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tasksource 14

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[ { "state_after": "no goals", "state_before": "ι : Type u_1\nI✝ J✝ : Box ι\nx y : ι → ℝ\nI J : WithBot (Box ι)\n⊢ ↑I = ↑J ↔ I = J", "tactic": "simp only [Subset.antisymm_iff, ← le_antisymm_iff, withBotCoe_subset_iff]" } ]

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[ { "state_after": "no goals", "state_before": "R : Type u_2\nB : Type u_1\nF : Type u_3\nE : B → Type ?u.410614\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_4\nZ : VectorBundleCore R B F ι\nb✝ : B\na : F\ni j : ι\nb : B\nhb : b ∈ baseSet Z i\nv : F\n⊢ Trivialization.symm (localTriv Z i) b v = ↑(coordChange Z i (indexAt Z b) b) v", "tactic": "apply (Z.localTriv i).symm_apply hb v" } ]

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[ { "state_after": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nh : lowerCentralSeries G n ≤ center G\n⊢ ∀ (y : G), y ∈ {x | ∃ p, p ∈ lowerCentralSeries G n ∧ ∃ q, q ∈ ⊤ ∧ p * q * p⁻¹ * q⁻¹ = x} → y = 1", "state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nh : lowerCentralSeries G n ≤ center G\n⊢ lowerCentralSeries G (n + 1) = ⊥", "tactic": "rw [lowerCentralSeries_succ, closure_eq_bot_iff, Set.subset_singleton_iff]" }, { "state_after": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nh : lowerCentralSeries G n ≤ center G\ny : G\nhy1 : y ∈ lowerCentralSeries G n\nz : G\n⊢ y * z * y⁻¹ * z⁻¹ = 1", "state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nh : lowerCentralSeries G n ≤ center G\n⊢ ∀ (y : G), y ∈ {x | ∃ p, p ∈ lowerCentralSeries G n ∧ ∃ q, q ∈ ⊤ ∧ p * q * p⁻¹ * q⁻¹ = x} → y = 1", "tactic": "rintro x ⟨y, hy1, z, ⟨⟩, rfl⟩" }, { "state_after": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nh : lowerCentralSeries G n ≤ center G\ny : G\nhy1 : y ∈ lowerCentralSeries G n\nz : G\n⊢ z * y = y * z", "state_before": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nh : lowerCentralSeries G n ≤ center G\ny : G\nhy1 : y ∈ lowerCentralSeries G n\nz : G\n⊢ y * z * y⁻¹ * z⁻¹ = 1", "tactic": "rw [mul_assoc, ← mul_inv_rev, mul_inv_eq_one, eq_comm]" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nh : lowerCentralSeries G n ≤ center G\ny : G\nhy1 : y ∈ lowerCentralSeries G n\nz : G\n⊢ z * y = y * z", "tactic": "exact mem_center_iff.mp (h hy1) z" } ]

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[ { "state_after": "no goals", "state_before": "α : Type u_2\nβ✝ : Type ?u.393825\nγ : Type u_3\nι : Type ?u.393831\ninst✝² : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace β✝\ninst✝ : TopologicalSpace γ\nf✝ g✝ : α → β✝\nβ : Type u_1\nf : α → β\nmα : MeasurableSpace α\nμa : Measure α\nmβ : MeasurableSpace β\nμb : Measure β\nhf : MeasurePreserving f\nh₂ : MeasurableEmbedding f\ng : β → γ\n⊢ AEStronglyMeasurable (g ∘ f) μa ↔ AEStronglyMeasurable g μb", "tactic": "rw [← hf.map_eq, h₂.aestronglyMeasurable_map_iff]" } ]

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[ { "state_after": "no goals", "state_before": "R : Type u\ninst✝¹ : CommMonoid R\nR' : Type v\ninst✝ : CommMonoidWithZero R'\nf : Rˣ →* R'ˣ\na : Rˣ\n⊢ ↑(ofUnitHom f) ↑a = ↑(↑f a)", "tactic": "simp [ofUnitHom]" } ]

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[ { "state_after": "n p : ℕ\nhp : Prime p\nhn : n ≠ 0\n⊢ ¬1 ≤ ↑(factorization (n / p ^ ↑(factorization n) p)) p", "state_before": "n p : ℕ\nhp : Prime p\nhn : n ≠ 0\n⊢ ¬p ∣ n / p ^ ↑(factorization n) p", "tactic": "rw [Nat.Prime.dvd_iff_one_le_factorization hp (ord_compl_pos p hn).ne']" }, { "state_after": "n p : ℕ\nhp : Prime p\nhn : n ≠ 0\n⊢ ¬1 ≤ ↑(factorization n - factorization (p ^ ↑(factorization n) p)) p", "state_before": "n p : ℕ\nhp : Prime p\nhn : n ≠ 0\n⊢ ¬1 ≤ ↑(factorization (n / p ^ ↑(factorization n) p)) p", "tactic": "rw [Nat.factorization_div (Nat.ord_proj_dvd n p)]" }, { "state_after": "no goals", "state_before": "n p : ℕ\nhp : Prime p\nhn : n ≠ 0\n⊢ ¬1 ≤ ↑(factorization n - factorization (p ^ ↑(factorization n) p)) p", "tactic": "simp [hp.factorization]" } ]

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[ { "state_after": "case intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s✝ : Set α\nε r✝ : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nr : ℝ≥0∞\nhr : r < ↑↑μ s\n⊢ ∃ K, K ⊆ s ∧ p K ∧ r < ↑↑μ K", "state_before": "α : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s : Set α\nε r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\n⊢ InnerRegular μ p fun s => MeasurableSet s ∧ ↑↑μ s ≠ ⊤", "tactic": "rintro s ⟨hs, hμs⟩ r hr" }, { "state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K", "state_before": "case intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s✝ : Set α\nε r✝ : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nr : ℝ≥0∞\nhr : r < ↑↑μ s\n⊢ ∃ K, K ⊆ s ∧ p K ∧ r < ↑↑μ K", "tactic": "obtain ⟨ε, hε, hεs, rfl⟩ : ∃ (ε : _)(_ : ε ≠ 0), ε + ε ≤ μ s ∧ r = μ s - (ε + ε) := by\n use (μ s - r) / 2\n simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]" }, { "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K", "state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K", "tactic": "rcases hs.exists_isOpen_diff_lt hμs hε with ⟨U, hsU, hUo, hUt, hμU⟩" }, { "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhsU' : U' ⊇ U \\ s\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K", "tactic": "rcases (U \\ s).exists_isOpen_lt_of_lt _ hμU with ⟨U', hsU', hU'o, hμU'⟩" }, { "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhsU' : U' ⊇ U \\ s\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K", "tactic": "replace hsU' := diff_subset_comm.1 hsU'" }, { "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K", "tactic": "rcases H.exists_subset_lt_add h0 hUo hUt.ne hε with ⟨K, hKU, hKc, hKr⟩" }, { "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ↑↑μ s < ↑↑μ (K \\ U') + (ε + ε)", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ∃ K, K ⊆ s ∧ p K ∧ ↑↑μ s - (ε + ε) < ↑↑μ K", "tactic": "refine' ⟨K \\ U', fun x hx => hsU' ⟨hKU hx.1, hx.2⟩, hd hKc hU'o, ENNReal.sub_lt_of_lt_add hεs _⟩" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ↑↑μ s < ↑↑μ (K \\ U') + (ε + ε)", "tactic": "calc\n μ s ≤ μ U := μ.mono hsU\n _ < μ K + ε := hKr\n _ ≤ μ (K \\ U') + μ U' + ε := (add_le_add_right (tsub_le_iff_right.1 le_measure_diff) _)\n _ ≤ μ (K \\ U') + ε + ε := by\n apply add_le_add_right; apply add_le_add_left\n exact hμU'.le\n _ = μ (K \\ U') + (ε + ε) := add_assoc _ _ _" }, { "state_after": "α : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s✝ : Set α\nε r✝ : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nr : ℝ≥0∞\nhr : r < ↑↑μ s\n⊢ ∃ x, (↑↑μ s - r) / 2 + (↑↑μ s - r) / 2 ≤ ↑↑μ s ∧ r = ↑↑μ s - ((↑↑μ s - r) / 2 + (↑↑μ s - r) / 2)", "state_before": "α : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s✝ : Set α\nε r✝ : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nr : ℝ≥0∞\nhr : r < ↑↑μ s\n⊢ ∃ ε x, ε + ε ≤ ↑↑μ s ∧ r = ↑↑μ s - (ε + ε)", "tactic": "use (μ s - r) / 2" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU s✝ : Set α\nε r✝ : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nr : ℝ≥0∞\nhr : r < ↑↑μ s\n⊢ ∃ x, (↑↑μ s - r) / 2 + (↑↑μ s - r) / 2 ≤ ↑↑μ s ∧ r = ↑↑μ s - ((↑↑μ s - r) / 2 + (↑↑μ s - r) / 2)", "tactic": "simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]" }, { "state_after": "case bc\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ↑↑μ (K \\ U') + ↑↑μ U' ≤ ↑↑μ (K \\ U') + ε", "state_before": "α : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ↑↑μ (K \\ U') + ↑↑μ U' + ε ≤ ↑↑μ (K \\ U') + ε + ε", "tactic": "apply add_le_add_right" }, { "state_after": "case bc.bc\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ↑↑μ U' ≤ ε", "state_before": "case bc\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ↑↑μ (K \\ U') + ↑↑μ U' ≤ ↑↑μ (K \\ U') + ε", "tactic": "apply add_le_add_left" }, { "state_after": "no goals", "state_before": "case bc.bc\nα : Type u_1\nβ : Type ?u.35639\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nμ : Measure α\np q : Set α → Prop\nU✝ s✝ : Set α\nε✝ r : ℝ≥0∞\ninst✝ : OuterRegular μ\nH : InnerRegular μ p IsOpen\nh0 : p ∅\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nε : ℝ≥0∞\nhεs : ε + ε ≤ ↑↑μ s\nhr : ↑↑μ s - (ε + ε) < ↑↑μ s\nhε : ε ≠ 0\nU : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhUt : ↑↑μ U < ⊤\nhμU : ↑↑μ (U \\ s) < ε\nU' : Set α\nhU'o : IsOpen U'\nhμU' : ↑↑μ U' < ε\nhsU' : U \\ U' ⊆ s\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhKr : ↑↑μ U < ↑↑μ K + ε\n⊢ ↑↑μ U' ≤ ε", "tactic": "exact hμU'.le" } ]

[ 370, 48 ]

[ 351, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.966398\nγ : Type ?u.966401\nδ : Type ?u.966404\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\nε : ℝ≥0∞\nhε : ε ≠ 0\nhε' : ε ≠ ⊤\n⊢ ε * ↑↑μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ", "state_before": "α : Type u_1\nβ : Type ?u.966398\nγ : Type ?u.966401\nδ : Type ?u.966404\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\nε : ℝ≥0∞\nhε : ε ≠ 0\nhε' : ε ≠ ⊤\n⊢ ↑↑μ {x | ε ≤ f x} * ε ≤ ∫⁻ (a : α), f a ∂μ", "tactic": "rw [mul_comm]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.966398\nγ : Type ?u.966401\nδ : Type ?u.966404\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\nε : ℝ≥0∞\nhε : ε ≠ 0\nhε' : ε ≠ ⊤\n⊢ ε * ↑↑μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ", "tactic": "exact mul_meas_ge_le_lintegral₀ hf ε" } ]

[ 852, 41 ]

[ 848, 1 ]

[ { "state_after": "no goals", "state_before": "θ : ℝ\n⊢ arg (cos ↑θ + sin ↑θ * I) - θ = 2 * π * ↑⌊(π - θ) / (2 * π)⌋", "tactic": "rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one]" } ]

[ 456, 82 ]

[ 454, 1 ]

[]

[ 248, 54 ]

[ 246, 1 ]

[]

[ 79, 44 ]

[ 78, 1 ]

[ { "state_after": "case a\np : ℕ\nR : Type ?u.1510099\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\n⊢ ↑(map (Int.castRingHom ℚ)) (↑(bind₁ (wittStructureInt p Φ)) (W_ ℤ n)) =\n ↑(map (Int.castRingHom ℚ)) (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℤ n)) Φ)", "state_before": "p : ℕ\nR : Type ?u.1510099\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\n⊢ ↑(bind₁ (wittStructureInt p Φ)) (W_ ℤ n) = ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℤ n)) Φ", "tactic": "apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective" }, { "state_after": "case a\np : ℕ\nR : Type ?u.1510099\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nthis :\n ↑(bind₁ (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ))) (W_ ℚ n) =\n ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ n)) (↑(map (Int.castRingHom ℚ)) Φ)\n⊢ ↑(map (Int.castRingHom ℚ)) (↑(bind₁ (wittStructureInt p Φ)) (W_ ℤ n)) =\n ↑(map (Int.castRingHom ℚ)) (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℤ n)) Φ)", "state_before": "case a\np : ℕ\nR : Type ?u.1510099\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\n⊢ ↑(map (Int.castRingHom ℚ)) (↑(bind₁ (wittStructureInt p Φ)) (W_ ℤ n)) =\n ↑(map (Int.castRingHom ℚ)) (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℤ n)) Φ)", "tactic": "have := wittStructureRat_prop p (map (Int.castRingHom ℚ) Φ) n" }, { "state_after": "no goals", "state_before": "case a\np : ℕ\nR : Type ?u.1510099\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nthis :\n ↑(bind₁ (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ))) (W_ ℚ n) =\n ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ n)) (↑(map (Int.castRingHom ℚ)) Φ)\n⊢ ↑(map (Int.castRingHom ℚ)) (↑(bind₁ (wittStructureInt p Φ)) (W_ ℤ n)) =\n ↑(map (Int.castRingHom ℚ)) (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℤ n)) Φ)", "tactic": "simpa only [map_bind₁, ← eval₂Hom_map_hom, eval₂Hom_C_left, map_rename, map_wittPolynomial,\n AlgHom.coe_toRingHom, map_wittStructureInt]" } ]

[ 312, 48 ]

[ 306, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.82447\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty α\nh : Preconnected (G □ H)\n⊢ Preconnected H", "tactic": "classical\nrintro b₁ b₂\nobtain ⟨w⟩ := h (Classical.arbitrary _, b₁) (Classical.arbitrary _, b₂)\nexact ⟨w.ofBoxProdRight⟩" }, { "state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.82447\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty α\nh : Preconnected (G □ H)\nb₁ b₂ : β\n⊢ Reachable H b₁ b₂", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.82447\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty α\nh : Preconnected (G □ H)\n⊢ Preconnected H", "tactic": "rintro b₁ b₂" }, { "state_after": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.82447\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty α\nh : Preconnected (G □ H)\nb₁ b₂ : β\nw : Walk (G □ H) (Classical.arbitrary α, b₁) (Classical.arbitrary α, b₂)\n⊢ Reachable H b₁ b₂", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.82447\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty α\nh : Preconnected (G □ H)\nb₁ b₂ : β\n⊢ Reachable H b₁ b₂", "tactic": "obtain ⟨w⟩ := h (Classical.arbitrary _, b₁) (Classical.arbitrary _, b₂)" }, { "state_after": "no goals", "state_before": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.82447\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty α\nh : Preconnected (G □ H)\nb₁ b₂ : β\nw : Walk (G □ H) (Classical.arbitrary α, b₁) (Classical.arbitrary α, b₂)\n⊢ Reachable H b₁ b₂", "tactic": "exact ⟨w.ofBoxProdRight⟩" } ]

[ 194, 27 ]

[ 189, 11 ]

[ { "state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.64809\nι : Sort ?u.64812\nπ : α → Type ?u.64817\ninst✝ : Nonempty α\ns : Set α\nf : α → β\na : α\nb : β\nh : ∃ a, a ∈ s ∧ f a = b\n⊢ Classical.choose h ∈ s ∧ f (Classical.choose h) = b", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.64809\nι : Sort ?u.64812\nπ : α → Type ?u.64817\ninst✝ : Nonempty α\ns : Set α\nf : α → β\na : α\nb : β\nh : ∃ a, a ∈ s ∧ f a = b\n⊢ invFunOn f s b ∈ s ∧ f (invFunOn f s b) = b", "tactic": "rw [invFunOn, dif_pos h]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.64809\nι : Sort ?u.64812\nπ : α → Type ?u.64817\ninst✝ : Nonempty α\ns : Set α\nf : α → β\na : α\nb : β\nh : ∃ a, a ∈ s ∧ f a = b\n⊢ Classical.choose h ∈ s ∧ f (Classical.choose h) = b", "tactic": "exact Classical.choose_spec h" } ]

[ 1215, 32 ]

[ 1213, 1 ]

[ { "state_after": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nu : ℕ → E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : ¬↑↑μ s = 0\n⊢ False", "state_before": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nu : ℕ → E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\n⊢ ↑↑μ s = 0", "tactic": "by_contra h" }, { "state_after": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nu : ℕ → E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : ¬↑↑μ s = 0\n⊢ ⊤ < ⊤", "state_before": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nu : ℕ → E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : ¬↑↑μ s = 0\n⊢ False", "tactic": "apply lt_irrefl ∞" }, { "state_after": "no goals", "state_before": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nu : ℕ → E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : ¬↑↑μ s = 0\n⊢ ⊤ < ⊤", "tactic": "calc\n ∞ = ∑' _ : ℕ, μ s := (ENNReal.tsum_const_eq_top_of_ne_zero h).symm\n _ = ∑' n : ℕ, μ ({u n} + s) := by\n congr 1; ext1 n; simp only [image_add_left, measure_preimage_add, singleton_add]\n _ = μ (⋃ n, {u n} + s) := Eq.symm <| measure_iUnion hs fun n => by\n simpa only [image_add_left, singleton_add] using measurable_id.const_add _ h's\n _ = μ (range u + s) := by rw [← iUnion_add, iUnion_singleton_eq_range]\n _ < ∞ := Bounded.measure_lt_top (hu.add sb)" }, { "state_after": "case e_f\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nu : ℕ → E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : ¬↑↑μ s = 0\n⊢ (fun x => ↑↑μ s) = fun n => ↑↑μ ({u n} + s)", "state_before": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nu : ℕ → E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : ¬↑↑μ s = 0\n⊢ (∑' (x : ℕ), ↑↑μ s) = ∑' (n : ℕ), ↑↑μ ({u n} + s)", "tactic": "congr 1" }, { "state_after": "case e_f.h\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nu : ℕ → E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : ¬↑↑μ s = 0\nn : ℕ\n⊢ ↑↑μ s = ↑↑μ ({u n} + s)", "state_before": "case e_f\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nu : ℕ → E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : ¬↑↑μ s = 0\n⊢ (fun x => ↑↑μ s) = fun n => ↑↑μ ({u n} + s)", "tactic": "ext1 n" }, { "state_after": "no goals", "state_before": "case e_f.h\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nu : ℕ → E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : ¬↑↑μ s = 0\nn : ℕ\n⊢ ↑↑μ s = ↑↑μ ({u n} + s)", "tactic": "simp only [image_add_left, measure_preimage_add, singleton_add]" }, { "state_after": "no goals", "state_before": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nu : ℕ → E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : ¬↑↑μ s = 0\nn : ℕ\n⊢ MeasurableSet ({u n} + s)", "tactic": "simpa only [image_add_left, singleton_add] using measurable_id.const_add _ h's" }, { "state_after": "no goals", "state_before": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nu : ℕ → E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : ¬↑↑μ s = 0\n⊢ ↑↑μ (⋃ (n : ℕ), {u n} + s) = ↑↑μ (range u + s)", "tactic": "rw [← iUnion_add, iUnion_singleton_eq_range]" } ]

[ 134, 48 ]

[ 121, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ IsCompact (U ∩ V)", "state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\n⊢ IsQuasiSeparated (f '' s)", "tactic": "intro U V hU hU' hU'' hV hV' hV''" }, { "state_after": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ U ∩ V = f '' (f ⁻¹' U ∩ f ⁻¹' V)\n\ncase convert_1\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ f ⁻¹' U ⊆ s\n\ncase convert_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ IsCompact (f ⁻¹' U)\n\ncase convert_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ f ⁻¹' V ⊆ s\n\ncase convert_4\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ IsCompact (f ⁻¹' V)", "state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ IsCompact (U ∩ V)", "tactic": "convert\n (H (f ⁻¹' U) (f ⁻¹' V)\n ?_ (h.continuous.1 _ hU') ?_ ?_ (h.continuous.1 _ hV') ?_).image h.continuous" }, { "state_after": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ f '' (f ⁻¹' U ∩ f ⁻¹' V) = U ∩ V", "state_before": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ U ∩ V = f '' (f ⁻¹' U ∩ f ⁻¹' V)", "tactic": "symm" }, { "state_after": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ U ∩ V ⊆ Set.range f", "state_before": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ f '' (f ⁻¹' U ∩ f ⁻¹' V) = U ∩ V", "tactic": "rw [← Set.preimage_inter, Set.image_preimage_eq_inter_range, Set.inter_eq_left_iff_subset]" }, { "state_after": "no goals", "state_before": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ U ∩ V ⊆ Set.range f", "tactic": "exact (Set.inter_subset_left _ _).trans (hU.trans (Set.image_subset_range _ _))" }, { "state_after": "case convert_1\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\nx : α\nhx : x ∈ f ⁻¹' U\n⊢ x ∈ s", "state_before": "case convert_1\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ f ⁻¹' U ⊆ s", "tactic": "intro x hx" }, { "state_after": "case convert_1\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\nx : α\nhx : x ∈ f ⁻¹' U\n⊢ f x ∈ f '' s", "state_before": "case convert_1\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\nx : α\nhx : x ∈ f ⁻¹' U\n⊢ x ∈ s", "tactic": "rw [← (h.inj.injOn _).mem_image_iff (Set.subset_univ _) trivial]" }, { "state_after": "no goals", "state_before": "case convert_1\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\nx : α\nhx : x ∈ f ⁻¹' U\n⊢ f x ∈ f '' s", "tactic": "exact hU hx" }, { "state_after": "case convert_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ IsCompact (f '' (f ⁻¹' U))", "state_before": "case convert_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ IsCompact (f ⁻¹' U)", "tactic": "rw [h.isCompact_iff_isCompact_image]" }, { "state_after": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ f '' (f ⁻¹' U) = U", "state_before": "case convert_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ IsCompact (f '' (f ⁻¹' U))", "tactic": "convert hU''" }, { "state_after": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ U ⊆ Set.range f", "state_before": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ f '' (f ⁻¹' U) = U", "tactic": "rw [Set.image_preimage_eq_inter_range, Set.inter_eq_left_iff_subset]" }, { "state_after": "no goals", "state_before": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ U ⊆ Set.range f", "tactic": "exact hU.trans (Set.image_subset_range _ _)" }, { "state_after": "case convert_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\nx : α\nhx : x ∈ f ⁻¹' V\n⊢ x ∈ s", "state_before": "case convert_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ f ⁻¹' V ⊆ s", "tactic": "intro x hx" }, { "state_after": "case convert_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\nx : α\nhx : x ∈ f ⁻¹' V\n⊢ f x ∈ f '' s", "state_before": "case convert_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\nx : α\nhx : x ∈ f ⁻¹' V\n⊢ x ∈ s", "tactic": "rw [← (h.inj.injOn _).mem_image_iff (Set.subset_univ _) trivial]" }, { "state_after": "no goals", "state_before": "case convert_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\nx : α\nhx : x ∈ f ⁻¹' V\n⊢ f x ∈ f '' s", "tactic": "exact hV hx" }, { "state_after": "case convert_4\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ IsCompact (f '' (f ⁻¹' V))", "state_before": "case convert_4\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ IsCompact (f ⁻¹' V)", "tactic": "rw [h.isCompact_iff_isCompact_image]" }, { "state_after": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ f '' (f ⁻¹' V) = V", "state_before": "case convert_4\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ IsCompact (f '' (f ⁻¹' V))", "tactic": "convert hV''" }, { "state_after": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ V ⊆ Set.range f", "state_before": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ f '' (f ⁻¹' V) = V", "tactic": "rw [Set.image_preimage_eq_inter_range, Set.inter_eq_left_iff_subset]" }, { "state_after": "no goals", "state_before": "case h.e'_3\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : Embedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen V\nhV'' : IsCompact V\n⊢ V ⊆ Set.range f", "tactic": "exact hV.trans (Set.image_subset_range _ _)" } ]

[ 91, 48 ]

[ 69, 1 ]

[]

[ 372, 39 ]

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[]

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[ { "state_after": "α : Type u_1\nβ : Type ?u.7617\ninst✝ : PartialOrder α\na✝ b : α\na : ↑(Iic b)\n⊢ a ⋖ ⊤ ↔ ↑a ⋖ b", "state_before": "α : Type u_1\nβ : Type ?u.7617\ninst✝ : PartialOrder α\na✝ b : α\na : ↑(Iic b)\n⊢ IsCoatom a ↔ ↑a ⋖ b", "tactic": "rw [← covby_top_iff]" }, { "state_after": "α : Type u_1\nβ : Type ?u.7617\ninst✝ : PartialOrder α\na✝ b : α\na : ↑(Iic b)\n⊢ OrdConnected (range ↑(OrderEmbedding.subtype fun c => c ≤ b))", "state_before": "α : Type u_1\nβ : Type ?u.7617\ninst✝ : PartialOrder α\na✝ b : α\na : ↑(Iic b)\n⊢ a ⋖ ⊤ ↔ ↑a ⋖ b", "tactic": "refine' (Set.OrdConnected.apply_covby_apply_iff (OrderEmbedding.subtype fun c => c ≤ b) _).symm" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.7617\ninst✝ : PartialOrder α\na✝ b : α\na : ↑(Iic b)\n⊢ OrdConnected (range ↑(OrderEmbedding.subtype fun c => c ≤ b))", "tactic": "simpa only [OrderEmbedding.subtype_apply, Subtype.range_coe_subtype] using Set.ordConnected_Iic" } ]

[ 196, 98 ]

[ 193, 1 ]

[ { "state_after": "case h\n𝕜 : Type ?u.66870\nE : Type u_1\nF : Type ?u.66876\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nabsorbs : Absorbent ℝ s\na : ℝ\nx✝ : E\n⊢ x✝ ∈ {x | gauge s x < a} ↔ x✝ ∈ ⋃ (r : ℝ) (_ : r ∈ Ioo 0 a), r • s", "state_before": "𝕜 : Type ?u.66870\nE : Type u_1\nF : Type ?u.66876\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nabsorbs : Absorbent ℝ s\na : ℝ\n⊢ {x | gauge s x < a} = ⋃ (r : ℝ) (_ : r ∈ Ioo 0 a), r • s", "tactic": "ext" }, { "state_after": "case h\n𝕜 : Type ?u.66870\nE : Type u_1\nF : Type ?u.66876\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nabsorbs : Absorbent ℝ s\na : ℝ\nx✝ : E\n⊢ gauge s x✝ < a ↔ ∃ i, 0 < i ∧ i < a ∧ x✝ ∈ i • s", "state_before": "case h\n𝕜 : Type ?u.66870\nE : Type u_1\nF : Type ?u.66876\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nabsorbs : Absorbent ℝ s\na : ℝ\nx✝ : E\n⊢ x✝ ∈ {x | gauge s x < a} ↔ x✝ ∈ ⋃ (r : ℝ) (_ : r ∈ Ioo 0 a), r • s", "tactic": "simp_rw [mem_setOf, mem_iUnion, exists_prop, mem_Ioo, and_assoc]" }, { "state_after": "no goals", "state_before": "case h\n𝕜 : Type ?u.66870\nE : Type u_1\nF : Type ?u.66876\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na✝ : ℝ\nabsorbs : Absorbent ℝ s\na : ℝ\nx✝ : E\n⊢ gauge s x✝ < a ↔ ∃ i, 0 < i ∧ i < a ∧ x✝ ∈ i • s", "tactic": "exact\n ⟨exists_lt_of_gauge_lt absorbs, fun ⟨r, hr₀, hr₁, hx⟩ =>\n (gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩" } ]

[ 188, 48 ]

[ 182, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nι : Sort u_1\nι' : Sort u_2\np : ι → Prop\np' : ι' → Prop\ns : ι → Set (α × α)\ns' : ι' → Set (β × β)\nh : HasBasis (𝓤 α) p s\nh' : HasBasis (𝓤 β) p' s'\nf : α → β\n⊢ UniformEmbedding f ↔\n Injective f ∧\n (∀ (i : ι'), p' i → ∃ j, p j ∧ ∀ (x y : α), (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧\n ∀ (j : ι), p j → ∃ i, p' i ∧ ∀ (x y : α), (f x, f y) ∈ s' i → (x, y) ∈ s j", "tactic": "rw [uniformEmbedding_iff, and_comm, h.uniformInducing_iff h']" } ]

[ 148, 64 ]

[ 143, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Type ?u.161852\nα : Type u_1\nβ : Type ?u.161858\ninst✝ : HeytingAlgebra α\na✝ b c a : α\n⊢ a ≤ ⊤ᶜ ↔ a ≤ ⊥", "tactic": "rw [le_compl_iff_disjoint_right, disjoint_top, le_bot_iff]" } ]

[ 891, 93 ]

[ 890, 1 ]

[]

[ 794, 6 ]

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[ { "state_after": "no goals", "state_before": "α : Type u_1\nx : α\nlt : α → α → Bool\nself : BinaryHeap α lt\ne : max self = some x\nh : ¬0 < Array.size self.arr\n⊢ False", "tactic": "simp [BinaryHeap.max, Array.get?, h] at e" } ]

[ 129, 95 ]

[ 128, 1 ]

[]

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[]

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[]

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[ { "state_after": "α : Type u_2\nβ : Type ?u.1911\nγ : Type ?u.1914\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : Encodable ι\ns : ι → Set α\nT : ι → Set (Set α)\nhTo : ∀ (i : ι) (t : Set α), t ∈ T i → IsOpen t\nhTc : ∀ (i : ι), Set.Countable (T i)\nhTs : ∀ (i : ι), s i = ⋂₀ T i\n⊢ IsGδ (⋂ (i : ι), s i)", "state_before": "α : Type u_2\nβ : Type ?u.1911\nγ : Type ?u.1914\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : Encodable ι\ns : ι → Set α\nhs : ∀ (i : ι), IsGδ (s i)\n⊢ IsGδ (⋂ (i : ι), s i)", "tactic": "choose T hTo hTc hTs using hs" }, { "state_after": "α : Type u_2\nβ : Type ?u.1911\nγ : Type ?u.1914\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : Encodable ι\nT : ι → Set (Set α)\nhTo : ∀ (i : ι) (t : Set α), t ∈ T i → IsOpen t\nhTc : ∀ (i : ι), Set.Countable (T i)\nhTs : ∀ (i : ι), (fun i => ⋂₀ T i) i = ⋂₀ T i\n⊢ IsGδ (⋂ (i : ι), (fun i => ⋂₀ T i) i)", "state_before": "α : Type u_2\nβ : Type ?u.1911\nγ : Type ?u.1914\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : Encodable ι\ns : ι → Set α\nT : ι → Set (Set α)\nhTo : ∀ (i : ι) (t : Set α), t ∈ T i → IsOpen t\nhTc : ∀ (i : ι), Set.Countable (T i)\nhTs : ∀ (i : ι), s i = ⋂₀ T i\n⊢ IsGδ (⋂ (i : ι), s i)", "tactic": "obtain rfl : s = fun i => ⋂₀ T i := funext hTs" }, { "state_after": "α : Type u_2\nβ : Type ?u.1911\nγ : Type ?u.1914\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : Encodable ι\nT : ι → Set (Set α)\nhTo : ∀ (i : ι) (t : Set α), t ∈ T i → IsOpen t\nhTc : ∀ (i : ι), Set.Countable (T i)\nhTs : ∀ (i : ι), (fun i => ⋂₀ T i) i = ⋂₀ T i\n⊢ ∀ (t : Set α), (t ∈ ⋃ (i : ι), T i) → IsOpen t", "state_before": "α : Type u_2\nβ : Type ?u.1911\nγ : Type ?u.1914\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : Encodable ι\nT : ι → Set (Set α)\nhTo : ∀ (i : ι) (t : Set α), t ∈ T i → IsOpen t\nhTc : ∀ (i : ι), Set.Countable (T i)\nhTs : ∀ (i : ι), (fun i => ⋂₀ T i) i = ⋂₀ T i\n⊢ IsGδ (⋂ (i : ι), (fun i => ⋂₀ T i) i)", "tactic": "refine' ⟨⋃ i, T i, _, countable_iUnion hTc, (sInter_iUnion _).symm⟩" }, { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type ?u.1911\nγ : Type ?u.1914\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : Encodable ι\nT : ι → Set (Set α)\nhTo : ∀ (i : ι) (t : Set α), t ∈ T i → IsOpen t\nhTc : ∀ (i : ι), Set.Countable (T i)\nhTs : ∀ (i : ι), (fun i => ⋂₀ T i) i = ⋂₀ T i\n⊢ ∀ (t : Set α), (t ∈ ⋃ (i : ι), T i) → IsOpen t", "tactic": "simpa [@forall_swap ι] using hTo" } ]

[ 88, 35 ]

[ 84, 1 ]

[]

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[ 219, 1 ]

[ { "state_after": "case a.h\nl : Type ?u.36987\nm : Type u_1\nn : Type u_2\no : Type ?u.36996\nm' : o → Type ?u.37001\nn' : o → Type ?u.37006\nR : Type ?u.37009\nS : Type ?u.37012\nα : Type v\nβ : Type w\nγ : Type ?u.37019\ninst✝¹ : Zero α\ninst✝ : Zero β\nf : α → β\nh : f 0 = 0\ni✝ : m\nx✝ : n\n⊢ map 0 f i✝ x✝ = OfNat.ofNat 0 i✝ x✝", "state_before": "l : Type ?u.36987\nm : Type u_1\nn : Type u_2\no : Type ?u.36996\nm' : o → Type ?u.37001\nn' : o → Type ?u.37006\nR : Type ?u.37009\nS : Type ?u.37012\nα : Type v\nβ : Type w\nγ : Type ?u.37019\ninst✝¹ : Zero α\ninst✝ : Zero β\nf : α → β\nh : f 0 = 0\n⊢ map 0 f = 0", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case a.h\nl : Type ?u.36987\nm : Type u_1\nn : Type u_2\no : Type ?u.36996\nm' : o → Type ?u.37001\nn' : o → Type ?u.37006\nR : Type ?u.37009\nS : Type ?u.37012\nα : Type v\nβ : Type w\nγ : Type ?u.37019\ninst✝¹ : Zero α\ninst✝ : Zero β\nf : α → β\nh : f 0 = 0\ni✝ : m\nx✝ : n\n⊢ map 0 f i✝ x✝ = OfNat.ofNat 0 i✝ x✝", "tactic": "simp [h]" } ]

[ 350, 11 ]

[ 347, 11 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.18761\nγ : Type ?u.18764\nδ : Type ?u.18767\ninst✝ : TopologicalSpace α\na : α\n⊢ 𝓝[{a}] a = pure a", "tactic": "rw [nhdsWithin, principal_singleton, inf_eq_right.2 (pure_le_nhds a)]" } ]

[ 282, 72 ]

[ 281, 1 ]

[]

[ 804, 84 ]

[ 800, 1 ]

[]

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[]

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[ 152, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type ?u.476654\nγ : Type ?u.476657\nι : Type ?u.476660\nM : Type u_3\nM' : Type ?u.476666\nN : Type u_1\nP : Type ?u.476672\nG : Type ?u.476675\nH : Type ?u.476678\nR : Type ?u.476681\nS : Type ?u.476684\ninst✝¹ : Zero M\np : α → Prop\nf : α →₀ M\ninst✝ : CommMonoid N\ng : α → M → N\n⊢ prod (filter p f) g * prod (filter (fun a => ¬p a) f) g = prod f g", "tactic": "classical simp_rw [prod_filter_index, support_filter, Finset.prod_filter_mul_prod_filter_not,\n Finsupp.prod]" }, { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type ?u.476654\nγ : Type ?u.476657\nι : Type ?u.476660\nM : Type u_3\nM' : Type ?u.476666\nN : Type u_1\nP : Type ?u.476672\nG : Type ?u.476675\nH : Type ?u.476678\nR : Type ?u.476681\nS : Type ?u.476684\ninst✝¹ : Zero M\np : α → Prop\nf : α →₀ M\ninst✝ : CommMonoid N\ng : α → M → N\n⊢ prod (filter p f) g * prod (filter (fun a => ¬p a) f) g = prod f g", "tactic": "simp_rw [prod_filter_index, support_filter, Finset.prod_filter_mul_prod_filter_not,\nFinsupp.prod]" } ]

[ 955, 18 ]

[ 952, 1 ]

[ { "state_after": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0 =\n bif f true false then m else 0", "state_before": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\n⊢ bitwise' f m 0 = bif f true false then m else 0", "tactic": "unfold bitwise'" }, { "state_after": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\n⊢ ∀ (b : Bool) (n : ℕ),\n binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) (bit b n) 0 =\n bif f true false then bit b n else 0", "state_before": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0 =\n bif f true false then m else 0", "tactic": "apply bitCasesOn m" }, { "state_after": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nb✝ : Bool\nn✝ : ℕ\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) (bit b✝ n✝) 0 =\n bif f true false then bit b✝ n✝ else 0", "state_before": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\n⊢ ∀ (b : Bool) (n : ℕ),\n binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) (bit b n) 0 =\n bif f true false then bit b n else 0", "tactic": "intros" }, { "state_after": "case h\nf : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nb✝ : Bool\nn✝ : ℕ\n⊢ (binaryRec (bif f true false then bit false 0 else 0) fun b n x => bit (f false b) (bif f false true then n else 0)) =\n fun n => bif f false true then n else 0", "state_before": "f : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nb✝ : Bool\nn✝ : ℕ\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) (bit b✝ n✝) 0 =\n bif f true false then bit b✝ n✝ else 0", "tactic": "rw [binaryRec_eq, binaryRec_zero]" }, { "state_after": "no goals", "state_before": "case h\nf : Bool → Bool → Bool\nh : f false false = false\nm : ℕ\nb✝ : Bool\nn✝ : ℕ\n⊢ (binaryRec (bif f true false then bit false 0 else 0) fun b n x => bit (f false b) (bif f false true then n else 0)) =\n fun n => bif f false true then n else 0", "tactic": "exact bitwise'_bit_aux h" } ]

[ 417, 29 ]

[ 414, 1 ]

[ { "state_after": "no goals", "state_before": "n m : Nat\n⊢ succ n - succ m = n - m", "tactic": "induction m with\n| zero => exact rfl\n| succ m ih => apply congrArg pred ih" }, { "state_after": "no goals", "state_before": "case zero\nn : Nat\n⊢ succ n - succ zero = n - zero", "tactic": "exact rfl" }, { "state_after": "no goals", "state_before": "case succ\nn m : Nat\nih : succ n - succ m = n - m\n⊢ succ n - succ (succ m) = n - succ m", "tactic": "apply congrArg pred ih" } ]

[ 221, 40 ]

[ 218, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\ni : ℕ\n⊢ approx (↑f) i ≤ approx (↑f) (Nat.succ i)", "tactic": "induction i with\n| zero => dsimp [approx]; apply @bot_le _ _ _ (f ⊥)\n| succ _ i_ih => intro ; apply f.monotone; apply i_ih" }, { "state_after": "case zero\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\n⊢ ⊥ ≤ ↑f ⊥", "state_before": "case zero\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\n⊢ approx (↑f) Nat.zero ≤ approx (↑f) (Nat.succ Nat.zero)", "tactic": "dsimp [approx]" }, { "state_after": "no goals", "state_before": "case zero\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\n⊢ ⊥ ≤ ↑f ⊥", "tactic": "apply @bot_le _ _ _ (f ⊥)" }, { "state_after": "case succ\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\nn✝ : ℕ\ni_ih : approx (↑f) n✝ ≤ approx (↑f) (Nat.succ n✝)\ni✝ : α\n⊢ approx (↑f) (Nat.succ n✝) i✝ ≤ approx (↑f) (Nat.succ (Nat.succ n✝)) i✝", "state_before": "case succ\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\nn✝ : ℕ\ni_ih : approx (↑f) n✝ ≤ approx (↑f) (Nat.succ n✝)\n⊢ approx (↑f) (Nat.succ n✝) ≤ approx (↑f) (Nat.succ (Nat.succ n✝))", "tactic": "intro" }, { "state_after": "case succ.a\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\nn✝ : ℕ\ni_ih : approx (↑f) n✝ ≤ approx (↑f) (Nat.succ n✝)\ni✝ : α\n⊢ approx (↑f) n✝ ≤ approx (↑f) (Nat.succ n✝)", "state_before": "case succ\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\nn✝ : ℕ\ni_ih : approx (↑f) n✝ ≤ approx (↑f) (Nat.succ n✝)\ni✝ : α\n⊢ approx (↑f) (Nat.succ n✝) i✝ ≤ approx (↑f) (Nat.succ (Nat.succ n✝)) i✝", "tactic": "apply f.monotone" }, { "state_after": "no goals", "state_before": "case succ.a\nα : Type u_1\nβ : α → Type u_2\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\nn✝ : ℕ\ni_ih : approx (↑f) n✝ ≤ approx (↑f) (Nat.succ n✝)\ni✝ : α\n⊢ approx (↑f) n✝ ≤ approx (↑f) (Nat.succ n✝)", "tactic": "apply i_ih" } ]

[ 63, 56 ]

[ 60, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.12733\ninst✝ : EDist α\nx y : α\ns t : Set α\nd : ℝ≥0∞\n⊢ einfsep s < d ↔ ∃ x x_1 y x_2 _h, edist x y < d", "tactic": "simp_rw [einfsep, iInf_lt_iff]" } ]

[ 88, 33 ]

[ 86, 1 ]

[ { "state_after": "α β : Type u\na b c : Cardinal\n⊢ (∀ (i : ↑(Iio b)), a ^ ↑i ≤ c) ↔ ∀ (x : Cardinal), x < b → a ^ x ≤ c\n\ncase h\nα β : Type u\na b c : Cardinal\n⊢ BddAbove (range fun c => a ^ ↑c)", "state_before": "α β : Type u\na b c : Cardinal\n⊢ a ^< b ≤ c ↔ ∀ (x : Cardinal), x < b → a ^ x ≤ c", "tactic": "rw [powerlt, ciSup_le_iff']" }, { "state_after": "no goals", "state_before": "α β : Type u\na b c : Cardinal\n⊢ (∀ (i : ↑(Iio b)), a ^ ↑i ≤ c) ↔ ∀ (x : Cardinal), x < b → a ^ x ≤ c", "tactic": "simp" }, { "state_after": "case h\nα β : Type u\na b c : Cardinal\n⊢ BddAbove (HPow.hPow a '' Iio b)", "state_before": "case h\nα β : Type u\na b c : Cardinal\n⊢ BddAbove (range fun c => a ^ ↑c)", "tactic": "rw [← image_eq_range]" }, { "state_after": "no goals", "state_before": "case h\nα β : Type u\na b c : Cardinal\n⊢ BddAbove (HPow.hPow a '' Iio b)", "tactic": "exact bddAbove_image.{u, u} _ bddAbove_Iio" } ]

[ 2291, 47 ]

[ 2287, 1 ]

[]

[ 76, 83 ]

[ 76, 9 ]

[]

[ 1807, 83 ]

[ 1804, 1 ]

[ { "state_after": "no goals", "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type ?u.909458\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nh : M ≅[L] N\n⊢ M ⊨ T ↔ N ⊨ T", "tactic": "rw [Theory.model_iff_subset_completeTheory, Theory.model_iff_subset_completeTheory,\n h.completeTheory_eq]" } ]

[ 1120, 25 ]

[ 1118, 1 ]

[]

[ 838, 56 ]

[ 836, 1 ]

[ { "state_after": "α : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b c a : α\n⊢ a + -a ≤ a + abs a", "state_before": "α : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b c a : α\n⊢ 0 ≤ a + abs a", "tactic": "rw [← add_right_neg a]" }, { "state_after": "case bc\nα : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b c a : α\n⊢ -a ≤ abs a", "state_before": "α : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b c a : α\n⊢ a + -a ≤ a + abs a", "tactic": "apply add_le_add_left" }, { "state_after": "no goals", "state_before": "case bc\nα : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na✝ b c a : α\n⊢ -a ≤ abs a", "tactic": "exact neg_le_abs_self a" } ]

[ 165, 26 ]

[ 162, 1 ]

[]

[ 326, 6 ]

[ 321, 1 ]

[]

[ 77, 6 ]

[ 76, 1 ]

[]

[ 419, 6 ]

[ 418, 1 ]

[ { "state_after": "no goals", "state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\n⊢ (𝟙 X - p) ≫ (𝟙 X - p) = 𝟙 X - p", "tactic": "simp only [comp_sub, sub_comp, id_comp, comp_id, hp, sub_self, sub_zero]" } ]

[ 104, 75 ]

[ 102, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\np q : α → Prop\nx : α\n⊢ {x | x ∈ s ∧ p x} = {x | x ∈ s ∧ q x} ↔ ∀ (x : α), x ∈ s → (p x ↔ q x)", "tactic": "simp_rw [ext_iff, mem_sep_iff, and_congr_right_iff]" } ]

[ 1421, 54 ]

[ 1420, 1 ]

[]

[ 469, 29 ]

[ 468, 1 ]

[]

[ 915, 39 ]

[ 914, 1 ]

[]

[ 337, 6 ]

[ 336, 1 ]

[]

[ 264, 58 ]

[ 262, 1 ]

[ { "state_after": "case h.e'_1\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))\n⊢ ↑(Int.gcd x y) ∣ x\n\ncase h.e'_2\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))\n⊢ ↑(Int.gcd x y) ∣ y\n\ncase h.e'_3\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))\n⊢ ↑(Int.gcd x y) ∣ z", "state_before": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))\n⊢ IsClassified h", "tactic": "convert h.normalize.mul_isClassified (Int.gcd x y)\n (isClassified_of_isPrimitiveClassified h.normalize hc) <;>\n rw [Int.mul_ediv_cancel']" }, { "state_after": "no goals", "state_before": "case h.e'_1\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))\n⊢ ↑(Int.gcd x y) ∣ x", "tactic": "exact Int.gcd_dvd_left x y" }, { "state_after": "no goals", "state_before": "case h.e'_2\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))\n⊢ ↑(Int.gcd x y) ∣ y", "tactic": "exact Int.gcd_dvd_right x y" }, { "state_after": "no goals", "state_before": "case h.e'_3\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : IsPrimitiveClassified (_ : PythagoreanTriple (x / ↑(Int.gcd x y)) (y / ↑(Int.gcd x y)) (z / ↑(Int.gcd x y)))\n⊢ ↑(Int.gcd x y) ∣ z", "tactic": "exact h.gcd_dvd" } ]

[ 225, 20 ]

[ 218, 1 ]

[]

[ 374, 26 ]

[ 373, 1 ]

[]

[ 201, 27 ]

[ 199, 1 ]

[]

[ 44, 39 ]

[ 43, 1 ]

[ { "state_after": "α : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ SimpleFunc.range f\n⊢ 0 ≤ ↑(T (↑f ⁻¹' {i})) i", "state_before": "α : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\n⊢ 0 ≤ setToSimpleFunc T f", "tactic": "refine' sum_nonneg fun i hi => _" }, { "state_after": "case pos\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ SimpleFunc.range f\nh0 : i = 0\n⊢ 0 ≤ ↑(T (↑f ⁻¹' {i})) i\n\ncase neg\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ SimpleFunc.range f\nh0 : ¬i = 0\n⊢ 0 ≤ ↑(T (↑f ⁻¹' {i})) i", "state_before": "α : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ SimpleFunc.range f\n⊢ 0 ≤ ↑(T (↑f ⁻¹' {i})) i", "tactic": "by_cases h0 : i = 0" }, { "state_after": "case neg\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ SimpleFunc.range f\nh0 : ¬i = 0\n⊢ 0 ≤ i", "state_before": "case neg\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ SimpleFunc.range f\nh0 : ¬i = 0\n⊢ 0 ≤ ↑(T (↑f ⁻¹' {i})) i", "tactic": "refine'\n hT_nonneg _ (measurableSet_fiber _ _) (measure_preimage_lt_top_of_integrable _ hfi h0) i _" }, { "state_after": "case neg\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ Set.range ↑f\nh0 : ¬i = 0\n⊢ 0 ≤ i", "state_before": "case neg\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ SimpleFunc.range f\nh0 : ¬i = 0\n⊢ 0 ≤ i", "tactic": "rw [mem_range] at hi" }, { "state_after": "case neg.intro\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ Set.range ↑f\nh0 : ¬i = 0\ny : α\nhy : ↑f y = i\n⊢ 0 ≤ i", "state_before": "case neg\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ Set.range ↑f\nh0 : ¬i = 0\n⊢ 0 ≤ i", "tactic": "obtain ⟨y, hy⟩ := Set.mem_range.mp hi" }, { "state_after": "case neg.intro\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ Set.range ↑f\nh0 : ¬i = 0\ny : α\nhy : ↑f y = i\n⊢ 0 ≤ ↑f y", "state_before": "case neg.intro\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ Set.range ↑f\nh0 : ¬i = 0\ny : α\nhy : ↑f y = i\n⊢ 0 ≤ i", "tactic": "rw [← hy]" }, { "state_after": "no goals", "state_before": "case neg.intro\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ Set.range ↑f\nh0 : ¬i = 0\ny : α\nhy : ↑f y = i\n⊢ 0 ≤ ↑f y", "tactic": "convert hf y" }, { "state_after": "no goals", "state_before": "case pos\nα : Type u_1\nE : Type ?u.361138\nF : Type ?u.361141\nF' : Type ?u.361144\nG : Type ?u.361147\n𝕜 : Type ?u.361150\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : NormedLatticeAddCommGroup G''\ninst✝² : NormedSpace ℝ G''\ninst✝¹ : NormedLatticeAddCommGroup G'\ninst✝ : NormedSpace ℝ G'\nT : Set α → G' →L[ℝ] G''\nhT_nonneg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ ↑(T s) x\nf : α →ₛ G'\nhf : 0 ≤ f\nhfi : Integrable ↑f\ni : G'\nhi : i ∈ SimpleFunc.range f\nh0 : i = 0\n⊢ 0 ≤ ↑(T (↑f ⁻¹' {i})) i", "tactic": "simp [h0]" } ]

[ 554, 15 ]

[ 543, 1 ]

[]

[ 1726, 78 ]

[ 1724, 1 ]

[ { "state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\na b ad bd a' b' gcd : α\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n⊢ 0 < ad * bd\n\nα : Type u_1\ninst✝ : LinearOrderedField α\na b ad bd a' b' gcd : α\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n⊢ 0 < 1 / gcd", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na b ad bd a' b' gcd : α\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n⊢ (a ≤ b) = (1 / gcd * (bd * a') ≤ 1 / gcd * (ad * b'))", "tactic": "rw [mul_le_mul_left, ← ha, ← hb, ← mul_assoc, ← mul_assoc, mul_comm bd, mul_le_mul_left]" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na b ad bd a' b' gcd : α\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n⊢ 0 < ad * bd", "tactic": "exact mul_pos had hbd" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na b ad bd a' b' gcd : α\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n⊢ 0 < 1 / gcd", "tactic": "exact one_div_pos.2 hgcd" } ]

[ 77, 29 ]

[ 72, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.78564\nι : Sort x\nf g : Filter α\ns t : Set α\n⊢ ⊤ ≤ 𝓟 univ", "tactic": "simp only [le_principal_iff, mem_top, eq_self_iff_true]" } ]

[ 669, 75 ]

[ 668, 9 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderClosedTopology α\na✝ b✝ : α\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace β\na b : α\nf : α → β\nh : a < b\n⊢ ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a", "tactic": "simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsWithin_Ici h]" } ]

[ 555, 69 ]

[ 553, 1 ]

[]

[ 330, 41 ]

[ 329, 1 ]

[]

[ 101, 46 ]

[ 100, 1 ]

[]

[ 854, 44 ]

[ 852, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.60622\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : β →ₛ γ\ng : α → β\nhgm : Measurable g\n⊢ ↑(SimpleFunc.range (comp f g hgm)) ⊆ ↑(SimpleFunc.range f)", "tactic": "simp only [coe_range, coe_comp, Set.range_comp_subset_range]" } ]

[ 358, 89 ]

[ 356, 1 ]

[]

[ 171, 6 ]

[ 170, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => a + x) ⁻¹' Ioo b c = Ioo (b - a) (c - a)", "tactic": "simp [← Ioi_inter_Iio]" } ]

[ 79, 25 ]

[ 78, 1 ]

[ { "state_after": "R : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : DecidableEq ι\nb : Basis ι R M\ninst✝ : _root_.Finite ι\ni j : ι\n⊢ ↑(↑(LinearMap.comp (toDual (dualBasis b)) (toDual b)) (↑b i)) (↑(dualBasis b) j) =\n ↑(↑(Dual.eval R M) (↑b i)) (↑(dualBasis b) j)", "state_before": "R : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : DecidableEq ι\nb : Basis ι R M\ninst✝ : _root_.Finite ι\n⊢ LinearMap.comp (toDual (dualBasis b)) (toDual b) = Dual.eval R M", "tactic": "refine' b.ext fun i => b.dualBasis.ext fun j => _" }, { "state_after": "no goals", "state_before": "R : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : DecidableEq ι\nb : Basis ι R M\ninst✝ : _root_.Finite ι\ni j : ι\n⊢ ↑(↑(LinearMap.comp (toDual (dualBasis b)) (toDual b)) (↑b i)) (↑(dualBasis b) j) =\n ↑(↑(Dual.eval R M) (↑b i)) (↑(dualBasis b) j)", "tactic": "rw [LinearMap.comp_apply, toDual_apply_left, coe_toDual_self, ← coe_dualBasis,\n Dual.eval_apply, Basis.repr_self, Finsupp.single_apply, dualBasis_apply_self]" } ]

[ 456, 82 ]

[ 453, 1 ]

[ { "state_after": "no goals", "state_before": "J : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd", "tactic": "simpa using ConcreteCategory.congr_hom pullback.condition x" }, { "state_after": "case a\nJ : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ ↑((forget TopCat).map (pullbackIsoProdSubtype f g).hom x) =\n ↑{ val := ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x),\n property :=\n (_ :\n (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd) }", "state_before": "J : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ (forget TopCat).map (pullbackIsoProdSubtype f g).hom x =\n { val := ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x),\n property :=\n (_ :\n (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd) }", "tactic": "apply Subtype.ext" }, { "state_after": "case a.h₁\nJ : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ (↑((forget TopCat).map (pullbackIsoProdSubtype f g).hom x)).fst =\n (↑{ val := ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x),\n property :=\n (_ :\n (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g\n ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd) }).fst\n\ncase a.h₂\nJ : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ (↑((forget TopCat).map (pullbackIsoProdSubtype f g).hom x)).snd =\n (↑{ val := ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x),\n property :=\n (_ :\n (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g\n ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd) }).snd", "state_before": "case a\nJ : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ ↑((forget TopCat).map (pullbackIsoProdSubtype f g).hom x) =\n ↑{ val := ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x),\n property :=\n (_ :\n (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd) }", "tactic": "apply Prod.ext" }, { "state_after": "no goals", "state_before": "case a.h₁\nJ : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ (↑((forget TopCat).map (pullbackIsoProdSubtype f g).hom x)).fst =\n (↑{ val := ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x),\n property :=\n (_ :\n (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g\n ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd) }).fst\n\ncase a.h₂\nJ : Type v\ninst✝ : SmallCategory J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ConcreteCategory.forget.obj (pullback f g)\n⊢ (↑((forget TopCat).map (pullbackIsoProdSubtype f g).hom x)).snd =\n (↑{ val := ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x),\n property :=\n (_ :\n (forget TopCat).map f ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).fst =\n (forget TopCat).map g\n ((forget TopCat).map pullback.fst x, (forget TopCat).map pullback.snd x).snd) }).snd", "tactic": "exacts [ConcreteCategory.congr_hom (pullbackIsoProdSubtype_hom_fst f g) x,\n ConcreteCategory.congr_hom (pullbackIsoProdSubtype_hom_snd f g) x]" } ]

[ 145, 71 ]

[ 138, 1 ]

[ { "state_after": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : C\nf : X ⟶ Y\nhf : Epi f\nZ : D\ng h : F.obj Y ⟶ Z\nH : F.map f ≫ g = F.map f ≫ h\n⊢ g = h", "state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : C\nf : X ⟶ Y\nhf : Epi f\n⊢ ∀ {Z : D} (g h : F.obj Y ⟶ Z), F.map f ≫ g = F.map f ≫ h → g = h", "tactic": "intro Z g h H" }, { "state_after": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : C\nf : X ⟶ Y\nhf : Epi f\nZ : D\ng h : F.obj Y ⟶ Z\nH : ↑(Adjunction.homEquiv adj X Z) (F.map f ≫ g) = ↑(Adjunction.homEquiv adj X Z) (F.map f ≫ h)\n⊢ g = h", "state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : C\nf : X ⟶ Y\nhf : Epi f\nZ : D\ng h : F.obj Y ⟶ Z\nH : F.map f ≫ g = F.map f ≫ h\n⊢ g = h", "tactic": "replace H := congr_arg (adj.homEquiv X Z) H" }, { "state_after": "no goals", "state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : C\nf : X ⟶ Y\nhf : Epi f\nZ : D\ng h : F.obj Y ⟶ Z\nH : ↑(Adjunction.homEquiv adj X Z) (F.map f ≫ g) = ↑(Adjunction.homEquiv adj X Z) (F.map f ≫ h)\n⊢ g = h", "tactic": "rwa [adj.homEquiv_naturality_left, adj.homEquiv_naturality_left, cancel_epi,\n Equiv.apply_eq_iff_eq] at H" } ]

[ 185, 41 ]

[ 178, 1 ]

[]

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[]

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[]

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[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.814900\nδ : Type ?u.814903\ninst✝¹ : MeasurableSpace α\nK : Type ?u.814909\ninst✝ : Zero β\nr : β\ns : Set α\nf : α →ₛ β\nhr : r ∈ SimpleFunc.range (restrict f s)\nh0 : r ≠ 0\nhs : MeasurableSet s\n⊢ r ∈ ↑f '' s", "tactic": "simpa [mem_restrict_range hs, h0, -mem_range] using hr" }, { "state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.814900\nδ : Type ?u.814903\ninst✝¹ : MeasurableSpace α\nK : Type ?u.814909\ninst✝ : Zero β\nr : β\ns : Set α\nf : α →ₛ β\nhr : r ∈ SimpleFunc.range 0\nh0 : r ≠ 0\nhs : ¬MeasurableSet s\n⊢ r ∈ ↑f '' s", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.814900\nδ : Type ?u.814903\ninst✝¹ : MeasurableSpace α\nK : Type ?u.814909\ninst✝ : Zero β\nr : β\ns : Set α\nf : α →ₛ β\nhr : r ∈ SimpleFunc.range (restrict f s)\nh0 : r ≠ 0\nhs : ¬MeasurableSet s\n⊢ r ∈ ↑f '' s", "tactic": "rw [restrict_of_not_measurable hs] at hr" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.814900\nδ : Type ?u.814903\ninst✝¹ : MeasurableSpace α\nK : Type ?u.814909\ninst✝ : Zero β\nr : β\ns : Set α\nf : α →ₛ β\nhr : r ∈ SimpleFunc.range 0\nh0 : r ≠ 0\nhs : ¬MeasurableSet s\n⊢ r ∈ ↑f '' s", "tactic": "exact (h0 <| eq_zero_of_mem_range_zero hr).elim" } ]

[ 807, 52 ]

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[]

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[ 197, 1 ]

[ { "state_after": "case hab\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nh : a ∣ b - c\n⊢ gcd a b ∣ c\n\ncase hba\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nh : a ∣ b - c\n⊢ gcd a c ∣ b", "state_before": "α : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nh : a ∣ b - c\n⊢ gcd a b = gcd a c", "tactic": "apply dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _) <;>\n rw [dvd_gcd_iff] <;>\n refine' ⟨gcd_dvd_left _ _, _⟩" }, { "state_after": "case hab.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\n⊢ gcd a b ∣ c", "state_before": "case hab\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nh : a ∣ b - c\n⊢ gcd a b ∣ c", "tactic": "rcases h with ⟨d, hd⟩" }, { "state_after": "case hab.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : b = gcd a b * e\n⊢ gcd a b ∣ c", "state_before": "case hab.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\n⊢ gcd a b ∣ c", "tactic": "rcases gcd_dvd_right a b with ⟨e, he⟩" }, { "state_after": "case hab.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : b = gcd a b * e\nf : α\nhf : a = gcd a b * f\n⊢ gcd a b ∣ c", "state_before": "case hab.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : b = gcd a b * e\n⊢ gcd a b ∣ c", "tactic": "rcases gcd_dvd_left a b with ⟨f, hf⟩" }, { "state_after": "case hab.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : b = gcd a b * e\nf : α\nhf : a = gcd a b * f\n⊢ c = gcd a b * (e - f * d)", "state_before": "case hab.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : b = gcd a b * e\nf : α\nhf : a = gcd a b * f\n⊢ gcd a b ∣ c", "tactic": "use e - f * d" }, { "state_after": "no goals", "state_before": "case hab.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : b = gcd a b * e\nf : α\nhf : a = gcd a b * f\n⊢ c = gcd a b * (e - f * d)", "tactic": "rw [mul_sub, ← he, ← mul_assoc, ← hf, ← hd, sub_sub_cancel]" }, { "state_after": "case hba.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\n⊢ gcd a c ∣ b", "state_before": "case hba\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nh : a ∣ b - c\n⊢ gcd a c ∣ b", "tactic": "rcases h with ⟨d, hd⟩" }, { "state_after": "case hba.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : c = gcd a c * e\n⊢ gcd a c ∣ b", "state_before": "case hba.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\n⊢ gcd a c ∣ b", "tactic": "rcases gcd_dvd_right a c with ⟨e, he⟩" }, { "state_after": "case hba.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : c = gcd a c * e\nf : α\nhf : a = gcd a c * f\n⊢ gcd a c ∣ b", "state_before": "case hba.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : c = gcd a c * e\n⊢ gcd a c ∣ b", "tactic": "rcases gcd_dvd_left a c with ⟨f, hf⟩" }, { "state_after": "case hba.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : c = gcd a c * e\nf : α\nhf : a = gcd a c * f\n⊢ b = gcd a c * (e + f * d)", "state_before": "case hba.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : c = gcd a c * e\nf : α\nhf : a = gcd a c * f\n⊢ gcd a c ∣ b", "tactic": "use e + f * d" }, { "state_after": "no goals", "state_before": "case hba.intro.intro.intro\nα : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : c = gcd a c * e\nf : α\nhf : a = gcd a c * f\n⊢ b = gcd a c * (e + f * d)", "tactic": "rw [mul_add, ← he, ← mul_assoc, ← hf, ← hd, ← add_sub_assoc, add_comm c b, add_sub_cancel]" } ]

[ 994, 95 ]

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[]

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[]

[ 1067, 40 ]

[ 1065, 1 ]

[ { "state_after": "M : Type ?u.2777832\nN : Type ?u.2777835\nG : Type ?u.2777838\nR : Type u_1\nS : Type ?u.2777844\nF : Type ?u.2777847\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh : IsPrimitiveRoot ζ✝ k\ninst✝ : IsDomain R\nζ : R\nhζ : IsPrimitiveRoot ζ k\nhk : 1 < k\n⊢ (1 - ζ) * ∑ i in range k, ζ ^ i = 0", "state_before": "M : Type ?u.2777832\nN : Type ?u.2777835\nG : Type ?u.2777838\nR : Type u_1\nS : Type ?u.2777844\nF : Type ?u.2777847\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh : IsPrimitiveRoot ζ✝ k\ninst✝ : IsDomain R\nζ : R\nhζ : IsPrimitiveRoot ζ k\nhk : 1 < k\n⊢ ∑ i in range k, ζ ^ i = 0", "tactic": "refine' eq_zero_of_ne_zero_of_mul_left_eq_zero (sub_ne_zero_of_ne (hζ.ne_one hk).symm) _" }, { "state_after": "no goals", "state_before": "M : Type ?u.2777832\nN : Type ?u.2777835\nG : Type ?u.2777838\nR : Type u_1\nS : Type ?u.2777844\nF : Type ?u.2777847\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh : IsPrimitiveRoot ζ✝ k\ninst✝ : IsDomain R\nζ : R\nhζ : IsPrimitiveRoot ζ k\nhk : 1 < k\n⊢ (1 - ζ) * ∑ i in range k, ζ ^ i = 0", "tactic": "rw [mul_neg_geom_sum, hζ.pow_eq_one, sub_self]" } ]

[ 668, 49 ]

[ 665, 1 ]

[]

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[]

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[]

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[]

[ 2001, 44 ]

[ 2000, 1 ]

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[ 429, 76 ]

[ 425, 1 ]

[]

[ 64, 6 ]

[ 63, 1 ]

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[ 204, 27 ]

[ 200, 1 ]

[]

[ 253, 24 ]

[ 251, 11 ]

[]

[ 96, 35 ]

[ 95, 1 ]

[]

[ 1991, 55 ]

[ 1990, 1 ]