∀ (x : { x // x ∈ U }), IsIso ((stalkFunctor C ↑x).map f.val) ⊢ Function.Bijective ((forget C).map (f.val.app (op U))) apply app_bijective_of_stalkFunctor_map_bijective case h C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X inst✝ : ∀ (x : { x // x ∈ U }), IsIso ((stalkFunctor C ↑x).map f.val) ⊢ ∀ (x : { x // x ∈ U }), Function.Bijective ↑((stalkFunctor C ↑x).map f.val) intro x case h C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X inst✝ : ∀ (x : { x // x ∈ U }), IsIso ((stalkFunctor C ↑x).map f.val) x : { x // x ∈ U } ⊢ Function.Bijective ↑((stalkFunctor C ↑x).map f.val) apply (isIso_iff_bijective _).mp case h C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X inst✝ : ∀ (x : { x // x ∈ U }), IsIso ((stalkFunctor C ↑x).map f.val) x : { x // x ∈ U } ⊢ IsIso ↑((stalkFunctor C ↑x).map f.val) exact Functor.map_isIso (forget C) ((stalkFunctor C x.1).map f.1) C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G U : Opens ↑X inst✝ : ∀ (x : { x // x ∈ U }), IsIso ((stalkFunctor C ↑x).map f.val) this : IsIso ((forget C).map (f.val.app (op U))) ⊢ IsIso (f.val.app (op U)) exact isIso_of_reflects_iso (f.1.app (op U)) (forget C) ** Qed
TopCat.Presheaf.isIso_of_stalkFunctor_map_iso C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G inst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val) ⊢ IsIso f suffices IsIso ((Sheaf.forget C X).map f) by exact isIso_of_fully_faithful (Sheaf.forget C X) f C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G inst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val) ⊢ IsIso ((Sheaf.forget C X).map f) suffices ∀ U : (Opens X)ᵒᵖ, IsIso (f.1.app U) by exact @NatIso.isIso_of_isIso_app _ _ _ _ F.1 G.1 f.1 this C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G inst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val) ⊢ ∀ (U : (Opens ↑X)ᵒᵖ), IsIso (f.val.app U) intro U C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G inst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val) U : (Opens ↑X)ᵒᵖ ⊢ IsIso (f.val.app U) induction U case h C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G inst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val) X✝ : Opens ↑X ⊢ IsIso (f.val.app (op X✝)) apply app_isIso_of_stalkFunctor_map_iso C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G inst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val) this : IsIso ((Sheaf.forget C X).map f) ⊢ IsIso f exact isIso_of_fully_faithful (Sheaf.forget C X) f C : Type u inst✝⁷ : Category.{v, u} C inst✝⁶ : HasColimits C X Y Z : TopCat inst✝⁵ : ConcreteCategory C inst✝⁴ : PreservesFilteredColimits (forget C) inst✝³ : HasLimits C inst✝² : PreservesLimits (forget C) inst✝¹ : ReflectsIsomorphisms (forget C) F G : Sheaf C X f : F ⟶ G inst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val) this : ∀ (U : (Opens ↑X)ᵒᵖ), IsIso (f.val.app U) ⊢ IsIso ((Sheaf.forget C X).map f) exact @NatIso.isIso_of_isIso_app _ _ _ _ F.1 G.1 f.1 this ** Qed
Ordinal.CNFRec_zero C : Ordinal.{u_2} → Sort u_1 b : Ordinal.{u_2} H0 : C 0 H : (o : Ordinal.{u_2}) → o ≠ 0 → C (o % b ^ log b o) → C o ⊢ CNFRec b H0 H 0 = H0 rw [CNFRec, dif_pos rfl] C : Ordinal.{u_2} → Sort u_1 b : Ordinal.{u_2} H0 : C 0 H : (o : Ordinal.{u_2}) → o ≠ 0 → C (o % b ^ log b o) → C o ⊢ Eq.mpr (_ : C 0 = C 0) H0 = H0 rfl ** Qed
Ordinal.CNFRec_pos b o : Ordinal.{u_2} C : Ordinal.{u_2} → Sort u_1 ho : o ≠ 0 H0 : C 0 H : (o : Ordinal.{u_2}) → o ≠ 0 → C (o % b ^ log b o) → C o ⊢ CNFRec b H0 H o = H o ho (CNFRec b H0 H (o % b ^ log b o)) rw [CNFRec, dif_neg ho] ** Qed
Ordinal.zero_CNF o : Ordinal.{u_1} ho : o ≠ 0 ⊢ CNF 0 o = [(0, o)] simp [CNF_ne_zero ho] ** Qed
Ordinal.one_CNF o : Ordinal.{u_1} ho : o ≠ 0 ⊢ CNF 1 o = [(0, o)] simp [CNF_ne_zero ho] ** Qed
Ordinal.CNF_of_le_one b o : Ordinal.{u_1} hb : b ≤ 1 ho : o ≠ 0 ⊢ CNF b o = [(0, o)] rcases le_one_iff.1 hb with (rfl \| rfl) case inl o : Ordinal.{u_1} ho : o ≠ 0 hb : 0 ≤ 1 ⊢ CNF 0 o = [(0, o)] exact zero_CNF ho case inr o : Ordinal.{u_1} ho : o ≠ 0 hb : 1 ≤ 1 ⊢ CNF 1 o = [(0, o)] exact one_CNF ho ** Qed
Ordinal.CNF_of_lt b o : Ordinal.{u_1} ho : o ≠ 0 hb : o < b ⊢ CNF b o = [(0, o)] simp only [CNF_ne_zero ho, log_eq_zero hb, opow_zero, div_one, mod_one, CNF_zero] ** Qed
Ordinal.CNF_foldr ** b o : Ordinal.{u_1} ⊢ foldr (fun p r => b ^ p.1 * p.2 + r) 0 (CNF b 0) = 0 rw [CNF_zero] b o : Ordinal.{u_1} ⊢ foldr (fun p r => b ^ p.1 * p.2 + r) 0 [] = 0 rfl b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : foldr (fun p r => b ^ p.1 * p.2 + r) 0 (CNF b (o % b ^ log b o)) = o % b ^ log b o ⊢ foldr (fun p r => b ^ p.1 * p.2 + r) 0 (CNF b o) = o rw [CNF_ne_zero ho, foldr_cons, IH, div_add_mod] Qed
Ordinal.CNF_fst_le_log b o : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} ⊢ x ∈ CNF b o → x.1 ≤ log b o refine' CNFRec b _ (fun o ho H ↦ _) o case refine'_1 b o : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} ⊢ x ∈ CNF b 0 → x.1 ≤ log b 0 rw [CNF_zero] case refine'_1 b o : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} ⊢ x ∈ [] → x.1 ≤ log b 0 intro contra case refine'_1 b o : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} contra : x ∈ [] ⊢ x.1 ≤ log b 0 contradiction case refine'_2 b o✝ : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o) ⊢ x ∈ CNF b o → x.1 ≤ log b o rw [CNF_ne_zero ho, mem_cons] case refine'_2 b o✝ : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o) ⊢ x = (log b o, o / b ^ log b o) ∨ x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b o rintro (rfl \| h) case refine'_2.inl b o✝ o : Ordinal.{u} ho : o ≠ 0 H : (log b o, o / b ^ log b o) ∈ CNF b (o % b ^ log b o) → (log b o, o / b ^ log b o).1 ≤ log b (o % b ^ log b o) ⊢ (log b o, o / b ^ log b o).1 ≤ log b o exact le_rfl case refine'_2.inr b o✝ : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o) h : x ∈ CNF b (o % b ^ log b o) ⊢ x.1 ≤ log b o exact (H h).trans (log_mono_right _ (mod_opow_log_lt_self b ho).le) ** Qed
Ordinal.CNF_lt_snd b o : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} ⊢ x ∈ CNF b o → 0 < x.2 refine' CNFRec b (by simp) (fun o ho IH ↦ _) o b o✝ : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → 0 < x.2 ⊢ x ∈ CNF b o → 0 < x.2 rw [CNF_ne_zero ho] b o✝ : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → 0 < x.2 ⊢ x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) → 0 < x.2 rintro (h \| ⟨_, h⟩) b o : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} ⊢ x ∈ CNF b 0 → 0 < x.2 simp case head b o✝ o : Ordinal.{u} ho : o ≠ 0 IH : (log b o, o / b ^ log b o) ∈ CNF b (o % b ^ log b o) → 0 < (log b o, o / b ^ log b o).2 ⊢ 0 < (log b o, o / b ^ log b o).2 exact div_opow_log_pos b ho case tail b o✝ : Ordinal.{u} x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → 0 < x.2 h : Mem x (CNF b (o % b ^ log b o)) ⊢ 0 < x.2 exact IH h ** Qed
Ordinal.CNF_snd_lt b o : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} ⊢ x ∈ CNF b o → x.2 < b refine' CNFRec b _ (fun o ho IH ↦ _) o case refine'_1 b o : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} ⊢ x ∈ CNF b 0 → x.2 < b simp only [CNF_zero, not_mem_nil, IsEmpty.forall_iff] case refine'_2 b o✝ : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b ⊢ x ∈ CNF b o → x.2 < b rw [CNF_ne_zero ho] case refine'_2 b o✝ : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b ⊢ x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) → x.2 < b intro h case refine'_2 b o✝ : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b h : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) ⊢ x.2 < b cases' (mem_cons.mp h) with h h case refine'_2.inl b o✝ : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) h : x = (log b o, o / b ^ log b o) ⊢ x.2 < b rw [h] case refine'_2.inl b o✝ : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) h : x = (log b o, o / b ^ log b o) ⊢ (log b o, o / b ^ log b o).2 < b simpa only using div_opow_log_lt o hb case refine'_2.inr b o✝ : Ordinal.{u} hb : 1 < b x : Ordinal.{u} × Ordinal.{u} o : Ordinal.{u} ho : o ≠ 0 IH : x ∈ CNF b (o % b ^ log b o) → x.2 < b h✝ : x ∈ (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) h : x ∈ CNF b (o % b ^ log b o) ⊢ x.2 < b exact IH h ** Qed
Ordinal.CNF_sorted b o : Ordinal.{u_1} ⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) refine' CNFRec b _ (fun o ho IH ↦ _) o case refine'_1 b o : Ordinal.{u_1} ⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b 0)) simp only [CNF_zero] case refine'_2 b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) ⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) cases' le_or_lt b 1 with hb hb case refine'_2.inl b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : b ≤ 1 ⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) simp only [CNF_of_le_one hb ho, map] case refine'_2.inr b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : 1 < b ⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) cases' lt_or_le o b with hob hbo case refine'_2.inr.inl b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : 1 < b hob : o < b ⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) simp only [CNF_of_lt ho hob, map] case refine'_2.inr.inr b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : 1 < b hbo : b ≤ o ⊢ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b o)) rw [CNF_ne_zero ho, map_cons, sorted_cons] case refine'_2.inr.inr b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : 1 < b hbo : b ≤ o ⊢ (∀ (b_1 : Ordinal.{u_1}), b_1 ∈ map Prod.fst (CNF b (o % b ^ log b o)) → (log b o, o / b ^ log b o).1 > b_1) ∧ Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) refine' ⟨fun a H ↦ _, IH⟩ case refine'_2.inr.inr b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : 1 < b hbo : b ≤ o a : Ordinal.{u_1} H : a ∈ map Prod.fst (CNF b (o % b ^ log b o)) ⊢ (log b o, o / b ^ log b o).1 > a rw [mem_map] at H case refine'_2.inr.inr b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : 1 < b hbo : b ≤ o a : Ordinal.{u_1} H : ∃ a_1, a_1 ∈ CNF b (o % b ^ log b o) ∧ a_1.1 = a ⊢ (log b o, o / b ^ log b o).1 > a rcases H with ⟨⟨a, a'⟩, H, rfl⟩ case refine'_2.inr.inr.intro.mk.intro b o✝ o : Ordinal.{u_1} ho : o ≠ 0 IH : Sorted (fun x x_1 => x > x_1) (map Prod.fst (CNF b (o % b ^ log b o))) hb : 1 < b hbo : b ≤ o a a' : Ordinal.{u_1} H : (a, a') ∈ CNF b (o % b ^ log b o) ⊢ (log b o, o / b ^ log b o).1 > (a, a').1 exact (CNF_fst_le_log H).trans_lt (log_mod_opow_log_lt_log_self hb ho hbo) ** Qed
Ordinal.nadd_def a✝ b✝ c a b : Ordinal.{u} ⊢ a ♯ b = max (blsub a fun a' x => a' ♯ b) (blsub b fun b' x => a ♯ b') rw [nadd] ** Qed
Ordinal.lt_nadd_iff a b c : Ordinal.{u} ⊢ a < b ♯ c ↔ (∃ b', b' < b ∧ a ≤ b' ♯ c) ∨ ∃ c', c' < c ∧ a ≤ b ♯ c' rw [nadd_def] a b c : Ordinal.{u} ⊢ a < max (blsub b fun a' x => a' ♯ c) (blsub c fun b' x => b ♯ b') ↔ (∃ b', b' < b ∧ a ≤ b' ♯ c) ∨ ∃ c', c' < c ∧ a ≤ b ♯ c' simp [lt_blsub_iff] ** Qed
Ordinal.nadd_le_iff a b c : Ordinal.{u} ⊢ b ♯ c ≤ a ↔ (∀ (b' : Ordinal.{u}), b' < b → b' ♯ c < a) ∧ ∀ (c' : Ordinal.{u}), c' < c → b ♯ c' < a rw [nadd_def] a b c : Ordinal.{u} ⊢ max (blsub b fun a' x => a' ♯ c) (blsub c fun b' x => b ♯ b') ≤ a ↔ (∀ (b' : Ordinal.{u}), b' < b → b' ♯ c < a) ∧ ∀ (c' : Ordinal.{u}), c' < c → b ♯ c' < a simp [blsub_le_iff] ** Qed
Ordinal.nadd_le_nadd_left a✝ b c : Ordinal.{u} h : b ≤ c a : Ordinal.{u} ⊢ a ♯ b ≤ a ♯ c rcases lt_or_eq_of_le h with (h \| rfl) case inl a✝ b c : Ordinal.{u} h✝ : b ≤ c a : Ordinal.{u} h : b < c ⊢ a ♯ b ≤ a ♯ c exact (nadd_lt_nadd_left h a).le case inr a✝ b a : Ordinal.{u} h : b ≤ b ⊢ a ♯ b ≤ a ♯ b exact le_rfl ** Qed
Ordinal.nadd_le_nadd_right a✝ b c : Ordinal.{u} h : b ≤ c a : Ordinal.{u} ⊢ b ♯ a ≤ c ♯ a rcases lt_or_eq_of_le h with (h \| rfl) case inl a✝ b c : Ordinal.{u} h✝ : b ≤ c a : Ordinal.{u} h : b < c ⊢ b ♯ a ≤ c ♯ a exact (nadd_lt_nadd_right h a).le case inr a✝ b a : Ordinal.{u} h : b ≤ b ⊢ b ♯ a ≤ b ♯ a exact le_rfl ** Qed
Ordinal.nadd_comm a✝ b✝ c : Ordinal.{u} a b : Ordinal.{u_1} ⊢ a ♯ b = b ♯ a rw [nadd_def, nadd_def, max_comm] a✝ b✝ c : Ordinal.{u} a b : Ordinal.{u_1} ⊢ max (blsub b fun b' x => a ♯ b') (blsub a fun a' x => a' ♯ b) = max (blsub b fun a' x => a' ♯ a) (blsub a fun b' x => b ♯ b') congr <;> ext <;> apply nadd_comm ** Qed
Ordinal.blsub_nadd_of_mono a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ blsub (a ♯ b) f = max (blsub a fun a' ha' => f (a' ♯ b) (_ : a' ♯ b < a ♯ b)) (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) apply (blsub_le_iff.2 fun i h => _).antisymm (max_le _ _) a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ ∀ (i : Ordinal.{u}) (h : i < a ♯ b), f i h < max (blsub a fun a' ha' => f (a' ♯ b) (_ : a' ♯ b < a ♯ b)) (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ (blsub a fun a' ha' => f (a' ♯ b) (_ : a' ♯ b < a ♯ b)) ≤ blsub (a ♯ b) f a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) ≤ blsub (a ♯ b) f intro i h a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ (blsub a fun a' ha' => f (a' ♯ b) (_ : a' ♯ b < a ♯ b)) ≤ blsub (a ♯ b) f a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) ≤ blsub (a ♯ b) f all_goals apply blsub_le_of_brange_subset.{u, u, v} rintro c ⟨d, hd, rfl⟩ apply mem_brange_self a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj i : Ordinal.{u} h : i < a ♯ b ⊢ f i h < max (blsub a fun a' ha' => f (a' ♯ b) (_ : a' ♯ b < a ♯ b)) (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) rcases lt_nadd_iff.1 h with (⟨a', ha', hi⟩ \| ⟨b', hb', hi⟩) case inl.intro.intro a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj i : Ordinal.{u} h : i < a ♯ b a' : Ordinal.{u} ha' : a' < a hi : i ≤ a' ♯ b ⊢ f i h < max (blsub a fun a' ha' => f (a' ♯ b) (_ : a' ♯ b < a ♯ b)) (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) exact lt_max_of_lt_left ((hf h (nadd_lt_nadd_right ha' b) hi).trans_lt (lt_blsub _ _ ha')) case inr.intro.intro a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj i : Ordinal.{u} h : i < a ♯ b b' : Ordinal.{u} hb' : b' < b hi : i ≤ a ♯ b' ⊢ f i h < max (blsub a fun a' ha' => f (a' ♯ b) (_ : a' ♯ b < a ♯ b)) (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) exact lt_max_of_lt_right ((hf h (nadd_lt_nadd_left hb' a) hi).trans_lt (lt_blsub _ _ hb')) a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ (blsub b fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) ≤ blsub (a ♯ b) f apply blsub_le_of_brange_subset.{u, u, v} a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj ⊢ (brange b fun a_1 ha => (fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) a_1 ha) ⊆ brange (a ♯ b) fun a_1 ha => f a_1 ha rintro c ⟨d, hd, rfl⟩ case intro.intro a b c : Ordinal.{u} f : (c : Ordinal.{u}) → c < a ♯ b → Ordinal.{max u v} hf : ∀ {i j : Ordinal.{u}} (hi : i < a ♯ b) (hj : j < a ♯ b), i ≤ j → f i hi ≤ f j hj d : Ordinal.{u} hd : d < b ⊢ (fun a_1 ha => (fun b' hb' => f (a ♯ b') (_ : a ♯ b' < a ♯ b)) a_1 ha) d hd ∈ brange (a ♯ b) fun a_1 ha => f a_1 ha apply mem_brange_self ** Qed
Ordinal.nadd_assoc a✝ b✝ c✝ : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ a ♯ b ♯ c = a ♯ (b ♯ c) rw [nadd_def a (b ♯ c), nadd_def, blsub_nadd_of_mono, blsub_nadd_of_mono, max_assoc] a✝ b✝ c✝ : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ max (blsub a fun a' ha' => a' ♯ b ♯ c) (max (blsub b fun b' hb' => a ♯ b' ♯ c) (blsub c fun b' x => a ♯ b ♯ b')) = max (blsub a fun a' x => a' ♯ (b ♯ c)) (max (blsub b fun a' ha' => a ♯ (a' ♯ c)) (blsub c fun b' hb' => a ♯ (b ♯ b'))) congr <;> ext (d hd) <;> apply nadd_assoc case hf a✝ b✝ c✝ : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ ∀ {i j : Ordinal.{u_1}}, i < b ♯ c → j < b ♯ c → i ≤ j → a ♯ i ≤ a ♯ j exact fun _ _ h => nadd_le_nadd_left h a case hf a✝ b✝ c✝ : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ ∀ {i j : Ordinal.{u_1}}, i < a ♯ b → j < a ♯ b → i ≤ j → i ♯ c ≤ j ♯ c exact fun _ _ h => nadd_le_nadd_right h c ** Qed
Ordinal.nadd_zero a b c : Ordinal.{u} ⊢ a ♯ 0 = a induction' a using Ordinal.induction with a IH case h a✝ b c a : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < a → k ♯ 0 = k ⊢ a ♯ 0 = a rw [nadd_def, blsub_zero, max_zero_right] case h a✝ b c a : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < a → k ♯ 0 = k ⊢ (blsub a fun a' x => a' ♯ 0) = a convert blsub_id a case h.e'_2.h.e'_2.h.h a✝ b c a : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < a → k ♯ 0 = k x✝¹ : Ordinal.{u} x✝ : x✝¹ < a ⊢ x✝¹ ♯ 0 = x✝¹ rename_i hb case h.e'_2.h.e'_2.h.h a✝ b c a : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < a → k ♯ 0 = k x✝ : Ordinal.{u} hb : x✝ < a ⊢ x✝ ♯ 0 = x✝ exact IH _ hb ** Qed
Ordinal.zero_nadd a b c : Ordinal.{u} ⊢ 0 ♯ a = a rw [nadd_comm, nadd_zero] ** Qed
Ordinal.nadd_one a b c : Ordinal.{u} ⊢ a ♯ 1 = succ a induction' a using Ordinal.induction with a IH case h a✝ b c a : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < a → k ♯ 1 = succ k ⊢ a ♯ 1 = succ a rw [nadd_def, blsub_one, nadd_zero, max_eq_right_iff, blsub_le_iff] case h a✝ b c a : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < a → k ♯ 1 = succ k ⊢ ∀ (i : Ordinal.{u}), i < a → i ♯ 1 < succ a intro i hi case h a✝ b c a : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < a → k ♯ 1 = succ k i : Ordinal.{u} hi : i < a ⊢ i ♯ 1 < succ a rwa [IH i hi, succ_lt_succ_iff] ** Qed
Ordinal.one_nadd a b c : Ordinal.{u} ⊢ 1 ♯ a = succ a rw [nadd_comm, nadd_one] ** Qed
Ordinal.nadd_succ a b c : Ordinal.{u} ⊢ a ♯ succ b = succ (a ♯ b) rw [← nadd_one (a ♯ b), nadd_assoc, nadd_one] ** Qed
Ordinal.succ_nadd a b c : Ordinal.{u} ⊢ succ a ♯ b = succ (a ♯ b) rw [← one_nadd (a ♯ b), ← nadd_assoc, one_nadd] ** Qed
Ordinal.nadd_nat a b c : Ordinal.{u} n : ℕ ⊢ a ♯ ↑n = a + ↑n induction' n with n hn case zero a b c : Ordinal.{u} ⊢ a ♯ ↑Nat.zero = a + ↑Nat.zero simp case succ a b c : Ordinal.{u} n : ℕ hn : a ♯ ↑n = a + ↑n ⊢ a ♯ ↑(Nat.succ n) = a + ↑(Nat.succ n) rw [Nat.cast_succ, add_one_eq_succ, nadd_succ, add_succ, hn] ** Qed
Ordinal.nat_nadd a b c : Ordinal.{u} n : ℕ ⊢ ↑n ♯ a = a + ↑n rw [nadd_comm, nadd_nat] ** Qed
Ordinal.add_le_nadd a b c : Ordinal.{u} ⊢ a + b ≤ a ♯ b induction b using limitRecOn with \| H₁ => simp \| H₂ c h => rwa [add_succ, nadd_succ, succ_le_succ_iff] \| H₃ c hc H => simp_rw [← IsNormal.blsub_eq.{u, u} (add_isNormal a) hc, blsub_le_iff] exact fun i hi => (H i hi).trans_lt (nadd_lt_nadd_left hi a) case H₁ a b c : Ordinal.{u} ⊢ a + 0 ≤ a ♯ 0 simp case H₂ a b c✝ c : Ordinal.{u} h : a + c ≤ a ♯ c ⊢ a + succ c ≤ a ♯ succ c rwa [add_succ, nadd_succ, succ_le_succ_iff] case H₃ a b c✝ c : Ordinal.{u} hc : IsLimit c H : ∀ (o' : Ordinal.{u}), o' < c → a + o' ≤ a ♯ o' ⊢ a + c ≤ a ♯ c simp_rw [← IsNormal.blsub_eq.{u, u} (add_isNormal a) hc, blsub_le_iff] case H₃ a b c✝ c : Ordinal.{u} hc : IsLimit c H : ∀ (o' : Ordinal.{u}), o' < c → a + o' ≤ a ♯ o' ⊢ ∀ (i : Ordinal.{u}), i < c → a + i < a ♯ c exact fun i hi => (H i hi).trans_lt (nadd_lt_nadd_left hi a) ** Qed
NatOrdinal.toOrdinal_cast_nat n : ℕ ⊢ ↑toOrdinal ↑n = ↑n induction' n with n hn case zero ⊢ ↑toOrdinal ↑Nat.zero = ↑Nat.zero rfl case succ n : ℕ hn : ↑toOrdinal ↑n = ↑n ⊢ ↑toOrdinal ↑(Nat.succ n) = ↑(Nat.succ n) change (toOrdinal n) ♯ 1 = n + 1 case succ n : ℕ hn : ↑toOrdinal ↑n = ↑n ⊢ ↑toOrdinal ↑n ♯ 1 = ↑n + 1 rw [hn] case succ n : ℕ hn : ↑toOrdinal ↑n = ↑n ⊢ ↑n ♯ 1 = ↑n + 1 exact nadd_one n ** Qed
Ordinal.toNatOrdinal_cast_nat n : ℕ ⊢ ↑toNatOrdinal ↑n = ↑n rw [← toOrdinal_cast_nat n] n : ℕ ⊢ ↑toNatOrdinal (↑toOrdinal ↑n) = ↑n rfl ** Qed
Ordinal.le_nadd_self a b : Ordinal.{u_1} ⊢ a ≤ b ♯ a simpa using nadd_le_nadd_right (Ordinal.zero_le b) a ** Qed
Ordinal.le_self_nadd a b : Ordinal.{u_1} ⊢ a ≤ a ♯ b simpa using nadd_le_nadd_left (Ordinal.zero_le b) a ** Qed
Ordinal.nmul_def a✝ b✝ c d : Ordinal.{u} a b : Ordinal.{u_1} ⊢ a ⨳ b = sInf {c \| ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} rw [nmul] ** Qed
Ordinal.nmul_nadd_lt a b c d a' b' : Ordinal.{u} ha : a' < a hb : b' < b ⊢ a' ⨳ b ♯ a ⨳ b' < a ⨳ b ♯ a' ⨳ b' rw [nmul_def a b] a b c d a' b' : Ordinal.{u} ha : a' < a hb : b' < b ⊢ a' ⨳ b ♯ a ⨳ b' < sInf {c \| ∀ (a' : Ordinal.{u}), a' < a → ∀ (b' : Ordinal.{u}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} ♯ a' ⨳ b' exact csInf_mem (nmul_nonempty a b) a' ha b' hb ** Qed
Ordinal.nmul_nadd_le a b c d a' b' : Ordinal.{u} ha : a' ≤ a hb : b' ≤ b ⊢ a' ⨳ b ♯ a ⨳ b' ≤ a ⨳ b ♯ a' ⨳ b' rcases lt_or_eq_of_le ha with (ha \| rfl) case inl a b c d a' b' : Ordinal.{u} ha✝ : a' ≤ a hb : b' ≤ b ha : a' < a ⊢ a' ⨳ b ♯ a ⨳ b' ≤ a ⨳ b ♯ a' ⨳ b' rcases lt_or_eq_of_le hb with (hb \| rfl) case inl.inl a b c d a' b' : Ordinal.{u} ha✝ : a' ≤ a hb✝ : b' ≤ b ha : a' < a hb : b' < b ⊢ a' ⨳ b ♯ a ⨳ b' ≤ a ⨳ b ♯ a' ⨳ b' exact (nmul_nadd_lt ha hb).le case inl.inr a c d a' b' : Ordinal.{u} ha✝ : a' ≤ a ha : a' < a hb : b' ≤ b' ⊢ a' ⨳ b' ♯ a ⨳ b' ≤ a ⨳ b' ♯ a' ⨳ b' rw [nadd_comm] case inr b c d a' b' : Ordinal.{u} hb : b' ≤ b ha : a' ≤ a' ⊢ a' ⨳ b ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a' ⨳ b' exact le_rfl ** Qed
Ordinal.lt_nmul_iff a b c d : Ordinal.{u} ⊢ c < a ⨳ b ↔ ∃ a', a' < a ∧ ∃ b', b' < b ∧ c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' refine' ⟨fun h => _, _⟩ case refine'_1 a b c d : Ordinal.{u} h : c < a ⨳ b ⊢ ∃ a', a' < a ∧ ∃ b', b' < b ∧ c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' rw [nmul] at h case refine'_1 a b c d : Ordinal.{u} h : c < sInf {c \| ∀ (a' : Ordinal.{u}), a' < a → ∀ (b' : Ordinal.{u}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} ⊢ ∃ a', a' < a ∧ ∃ b', b' < b ∧ c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' simpa using not_mem_of_lt_csInf h ⟨0, fun _ _ => bot_le⟩ case refine'_2 a b c d : Ordinal.{u} ⊢ (∃ a', a' < a ∧ ∃ b', b' < b ∧ c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b') → c < a ⨳ b rintro ⟨a', ha, b', hb, h⟩ case refine'_2.intro.intro.intro.intro a b c d a' : Ordinal.{u} ha : a' < a b' : Ordinal.{u} hb : b' < b h : c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' ⊢ c < a ⨳ b have := h.trans_lt (nmul_nadd_lt ha hb) case refine'_2.intro.intro.intro.intro a b c d a' : Ordinal.{u} ha : a' < a b' : Ordinal.{u} hb : b' < b h : c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' this : c ♯ a' ⨳ b' < a ⨳ b ♯ a' ⨳ b' ⊢ c < a ⨳ b rwa [nadd_lt_nadd_iff_right] at this ** Qed
Ordinal.nmul_le_iff a b c d : Ordinal.{u} ⊢ a ⨳ b ≤ c ↔ ∀ (a' : Ordinal.{u}), a' < a → ∀ (b' : Ordinal.{u}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b' rw [← not_iff_not] a b c d : Ordinal.{u} ⊢ ¬a ⨳ b ≤ c ↔ ¬∀ (a' : Ordinal.{u}), a' < a → ∀ (b' : Ordinal.{u}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b' simp [lt_nmul_iff] ** Qed
Ordinal.nmul_comm a✝ b✝ c d : Ordinal.{u} a b : Ordinal.{u_1} ⊢ a ⨳ b = b ⨳ a rw [nmul, nmul] a✝ b✝ c d : Ordinal.{u} a b : Ordinal.{u_1} ⊢ sInf {c \| ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} = sInf {c \| ∀ (a' : Ordinal.{u_1}), a' < b → ∀ (b' : Ordinal.{u_1}), b' < a → a' ⨳ a ♯ b ⨳ b' < c ♯ a' ⨳ b'} congr case e_a a✝ b✝ c d : Ordinal.{u} a b : Ordinal.{u_1} ⊢ {c \| ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} = {c \| ∀ (a' : Ordinal.{u_1}), a' < b → ∀ (b' : Ordinal.{u_1}), b' < a → a' ⨳ a ♯ b ⨳ b' < c ♯ a' ⨳ b'} ext x case e_a.h a✝ b✝ c d : Ordinal.{u} a b x : Ordinal.{u_1} ⊢ x ∈ {c \| ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} ↔ x ∈ {c \| ∀ (a' : Ordinal.{u_1}), a' < b → ∀ (b' : Ordinal.{u_1}), b' < a → a' ⨳ a ♯ b ⨳ b' < c ♯ a' ⨳ b'} constructor <;> intro H c hc d hd case e_a.h.mp a✝ b✝ c✝ d✝ : Ordinal.{u} a b x : Ordinal.{u_1} H : x ∈ {c \| ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} c : Ordinal.{u_1} hc : c < b d : Ordinal.{u_1} hd : d < a ⊢ c ⨳ a ♯ b ⨳ d < x ♯ c ⨳ d rw [nadd_comm, ← nmul_comm d b, ← nmul_comm a c, ← nmul_comm d] case e_a.h.mp a✝ b✝ c✝ d✝ : Ordinal.{u} a b x : Ordinal.{u_1} H : x ∈ {c \| ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < b → a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} c : Ordinal.{u_1} hc : c < b d : Ordinal.{u_1} hd : d < a ⊢ d ⨳ b ♯ a ⨳ c < x ♯ d ⨳ c exact H _ hd _ hc case e_a.h.mpr a✝ b✝ c✝ d✝ : Ordinal.{u} a b x : Ordinal.{u_1} H : x ∈ {c \| ∀ (a' : Ordinal.{u_1}), a' < b → ∀ (b' : Ordinal.{u_1}), b' < a → a' ⨳ a ♯ b ⨳ b' < c ♯ a' ⨳ b'} c : Ordinal.{u_1} hc : c < a d : Ordinal.{u_1} hd : d < b ⊢ c ⨳ b ♯ a ⨳ d < x ♯ c ⨳ d rw [nadd_comm, nmul_comm a d, nmul_comm c, nmul_comm c] case e_a.h.mpr a✝ b✝ c✝ d✝ : Ordinal.{u} a b x : Ordinal.{u_1} H : x ∈ {c \| ∀ (a' : Ordinal.{u_1}), a' < b → ∀ (b' : Ordinal.{u_1}), b' < a → a' ⨳ a ♯ b ⨳ b' < c ♯ a' ⨳ b'} c : Ordinal.{u_1} hc : c < a d : Ordinal.{u_1} hd : d < b ⊢ d ⨳ a ♯ b ⨳ c < x ♯ d ⨳ c exact H _ hd _ hc ** Qed
Ordinal.nmul_zero a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ a ⨳ 0 = 0 rw [← Ordinal.le_zero, nmul_le_iff] a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < 0 → a' ⨳ 0 ♯ a ⨳ b' < 0 ♯ a' ⨳ b' exact fun _ _ a ha => (Ordinal.not_lt_zero a ha).elim ** Qed
Ordinal.zero_nmul a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ 0 ⨳ a = 0 rw [nmul_comm, nmul_zero] ** Qed
Ordinal.nmul_one a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ a ⨳ 1 = a rw [nmul] a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ sInf {c \| ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < 1 → a' ⨳ 1 ♯ a ⨳ b' < c ♯ a' ⨳ b'} = a simp only [lt_one_iff_zero, forall_eq, nmul_zero, nadd_zero] a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ sInf {c \| ∀ (a' : Ordinal.{u_1}), a' < a → a' ⨳ 1 < c} = a convert csInf_Ici (α := Ordinal) case h.e'_2.h.e'_3 a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ {c \| ∀ (a' : Ordinal.{u_1}), a' < a → a' ⨳ 1 < c} = Set.Ici a ext b case h.e'_2.h.e'_3.h a✝ b✝ c d : Ordinal.{u} a b : Ordinal.{u_1} ⊢ b ∈ {c \| ∀ (a' : Ordinal.{u_1}), a' < a → a' ⨳ 1 < c} ↔ b ∈ Set.Ici a simp only [Set.mem_setOf_eq, Set.mem_Ici] case h.e'_2.h.e'_3.h a✝ b✝ c d : Ordinal.{u} a b : Ordinal.{u_1} ⊢ (∀ (a' : Ordinal.{u_1}), a' < a → a' ⨳ 1 < b) ↔ a ≤ b refine' ⟨fun H => le_of_forall_lt fun c hc => _, fun ha c hc => _⟩ case h.e'_2.h.e'_3.h.refine'_1 a✝ b✝ c✝ d : Ordinal.{u} a b : Ordinal.{u_1} H : ∀ (a' : Ordinal.{u_1}), a' < a → a' ⨳ 1 < b c : Ordinal.{u_1} hc : c < a ⊢ c < b simpa only [nmul_one c] using H c hc case h.e'_2.h.e'_3.h.refine'_2 a✝ b✝ c✝ d : Ordinal.{u} a b : Ordinal.{u_1} ha : a ≤ b c : Ordinal.{u_1} hc : c < a ⊢ c ⨳ 1 < b simpa only [nmul_one c] using hc.trans_le ha ** Qed
Ordinal.one_nmul a✝ b c d : Ordinal.{u} a : Ordinal.{u_1} ⊢ 1 ⨳ a = a rw [nmul_comm, nmul_one] ** Qed
Ordinal.nmul_lt_nmul_of_pos_left a b c d : Ordinal.{u} h₁ : a < b h₂ : 0 < c ⊢ c ⨳ a ♯ 0 ⨳ a ≤ 0 ⨳ b ♯ c ⨳ a simp ** Qed
Ordinal.nmul_lt_nmul_of_pos_right a b c d : Ordinal.{u} h₁ : a < b h₂ : 0 < c ⊢ a ⨳ c ♯ a ⨳ 0 ≤ a ⨳ c ♯ b ⨳ 0 simp ** Qed
Ordinal.nmul_le_nmul_of_nonneg_left a b c d : Ordinal.{u} h₁ : a ≤ b h₂ : 0 ≤ c ⊢ c ⨳ a ≤ c ⨳ b rcases lt_or_eq_of_le h₁ with (h₁ \| rfl) <;> rcases lt_or_eq_of_le h₂ with (h₂ \| rfl) case inl.inr a b d : Ordinal.{u} h₁✝ : a ≤ b h₁ : a < b h₂ : 0 ≤ 0 ⊢ 0 ⨳ a ≤ 0 ⨳ b case inr.inl a c d : Ordinal.{u} h₂✝ : 0 ≤ c h₁ : a ≤ a h₂ : 0 < c ⊢ c ⨳ a ≤ c ⨳ a case inr.inr a d : Ordinal.{u} h₁ : a ≤ a h₂ : 0 ≤ 0 ⊢ 0 ⨳ a ≤ 0 ⨳ a all_goals simp case inl.inl a b c d : Ordinal.{u} h₁✝ : a ≤ b h₂✝ : 0 ≤ c h₁ : a < b h₂ : 0 < c ⊢ c ⨳ a ≤ c ⨳ b exact (nmul_lt_nmul_of_pos_left h₁ h₂).le case inr.inr a d : Ordinal.{u} h₁ : a ≤ a h₂ : 0 ≤ 0 ⊢ 0 ⨳ a ≤ 0 ⨳ a simp ** Qed
Ordinal.nmul_le_nmul_of_nonneg_right a b c d : Ordinal.{u} h₁ : a ≤ b h₂ : 0 ≤ c ⊢ a ⨳ c ≤ b ⨳ c rw [nmul_comm, nmul_comm b] a b c d : Ordinal.{u} h₁ : a ≤ b h₂ : 0 ≤ c ⊢ c ⨳ a ≤ c ⨳ b exact nmul_le_nmul_of_nonneg_left h₁ h₂ ** Qed
Ordinal.nmul_nadd a✝ b✝ c✝ d : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ a ⨳ (b ♯ c) = a ⨳ b ♯ a ⨳ c refine le_antisymm (nmul_le_iff.2 fun a' ha d hd => ?_) (nadd_le_iff.2 ⟨fun d hd => ?_, fun d hd => ?_⟩) case refine_1 a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd : d < b ♯ c ⊢ a' ⨳ (b ♯ c) ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d rw [nmul_nadd a' b c] case refine_1 a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd : d < b ♯ c ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d rcases lt_nadd_iff.1 hd with (⟨b', hb, hd⟩ \| ⟨c', hc, hd⟩) case refine_1.inl.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c b' : Ordinal.{u_1} hb : b' < b hd : d ≤ b' ♯ c ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d have := nadd_lt_nadd_of_lt_of_le (nmul_nadd_lt ha hb) (nmul_nadd_le ha.le hd) case refine_1.inl.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c b' : Ordinal.{u_1} hb : b' < b hd : d ≤ b' ♯ c this : a' ⨳ b ♯ a ⨳ b' ♯ (a' ⨳ (b' ♯ c) ♯ a ⨳ d) < a ⨳ b ♯ a' ⨳ b' ♯ (a ⨳ (b' ♯ c) ♯ a' ⨳ d) ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d rw [nmul_nadd a' b' c, nmul_nadd a b' c] at this case refine_1.inl.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c b' : Ordinal.{u_1} hb : b' < b hd : d ≤ b' ♯ c this : a' ⨳ b ♯ a ⨳ b' ♯ (a' ⨳ b' ♯ a' ⨳ c ♯ a ⨳ d) < a ⨳ b ♯ a' ⨳ b' ♯ (a ⨳ b' ♯ a ⨳ c ♯ a' ⨳ d) ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d simp only [nadd_assoc] at this case refine_1.inl.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c b' : Ordinal.{u_1} hb : b' < b hd : d ≤ b' ♯ c this : a' ⨳ b ♯ (a ⨳ b' ♯ (a' ⨳ b' ♯ (a' ⨳ c ♯ a ⨳ d))) < a ⨳ b ♯ (a' ⨳ b' ♯ (a ⨳ b' ♯ (a ⨳ c ♯ a' ⨳ d))) ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d rwa [nadd_left_comm, nadd_left_comm _ (a ⨳ b'), nadd_left_comm (a ⨳ b), nadd_lt_nadd_iff_left, nadd_left_comm (a' ⨳ b), nadd_left_comm (a ⨳ b), nadd_lt_nadd_iff_left, ← nadd_assoc, ← nadd_assoc] at this case refine_1.inr.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c c' : Ordinal.{u_1} hc : c' < c hd : d ≤ b ♯ c' ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d have := nadd_lt_nadd_of_le_of_lt (nmul_nadd_le ha.le hd) (nmul_nadd_lt ha hc) case refine_1.inr.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c c' : Ordinal.{u_1} hc : c' < c hd : d ≤ b ♯ c' this : a' ⨳ (b ♯ c') ♯ a ⨳ d ♯ (a' ⨳ c ♯ a ⨳ c') < a ⨳ (b ♯ c') ♯ a' ⨳ d ♯ (a ⨳ c ♯ a' ⨳ c') ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d rw [nmul_nadd a' b c', nmul_nadd a b c'] at this case refine_1.inr.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c c' : Ordinal.{u_1} hc : c' < c hd : d ≤ b ♯ c' this : a' ⨳ b ♯ a' ⨳ c' ♯ a ⨳ d ♯ (a' ⨳ c ♯ a ⨳ c') < a ⨳ b ♯ a ⨳ c' ♯ a' ⨳ d ♯ (a ⨳ c ♯ a' ⨳ c') ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d simp only [nadd_assoc] at this case refine_1.inr.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a d : Ordinal.{u_1} hd✝ : d < b ♯ c c' : Ordinal.{u_1} hc : c' < c hd : d ≤ b ♯ c' this : a' ⨳ b ♯ (a' ⨳ c' ♯ (a ⨳ d ♯ (a' ⨳ c ♯ a ⨳ c'))) < a ⨳ b ♯ (a ⨳ c' ♯ (a' ⨳ d ♯ (a ⨳ c ♯ a' ⨳ c'))) ⊢ a' ⨳ b ♯ a' ⨳ c ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d rwa [nadd_left_comm, nadd_comm (a ⨳ c), nadd_left_comm (a' ⨳ d), nadd_left_comm (a ⨳ c'), nadd_left_comm (a ⨳ b), nadd_lt_nadd_iff_left, nadd_comm (a' ⨳ c), nadd_left_comm (a ⨳ d), nadd_left_comm (a' ⨳ b), nadd_left_comm (a ⨳ b), nadd_lt_nadd_iff_left, nadd_comm (a ⨳ d), nadd_comm (a' ⨳ d), ← nadd_assoc, ← nadd_assoc] at this case refine_2 a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd : d < a ⨳ b ⊢ d ♯ a ⨳ c < a ⨳ (b ♯ c) rcases lt_nmul_iff.1 hd with ⟨a', ha, b', hb, hd⟩ case refine_2.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ b a' : Ordinal.{u_1} ha : a' < a b' : Ordinal.{u_1} hb : b' < b hd : d ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' ⊢ d ♯ a ⨳ c < a ⨳ (b ♯ c) have := nadd_lt_nadd_of_le_of_lt hd (nmul_nadd_lt ha (nadd_lt_nadd_right hb c)) case refine_2.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ b a' : Ordinal.{u_1} ha : a' < a b' : Ordinal.{u_1} hb : b' < b hd : d ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' this : d ♯ a' ⨳ b' ♯ (a' ⨳ (b ♯ c) ♯ a ⨳ (b' ♯ c)) < a' ⨳ b ♯ a ⨳ b' ♯ (a ⨳ (b ♯ c) ♯ a' ⨳ (b' ♯ c)) ⊢ d ♯ a ⨳ c < a ⨳ (b ♯ c) rw [nmul_nadd a' b c, nmul_nadd a b' c, nmul_nadd a'] at this case refine_2.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ b a' : Ordinal.{u_1} ha : a' < a b' : Ordinal.{u_1} hb : b' < b hd : d ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' this : d ♯ a' ⨳ b' ♯ (a' ⨳ b ♯ a' ⨳ c ♯ (a ⨳ b' ♯ a ⨳ c)) < a' ⨳ b ♯ a ⨳ b' ♯ (a ⨳ (b ♯ c) ♯ (a' ⨳ b' ♯ a' ⨳ c)) ⊢ d ♯ a ⨳ c < a ⨳ (b ♯ c) simp only [nadd_assoc] at this case refine_2.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ b a' : Ordinal.{u_1} ha : a' < a b' : Ordinal.{u_1} hb : b' < b hd : d ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' this : d ♯ (a' ⨳ b' ♯ (a' ⨳ b ♯ (a' ⨳ c ♯ (a ⨳ b' ♯ a ⨳ c)))) < a' ⨳ b ♯ (a ⨳ b' ♯ (a ⨳ (b ♯ c) ♯ (a' ⨳ b' ♯ a' ⨳ c))) ⊢ d ♯ a ⨳ c < a ⨳ (b ♯ c) rwa [nadd_left_comm (a' ⨳ b'), nadd_left_comm, nadd_lt_nadd_iff_left, nadd_left_comm, nadd_left_comm _ (a' ⨳ b'), nadd_left_comm (a ⨳ b'), nadd_lt_nadd_iff_left, nadd_left_comm (a' ⨳ c), nadd_left_comm, nadd_lt_nadd_iff_left, nadd_left_comm, nadd_comm _ (a' ⨳ c), nadd_lt_nadd_iff_left] at this case refine_3 a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd : d < a ⨳ c ⊢ a ⨳ b ♯ d < a ⨳ (b ♯ c) rcases lt_nmul_iff.1 hd with ⟨a', ha, c', hc, hd⟩ case refine_3.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ c a' : Ordinal.{u_1} ha : a' < a c' : Ordinal.{u_1} hc : c' < c hd : d ♯ a' ⨳ c' ≤ a' ⨳ c ♯ a ⨳ c' ⊢ a ⨳ b ♯ d < a ⨳ (b ♯ c) have := nadd_lt_nadd_of_lt_of_le (nmul_nadd_lt ha (nadd_lt_nadd_left hc b)) hd case refine_3.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ c a' : Ordinal.{u_1} ha : a' < a c' : Ordinal.{u_1} hc : c' < c hd : d ♯ a' ⨳ c' ≤ a' ⨳ c ♯ a ⨳ c' this : a' ⨳ (b ♯ c) ♯ a ⨳ (b ♯ c') ♯ (d ♯ a' ⨳ c') < a ⨳ (b ♯ c) ♯ a' ⨳ (b ♯ c') ♯ (a' ⨳ c ♯ a ⨳ c') ⊢ a ⨳ b ♯ d < a ⨳ (b ♯ c) rw [nmul_nadd a' b c, nmul_nadd a b c', nmul_nadd a'] at this case refine_3.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ c a' : Ordinal.{u_1} ha : a' < a c' : Ordinal.{u_1} hc : c' < c hd : d ♯ a' ⨳ c' ≤ a' ⨳ c ♯ a ⨳ c' this : a' ⨳ b ♯ a' ⨳ c ♯ (a ⨳ b ♯ a ⨳ c') ♯ (d ♯ a' ⨳ c') < a ⨳ (b ♯ c) ♯ (a' ⨳ b ♯ a' ⨳ c') ♯ (a' ⨳ c ♯ a ⨳ c') ⊢ a ⨳ b ♯ d < a ⨳ (b ♯ c) simp only [nadd_assoc] at this case refine_3.intro.intro.intro.intro a✝ b✝ c✝ d✝ : Ordinal.{u} a b c d : Ordinal.{u_1} hd✝ : d < a ⨳ c a' : Ordinal.{u_1} ha : a' < a c' : Ordinal.{u_1} hc : c' < c hd : d ♯ a' ⨳ c' ≤ a' ⨳ c ♯ a ⨳ c' this : a' ⨳ b ♯ (a' ⨳ c ♯ (a ⨳ b ♯ (a ⨳ c' ♯ (d ♯ a' ⨳ c')))) < a ⨳ (b ♯ c) ♯ (a' ⨳ b ♯ (a' ⨳ c' ♯ (a' ⨳ c ♯ a ⨳ c'))) ⊢ a ⨳ b ♯ d < a ⨳ (b ♯ c) rwa [nadd_left_comm _ (a' ⨳ b), nadd_lt_nadd_iff_left, nadd_left_comm (a' ⨳ c'), nadd_left_comm _ (a' ⨳ c), nadd_lt_nadd_iff_left, nadd_left_comm, nadd_comm (a' ⨳ c'), nadd_left_comm _ (a ⨳ c'), nadd_lt_nadd_iff_left, nadd_comm _ (a' ⨳ c'), nadd_comm _ (a' ⨳ c'), nadd_left_comm, nadd_lt_nadd_iff_left] at this ** Qed
Ordinal.nadd_nmul a✝ b✝ c✝ d : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ (a ♯ b) ⨳ c = a ⨳ c ♯ b ⨳ c rw [nmul_comm, nmul_nadd, nmul_comm, nmul_comm c] ** Qed
Ordinal.nmul_nadd_lt₃ a b c d a' b' c' : Ordinal.{u} ha : a' < a hb : b' < b hc : c' < c ⊢ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' < a ⨳ b ⨳ c ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' simpa only [nadd_nmul, ← nadd_assoc] using nmul_nadd_lt (nmul_nadd_lt ha hb) hc ** Qed
Ordinal.nmul_nadd_le₃ a b c d a' b' c' : Ordinal.{u} ha : a' ≤ a hb : b' ≤ b hc : c' ≤ c ⊢ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' ≤ a ⨳ b ⨳ c ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' simpa only [nadd_nmul, ← nadd_assoc] using nmul_nadd_le (nmul_nadd_le ha hb) hc ** Qed
Ordinal.nmul_nadd_lt₃' a b c d a' b' c' : Ordinal.{u} ha : a' < a hb : b' < b hc : c' < c ⊢ a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') < a ⨳ (b ⨳ c) ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') simp only [nmul_comm _ (_ ⨳ _)] case h.e'_4 a b c d a' b' c' : Ordinal.{u} ha : a' < a hb : b' < b hc : c' < c ⊢ b ⨳ c ⨳ a ♯ b' ⨳ c ⨳ a' ♯ b ⨳ c' ⨳ a' ♯ b' ⨳ c' ⨳ a = b ⨳ c ⨳ a ♯ b' ⨳ c' ⨳ a ♯ b' ⨳ c ⨳ a' ♯ b ⨳ c' ⨳ a' simp only [nadd_eq_add, NatOrdinal.toOrdinal_toNatOrdinal] case h.e'_4 a b c d a' b' c' : Ordinal.{u} ha : a' < a hb : b' < b hc : c' < c ⊢ ↑toOrdinal (↑toNatOrdinal (b ⨳ c ⨳ a) + ↑toNatOrdinal (b' ⨳ c ⨳ a') + ↑toNatOrdinal (b ⨳ c' ⨳ a') + ↑toNatOrdinal (b' ⨳ c' ⨳ a)) = ↑toOrdinal (↑toNatOrdinal (b ⨳ c ⨳ a) + ↑toNatOrdinal (b' ⨳ c' ⨳ a) + ↑toNatOrdinal (b' ⨳ c ⨳ a') + ↑toNatOrdinal (b ⨳ c' ⨳ a')) abel_nf ** Qed
Ordinal.nmul_nadd_le₃' a b c d a' b' c' : Ordinal.{u} ha : a' ≤ a hb : b' ≤ b hc : c' ≤ c ⊢ a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') ≤ a ⨳ (b ⨳ c) ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') simp only [nmul_comm _ (_ ⨳ _)] case h.e'_4 a b c d a' b' c' : Ordinal.{u} ha : a' ≤ a hb : b' ≤ b hc : c' ≤ c ⊢ b ⨳ c ⨳ a ♯ b' ⨳ c ⨳ a' ♯ b ⨳ c' ⨳ a' ♯ b' ⨳ c' ⨳ a = b ⨳ c ⨳ a ♯ b' ⨳ c' ⨳ a ♯ b' ⨳ c ⨳ a' ♯ b ⨳ c' ⨳ a' simp only [nadd_eq_add, NatOrdinal.toOrdinal_toNatOrdinal] case h.e'_4 a b c d a' b' c' : Ordinal.{u} ha : a' ≤ a hb : b' ≤ b hc : c' ≤ c ⊢ ↑toOrdinal (↑toNatOrdinal (b ⨳ c ⨳ a) + ↑toNatOrdinal (b' ⨳ c ⨳ a') + ↑toNatOrdinal (b ⨳ c' ⨳ a') + ↑toNatOrdinal (b' ⨳ c' ⨳ a)) = ↑toOrdinal (↑toNatOrdinal (b ⨳ c ⨳ a) + ↑toNatOrdinal (b' ⨳ c' ⨳ a) + ↑toNatOrdinal (b' ⨳ c ⨳ a') + ↑toNatOrdinal (b ⨳ c' ⨳ a')) abel_nf ** Qed