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Mathlib/Data/Multiset/Basic.lean
Multiset.map_comp_cons
[ { "state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.114268\nf : α → β\nt : α\nx✝ : Multiset α\n⊢ (map f ∘ cons t) x✝ = (cons (f t) ∘ map f) x✝", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.114268\nf : α → β\nt : α\n⊢ map f ∘ cons t = cons (f t) ∘ map f", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.114268\nf : α → β\nt : α\nx✝ : Multiset α\n⊢ (map f ∘ cons t) x✝ = (cons (f t) ∘ map f) x✝", "tactic": "simp" } ]
[ 1182, 7 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1180, 1 ]
Mathlib/MeasureTheory/Function/UniformIntegrable.lean
MeasureTheory.uniformIntegrable_finite
[ { "state_after": "case intro\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nhf : ∀ (i : ι), Memℒp (f i) p\nval✝ : Fintype ι\n⊢ UniformIntegrable f p μ", "state_before": "α : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nhf : ∀ (i : ι), Memℒp (f i) p\n⊢ UniformIntegrable f p μ", "tactic": "cases nonempty_fintype ι" }, { "state_after": "case intro\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nhf : ∀ (i : ι), Memℒp (f i) p\nval✝ : Fintype ι\n⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C", "state_before": "case intro\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nhf : ∀ (i : ι), Memℒp (f i) p\nval✝ : Fintype ι\n⊢ UniformIntegrable f p μ", "tactic": "refine' ⟨fun n => (hf n).1, unifIntegrable_finite μ hp_one hp_top hf, _⟩" }, { "state_after": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nhf : ∀ (i : ι), Memℒp (f i) p\nval✝ : Fintype ι\nhι : Nonempty ι\n⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C\n\ncase neg\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nhf : ∀ (i : ι), Memℒp (f i) p\nval✝ : Fintype ι\nhι : ¬Nonempty ι\n⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C", "state_before": "case intro\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nhf : ∀ (i : ι), Memℒp (f i) p\nval✝ : Fintype ι\n⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C", "tactic": "by_cases hι : Nonempty ι" }, { "state_after": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\n⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C", "state_before": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nhf : ∀ (i : ι), Memℒp (f i) p\nval✝ : Fintype ι\nhι : Nonempty ι\n⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C", "tactic": "choose _ hf using hf" }, { "state_after": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\nC : ℝ≥0∞ :=\n Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C", "state_before": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\n⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C", "tactic": "set C := (Finset.univ.image fun i : ι => snorm (f i) p μ).max'\n ⟨snorm (f hι.some) p μ, Finset.mem_image.2 ⟨hι.some, Finset.mem_univ _, rfl⟩⟩" }, { "state_after": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\nC : ℝ≥0∞ :=\n Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)\ni : ι\n⊢ snorm (f i) p μ ≤ ↑(ENNReal.toNNReal C)", "state_before": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\nC : ℝ≥0∞ :=\n Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C", "tactic": "refine' ⟨C.toNNReal, fun i => _⟩" }, { "state_after": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\nC : ℝ≥0∞ :=\n Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)\ni : ι\n⊢ snorm (f i) p μ ≤ C\n\ncase pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\nC : ℝ≥0∞ :=\n Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)\ni : ι\n⊢ C ≠ ⊤", "state_before": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\nC : ℝ≥0∞ :=\n Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)\ni : ι\n⊢ snorm (f i) p μ ≤ ↑(ENNReal.toNNReal C)", "tactic": "rw [ENNReal.coe_toNNReal]" }, { "state_after": "no goals", "state_before": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\nC : ℝ≥0∞ :=\n Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)\ni : ι\n⊢ snorm (f i) p μ ≤ C", "tactic": "exact Finset.le_max' (α := ℝ≥0∞) _ _ (Finset.mem_image.2 ⟨i, Finset.mem_univ _, rfl⟩)" }, { "state_after": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\nC : ℝ≥0∞ :=\n Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)\ni : ι\ny : ℝ≥0∞\nhy : y ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ\n⊢ y < ⊤", "state_before": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\nC : ℝ≥0∞ :=\n Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)\ni : ι\n⊢ C ≠ ⊤", "tactic": "refine' ne_of_lt ((Finset.max'_lt_iff _ _).2 fun y hy => _)" }, { "state_after": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\nC : ℝ≥0∞ :=\n Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)\ni : ι\ny : ℝ≥0∞\nhy : ∃ a, a ∈ Finset.univ ∧ snorm (f a) p μ = y\n⊢ y < ⊤", "state_before": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\nC : ℝ≥0∞ :=\n Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)\ni : ι\ny : ℝ≥0∞\nhy : y ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ\n⊢ y < ⊤", "tactic": "rw [Finset.mem_image] at hy" }, { "state_after": "case pos.intro.intro\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\nC : ℝ≥0∞ :=\n Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)\ni✝ i : ι\n⊢ snorm (f i) p μ < ⊤", "state_before": "case pos\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\nC : ℝ≥0∞ :=\n Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)\ni : ι\ny : ℝ≥0∞\nhy : ∃ a, a ∈ Finset.univ ∧ snorm (f a) p μ = y\n⊢ y < ⊤", "tactic": "obtain ⟨i, -, rfl⟩ := hy" }, { "state_after": "no goals", "state_before": "case pos.intro.intro\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), snorm (f i) p μ < ⊤\nC : ℝ≥0∞ :=\n Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)\n (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)\ni✝ i : ι\n⊢ snorm (f i) p μ < ⊤", "tactic": "exact hf i" }, { "state_after": "no goals", "state_before": "case neg\nα : Type u_3\nβ : Type u_2\nι : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ⊤\nhf : ∀ (i : ι), Memℒp (f i) p\nval✝ : Fintype ι\nhι : ¬Nonempty ι\n⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C", "tactic": "exact ⟨0, fun i => False.elim <| hι <| Nonempty.intro i⟩" } ]
[ 779, 61 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 764, 1 ]
Mathlib/Dynamics/OmegaLimit.lean
omegaLimit_image_eq
[ { "state_after": "no goals", "state_before": "τ : Type u_2\nα : Type u_4\nβ : Type u_3\nι : Type ?u.26859\ninst✝ : TopologicalSpace β\nf✝ : Filter τ\nϕ✝ : τ → α → β\ns s₁ s₂ : Set α\nα' : Type u_1\nϕ : τ → α' → β\nf : Filter τ\ng : α → α'\n⊢ ω f ϕ (g '' s) = ω f (fun t x => ϕ t (g x)) s", "tactic": "simp only [omegaLimit, image2_image_right]" } ]
[ 116, 98 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 115, 1 ]
Mathlib/Geometry/Manifold/ChartedSpace.lean
prodChartedSpace_chartAt
[]
[ 726, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 724, 1 ]
Mathlib/Topology/MetricSpace/Basic.lean
Metric.dist_le_diam_of_mem'
[ { "state_after": "α : Type u\nβ : Type v\nX : Type ?u.532598\nι : Type ?u.532601\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nh : EMetric.diam s ≠ ⊤\nhx : x ∈ s\nhy : y ∈ s\n⊢ ENNReal.toReal (edist x y) ≤ ENNReal.toReal (EMetric.diam s)", "state_before": "α : Type u\nβ : Type v\nX : Type ?u.532598\nι : Type ?u.532601\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nh : EMetric.diam s ≠ ⊤\nhx : x ∈ s\nhy : y ∈ s\n⊢ dist x y ≤ diam s", "tactic": "rw [diam, dist_edist]" }, { "state_after": "α : Type u\nβ : Type v\nX : Type ?u.532598\nι : Type ?u.532601\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nh : EMetric.diam s ≠ ⊤\nhx : x ∈ s\nhy : y ∈ s\n⊢ edist x y ≤ EMetric.diam s", "state_before": "α : Type u\nβ : Type v\nX : Type ?u.532598\nι : Type ?u.532601\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nh : EMetric.diam s ≠ ⊤\nhx : x ∈ s\nhy : y ∈ s\n⊢ ENNReal.toReal (edist x y) ≤ ENNReal.toReal (EMetric.diam s)", "tactic": "rw [ENNReal.toReal_le_toReal (edist_ne_top _ _) h]" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nX : Type ?u.532598\nι : Type ?u.532601\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\nh : EMetric.diam s ≠ ⊤\nhx : x ∈ s\nhy : y ∈ s\n⊢ edist x y ≤ EMetric.diam s", "tactic": "exact EMetric.edist_le_diam_of_mem hx hy" } ]
[ 2642, 43 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 2638, 1 ]
Mathlib/Analysis/SpecialFunctions/Exp.lean
Real.isTheta_exp_comp_one
[ { "state_after": "no goals", "state_before": "α : Type u_1\nx y z : ℝ\nl : Filter α\nf : α → ℝ\n⊢ ((fun x => exp (f x)) =Θ[l] fun x => 1) ↔ IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun x => abs (f x)", "tactic": "simp only [← exp_zero, isTheta_exp_comp_exp_comp, sub_zero]" } ]
[ 418, 62 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 416, 1 ]
Mathlib/Analysis/NormedSpace/Basic.lean
rescale_to_shell_semi_normed_zpow
[ { "state_after": "α : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\n⊢ ∃ n, c ^ n ≠ 0 ∧ ‖c ^ n • x‖ < ε ∧ ε / ‖c‖ ≤ ‖c ^ n • x‖ ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "state_before": "α : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\n⊢ ∃ n, c ^ n ≠ 0 ∧ ‖c ^ n • x‖ < ε ∧ ε / ‖c‖ ≤ ‖c ^ n • x‖ ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "tactic": "have xεpos : 0 < ‖x‖ / ε := div_pos ((Ne.symm hx).le_iff_lt.1 (norm_nonneg x)) εpos" }, { "state_after": "case intro\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\n⊢ ∃ n, c ^ n ≠ 0 ∧ ‖c ^ n • x‖ < ε ∧ ε / ‖c‖ ≤ ‖c ^ n • x‖ ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "state_before": "α : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\n⊢ ∃ n, c ^ n ≠ 0 ∧ ‖c ^ n • x‖ < ε ∧ ε / ‖c‖ ≤ ‖c ^ n • x‖ ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "tactic": "rcases exists_mem_Ico_zpow xεpos hc with ⟨n, hn⟩" }, { "state_after": "case intro\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\n⊢ ∃ n, c ^ n ≠ 0 ∧ ‖c ^ n • x‖ < ε ∧ ε / ‖c‖ ≤ ‖c ^ n • x‖ ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "state_before": "case intro\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\n⊢ ∃ n, c ^ n ≠ 0 ∧ ‖c ^ n • x‖ < ε ∧ ε / ‖c‖ ≤ ‖c ^ n • x‖ ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "tactic": "have cpos : 0 < ‖c‖ := lt_trans (zero_lt_one : (0 : ℝ) < 1) hc" }, { "state_after": "case intro\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ∃ n, c ^ n ≠ 0 ∧ ‖c ^ n • x‖ < ε ∧ ε / ‖c‖ ≤ ‖c ^ n • x‖ ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "state_before": "case intro\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\n⊢ ∃ n, c ^ n ≠ 0 ∧ ‖c ^ n • x‖ < ε ∧ ε / ‖c‖ ≤ ‖c ^ n • x‖ ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "tactic": "have cnpos : 0 < ‖c ^ (n + 1)‖ := by\n rw [norm_zpow]\n exact lt_trans xεpos hn.2" }, { "state_after": "case intro.refine'_1\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ c ^ (-(n + 1)) ≠ 0\n\ncase intro.refine'_2\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1)) • x‖ < ε\n\ncase intro.refine'_3\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ε / ‖c‖ ≤ ‖c ^ (-(n + 1)) • x‖\n\ncase intro.refine'_4\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1))‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "state_before": "case intro\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ∃ n, c ^ n ≠ 0 ∧ ‖c ^ n • x‖ < ε ∧ ε / ‖c‖ ≤ ‖c ^ n • x‖ ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "tactic": "refine' ⟨-(n + 1), _, _, _, _⟩" }, { "state_after": "case intro.refine'_1\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ c ^ (-(n + 1)) ≠ 0\n\ncase intro.refine'_2\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1)) • x‖ < ε\n\ncase intro.refine'_3\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ε / ‖c‖ ≤ ‖c ^ (-(n + 1)) • x‖\n\ncase intro.refine'_4\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1))‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "state_before": "case intro.refine'_1\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ c ^ (-(n + 1)) ≠ 0\n\ncase intro.refine'_2\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1)) • x‖ < ε\n\ncase intro.refine'_3\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ε / ‖c‖ ≤ ‖c ^ (-(n + 1)) • x‖\n\ncase intro.refine'_4\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1))‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "tactic": "show c ^ (-(n + 1)) ≠ 0" }, { "state_after": "case intro.refine'_2\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1)) • x‖ < ε\n\ncase intro.refine'_3\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ε / ‖c‖ ≤ ‖c ^ (-(n + 1)) • x‖\n\ncase intro.refine'_4\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1))‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "state_before": "case intro.refine'_1\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ c ^ (-(n + 1)) ≠ 0\n\ncase intro.refine'_2\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1)) • x‖ < ε\n\ncase intro.refine'_3\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ε / ‖c‖ ≤ ‖c ^ (-(n + 1)) • x‖\n\ncase intro.refine'_4\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1))‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "tactic": "exact zpow_ne_zero _ (norm_pos_iff.1 cpos)" }, { "state_after": "case intro.refine'_2\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1)) • x‖ < ε\n\ncase intro.refine'_3\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ε / ‖c‖ ≤ ‖c ^ (-(n + 1)) • x‖\n\ncase intro.refine'_4\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1))‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "state_before": "case intro.refine'_2\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1)) • x‖ < ε\n\ncase intro.refine'_3\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ε / ‖c‖ ≤ ‖c ^ (-(n + 1)) • x‖\n\ncase intro.refine'_4\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1))‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "tactic": "show ‖c ^ (-(n + 1)) • x‖ < ε" }, { "state_after": "case intro.refine'_3\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ε / ‖c‖ ≤ ‖c ^ (-(n + 1)) • x‖\n\ncase intro.refine'_4\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1))‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "state_before": "case intro.refine'_3\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ε / ‖c‖ ≤ ‖c ^ (-(n + 1)) • x‖\n\ncase intro.refine'_4\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1))‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "tactic": "show ε / ‖c‖ ≤ ‖c ^ (-(n + 1)) • x‖" }, { "state_after": "case intro.refine'_4\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1))‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "state_before": "case intro.refine'_4\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1))‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "tactic": "show ‖c ^ (-(n + 1))‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖" }, { "state_after": "α : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\n⊢ 0 < ‖c‖ ^ (n + 1)", "state_before": "α : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\n⊢ 0 < ‖c ^ (n + 1)‖", "tactic": "rw [norm_zpow]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\n⊢ 0 < ‖c‖ ^ (n + 1)", "tactic": "exact lt_trans xεpos hn.2" }, { "state_after": "case intro.refine'_2\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖x‖ < ‖c‖ ^ (n + 1) * ε", "state_before": "case intro.refine'_2\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1)) • x‖ < ε", "tactic": "rw [norm_smul, zpow_neg, norm_inv, ← _root_.div_eq_inv_mul, div_lt_iff cnpos, mul_comm,\n norm_zpow]" }, { "state_after": "no goals", "state_before": "case intro.refine'_2\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖x‖ < ‖c‖ ^ (n + 1) * ε", "tactic": "exact (div_lt_iff εpos).1 hn.2" }, { "state_after": "case intro.refine'_3\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c‖ ^ n * ε ≤ ‖x‖", "state_before": "case intro.refine'_3\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ε / ‖c‖ ≤ ‖c ^ (-(n + 1)) • x‖", "tactic": "rw [zpow_neg, div_le_iff cpos, norm_smul, norm_inv, norm_zpow, zpow_add₀ (ne_of_gt cpos),\n zpow_one, mul_inv_rev, mul_comm, ← mul_assoc, ← mul_assoc, mul_inv_cancel (ne_of_gt cpos),\n one_mul, ← _root_.div_eq_inv_mul, le_div_iff (zpow_pos_of_pos cpos _), mul_comm]" }, { "state_after": "no goals", "state_before": "case intro.refine'_3\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c‖ ^ n * ε ≤ ‖x‖", "tactic": "exact (le_div_iff εpos).1 hn.1" }, { "state_after": "case intro.refine'_4\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c‖ ^ n * ‖c‖ ≤ ‖x‖ / ε * ‖c‖", "state_before": "case intro.refine'_4\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c ^ (-(n + 1))‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖", "tactic": "rw [zpow_neg, norm_inv, inv_inv, norm_zpow, zpow_add₀ cpos.ne', zpow_one, mul_right_comm, ←\n _root_.div_eq_inv_mul]" }, { "state_after": "no goals", "state_before": "case intro.refine'_4\nα : Type u_1\nβ : Type ?u.220958\nγ : Type ?u.220961\nι : Type ?u.220964\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_2\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.220995\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : ‖x‖ ≠ 0\nxεpos : 0 < ‖x‖ / ε\nn : ℤ\nhn : ‖x‖ / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n⊢ ‖c‖ ^ n * ‖c‖ ≤ ‖x‖ / ε * ‖c‖", "tactic": "exact mul_le_mul_of_nonneg_right hn.1 (norm_nonneg _)" } ]
[ 288, 58 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 265, 1 ]
Mathlib/Data/Seq/Computation.lean
Computation.liftRel_def
[ { "state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nca : Computation α\ncb : Computation β\nh : LiftRel R ca cb\na : α\nb : β\nma : a ∈ ca\nmb : b ∈ cb\nb' : β\nmb' : b' ∈ cb\nab : R a b'\n⊢ R a b", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nca : Computation α\ncb : Computation β\nh : LiftRel R ca cb\na : α\nb : β\nma : a ∈ ca\nmb : b ∈ cb\n⊢ R a b", "tactic": "let ⟨b', mb', ab⟩ := h.left ma" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nca : Computation α\ncb : Computation β\nh : LiftRel R ca cb\na : α\nb : β\nma : a ∈ ca\nmb : b ∈ cb\nb' : β\nmb' : b' ∈ cb\nab : R a b'\n⊢ R a b", "tactic": "rwa [mem_unique mb mb']" } ]
[ 1157, 25 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1145, 1 ]
Std/Data/List/Lemmas.lean
List.drop_sublist
[]
[ 1725, 28 ]
e68aa8f5fe47aad78987df45f99094afbcb5e936
https://github.com/leanprover/std4
[ 1724, 1 ]
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
Subalgebra.mul_mem
[]
[ 133, 16 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 132, 11 ]
Mathlib/CategoryTheory/Subobject/Lattice.lean
CategoryTheory.Subobject.underlyingIso_inv_top_arrow
[]
[ 249, 24 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 247, 1 ]
Mathlib/Analysis/Calculus/Taylor.lean
hasDerivWithinAt_taylor_coeff_within
[ { "state_after": "𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\n⊢ HasDerivWithinAt (fun z => (((↑k + 1) * ↑k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z)\n ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y -\n ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y)\n t y", "state_before": "𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y\n⊢ HasDerivWithinAt (fun z => (((↑k + 1) * ↑k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z)\n ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y -\n ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y)\n t y", "tactic": "replace hf :\n HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by\n convert (hf.mono_of_mem hs).hasDerivWithinAt using 1\n rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]\n exact (derivWithin_of_mem hs ht hf).symm" }, { "state_after": "case h.e'_7\n𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\nthis : HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y\n⊢ (((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y -\n ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y =\n (((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y +\n -((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y", "state_before": "𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\nthis : HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y\n⊢ HasDerivWithinAt (fun z => (((↑k + 1) * ↑k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z)\n ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y -\n ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y)\n t y", "tactic": "convert this.smul hf using 1" }, { "state_after": "case h.e'_7\n𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\nthis : HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y\n⊢ ((x - y) ^ (k + 1) / ((↑k + 1) * ↑k !)) • iteratedDerivWithin (k + 2) f s y -\n ((x - y) ^ k / ↑k !) • iteratedDerivWithin (k + 1) f s y =\n ((x - y) ^ (k + 1) / ((↑k + 1) * ↑k !)) • iteratedDerivWithin (k + 2) f s y +\n (-(x - y) ^ k / ↑k !) • iteratedDerivWithin (k + 1) f s y", "state_before": "case h.e'_7\n𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\nthis : HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y\n⊢ (((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y -\n ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y =\n (((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y +\n -((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y", "tactic": "field_simp [Nat.cast_add_one_ne_zero k, Nat.factorial_ne_zero k]" }, { "state_after": "no goals", "state_before": "case h.e'_7\n𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\nthis : HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y\n⊢ ((x - y) ^ (k + 1) / ((↑k + 1) * ↑k !)) • iteratedDerivWithin (k + 2) f s y -\n ((x - y) ^ k / ↑k !) • iteratedDerivWithin (k + 1) f s y =\n ((x - y) ^ (k + 1) / ((↑k + 1) * ↑k !)) • iteratedDerivWithin (k + 2) f s y +\n (-(x - y) ^ k / ↑k !) • iteratedDerivWithin (k + 1) f s y", "tactic": "rw [neg_div, neg_smul, sub_eq_add_neg]" }, { "state_after": "case h.e'_7\n𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y\n⊢ iteratedDerivWithin (k + 2) f s y = derivWithin (iteratedDerivWithin (k + 1) f s) t y", "state_before": "𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y\n⊢ HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y", "tactic": "convert (hf.mono_of_mem hs).hasDerivWithinAt using 1" }, { "state_after": "case h.e'_7\n𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y\n⊢ derivWithin (iteratedDerivWithin (k + 1) f s) s y = derivWithin (iteratedDerivWithin (k + 1) f s) t y", "state_before": "case h.e'_7\n𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y\n⊢ iteratedDerivWithin (k + 2) f s y = derivWithin (iteratedDerivWithin (k + 1) f s) t y", "tactic": "rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]" }, { "state_after": "no goals", "state_before": "case h.e'_7\n𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y\n⊢ derivWithin (iteratedDerivWithin (k + 1) f s) s y = derivWithin (iteratedDerivWithin (k + 1) f s) t y", "tactic": "exact (derivWithin_of_mem hs ht hf).symm" }, { "state_after": "𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\nthis : -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k)\n⊢ HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k))\n t y", "state_before": "𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\nthis : -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k)\n⊢ HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y", "tactic": "rw [this]" }, { "state_after": "no goals", "state_before": "𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\nthis : -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k)\n⊢ HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k))\n t y", "tactic": "exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _" }, { "state_after": "𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\n⊢ -(x - y) ^ k = (-1 + -↑k) * (x - y) ^ k * ↑k ! / ((↑k + 1) * ↑k !)", "state_before": "𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\n⊢ -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k)", "tactic": "field_simp [k.factorial_ne_zero]" }, { "state_after": "case ha\n𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\n⊢ ↑k + 1 ≠ 0\n\ncase hc\n𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\n⊢ ↑k ! ≠ 0", "state_before": "𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\n⊢ -(x - y) ^ k = (-1 + -↑k) * (x - y) ^ k * ↑k ! / ((↑k + 1) * ↑k !)", "tactic": "rw [mul_div_mul_right, ← neg_add_rev, neg_mul, neg_div, mul_div_cancel_left]" }, { "state_after": "no goals", "state_before": "case ha\n𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\n⊢ ↑k + 1 ≠ 0", "tactic": "exact k.cast_add_one_ne_zero" }, { "state_after": "no goals", "state_before": "case hc\n𝕜 : Type ?u.264755\nE : Type u_1\nF : Type ?u.264761\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y\n⊢ ↑k ! ≠ 0", "tactic": "simp [k.factorial_ne_zero]" } ]
[ 176, 41 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 151, 1 ]
Mathlib/Algebra/GeomSum.lean
RingHom.map_geom_sum
[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Semiring β\nx : α\nn : ℕ\nf : α →+* β\n⊢ ↑f (∑ i in range n, x ^ i) = ∑ i in range n, ↑f x ^ i", "tactic": "simp [f.map_sum]" } ]
[ 397, 79 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 396, 1 ]
Mathlib/Algebra/Module/Submodule/Lattice.lean
Submodule.inf_coe
[]
[ 234, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 233, 1 ]
Mathlib/CategoryTheory/Limits/HasLimits.lean
CategoryTheory.Limits.limit.lift_map
[ { "state_after": "case w\nJ : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F G : J ⥤ C\ninst✝¹ : HasLimit F\ninst✝ : HasLimit G\nc : Cone F\nα : F ⟶ G\nj✝ : J\n⊢ (lift F c ≫ limMap α) ≫ π G j✝ = lift G ((Cones.postcompose α).obj c) ≫ π G j✝", "state_before": "J : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F G : J ⥤ C\ninst✝¹ : HasLimit F\ninst✝ : HasLimit G\nc : Cone F\nα : F ⟶ G\n⊢ lift F c ≫ limMap α = lift G ((Cones.postcompose α).obj c)", "tactic": "ext" }, { "state_after": "case w\nJ : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F G : J ⥤ C\ninst✝¹ : HasLimit F\ninst✝ : HasLimit G\nc : Cone F\nα : F ⟶ G\nj✝ : J\n⊢ c.π.app j✝ ≫ α.app j✝ = ((Cones.postcompose α).obj c).π.app j✝", "state_before": "case w\nJ : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F G : J ⥤ C\ninst✝¹ : HasLimit F\ninst✝ : HasLimit G\nc : Cone F\nα : F ⟶ G\nj✝ : J\n⊢ (lift F c ≫ limMap α) ≫ π G j✝ = lift G ((Cones.postcompose α).obj c) ≫ π G j✝", "tactic": "rw [assoc, limMap_π, limit.lift_π_assoc, limit.lift_π]" }, { "state_after": "no goals", "state_before": "case w\nJ : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F G : J ⥤ C\ninst✝¹ : HasLimit F\ninst✝ : HasLimit G\nc : Cone F\nα : F ⟶ G\nj✝ : J\n⊢ c.π.app j✝ ≫ α.app j✝ = ((Cones.postcompose α).obj c).π.app j✝", "tactic": "rfl" } ]
[ 278, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 274, 1 ]
Mathlib/RingTheory/PrincipalIdealDomain.lean
exists_associated_pow_of_mul_eq_pow'
[]
[ 462, 71 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 460, 1 ]
Mathlib/Tactic/PushNeg.lean
Mathlib.Tactic.PushNeg.not_forall_eq
[]
[ 23, 74 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 23, 1 ]
Mathlib/LinearAlgebra/SesquilinearForm.lean
LinearMap.IsRefl.eq_zero
[]
[ 178, 70 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 178, 1 ]
Mathlib/FieldTheory/IntermediateField.lean
IntermediateField.fieldRange_val
[]
[ 513, 33 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 512, 1 ]
Mathlib/Analysis/NormedSpace/Exponential.lean
algebraMap_exp_comm_of_mem_ball
[]
[ 326, 57 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 323, 1 ]
Mathlib/FieldTheory/Subfield.lean
RingHom.map_fieldRange
[ { "state_after": "no goals", "state_before": "K : Type u\nL : Type v\nM : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field M\ng : L →+* M\nf : K →+* L\n⊢ Subfield.map g (fieldRange f) = fieldRange (comp g f)", "tactic": "simpa only [fieldRange_eq_map] using (⊤ : Subfield K).map_map g f" } ]
[ 569, 68 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 568, 1 ]
Mathlib/Order/Filter/Extr.lean
isMaxOn_iff
[]
[ 139, 10 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 138, 1 ]
Std/Data/String/Lemmas.lean
Substring.Valid.take
[]
[ 1026, 64 ]
e68aa8f5fe47aad78987df45f99094afbcb5e936
https://github.com/leanprover/std4
[ 1025, 1 ]
Mathlib/Topology/Instances/Int.lean
Int.ball_eq_Ioo
[ { "state_after": "no goals", "state_before": "x : ℤ\nr : ℝ\n⊢ ball x r = Ioo ⌊↑x - r⌋ ⌈↑x + r⌉", "tactic": "rw [← preimage_ball, Real.ball_eq_Ioo, preimage_Ioo]" } ]
[ 63, 55 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 62, 1 ]
Mathlib/Data/Real/Cardinality.lean
Cardinal.cantorFunctionAux_false
[ { "state_after": "no goals", "state_before": "c : ℝ\nf g : ℕ → Bool\nn : ℕ\nh : f n = false\n⊢ cantorFunctionAux c f n = 0", "tactic": "simp [cantorFunctionAux, h]" } ]
[ 73, 30 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 72, 1 ]
Mathlib/Analysis/SpecialFunctions/Bernstein.lean
bernsteinApproximation_uniform
[ { "state_after": "f : C(↑I, ℝ)\n⊢ ∀ (i : ℝ), 0 < i → ∀ᶠ (x : ℕ) in atTop, ‖bernsteinApproximation x f - f‖ < i", "state_before": "f : C(↑I, ℝ)\n⊢ Tendsto (fun n => bernsteinApproximation n f) atTop (𝓝 f)", "tactic": "simp only [Metric.nhds_basis_ball.tendsto_right_iff, Metric.mem_ball, dist_eq_norm]" }, { "state_after": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖bernsteinApproximation x f - f‖ < ε", "state_before": "f : C(↑I, ℝ)\n⊢ ∀ (i : ℝ), 0 < i → ∀ᶠ (x : ℕ) in atTop, ‖bernsteinApproximation x f - f‖ < i", "tactic": "intro ε h" }, { "state_after": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖bernsteinApproximation x f - f‖ < ε", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖bernsteinApproximation x f - f‖ < ε", "tactic": "let δ := δ f ε h" }, { "state_after": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖bernsteinApproximation x f - f‖ < ε", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖bernsteinApproximation x f - f‖ < ε", "tactic": "have nhds_zero := tendsto_const_div_atTop_nhds_0_nat (2 * ‖f‖ * δ ^ (-2 : ℤ))" }, { "state_after": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\n⊢ ‖bernsteinApproximation n f - f‖ < ε", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖bernsteinApproximation x f - f‖ < ε", "tactic": "filter_upwards [nhds_zero.eventually (gt_mem_nhds (half_pos h)), eventually_gt_atTop 0] with n nh\n npos'" }, { "state_after": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\n⊢ ‖bernsteinApproximation n f - f‖ < ε", "state_before": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\n⊢ ‖bernsteinApproximation n f - f‖ < ε", "tactic": "have npos : 0 < (n : ℝ) := by positivity" }, { "state_after": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\n⊢ ‖bernsteinApproximation n f - f‖ < ε", "state_before": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\n⊢ ‖bernsteinApproximation n f - f‖ < ε", "tactic": "have w₂ : 0 ≤ δ ^ (-2:ℤ) := zpow_neg_two_nonneg _" }, { "state_after": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\n⊢ ∀ (x : ↑I), ‖↑(bernsteinApproximation n f - f) x‖ < ε", "state_before": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\n⊢ ‖bernsteinApproximation n f - f‖ < ε", "tactic": "rw [ContinuousMap.norm_lt_iff _ h]" }, { "state_after": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\n⊢ ‖↑(bernsteinApproximation n f - f) x‖ < ε", "state_before": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\n⊢ ∀ (x : ↑I), ‖↑(bernsteinApproximation n f - f) x‖ < ε", "tactic": "intro x" }, { "state_after": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ‖↑(bernsteinApproximation n f - f) x‖ < ε", "state_before": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\n⊢ ‖↑(bernsteinApproximation n f - f) x‖ < ε", "tactic": "let S := S f ε h n x" }, { "state_after": "case h.calc_1\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ∑ k in S, abs (↑f k/ₙ - ↑f x) * ↑(bernstein n ↑k) x ≤ ε / 2\n\ncase h.calc_2\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ∑ k in Sᶜ, abs (↑f k/ₙ - ↑f x) * ↑(bernstein n ↑k) x < ε / 2", "state_before": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ‖↑(bernsteinApproximation n f - f) x‖ < ε", "tactic": "calc\n |(bernsteinApproximation n f - f) x| = |bernsteinApproximation n f x - f x| := rfl\n _ = |bernsteinApproximation n f x - f x * 1| := by rw [mul_one]\n _ = |bernsteinApproximation n f x - f x * ∑ k : Fin (n + 1), bernstein n k x| := by\n rw [bernstein.probability]\n _ = |∑ k : Fin (n + 1), (f k/ₙ - f x) * bernstein n k x| := by\n simp [bernsteinApproximation, Finset.mul_sum, sub_mul]\n _ ≤ ∑ k : Fin (n + 1), |(f k/ₙ - f x) * bernstein n k x| := (Finset.abs_sum_le_sum_abs _ _)\n _ = ∑ k : Fin (n + 1), |f k/ₙ - f x| * bernstein n k x := by\n simp_rw [abs_mul, abs_eq_self.mpr bernstein_nonneg]\n _ = (∑ k in S, |f k/ₙ - f x| * bernstein n k x) + ∑ k in Sᶜ, |f k/ₙ - f x| * bernstein n k x :=\n (S.sum_add_sum_compl _).symm\n _ < ε / 2 + ε / 2 :=\n (add_lt_add_of_le_of_lt ?_ ?_)\n _ = ε := add_halves ε" }, { "state_after": "no goals", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\n⊢ 0 < ↑n", "tactic": "positivity" }, { "state_after": "no goals", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ abs (↑(bernsteinApproximation n f) x - ↑f x) = abs (↑(bernsteinApproximation n f) x - ↑f x * 1)", "tactic": "rw [mul_one]" }, { "state_after": "no goals", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ abs (↑(bernsteinApproximation n f) x - ↑f x * 1) =\n abs (↑(bernsteinApproximation n f) x - ↑f x * ∑ k : Fin (n + 1), ↑(bernstein n ↑k) x)", "tactic": "rw [bernstein.probability]" }, { "state_after": "no goals", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ abs (↑(bernsteinApproximation n f) x - ↑f x * ∑ k : Fin (n + 1), ↑(bernstein n ↑k) x) =\n abs (∑ k : Fin (n + 1), (↑f k/ₙ - ↑f x) * ↑(bernstein n ↑k) x)", "tactic": "simp [bernsteinApproximation, Finset.mul_sum, sub_mul]" }, { "state_after": "no goals", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ∑ k : Fin (n + 1), abs ((↑f k/ₙ - ↑f x) * ↑(bernstein n ↑k) x) =\n ∑ k : Fin (n + 1), abs (↑f k/ₙ - ↑f x) * ↑(bernstein n ↑k) x", "tactic": "simp_rw [abs_mul, abs_eq_self.mpr bernstein_nonneg]" }, { "state_after": "no goals", "state_before": "case h.calc_1\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ∑ k in S, abs (↑f k/ₙ - ↑f x) * ↑(bernstein n ↑k) x ≤ ε / 2", "tactic": "calc\n (∑ k in S, |f k/ₙ - f x| * bernstein n k x) ≤ ∑ k in S, ε / 2 * bernstein n k x := by\n gcongr with _ m\n exact le_of_lt (lt_of_mem_S m)\n _ = ε / 2 * ∑ k in S, bernstein n k x := by rw [Finset.mul_sum]\n _ ≤ ε / 2 * ∑ k : Fin (n + 1), bernstein n k x := by\n gcongr\n exact Finset.sum_le_univ_sum_of_nonneg fun k => bernstein_nonneg\n _ = ε / 2 := by rw [bernstein.probability, mul_one]" }, { "state_after": "case h.h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\ni✝ : Fin (n + 1)\nm : i✝ ∈ S\n⊢ abs (↑f i✝/ₙ - ↑f x) ≤ ε / 2", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ∑ k in S, abs (↑f k/ₙ - ↑f x) * ↑(bernstein n ↑k) x ≤ ∑ k in S, ε / 2 * ↑(bernstein n ↑k) x", "tactic": "gcongr with _ m" }, { "state_after": "no goals", "state_before": "case h.h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\ni✝ : Fin (n + 1)\nm : i✝ ∈ S\n⊢ abs (↑f i✝/ₙ - ↑f x) ≤ ε / 2", "tactic": "exact le_of_lt (lt_of_mem_S m)" }, { "state_after": "no goals", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ∑ k in S, ε / 2 * ↑(bernstein n ↑k) x = ε / 2 * ∑ k in S, ↑(bernstein n ↑k) x", "tactic": "rw [Finset.mul_sum]" }, { "state_after": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ∑ k in S, ↑(bernstein n ↑k) x ≤ ∑ k : Fin (n + 1), ↑(bernstein n ↑k) x", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ε / 2 * ∑ k in S, ↑(bernstein n ↑k) x ≤ ε / 2 * ∑ k : Fin (n + 1), ↑(bernstein n ↑k) x", "tactic": "gcongr" }, { "state_after": "no goals", "state_before": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ∑ k in S, ↑(bernstein n ↑k) x ≤ ∑ k : Fin (n + 1), ↑(bernstein n ↑k) x", "tactic": "exact Finset.sum_le_univ_sum_of_nonneg fun k => bernstein_nonneg" }, { "state_after": "no goals", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ε / 2 * ∑ k : Fin (n + 1), ↑(bernstein n ↑k) x = ε / 2", "tactic": "rw [bernstein.probability, mul_one]" }, { "state_after": "no goals", "state_before": "case h.calc_2\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ∑ k in Sᶜ, abs (↑f k/ₙ - ↑f x) * ↑(bernstein n ↑k) x < ε / 2", "tactic": "calc\n (∑ k in Sᶜ, |f k/ₙ - f x| * bernstein n k x) ≤ ∑ k in Sᶜ, 2 * ‖f‖ * bernstein n k x := by\n gcongr\n apply f.dist_le_two_norm\n _ = 2 * ‖f‖ * ∑ k in Sᶜ, bernstein n k x := by rw [Finset.mul_sum]\n _ ≤ 2 * ‖f‖ * ∑ k in Sᶜ, δ ^ (-2 : ℤ) * ((x : ℝ) - k/ₙ) ^ 2 * bernstein n k x := by\n gcongr with _ m\n conv_lhs => rw [← one_mul (bernstein _ _ _)]\n gcongr\n exact le_of_mem_S_compl m\n _ ≤ 2 * ‖f‖ * ∑ k : Fin (n + 1), δ ^ (-2 : ℤ) * ((x : ℝ) - k/ₙ) ^ 2 * bernstein n k x := by\n gcongr\n refine Finset.sum_le_univ_sum_of_nonneg <| fun k => ?_\n positivity\n _ = 2 * ‖f‖ * δ ^ (-2 : ℤ) * ∑ k : Fin (n + 1), ((x : ℝ) - k/ₙ) ^ 2 * bernstein n k x := by\n conv_rhs =>\n rw [mul_assoc, Finset.mul_sum]\n simp only [← mul_assoc]\n _ = 2 * ‖f‖ * δ ^ (-2 : ℤ) * x * (1 - x) / n := by rw [variance npos]; ring\n _ ≤ 2 * ‖f‖ * δ ^ (-2 : ℤ) * 1 * 1 / n := by gcongr <;> unit_interval\n _ < ε / 2 := by simp only [mul_one]; exact nh" }, { "state_after": "case h.h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\ni✝ : Fin (n + 1)\na✝ : i✝ ∈ Sᶜ\n⊢ abs (↑f i✝/ₙ - ↑f x) ≤ 2 * ‖f‖", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ∑ k in Sᶜ, abs (↑f k/ₙ - ↑f x) * ↑(bernstein n ↑k) x ≤ ∑ k in Sᶜ, 2 * ‖f‖ * ↑(bernstein n ↑k) x", "tactic": "gcongr" }, { "state_after": "no goals", "state_before": "case h.h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\ni✝ : Fin (n + 1)\na✝ : i✝ ∈ Sᶜ\n⊢ abs (↑f i✝/ₙ - ↑f x) ≤ 2 * ‖f‖", "tactic": "apply f.dist_le_two_norm" }, { "state_after": "no goals", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ∑ k in Sᶜ, 2 * ‖f‖ * ↑(bernstein n ↑k) x = 2 * ‖f‖ * ∑ k in Sᶜ, ↑(bernstein n ↑k) x", "tactic": "rw [Finset.mul_sum]" }, { "state_after": "case h.h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\ni✝ : Fin (n + 1)\nm : i✝ ∈ Sᶜ\n⊢ ↑(bernstein n ↑i✝) x ≤ δ ^ (-2) * (↑x - ↑i✝/ₙ) ^ 2 * ↑(bernstein n ↑i✝) x", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ 2 * ‖f‖ * ∑ k in Sᶜ, ↑(bernstein n ↑k) x ≤ 2 * ‖f‖ * ∑ k in Sᶜ, δ ^ (-2) * (↑x - ↑k/ₙ) ^ 2 * ↑(bernstein n ↑k) x", "tactic": "gcongr with _ m" }, { "state_after": "case h.h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\ni✝ : Fin (n + 1)\nm : i✝ ∈ Sᶜ\n⊢ 1 * ↑(bernstein n ↑i✝) x ≤ δ ^ (-2) * (↑x - ↑i✝/ₙ) ^ 2 * ↑(bernstein n ↑i✝) x", "state_before": "case h.h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\ni✝ : Fin (n + 1)\nm : i✝ ∈ Sᶜ\n⊢ ↑(bernstein n ↑i✝) x ≤ δ ^ (-2) * (↑x - ↑i✝/ₙ) ^ 2 * ↑(bernstein n ↑i✝) x", "tactic": "conv_lhs => rw [← one_mul (bernstein _ _ _)]" }, { "state_after": "case h.h.h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\ni✝ : Fin (n + 1)\nm : i✝ ∈ Sᶜ\n⊢ 1 ≤ δ ^ (-2) * (↑x - ↑i✝/ₙ) ^ 2", "state_before": "case h.h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\ni✝ : Fin (n + 1)\nm : i✝ ∈ Sᶜ\n⊢ 1 * ↑(bernstein n ↑i✝) x ≤ δ ^ (-2) * (↑x - ↑i✝/ₙ) ^ 2 * ↑(bernstein n ↑i✝) x", "tactic": "gcongr" }, { "state_after": "no goals", "state_before": "case h.h.h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\ni✝ : Fin (n + 1)\nm : i✝ ∈ Sᶜ\n⊢ 1 ≤ δ ^ (-2) * (↑x - ↑i✝/ₙ) ^ 2", "tactic": "exact le_of_mem_S_compl m" }, { "state_after": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ∑ k in Sᶜ, δ ^ (-2) * (↑x - ↑k/ₙ) ^ 2 * ↑(bernstein n ↑k) x ≤\n ∑ k : Fin (n + 1), δ ^ (-2) * (↑x - ↑k/ₙ) ^ 2 * ↑(bernstein n ↑k) x", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ 2 * ‖f‖ * ∑ k in Sᶜ, δ ^ (-2) * (↑x - ↑k/ₙ) ^ 2 * ↑(bernstein n ↑k) x ≤\n 2 * ‖f‖ * ∑ k : Fin (n + 1), δ ^ (-2) * (↑x - ↑k/ₙ) ^ 2 * ↑(bernstein n ↑k) x", "tactic": "gcongr" }, { "state_after": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\nk : Fin (n + 1)\n⊢ 0 ≤ δ ^ (-2) * (↑x - ↑k/ₙ) ^ 2 * ↑(bernstein n ↑k) x", "state_before": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ ∑ k in Sᶜ, δ ^ (-2) * (↑x - ↑k/ₙ) ^ 2 * ↑(bernstein n ↑k) x ≤\n ∑ k : Fin (n + 1), δ ^ (-2) * (↑x - ↑k/ₙ) ^ 2 * ↑(bernstein n ↑k) x", "tactic": "refine Finset.sum_le_univ_sum_of_nonneg <| fun k => ?_" }, { "state_after": "no goals", "state_before": "case h\nf : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\nk : Fin (n + 1)\n⊢ 0 ≤ δ ^ (-2) * (↑x - ↑k/ₙ) ^ 2 * ↑(bernstein n ↑k) x", "tactic": "positivity" }, { "state_after": "no goals", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ 2 * ‖f‖ * ∑ k : Fin (n + 1), δ ^ (-2) * (↑x - ↑k/ₙ) ^ 2 * ↑(bernstein n ↑k) x =\n 2 * ‖f‖ * δ ^ (-2) * ∑ k : Fin (n + 1), (↑x - ↑k/ₙ) ^ 2 * ↑(bernstein n ↑k) x", "tactic": "conv_rhs =>\n rw [mul_assoc, Finset.mul_sum]\n simp only [← mul_assoc]" }, { "state_after": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ 2 * ‖f‖ * δ ^ (-2) * (↑x * (1 - ↑x) / ↑n) = 2 * ‖f‖ * δ ^ (-2) * ↑x * (1 - ↑x) / ↑n", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ 2 * ‖f‖ * δ ^ (-2) * ∑ k : Fin (n + 1), (↑x - ↑k/ₙ) ^ 2 * ↑(bernstein n ↑k) x =\n 2 * ‖f‖ * δ ^ (-2) * ↑x * (1 - ↑x) / ↑n", "tactic": "rw [variance npos]" }, { "state_after": "no goals", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ 2 * ‖f‖ * δ ^ (-2) * (↑x * (1 - ↑x) / ↑n) = 2 * ‖f‖ * δ ^ (-2) * ↑x * (1 - ↑x) / ↑n", "tactic": "ring" }, { "state_after": "no goals", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ 2 * ‖f‖ * δ ^ (-2) * ↑x * (1 - ↑x) / ↑n ≤ 2 * ‖f‖ * δ ^ (-2) * 1 * 1 / ↑n", "tactic": "gcongr <;> unit_interval" }, { "state_after": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ 2 * ‖f‖ * δ ^ (-2) * 1 * 1 / ↑n < ε / 2", "tactic": "simp only [mul_one]" }, { "state_after": "no goals", "state_before": "f : C(↑I, ℝ)\nε : ℝ\nh : 0 < ε\nδ : ℝ := bernsteinApproximation.δ f ε h\nnhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)\nn : ℕ\nnh : 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2\nnpos' : 0 < n\nnpos : 0 < ↑n\nw₂ : 0 ≤ δ ^ (-2)\nx : ↑I\nS : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x\n⊢ 2 * ‖f‖ * bernsteinApproximation.δ f ε h ^ (-2) / ↑n < ε / 2", "tactic": "exact nh" } ]
[ 304, 52 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 234, 1 ]
Mathlib/Data/Finset/NAry.lean
Finset.image₂_inter_union_subset_union
[ { "state_after": "α : Type u_2\nα' : Type ?u.100378\nβ : Type u_3\nβ' : Type ?u.100384\nγ : Type u_1\nγ' : Type ?u.100390\nδ : Type ?u.100393\nδ' : Type ?u.100396\nε : Type ?u.100399\nε' : Type ?u.100402\nζ : Type ?u.100405\nζ' : Type ?u.100408\nν : Type ?u.100411\ninst✝⁹ : DecidableEq α'\ninst✝⁸ : DecidableEq β'\ninst✝⁷ : DecidableEq γ\ninst✝⁶ : DecidableEq γ'\ninst✝⁵ : DecidableEq δ\ninst✝⁴ : DecidableEq δ'\ninst✝³ : DecidableEq ε\ninst✝² : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\n⊢ image2 f (↑s ∩ ↑s') (↑t ∪ ↑t') ⊆ image2 f ↑s ↑t ∪ image2 f ↑s' ↑t'", "state_before": "α : Type u_2\nα' : Type ?u.100378\nβ : Type u_3\nβ' : Type ?u.100384\nγ : Type u_1\nγ' : Type ?u.100390\nδ : Type ?u.100393\nδ' : Type ?u.100396\nε : Type ?u.100399\nε' : Type ?u.100402\nζ : Type ?u.100405\nζ' : Type ?u.100408\nν : Type ?u.100411\ninst✝⁹ : DecidableEq α'\ninst✝⁸ : DecidableEq β'\ninst✝⁷ : DecidableEq γ\ninst✝⁶ : DecidableEq γ'\ninst✝⁵ : DecidableEq δ\ninst✝⁴ : DecidableEq δ'\ninst✝³ : DecidableEq ε\ninst✝² : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\n⊢ ↑(image₂ f (s ∩ s') (t ∪ t')) ⊆ ↑(image₂ f s t ∪ image₂ f s' t')", "tactic": "push_cast" }, { "state_after": "no goals", "state_before": "α : Type u_2\nα' : Type ?u.100378\nβ : Type u_3\nβ' : Type ?u.100384\nγ : Type u_1\nγ' : Type ?u.100390\nδ : Type ?u.100393\nδ' : Type ?u.100396\nε : Type ?u.100399\nε' : Type ?u.100402\nζ : Type ?u.100405\nζ' : Type ?u.100408\nν : Type ?u.100411\ninst✝⁹ : DecidableEq α'\ninst✝⁸ : DecidableEq β'\ninst✝⁷ : DecidableEq γ\ninst✝⁶ : DecidableEq γ'\ninst✝⁵ : DecidableEq δ\ninst✝⁴ : DecidableEq δ'\ninst✝³ : DecidableEq ε\ninst✝² : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\n⊢ image2 f (↑s ∩ ↑s') (↑t ∪ ↑t') ⊆ image2 f ↑s ↑t ∪ image2 f ↑s' ↑t'", "tactic": "exact Set.image2_inter_union_subset_union" } ]
[ 527, 46 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 523, 1 ]
Mathlib/Order/Hom/Lattice.lean
SupHom.coe_copy
[]
[ 370, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 369, 1 ]
Mathlib/ModelTheory/Basic.lean
FirstOrder.Language.Hom.comp_assoc
[]
[ 586, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 584, 1 ]
Mathlib/Analysis/ODE/Gronwall.lean
gronwallBound_K0
[]
[ 52, 29 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 51, 1 ]
Mathlib/MeasureTheory/Group/FundamentalDomain.lean
MeasureTheory.IsFundamentalDomain.restrict_restrict
[]
[ 204, 74 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 202, 1 ]
Mathlib/MeasureTheory/Group/Arithmetic.lean
AEMeasurable.const_smul
[]
[ 667, 19 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 666, 1 ]
Mathlib/Data/Finset/Basic.lean
Finset.disjoint_empty_right
[]
[ 954, 21 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 953, 1 ]
Mathlib/Order/Filter/Basic.lean
Filter.comap_eval_neBot
[]
[ 2419, 29 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 2417, 1 ]
Mathlib/CategoryTheory/Simple.lean
CategoryTheory.Simple.of_iso
[ { "state_after": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\n⊢ IsIso f ↔ f ≠ 0", "state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\n⊢ IsIso f ↔ f ≠ 0", "tactic": "haveI : Mono (f ≫ i.hom) := mono_comp _ _" }, { "state_after": "case mp\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\n⊢ IsIso f → f ≠ 0\n\ncase mpr\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\n⊢ f ≠ 0 → IsIso f", "state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\n⊢ IsIso f ↔ f ≠ 0", "tactic": "constructor" }, { "state_after": "case mp\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : IsIso f\nw : f = 0\n⊢ False", "state_before": "case mp\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\n⊢ IsIso f → f ≠ 0", "tactic": "intro h w" }, { "state_after": "case j\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : IsIso f\nw : f = 0\n⊢ IsIso (f ≫ i.hom)\n\ncase mp\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : IsIso f\nw : f = 0\nj : IsIso (f ≫ i.hom)\n⊢ False", "state_before": "case mp\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : IsIso f\nw : f = 0\n⊢ False", "tactic": "have j : IsIso (f ≫ i.hom)" }, { "state_after": "case mp\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : IsIso f\nw : f = 0\nj : IsIso (f ≫ i.hom)\n⊢ False", "state_before": "case j\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : IsIso f\nw : f = 0\n⊢ IsIso (f ≫ i.hom)\n\ncase mp\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : IsIso f\nw : f = 0\nj : IsIso (f ≫ i.hom)\n⊢ False", "tactic": "infer_instance" }, { "state_after": "case mp\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : IsIso f\nw : f = 0\nj✝ : IsIso (f ≫ i.hom)\nj : f ≫ i.hom ≠ 0\n⊢ False", "state_before": "case mp\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : IsIso f\nw : f = 0\nj : IsIso (f ≫ i.hom)\n⊢ False", "tactic": "rw [Simple.mono_isIso_iff_nonzero] at j" }, { "state_after": "case mp\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nm : Mono 0\nthis : Mono (0 ≫ i.hom)\nh : IsIso 0\nj✝ : IsIso (0 ≫ i.hom)\nj : 0 ≫ i.hom ≠ 0\n⊢ False", "state_before": "case mp\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : IsIso f\nw : f = 0\nj✝ : IsIso (f ≫ i.hom)\nj : f ≫ i.hom ≠ 0\n⊢ False", "tactic": "subst w" }, { "state_after": "no goals", "state_before": "case mp\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nm : Mono 0\nthis : Mono (0 ≫ i.hom)\nh : IsIso 0\nj✝ : IsIso (0 ≫ i.hom)\nj : 0 ≫ i.hom ≠ 0\n⊢ False", "tactic": "simp at j" }, { "state_after": "case mpr\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : f ≠ 0\n⊢ IsIso f", "state_before": "case mpr\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\n⊢ f ≠ 0 → IsIso f", "tactic": "intro h" }, { "state_after": "case mpr\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : f ≠ 0\nj : IsIso (f ≫ i.hom)\n⊢ IsIso f", "state_before": "case mpr\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : f ≠ 0\n⊢ IsIso f", "tactic": "have j : IsIso (f ≫ i.hom) := by\n apply isIso_of_mono_of_nonzero\n intro w\n apply h\n simpa using (cancel_mono i.inv).2 w" }, { "state_after": "case mpr\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : f ≠ 0\nj : IsIso (f ≫ i.hom)\n⊢ IsIso ((f ≫ i.hom) ≫ i.inv)", "state_before": "case mpr\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : f ≠ 0\nj : IsIso (f ≫ i.hom)\n⊢ IsIso f", "tactic": "rw [← Category.comp_id f, ← i.hom_inv_id, ← Category.assoc]" }, { "state_after": "no goals", "state_before": "case mpr\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : f ≠ 0\nj : IsIso (f ≫ i.hom)\n⊢ IsIso ((f ≫ i.hom) ≫ i.inv)", "tactic": "infer_instance" }, { "state_after": "case w\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : f ≠ 0\n⊢ f ≫ i.hom ≠ 0", "state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : f ≠ 0\n⊢ IsIso (f ≫ i.hom)", "tactic": "apply isIso_of_mono_of_nonzero" }, { "state_after": "case w\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : f ≠ 0\nw : f ≫ i.hom = 0\n⊢ False", "state_before": "case w\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : f ≠ 0\n⊢ f ≫ i.hom ≠ 0", "tactic": "intro w" }, { "state_after": "case w\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : f ≠ 0\nw : f ≫ i.hom = 0\n⊢ f = 0", "state_before": "case w\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : f ≠ 0\nw : f ≫ i.hom = 0\n⊢ False", "tactic": "apply h" }, { "state_after": "no goals", "state_before": "case w\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nthis : Mono (f ≫ i.hom)\nh : f ≠ 0\nw : f ≫ i.hom = 0\n⊢ f = 0", "tactic": "simpa using (cancel_mono i.inv).2 w" } ]
[ 81, 25 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 64, 1 ]
Mathlib/Analysis/InnerProductSpace/Projection.lean
reflection_mem_subspace_orthogonalComplement_eq_neg
[ { "state_after": "no goals", "state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.834026\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv : E\nhv : v ∈ Kᗮ\n⊢ ↑(reflection K) v = -v", "tactic": "simp [reflection_apply, orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero hv]" } ]
[ 862, 93 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 860, 1 ]
Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean
Complex.differentiable_exp
[]
[ 45, 56 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 44, 1 ]
Mathlib/LinearAlgebra/TensorProduct.lean
TensorProduct.tmul_sub
[]
[ 1252, 27 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1251, 1 ]
Std/Data/PairingHeap.lean
Std.PairingHeapImp.Heap.noSibling_tail
[ { "state_after": "α : Type u_1\nle : α → α → Bool\ns : Heap α\n⊢ NoSibling (Option.getD (tail? le s) nil)", "state_before": "α : Type u_1\nle : α → α → Bool\ns : Heap α\n⊢ NoSibling (tail le s)", "tactic": "simp only [Heap.tail]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nle : α → α → Bool\ns : Heap α\n⊢ NoSibling (Option.getD (tail? le s) nil)", "tactic": "match eq : s.tail? le with\n| none => cases s with cases eq | nil => constructor\n| some tl => exact Heap.noSibling_tail? eq" }, { "state_after": "no goals", "state_before": "α : Type u_1\nle : α → α → Bool\ns : Heap α\neq : tail? le s = none\n⊢ NoSibling (Option.getD none nil)", "tactic": "cases s with cases eq | nil => constructor" }, { "state_after": "no goals", "state_before": "case nil.refl\nα : Type u_1\nle : α → α → Bool\n⊢ NoSibling (Option.getD none nil)", "tactic": "constructor" }, { "state_after": "no goals", "state_before": "α : Type u_1\nle : α → α → Bool\ns tl : Heap α\neq : tail? le s = some tl\n⊢ NoSibling (Option.getD (some tl) nil)", "tactic": "exact Heap.noSibling_tail? eq" } ]
[ 118, 45 ]
e68aa8f5fe47aad78987df45f99094afbcb5e936
https://github.com/leanprover/std4
[ 114, 1 ]
Mathlib/Data/Sym/Sym2.lean
Sym2.diag_injective
[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.40168\nγ : Type ?u.40171\nx y : α\nh : diag x = diag y\n⊢ x = y", "tactic": "cases Quotient.exact h <;> rfl" } ]
[ 431, 33 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 430, 1 ]
Mathlib/Combinatorics/SimpleGraph/Basic.lean
SimpleGraph.edge_other_incident_set
[ { "state_after": "ι : Sort ?u.149788\n𝕜 : Type ?u.149791\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne✝ : Sym2 V\ninst✝ : DecidableEq V\nv : V\ne : Sym2 V\nh : e ∈ incidenceSet G v\n⊢ otherVertexOfIncident G h ∈ e", "state_before": "ι : Sort ?u.149788\n𝕜 : Type ?u.149791\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne✝ : Sym2 V\ninst✝ : DecidableEq V\nv : V\ne : Sym2 V\nh : e ∈ incidenceSet G v\n⊢ e ∈ incidenceSet G (otherVertexOfIncident G h)", "tactic": "use h.1" }, { "state_after": "no goals", "state_before": "ι : Sort ?u.149788\n𝕜 : Type ?u.149791\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne✝ : Sym2 V\ninst✝ : DecidableEq V\nv : V\ne : Sym2 V\nh : e ∈ incidenceSet G v\n⊢ otherVertexOfIncident G h ∈ e", "tactic": "simp [otherVertexOfIncident, Sym2.other_mem']" } ]
[ 1065, 48 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1062, 1 ]
Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean
DiscreteValuationRing.TFAE
[ { "state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), FiniteDimensional.finrank (ResidueField R) (CotangentSpace R) = 1,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), FiniteDimensional.finrank (ResidueField R) (CotangentSpace R) = 1,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "tactic": "have ne_bot := Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h" }, { "state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), FiniteDimensional.finrank (ResidueField R) (CotangentSpace R) = 1,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "tactic": "rw [finrank_eq_one_iff']" }, { "state_after": "case tfae_1_to_2\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\n⊢ DiscreteValuationRing R → ValuationRing R\n\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "tactic": "tfae_have 1 → 2" }, { "state_after": "case tfae_2_to_1\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n⊢ ValuationRing R → DiscreteValuationRing R\n\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "tactic": "tfae_have 2 → 1" }, { "state_after": "case tfae_1_to_4\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\n⊢ DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\n\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "tactic": "tfae_have 1 → 4" }, { "state_after": "case tfae_4_to_3\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\n⊢ (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\n\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "tactic": "tfae_have 4 → 3" }, { "state_after": "case tfae_3_to_5\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\n⊢ IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\n\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "tactic": "tfae_have 3 → 5" }, { "state_after": "case tfae_5_to_6\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\n⊢ Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\n\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "tactic": "tfae_have 5 → 6" }, { "state_after": "case tfae_6_to_5\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\n⊢ (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\n\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "tactic": "tfae_have 6 → 5" }, { "state_after": "case tfae_5_to_7\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\n⊢ Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\n\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "tactic": "tfae_have 5 → 7" }, { "state_after": "case tfae_7_to_2\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\n⊢ (∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n) → ValuationRing R\n\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\ntfae_7_to_2 : (∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n) → ValuationRing R\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "tactic": "tfae_have 7 → 2" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\ntfae_7_to_2 : (∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n) → ValuationRing R\n⊢ List.TFAE\n [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P,\n Submodule.IsPrincipal (maximalIdeal R), ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w,\n ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n]", "tactic": "tfae_finish" }, { "state_after": "case tfae_1_to_2\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\n✝ : DiscreteValuationRing R\n⊢ ValuationRing R", "state_before": "case tfae_1_to_2\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\n⊢ DiscreteValuationRing R → ValuationRing R", "tactic": "intro" }, { "state_after": "no goals", "state_before": "case tfae_1_to_2\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\n✝ : DiscreteValuationRing R\n⊢ ValuationRing R", "tactic": "infer_instance" }, { "state_after": "case tfae_2_to_1\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n✝ : ValuationRing R\n⊢ DiscreteValuationRing R", "state_before": "case tfae_2_to_1\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n⊢ ValuationRing R → DiscreteValuationRing R", "tactic": "intro" }, { "state_after": "case tfae_2_to_1\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n✝ : ValuationRing R\nthis : GCDMonoid R\n⊢ DiscreteValuationRing R", "state_before": "case tfae_2_to_1\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n✝ : ValuationRing R\n⊢ DiscreteValuationRing R", "tactic": "haveI := IsBezout.toGCDDomain R" }, { "state_after": "case tfae_2_to_1\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n✝ : ValuationRing R\nthis✝ : GCDMonoid R\nthis : UniqueFactorizationMonoid R\n⊢ DiscreteValuationRing R", "state_before": "case tfae_2_to_1\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n✝ : ValuationRing R\nthis : GCDMonoid R\n⊢ DiscreteValuationRing R", "tactic": "haveI : UniqueFactorizationMonoid R := ufm_of_gcd_of_wfDvdMonoid" }, { "state_after": "case tfae_2_to_1.h₁\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n✝ : ValuationRing R\nthis✝ : GCDMonoid R\nthis : UniqueFactorizationMonoid R\n⊢ ∃ p, Irreducible p\n\ncase tfae_2_to_1.h₂\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n✝ : ValuationRing R\nthis✝ : GCDMonoid R\nthis : UniqueFactorizationMonoid R\n⊢ ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q", "state_before": "case tfae_2_to_1\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n✝ : ValuationRing R\nthis✝ : GCDMonoid R\nthis : UniqueFactorizationMonoid R\n⊢ DiscreteValuationRing R", "tactic": "apply DiscreteValuationRing.of_ufd_of_unique_irreducible" }, { "state_after": "case tfae_2_to_1.h₁.intro.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n✝ : ValuationRing R\nthis✝ : GCDMonoid R\nthis : UniqueFactorizationMonoid R\nx : R\nhx₁ : x ≠ 0\nhx₂ : ¬IsUnit x\n⊢ ∃ p, Irreducible p", "state_before": "case tfae_2_to_1.h₁\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n✝ : ValuationRing R\nthis✝ : GCDMonoid R\nthis : UniqueFactorizationMonoid R\n⊢ ∃ p, Irreducible p", "tactic": "obtain ⟨x, hx₁, hx₂⟩ := Ring.exists_not_isUnit_of_not_isField h" }, { "state_after": "case tfae_2_to_1.h₁.intro.intro.intro.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n✝ : ValuationRing R\nthis✝ : GCDMonoid R\nthis : UniqueFactorizationMonoid R\nx : R\nhx₁ : x ≠ 0\nhx₂ : ¬IsUnit x\np : R\nhp₁ : Irreducible p\n⊢ ∃ p, Irreducible p", "state_before": "case tfae_2_to_1.h₁.intro.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n✝ : ValuationRing R\nthis✝ : GCDMonoid R\nthis : UniqueFactorizationMonoid R\nx : R\nhx₁ : x ≠ 0\nhx₂ : ¬IsUnit x\n⊢ ∃ p, Irreducible p", "tactic": "obtain ⟨p, hp₁, -⟩ := WfDvdMonoid.exists_irreducible_factor hx₂ hx₁" }, { "state_after": "no goals", "state_before": "case tfae_2_to_1.h₁.intro.intro.intro.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n✝ : ValuationRing R\nthis✝ : GCDMonoid R\nthis : UniqueFactorizationMonoid R\nx : R\nhx₁ : x ≠ 0\nhx₂ : ¬IsUnit x\np : R\nhp₁ : Irreducible p\n⊢ ∃ p, Irreducible p", "tactic": "exact ⟨p, hp₁⟩" }, { "state_after": "no goals", "state_before": "case tfae_2_to_1.h₂\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\n✝ : ValuationRing R\nthis✝ : GCDMonoid R\nthis : UniqueFactorizationMonoid R\n⊢ ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q", "tactic": "exact ValuationRing.unique_irreducible" }, { "state_after": "case tfae_1_to_4\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\nH : DiscreteValuationRing R\n⊢ IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P", "state_before": "case tfae_1_to_4\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\n⊢ DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P", "tactic": "intro H" }, { "state_after": "no goals", "state_before": "case tfae_1_to_4\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\nH : DiscreteValuationRing R\n⊢ IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P", "tactic": "exact ⟨inferInstance, ((DiscreteValuationRing.iff_pid_with_one_nonzero_prime R).mp H).2⟩" }, { "state_after": "case tfae_4_to_3.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\nh₁ : IsIntegrallyClosed R\nh₂ : ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\n⊢ IsDedekindDomain R", "state_before": "case tfae_4_to_3\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\n⊢ (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R", "tactic": "rintro ⟨h₁, h₂⟩" }, { "state_after": "no goals", "state_before": "case tfae_4_to_3.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\nh₁ : IsIntegrallyClosed R\nh₂ : ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\n⊢ IsDedekindDomain R", "tactic": "exact\n ⟨inferInstance, fun I hI hI' =>\n ExistsUnique.unique h₂ ⟨ne_bot, inferInstance⟩ ⟨hI, hI'⟩ ▸ maximalIdeal.isMaximal R, h₁⟩" }, { "state_after": "case tfae_3_to_5\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh✝ : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\nh : IsDedekindDomain R\n⊢ Submodule.IsPrincipal (maximalIdeal R)", "state_before": "case tfae_3_to_5\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\n⊢ IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)", "tactic": "intro h" }, { "state_after": "no goals", "state_before": "case tfae_3_to_5\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh✝ : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\nh : IsDedekindDomain R\n⊢ Submodule.IsPrincipal (maximalIdeal R)", "tactic": "exact maximalIdeal_isPrincipal_of_isDedekindDomain R" }, { "state_after": "case tfae_5_to_6.mk.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\n⊢ ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w", "state_before": "case tfae_5_to_6\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\n⊢ Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w", "tactic": "rintro ⟨x, hx⟩" }, { "state_after": "case tfae_5_to_6.mk.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\n⊢ ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w", "state_before": "case tfae_5_to_6.mk.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\n⊢ ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w", "tactic": "have : x ∈ maximalIdeal R := by rw [hx]; exact Submodule.subset_span (Set.mem_singleton x)" }, { "state_after": "case tfae_5_to_6.mk.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\n⊢ ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w", "state_before": "case tfae_5_to_6.mk.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\n⊢ ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w", "tactic": "let x' : maximalIdeal R := ⟨x, this⟩" }, { "state_after": "case tfae_5_to_6.mk.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\n⊢ ∃ _n, ∀ (w : CotangentSpace R), ∃ c, c • Submodule.Quotient.mk x' = w", "state_before": "case tfae_5_to_6.mk.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\n⊢ ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w", "tactic": "use Submodule.Quotient.mk x'" }, { "state_after": "case tfae_5_to_6.mk.intro.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\n⊢ ∀ (w : CotangentSpace R), ∃ c, c • Submodule.Quotient.mk x' = w\n\ncase tfae_5_to_6.mk.intro.w\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\n⊢ Submodule.Quotient.mk x' ≠ 0", "state_before": "case tfae_5_to_6.mk.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\n⊢ ∃ _n, ∀ (w : CotangentSpace R), ∃ c, c • Submodule.Quotient.mk x' = w", "tactic": "constructor" }, { "state_after": "case tfae_5_to_6.mk.intro.w\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\n⊢ Submodule.Quotient.mk x' ≠ 0\n\ncase tfae_5_to_6.mk.intro.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\n⊢ ∀ (w : CotangentSpace R), ∃ c, c • Submodule.Quotient.mk x' = w", "state_before": "case tfae_5_to_6.mk.intro.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\n⊢ ∀ (w : CotangentSpace R), ∃ c, c • Submodule.Quotient.mk x' = w\n\ncase tfae_5_to_6.mk.intro.w\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\n⊢ Submodule.Quotient.mk x' ≠ 0", "tactic": "swap" }, { "state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\n⊢ x ∈ Submodule.span R {x}", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\n⊢ x ∈ maximalIdeal R", "tactic": "rw [hx]" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\n⊢ x ∈ Submodule.span R {x}", "tactic": "exact Submodule.subset_span (Set.mem_singleton x)" }, { "state_after": "case tfae_5_to_6.mk.intro.w\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : Submodule.Quotient.mk x' = 0\n⊢ False", "state_before": "case tfae_5_to_6.mk.intro.w\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\n⊢ Submodule.Quotient.mk x' ≠ 0", "tactic": "intro e" }, { "state_after": "case tfae_5_to_6.mk.intro.w\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : x' ∈ maximalIdeal R • ⊤\n⊢ False", "state_before": "case tfae_5_to_6.mk.intro.w\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : Submodule.Quotient.mk x' = 0\n⊢ False", "tactic": "rw [Submodule.Quotient.mk_eq_zero] at e" }, { "state_after": "case tfae_5_to_6.mk.intro.w\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : x' ∈ maximalIdeal R • ⊤\n⊢ maximalIdeal R = ⊥", "state_before": "case tfae_5_to_6.mk.intro.w\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : x' ∈ maximalIdeal R • ⊤\n⊢ False", "tactic": "apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h" }, { "state_after": "case tfae_5_to_6.mk.intro.w.hN\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : x' ∈ maximalIdeal R • ⊤\n⊢ Submodule.FG (maximalIdeal R)\n\ncase tfae_5_to_6.mk.intro.w.hIN\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : x' ∈ maximalIdeal R • ⊤\n⊢ maximalIdeal R ≤ maximalIdeal R • maximalIdeal R\n\ncase tfae_5_to_6.mk.intro.w.hIjac\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : x' ∈ maximalIdeal R • ⊤\n⊢ maximalIdeal R ≤ Ideal.jacobson ⊥", "state_before": "case tfae_5_to_6.mk.intro.w\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : x' ∈ maximalIdeal R • ⊤\n⊢ maximalIdeal R = ⊥", "tactic": "apply Submodule.eq_bot_of_le_smul_of_le_jacobson_bot (maximalIdeal R)" }, { "state_after": "no goals", "state_before": "case tfae_5_to_6.mk.intro.w.hN\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : x' ∈ maximalIdeal R • ⊤\n⊢ Submodule.FG (maximalIdeal R)", "tactic": "exact ⟨{x}, (Finset.coe_singleton x).symm ▸ hx.symm⟩" }, { "state_after": "case tfae_5_to_6.mk.intro.w.hIN\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : x' ∈ maximalIdeal R • ⊤\n⊢ Submodule.span R {x} ≤ maximalIdeal R • maximalIdeal R", "state_before": "case tfae_5_to_6.mk.intro.w.hIN\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : x' ∈ maximalIdeal R • ⊤\n⊢ maximalIdeal R ≤ maximalIdeal R • maximalIdeal R", "tactic": "conv_lhs => rw [hx]" }, { "state_after": "case tfae_5_to_6.mk.intro.w.hIN\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : ↑x' ∈ maximalIdeal R • maximalIdeal R\n⊢ Submodule.span R {x} ≤ maximalIdeal R • maximalIdeal R", "state_before": "case tfae_5_to_6.mk.intro.w.hIN\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : x' ∈ maximalIdeal R • ⊤\n⊢ Submodule.span R {x} ≤ maximalIdeal R • maximalIdeal R", "tactic": "rw [Submodule.mem_smul_top_iff] at e" }, { "state_after": "no goals", "state_before": "case tfae_5_to_6.mk.intro.w.hIN\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : ↑x' ∈ maximalIdeal R • maximalIdeal R\n⊢ Submodule.span R {x} ≤ maximalIdeal R • maximalIdeal R", "tactic": "rwa [Submodule.span_le, Set.singleton_subset_iff]" }, { "state_after": "no goals", "state_before": "case tfae_5_to_6.mk.intro.w.hIjac\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\ne : x' ∈ maximalIdeal R • ⊤\n⊢ maximalIdeal R ≤ Ideal.jacobson ⊥", "tactic": "rw [LocalRing.jacobson_eq_maximalIdeal (⊥ : Ideal R) bot_ne_top]" }, { "state_after": "case tfae_5_to_6.mk.intro.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\nw : CotangentSpace R\ny : { x // x ∈ maximalIdeal R }\n⊢ ∃ c, c • Submodule.Quotient.mk x' = Quotient.mk'' y", "state_before": "case tfae_5_to_6.mk.intro.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\n⊢ ∀ (w : CotangentSpace R), ∃ c, c • Submodule.Quotient.mk x' = w", "tactic": "refine' fun w => Quotient.inductionOn' w fun y => _" }, { "state_after": "case tfae_5_to_6.mk.intro.h.mk\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\nw : CotangentSpace R\ny : R\nhy : y ∈ maximalIdeal R\n⊢ ∃ c, c • Submodule.Quotient.mk x' = Quotient.mk'' { val := y, property := hy }", "state_before": "case tfae_5_to_6.mk.intro.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\nw : CotangentSpace R\ny : { x // x ∈ maximalIdeal R }\n⊢ ∃ c, c • Submodule.Quotient.mk x' = Quotient.mk'' y", "tactic": "obtain ⟨y, hy⟩ := y" }, { "state_after": "case tfae_5_to_6.mk.intro.h.mk\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\nw : CotangentSpace R\ny : R\nhy✝ : y ∈ maximalIdeal R\nhy : ∃ a, a • x = y\n⊢ ∃ c, c • Submodule.Quotient.mk x' = Quotient.mk'' { val := y, property := hy✝ }", "state_before": "case tfae_5_to_6.mk.intro.h.mk\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\nw : CotangentSpace R\ny : R\nhy : y ∈ maximalIdeal R\n⊢ ∃ c, c • Submodule.Quotient.mk x' = Quotient.mk'' { val := y, property := hy }", "tactic": "rw [hx, Submodule.mem_span_singleton] at hy" }, { "state_after": "case tfae_5_to_6.mk.intro.h.mk.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\nw : CotangentSpace R\na : R\nhy : a • x ∈ maximalIdeal R\n⊢ ∃ c, c • Submodule.Quotient.mk x' = Quotient.mk'' { val := a • x, property := hy }", "state_before": "case tfae_5_to_6.mk.intro.h.mk\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\nw : CotangentSpace R\ny : R\nhy✝ : y ∈ maximalIdeal R\nhy : ∃ a, a • x = y\n⊢ ∃ c, c • Submodule.Quotient.mk x' = Quotient.mk'' { val := y, property := hy✝ }", "tactic": "obtain ⟨a, rfl⟩ := hy" }, { "state_after": "no goals", "state_before": "case tfae_5_to_6.mk.intro.h.mk.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x ∈ maximalIdeal R\nx' : { x // x ∈ maximalIdeal R } := { val := x, property := this }\nw : CotangentSpace R\na : R\nhy : a • x ∈ maximalIdeal R\n⊢ ∃ c, c • Submodule.Quotient.mk x' = Quotient.mk'' { val := a • x, property := hy }", "tactic": "exact ⟨Ideal.Quotient.mk _ a, rfl⟩" }, { "state_after": "case tfae_6_to_5.intro.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : CotangentSpace R\nhx : x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • x = w\n⊢ Submodule.IsPrincipal (maximalIdeal R)", "state_before": "case tfae_6_to_5\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\n⊢ (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)", "tactic": "rintro ⟨x, hx, hx'⟩" }, { "state_after": "case tfae_6_to_5.intro.intro.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ Submodule.IsPrincipal (maximalIdeal R)", "state_before": "case tfae_6_to_5.intro.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : CotangentSpace R\nhx : x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • x = w\n⊢ Submodule.IsPrincipal (maximalIdeal R)", "tactic": "induction x using Quotient.inductionOn' with | h x => ?_" }, { "state_after": "case tfae_6_to_5.intro.intro.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ maximalIdeal R = Submodule.span R {↑x}", "state_before": "case tfae_6_to_5.intro.intro.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ Submodule.IsPrincipal (maximalIdeal R)", "tactic": "use x" }, { "state_after": "case tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ maximalIdeal R ≤ Submodule.span R {↑x}\n\ncase tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ Submodule.span R {↑x} ≤ maximalIdeal R", "state_before": "case tfae_6_to_5.intro.intro.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ maximalIdeal R = Submodule.span R {↑x}", "tactic": "apply le_antisymm" }, { "state_after": "case tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ Submodule.span R {↑x} ≤ maximalIdeal R\n\ncase tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ maximalIdeal R ≤ Submodule.span R {↑x}", "state_before": "case tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ maximalIdeal R ≤ Submodule.span R {↑x}\n\ncase tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ Submodule.span R {↑x} ≤ maximalIdeal R", "tactic": "swap" }, { "state_after": "case tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nh₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R\nh₂ : maximalIdeal R ≤ Ideal.jacobson ⊥\n⊢ maximalIdeal R ≤ Submodule.span R {↑x}", "state_before": "case tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nh₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R\n⊢ maximalIdeal R ≤ Submodule.span R {↑x}", "tactic": "have h₂ : maximalIdeal R ≤ (⊥ : Ideal R).jacobson := by\n rw [LocalRing.jacobson_eq_maximalIdeal]\n exact bot_ne_top" }, { "state_after": "case tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nh₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R\nh₂ : maximalIdeal R ≤ Ideal.jacobson ⊥\nthis : Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R = Ideal.span {↑x} ⊔ ⊥ • maximalIdeal R\n⊢ maximalIdeal R ≤ Submodule.span R {↑x}", "state_before": "case tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nh₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R\nh₂ : maximalIdeal R ≤ Ideal.jacobson ⊥\n⊢ maximalIdeal R ≤ Submodule.span R {↑x}", "tactic": "have :=\n Submodule.smul_sup_eq_smul_sup_of_le_smul_of_le_jacobson (IsNoetherian.noetherian _) h₂ h₁" }, { "state_after": "case tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nh₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R\nh₂ : maximalIdeal R ≤ Ideal.jacobson ⊥\nthis : Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R = Ideal.span {↑x}\n⊢ maximalIdeal R ≤ Submodule.span R {↑x}", "state_before": "case tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nh₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R\nh₂ : maximalIdeal R ≤ Ideal.jacobson ⊥\nthis : Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R = Ideal.span {↑x} ⊔ ⊥ • maximalIdeal R\n⊢ maximalIdeal R ≤ Submodule.span R {↑x}", "tactic": "rw [Submodule.bot_smul, sup_bot_eq] at this" }, { "state_after": "case tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nh₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R\nh₂ : maximalIdeal R ≤ Ideal.jacobson ⊥\nthis : Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R = Ideal.span {↑x}\n⊢ Submodule.span R {↑x} = Submodule.span R {↑x} ⊔ maximalIdeal R", "state_before": "case tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nh₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R\nh₂ : maximalIdeal R ≤ Ideal.jacobson ⊥\nthis : Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R = Ideal.span {↑x}\n⊢ maximalIdeal R ≤ Submodule.span R {↑x}", "tactic": "rw [← sup_eq_left, eq_comm]" }, { "state_after": "no goals", "state_before": "case tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nh₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R\nh₂ : maximalIdeal R ≤ Ideal.jacobson ⊥\nthis : Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R = Ideal.span {↑x}\n⊢ Submodule.span R {↑x} = Submodule.span R {↑x} ⊔ maximalIdeal R", "tactic": "exact le_sup_left.antisymm (h₁.trans <| le_of_eq this)" }, { "state_after": "case tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ ↑x ∈ ↑(maximalIdeal R)", "state_before": "case tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ Submodule.span R {↑x} ≤ maximalIdeal R", "tactic": "rw [Submodule.span_le, Set.singleton_subset_iff]" }, { "state_after": "no goals", "state_before": "case tfae_6_to_5.intro.intro.h.a\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ ↑x ∈ ↑(maximalIdeal R)", "tactic": "exact x.prop" }, { "state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "tactic": "refine' sup_le le_sup_left _" }, { "state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\n⊢ m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\n⊢ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "tactic": "rintro m hm" }, { "state_after": "case intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\nc : ResidueField R\nhc : c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n⊢ m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\n⊢ m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "tactic": "obtain ⟨c, hc⟩ := hx' (Submodule.Quotient.mk ⟨m, hm⟩)" }, { "state_after": "case intro.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\nc : R\nhc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n⊢ m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "state_before": "case intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\nc : ResidueField R\nhc : c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n⊢ m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "tactic": "induction c using Quotient.inductionOn' with | h c => ?_" }, { "state_after": "case intro.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\nc : R\nhc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n⊢ c * ↑x - (c * ↑x - m) ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "state_before": "case intro.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\nc : R\nhc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n⊢ m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "tactic": "rw [← sub_sub_cancel (c * x) m]" }, { "state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\nc : R\nhc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n⊢ c * ↑x ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R\n\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\nc : R\nhc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n⊢ c * ↑x - m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "state_before": "case intro.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\nc : R\nhc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n⊢ c * ↑x - (c * ↑x - m) ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "tactic": "apply sub_mem _ _" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\nc : R\nhc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n⊢ c * ↑x ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "tactic": "refine' Ideal.mem_sup_left (Ideal.mem_span_singleton'.mpr ⟨c, rfl⟩)" }, { "state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\nc : R\nhc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\nthis : (fun x x_1 => x • x_1) c x - { val := m, property := hm } ∈ maximalIdeal R • ⊤\n⊢ c * ↑x - m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\nc : R\nhc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n⊢ c * ↑x - m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "tactic": "have := (Submodule.Quotient.eq _).mp hc" }, { "state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\nc : R\nhc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\nthis : ↑((fun x x_1 => x • x_1) c x - { val := m, property := hm }) ∈ maximalIdeal R • maximalIdeal R\n⊢ c * ↑x - m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\nc : R\nhc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\nthis : (fun x x_1 => x • x_1) c x - { val := m, property := hm } ∈ maximalIdeal R • ⊤\n⊢ c * ↑x - m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "tactic": "rw [Submodule.mem_smul_top_iff] at this" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nm : R\nhm : m ∈ maximalIdeal R\nc : R\nhc : Quotient.mk'' c • Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\nthis : ↑((fun x x_1 => x • x_1) c x - { val := m, property := hm }) ∈ maximalIdeal R • maximalIdeal R\n⊢ c * ↑x - m ∈ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R", "tactic": "exact Ideal.mem_sup_right this" }, { "state_after": "case h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nh₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R\n⊢ ⊥ ≠ ⊤", "state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nh₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R\n⊢ maximalIdeal R ≤ Ideal.jacobson ⊥", "tactic": "rw [LocalRing.jacobson_eq_maximalIdeal]" }, { "state_after": "no goals", "state_before": "case h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\nx : { x // x ∈ maximalIdeal R }\nhx : Quotient.mk'' x ≠ 0\nhx' : ∀ (w : CotangentSpace R), ∃ c, c • Quotient.mk'' x = w\nh₁ : Ideal.span {↑x} ⊔ maximalIdeal R ≤ Ideal.span {↑x} ⊔ maximalIdeal R • maximalIdeal R\n⊢ ⊥ ≠ ⊤", "tactic": "exact bot_ne_top" }, { "state_after": "no goals", "state_before": "case tfae_5_to_7\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\n⊢ Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n", "tactic": "exact exists_maximalIdeal_pow_eq_of_principal R h" }, { "state_after": "case tfae_7_to_2\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\n⊢ (∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n) → IsTotal (Ideal R) fun x x_1 => x ≤ x_1", "state_before": "case tfae_7_to_2\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\n⊢ (∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n) → ValuationRing R", "tactic": "rw [ValuationRing.iff_ideal_total]" }, { "state_after": "case tfae_7_to_2\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\n⊢ IsTotal (Ideal R) fun x x_1 => x ≤ x_1", "state_before": "case tfae_7_to_2\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\n⊢ (∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n) → IsTotal (Ideal R) fun x x_1 => x ≤ x_1", "tactic": "intro H" }, { "state_after": "case tfae_7_to_2.total\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\n⊢ ∀ (a b : Ideal R), a ≤ b ∨ b ≤ a", "state_before": "case tfae_7_to_2\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\n⊢ IsTotal (Ideal R) fun x x_1 => x ≤ x_1", "tactic": "constructor" }, { "state_after": "case tfae_7_to_2.total\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nI J : Ideal R\n⊢ I ≤ J ∨ J ≤ I", "state_before": "case tfae_7_to_2.total\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\n⊢ ∀ (a b : Ideal R), a ≤ b ∨ b ≤ a", "tactic": "intro I J" }, { "state_after": "case pos\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nI J : Ideal R\nhI : I = ⊥\n⊢ I ≤ J ∨ J ≤ I\n\ncase neg\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nI J : Ideal R\nhI : ¬I = ⊥\n⊢ I ≤ J ∨ J ≤ I", "state_before": "case tfae_7_to_2.total\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nI J : Ideal R\n⊢ I ≤ J ∨ J ≤ I", "tactic": "by_cases hI : I = ⊥" }, { "state_after": "case pos\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nI J : Ideal R\nhI : ¬I = ⊥\nhJ : J = ⊥\n⊢ I ≤ J ∨ J ≤ I\n\ncase neg\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nI J : Ideal R\nhI : ¬I = ⊥\nhJ : ¬J = ⊥\n⊢ I ≤ J ∨ J ≤ I", "state_before": "case neg\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nI J : Ideal R\nhI : ¬I = ⊥\n⊢ I ≤ J ∨ J ≤ I", "tactic": "by_cases hJ : J = ⊥" }, { "state_after": "case neg.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nJ : Ideal R\nhJ : ¬J = ⊥\nn : ℕ\nhI : ¬maximalIdeal R ^ n = ⊥\n⊢ maximalIdeal R ^ n ≤ J ∨ J ≤ maximalIdeal R ^ n", "state_before": "case neg\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nI J : Ideal R\nhI : ¬I = ⊥\nhJ : ¬J = ⊥\n⊢ I ≤ J ∨ J ≤ I", "tactic": "obtain ⟨n, rfl⟩ := H I hI" }, { "state_after": "case neg.intro.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nn : ℕ\nhI : ¬maximalIdeal R ^ n = ⊥\nm : ℕ\nhJ : ¬maximalIdeal R ^ m = ⊥\n⊢ maximalIdeal R ^ n ≤ maximalIdeal R ^ m ∨ maximalIdeal R ^ m ≤ maximalIdeal R ^ n", "state_before": "case neg.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nJ : Ideal R\nhJ : ¬J = ⊥\nn : ℕ\nhI : ¬maximalIdeal R ^ n = ⊥\n⊢ maximalIdeal R ^ n ≤ J ∨ J ≤ maximalIdeal R ^ n", "tactic": "obtain ⟨m, rfl⟩ := H J hJ" }, { "state_after": "case neg.intro.intro.inl\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nn : ℕ\nhI : ¬maximalIdeal R ^ n = ⊥\nm : ℕ\nhJ : ¬maximalIdeal R ^ m = ⊥\nh' : m ≤ n\n⊢ maximalIdeal R ^ n ≤ maximalIdeal R ^ m ∨ maximalIdeal R ^ m ≤ maximalIdeal R ^ n\n\ncase neg.intro.intro.inr\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nn : ℕ\nhI : ¬maximalIdeal R ^ n = ⊥\nm : ℕ\nhJ : ¬maximalIdeal R ^ m = ⊥\nh' : n ≤ m\n⊢ maximalIdeal R ^ n ≤ maximalIdeal R ^ m ∨ maximalIdeal R ^ m ≤ maximalIdeal R ^ n", "state_before": "case neg.intro.intro\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nn : ℕ\nhI : ¬maximalIdeal R ^ n = ⊥\nm : ℕ\nhJ : ¬maximalIdeal R ^ m = ⊥\n⊢ maximalIdeal R ^ n ≤ maximalIdeal R ^ m ∨ maximalIdeal R ^ m ≤ maximalIdeal R ^ n", "tactic": "cases' le_total m n with h' h'" }, { "state_after": "case pos\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nJ : Ideal R\n⊢ ⊥ ≤ J ∨ J ≤ ⊥", "state_before": "case pos\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nI J : Ideal R\nhI : I = ⊥\n⊢ I ≤ J ∨ J ≤ I", "tactic": "subst hI" }, { "state_after": "case pos.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nJ : Ideal R\n⊢ ⊥ ≤ J", "state_before": "case pos\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nJ : Ideal R\n⊢ ⊥ ≤ J ∨ J ≤ ⊥", "tactic": "left" }, { "state_after": "no goals", "state_before": "case pos.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nJ : Ideal R\n⊢ ⊥ ≤ J", "tactic": "exact bot_le" }, { "state_after": "case pos\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nI : Ideal R\nhI : ¬I = ⊥\n⊢ I ≤ ⊥ ∨ ⊥ ≤ I", "state_before": "case pos\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nI J : Ideal R\nhI : ¬I = ⊥\nhJ : J = ⊥\n⊢ I ≤ J ∨ J ≤ I", "tactic": "subst hJ" }, { "state_after": "case pos.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nI : Ideal R\nhI : ¬I = ⊥\n⊢ ⊥ ≤ I", "state_before": "case pos\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nI : Ideal R\nhI : ¬I = ⊥\n⊢ I ≤ ⊥ ∨ ⊥ ≤ I", "tactic": "right" }, { "state_after": "no goals", "state_before": "case pos.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nI : Ideal R\nhI : ¬I = ⊥\n⊢ ⊥ ≤ I", "tactic": "exact bot_le" }, { "state_after": "case neg.intro.intro.inl.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nn : ℕ\nhI : ¬maximalIdeal R ^ n = ⊥\nm : ℕ\nhJ : ¬maximalIdeal R ^ m = ⊥\nh' : m ≤ n\n⊢ maximalIdeal R ^ n ≤ maximalIdeal R ^ m", "state_before": "case neg.intro.intro.inl\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nn : ℕ\nhI : ¬maximalIdeal R ^ n = ⊥\nm : ℕ\nhJ : ¬maximalIdeal R ^ m = ⊥\nh' : m ≤ n\n⊢ maximalIdeal R ^ n ≤ maximalIdeal R ^ m ∨ maximalIdeal R ^ m ≤ maximalIdeal R ^ n", "tactic": "left" }, { "state_after": "no goals", "state_before": "case neg.intro.intro.inl.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nn : ℕ\nhI : ¬maximalIdeal R ^ n = ⊥\nm : ℕ\nhJ : ¬maximalIdeal R ^ m = ⊥\nh' : m ≤ n\n⊢ maximalIdeal R ^ n ≤ maximalIdeal R ^ m", "tactic": "exact Ideal.pow_le_pow h'" }, { "state_after": "case neg.intro.intro.inr.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nn : ℕ\nhI : ¬maximalIdeal R ^ n = ⊥\nm : ℕ\nhJ : ¬maximalIdeal R ^ m = ⊥\nh' : n ≤ m\n⊢ maximalIdeal R ^ m ≤ maximalIdeal R ^ n", "state_before": "case neg.intro.intro.inr\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nn : ℕ\nhI : ¬maximalIdeal R ^ n = ⊥\nm : ℕ\nhJ : ¬maximalIdeal R ^ m = ⊥\nh' : n ≤ m\n⊢ maximalIdeal R ^ n ≤ maximalIdeal R ^ m ∨ maximalIdeal R ^ m ≤ maximalIdeal R ^ n", "tactic": "right" }, { "state_after": "no goals", "state_before": "case neg.intro.intro.inr.h\nR : Type u_1\ninst✝⁶ : CommRing R\nK : Type ?u.224397\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsNoetherianRing R\ninst✝¹ : LocalRing R\ninst✝ : IsDomain R\nh : ¬IsField R\nne_bot : maximalIdeal R ≠ ⊥\ntfae_1_to_2 : DiscreteValuationRing R → ValuationRing R\ntfae_2_to_1 : ValuationRing R → DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R → IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R ∧ ∃! P, P ≠ ⊥ ∧ Ideal.IsPrime P) → IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) → ∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w\ntfae_6_to_5 : (∃ v _n, ∀ (w : CotangentSpace R), ∃ c, c • v = w) → Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) → ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nH : ∀ (I : Ideal R), I ≠ ⊥ → ∃ n, I = maximalIdeal R ^ n\nn : ℕ\nhI : ¬maximalIdeal R ^ n = ⊥\nm : ℕ\nhJ : ¬maximalIdeal R ^ m = ⊥\nh' : n ≤ m\n⊢ maximalIdeal R ^ m ≤ maximalIdeal R ^ n", "tactic": "exact Ideal.pow_le_pow h'" } ]
[ 249, 14 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 156, 1 ]
Mathlib/Topology/Algebra/Order/ProjIcc.lean
Filter.Tendsto.IccExtend'
[]
[ 36, 17 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 33, 1 ]
Mathlib/Analysis/Calculus/FDeriv/Star.lean
DifferentiableWithinAt.star
[]
[ 61, 50 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 59, 1 ]
Mathlib/RingTheory/Finiteness.lean
Submodule.fg_top
[ { "state_after": "no goals", "state_before": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.199269\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf : M →ₗ[R] P\nN : Submodule R M\nh : FG N\n⊢ FG (map (Submodule.subtype N) ⊤)", "tactic": "rwa [map_top, range_subtype]" } ]
[ 234, 95 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 232, 1 ]
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
MeasureTheory.integrableOn_union
[]
[ 190, 73 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 189, 1 ]
Mathlib/Data/Dfinsupp/Basic.lean
Dfinsupp.prod_zero_index
[]
[ 1730, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1728, 1 ]
Mathlib/Data/Polynomial/Identities.lean
Polynomial.poly_binom_aux2
[ { "state_after": "R : Type u\nS : Type v\nT : Type w\nι : Type x\nk : Type y\nA : Type z\na b : R\nm n : ℕ\ninst✝ : CommRing R\nf : R[X]\nx y : R\n⊢ eval₂ (RingHom.id R) (x + y) f =\n sum f fun e a => a * (x ^ e + ↑e * x ^ (e - 1) * y + ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2)", "state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type x\nk : Type y\nA : Type z\na b : R\nm n : ℕ\ninst✝ : CommRing R\nf : R[X]\nx y : R\n⊢ eval (x + y) f = sum f fun e a => a * (x ^ e + ↑e * x ^ (e - 1) * y + ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2)", "tactic": "unfold eval" }, { "state_after": "R : Type u\nS : Type v\nT : Type w\nι : Type x\nk : Type y\nA : Type z\na b : R\nm n : ℕ\ninst✝ : CommRing R\nf : R[X]\nx y : R\n⊢ (sum f fun e a => ↑(RingHom.id R) a * (x + y) ^ e) =\n sum f fun e a => a * (x ^ e + ↑e * x ^ (e - 1) * y + ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2)", "state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type x\nk : Type y\nA : Type z\na b : R\nm n : ℕ\ninst✝ : CommRing R\nf : R[X]\nx y : R\n⊢ eval₂ (RingHom.id R) (x + y) f =\n sum f fun e a => a * (x ^ e + ↑e * x ^ (e - 1) * y + ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2)", "tactic": "rw [eval₂_eq_sum]" }, { "state_after": "case e_f.h.h\nR : Type u\nS : Type v\nT : Type w\nι : Type x\nk : Type y\nA : Type z\na b : R\nm n✝ : ℕ\ninst✝ : CommRing R\nf : R[X]\nx y : R\nn : ℕ\nz : R\n⊢ ↑(RingHom.id R) z * (x + y) ^ n = z * (x ^ n + ↑n * x ^ (n - 1) * y + ↑(Polynomial.polyBinomAux1 x y n z) * y ^ 2)", "state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type x\nk : Type y\nA : Type z\na b : R\nm n : ℕ\ninst✝ : CommRing R\nf : R[X]\nx y : R\n⊢ (sum f fun e a => ↑(RingHom.id R) a * (x + y) ^ e) =\n sum f fun e a => a * (x ^ e + ↑e * x ^ (e - 1) * y + ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2)", "tactic": "congr with (n z)" }, { "state_after": "no goals", "state_before": "case e_f.h.h\nR : Type u\nS : Type v\nT : Type w\nι : Type x\nk : Type y\nA : Type z\na b : R\nm n✝ : ℕ\ninst✝ : CommRing R\nf : R[X]\nx y : R\nn : ℕ\nz : R\n⊢ ↑(RingHom.id R) z * (x + y) ^ n = z * (x ^ n + ↑n * x ^ (n - 1) * y + ↑(Polynomial.polyBinomAux1 x y n z) * y ^ 2)", "tactic": "apply (polyBinomAux1 x y _ _).property" } ]
[ 69, 41 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 65, 9 ]
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
Algebra.top_toSubsemiring
[]
[ 794, 74 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 794, 1 ]
Mathlib/Combinatorics/SimpleGraph/Trails.lean
SimpleGraph.Walk.IsEulerian.isTrail
[ { "state_after": "V : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\n⊢ ∀ (a : Sym2 V), List.count a (edges p) ≤ 1", "state_before": "V : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\n⊢ IsTrail p", "tactic": "rw [isTrail_def, List.nodup_iff_count_le_one]" }, { "state_after": "V : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\ne : Sym2 V\n⊢ List.count e (edges p) ≤ 1", "state_before": "V : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\n⊢ ∀ (a : Sym2 V), List.count a (edges p) ≤ 1", "tactic": "intro e" }, { "state_after": "case pos\nV : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\ne : Sym2 V\nhe : e ∈ edges p\n⊢ List.count e (edges p) ≤ 1\n\ncase neg\nV : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\ne : Sym2 V\nhe : ¬e ∈ edges p\n⊢ List.count e (edges p) ≤ 1", "state_before": "V : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\ne : Sym2 V\n⊢ List.count e (edges p) ≤ 1", "tactic": "by_cases he : e ∈ p.edges" }, { "state_after": "no goals", "state_before": "case pos\nV : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\ne : Sym2 V\nhe : e ∈ edges p\n⊢ List.count e (edges p) ≤ 1", "tactic": "exact (h e (edges_subset_edgeSet _ he)).le" }, { "state_after": "no goals", "state_before": "case neg\nV : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\ne : Sym2 V\nhe : ¬e ∈ edges p\n⊢ List.count e (edges p) ≤ 1", "tactic": "simp [he]" } ]
[ 102, 14 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 97, 1 ]
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
MeasureTheory.one_le_prob_iff
[]
[ 3243, 63 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 3242, 1 ]
Mathlib/ModelTheory/FinitelyGenerated.lean
FirstOrder.Language.Structure.FG.map_of_surjective
[ { "state_after": "L : Language\nM : Type u_4\ninst✝¹ : Structure L M\nN : Type u_1\ninst✝ : Structure L N\nh : FG L M\nf : M →[L] N\nhs : Hom.range f = ⊤\n⊢ FG L N", "state_before": "L : Language\nM : Type u_4\ninst✝¹ : Structure L M\nN : Type u_1\ninst✝ : Structure L N\nh : FG L M\nf : M →[L] N\nhs : Function.Surjective ↑f\n⊢ FG L N", "tactic": "rw [← Hom.range_eq_top] at hs" }, { "state_after": "L : Language\nM : Type u_4\ninst✝¹ : Structure L M\nN : Type u_1\ninst✝ : Structure L N\nh : FG L M\nf : M →[L] N\nhs : Hom.range f = ⊤\n⊢ Substructure.FG (Hom.range f)", "state_before": "L : Language\nM : Type u_4\ninst✝¹ : Structure L M\nN : Type u_1\ninst✝ : Structure L N\nh : FG L M\nf : M →[L] N\nhs : Hom.range f = ⊤\n⊢ FG L N", "tactic": "rw [fg_def, ← hs]" }, { "state_after": "no goals", "state_before": "L : Language\nM : Type u_4\ninst✝¹ : Structure L M\nN : Type u_1\ninst✝ : Structure L N\nh : FG L M\nf : M →[L] N\nhs : Hom.range f = ⊤\n⊢ Substructure.FG (Hom.range f)", "tactic": "exact h.range f" } ]
[ 226, 18 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 222, 1 ]
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
intervalIntegral.abs_integral_le_integral_abs
[ { "state_after": "no goals", "state_before": "ι : Type ?u.20745139\n𝕜 : Type ?u.20745142\nE : Type ?u.20745145\nF : Type ?u.20745148\nA : Type ?u.20745151\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf g : ℝ → ℝ\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nhab : a ≤ b\n⊢ abs (∫ (x : ℝ) in a..b, f x ∂μ) ≤ ∫ (x : ℝ) in a..b, abs (f x) ∂μ", "tactic": "simpa only [← Real.norm_eq_abs] using norm_integral_le_integral_norm hab" } ]
[ 1376, 75 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1374, 1 ]
Mathlib/LinearAlgebra/Dimension.lean
Submodule.rank_add_le_rank_add_rank
[ { "state_after": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1021823\ninst✝¹⁰ : DivisionRing K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\ninst✝⁷ : AddCommGroup V'\ninst✝⁶ : Module K V'\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ns t : Submodule K V\n⊢ Module.rank K { x // x ∈ s ⊔ t } ≤ Module.rank K { x // x ∈ s ⊔ t } + Module.rank K { x // x ∈ s ⊓ t }", "state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1021823\ninst✝¹⁰ : DivisionRing K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\ninst✝⁷ : AddCommGroup V'\ninst✝⁶ : Module K V'\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ns t : Submodule K V\n⊢ Module.rank K { x // x ∈ s ⊔ t } ≤ Module.rank K { x // x ∈ s } + Module.rank K { x // x ∈ t }", "tactic": "rw [← Submodule.rank_sup_add_rank_inf_eq]" }, { "state_after": "no goals", "state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1021823\ninst✝¹⁰ : DivisionRing K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\ninst✝⁷ : AddCommGroup V'\ninst✝⁶ : Module K V'\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ns t : Submodule K V\n⊢ Module.rank K { x // x ∈ s ⊔ t } ≤ Module.rank K { x // x ∈ s ⊔ t } + Module.rank K { x // x ∈ s ⊓ t }", "tactic": "exact self_le_add_right _ _" } ]
[ 1158, 30 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1155, 1 ]
Mathlib/Analysis/SpecialFunctions/Exp.lean
Complex.comap_exp_nhds_zero
[ { "state_after": "no goals", "state_before": "⊢ comap re (comap Real.exp (𝓝 0)) = comap re atBot", "tactic": "rw [Real.comap_exp_nhds_zero]" } ]
[ 438, 59 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 434, 1 ]
Mathlib/Data/Num/Lemmas.lean
PosNum.add_succ
[ { "state_after": "no goals", "state_before": "α : Type ?u.79617\nb : PosNum\n⊢ 1 + succ b = succ (1 + b)", "tactic": "simp [one_add]" } ]
[ 112, 52 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 105, 1 ]
Mathlib/Topology/Basic.lean
Finset.interior_iInter
[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nι✝ : Sort w\na : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\nι : Type u_1\ns : Finset ι\nf : ι → Set α\n⊢ interior (⋂ (i : ι) (_ : i ∈ s), f i) = ⋂ (i : ι) (_ : i ∈ s), interior (f i)", "tactic": "classical\n refine' s.induction_on (by simp) _\n intro i s _ h₂\n simp [h₂]" }, { "state_after": "α : Type u\nβ : Type v\nι✝ : Sort w\na : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\nι : Type u_1\ns : Finset ι\nf : ι → Set α\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n (interior (⋂ (i : ι) (_ : i ∈ s), f i) = ⋂ (i : ι) (_ : i ∈ s), interior (f i)) →\n interior (⋂ (i : ι) (_ : i ∈ insert a s), f i) = ⋂ (i : ι) (_ : i ∈ insert a s), interior (f i)", "state_before": "α : Type u\nβ : Type v\nι✝ : Sort w\na : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\nι : Type u_1\ns : Finset ι\nf : ι → Set α\n⊢ interior (⋂ (i : ι) (_ : i ∈ s), f i) = ⋂ (i : ι) (_ : i ∈ s), interior (f i)", "tactic": "refine' s.induction_on (by simp) _" }, { "state_after": "α : Type u\nβ : Type v\nι✝ : Sort w\na : α\ns✝¹ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\nι : Type u_1\ns✝ : Finset ι\nf : ι → Set α\ni : ι\ns : Finset ι\na✝ : ¬i ∈ s\nh₂ : interior (⋂ (i : ι) (_ : i ∈ s), f i) = ⋂ (i : ι) (_ : i ∈ s), interior (f i)\n⊢ interior (⋂ (i_1 : ι) (_ : i_1 ∈ insert i s), f i_1) = ⋂ (i_1 : ι) (_ : i_1 ∈ insert i s), interior (f i_1)", "state_before": "α : Type u\nβ : Type v\nι✝ : Sort w\na : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\nι : Type u_1\ns : Finset ι\nf : ι → Set α\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n (interior (⋂ (i : ι) (_ : i ∈ s), f i) = ⋂ (i : ι) (_ : i ∈ s), interior (f i)) →\n interior (⋂ (i : ι) (_ : i ∈ insert a s), f i) = ⋂ (i : ι) (_ : i ∈ insert a s), interior (f i)", "tactic": "intro i s _ h₂" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nι✝ : Sort w\na : α\ns✝¹ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\nι : Type u_1\ns✝ : Finset ι\nf : ι → Set α\ni : ι\ns : Finset ι\na✝ : ¬i ∈ s\nh₂ : interior (⋂ (i : ι) (_ : i ∈ s), f i) = ⋂ (i : ι) (_ : i ∈ s), interior (f i)\n⊢ interior (⋂ (i_1 : ι) (_ : i_1 ∈ insert i s), f i_1) = ⋂ (i_1 : ι) (_ : i_1 ∈ insert i s), interior (f i_1)", "tactic": "simp [h₂]" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nι✝ : Sort w\na : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\nι : Type u_1\ns : Finset ι\nf : ι → Set α\n⊢ interior (⋂ (i : ι) (_ : i ∈ ∅), f i) = ⋂ (i : ι) (_ : i ∈ ∅), interior (f i)", "tactic": "simp" } ]
[ 362, 14 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 357, 1 ]
Mathlib/Data/Fintype/Card.lean
Fintype.card_eq_one_of_forall_eq
[]
[ 589, 35 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 588, 1 ]
Mathlib/Topology/Inseparable.lean
SeparationQuotient.continuous_lift₂
[ { "state_after": "no goals", "state_before": "X : Type u_1\nY : Type u_2\nZ : Type u_3\nα : Type ?u.113085\nι : Type ?u.113088\nπ : ι → Type ?u.113093\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\nf✝ : X → Y\nt : Set (SeparationQuotient X)\nf : X → Y → Z\nhf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d\n⊢ Continuous (uncurry (lift₂ f hf)) ↔ Continuous (uncurry f)", "tactic": "simp only [continuous_iff_continuousOn_univ, continuousOn_lift₂, preimage_univ]" } ]
[ 634, 82 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 632, 1 ]
Mathlib/Analysis/Complex/Schwarz.lean
Complex.schwarz_aux
[ { "state_after": "E : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\n⊢ ‖dslope f c z‖ ≤ R₂ / R₁", "state_before": "E : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : z ∈ ball c R₁\n⊢ ‖dslope f c z‖ ≤ R₂ / R₁", "tactic": "have hR₁ : 0 < R₁ := nonempty_ball.1 ⟨z, hz⟩" }, { "state_after": "E : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\n⊢ ∀ᶠ (r : ℝ) in 𝓝[Iio R₁] R₁, ‖dslope f c z‖ ≤ R₂ / r", "state_before": "E : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\n⊢ ‖dslope f c z‖ ≤ R₂ / R₁", "tactic": "suffices ∀ᶠ r in 𝓝[<] R₁, ‖dslope f c z‖ ≤ R₂ / r by\n refine' ge_of_tendsto _ this\n exact (tendsto_const_nhds.div tendsto_id hR₁.ne').mono_left nhdsWithin_le_nhds" }, { "state_after": "E : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\n⊢ ∀ᶠ (r : ℝ) in 𝓝[Iio R₁] R₁, ‖dslope f c z‖ ≤ R₂ / r", "state_before": "E : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\n⊢ ∀ᶠ (r : ℝ) in 𝓝[Iio R₁] R₁, ‖dslope f c z‖ ≤ R₂ / r", "tactic": "rw [mem_ball] at hz" }, { "state_after": "case h\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\n⊢ ‖dslope f c z‖ ≤ R₂ / r", "state_before": "E : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\n⊢ ∀ᶠ (r : ℝ) in 𝓝[Iio R₁] R₁, ‖dslope f c z‖ ≤ R₂ / r", "tactic": "filter_upwards [Ioo_mem_nhdsWithin_Iio ⟨hz, le_rfl⟩] with r hr" }, { "state_after": "case h\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\n⊢ ‖dslope f c z‖ ≤ R₂ / r", "state_before": "case h\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\n⊢ ‖dslope f c z‖ ≤ R₂ / r", "tactic": "have hr₀ : 0 < r := dist_nonneg.trans_lt hr.1" }, { "state_after": "case hd\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\n⊢ DiffContOnCl ℂ (dslope f c) (ball c r)\n\ncase h\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\n⊢ ‖dslope f c z‖ ≤ R₂ / r", "state_before": "case h\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\n⊢ ‖dslope f c z‖ ≤ R₂ / r", "tactic": "replace hd : DiffContOnCl ℂ (dslope f c) (ball c r)" }, { "state_after": "case h.refine'_1\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\n⊢ ∀ (z : ℂ), z ∈ frontier (ball c r) → ‖dslope f c z‖ ≤ R₂ / r\n\ncase h.refine'_2\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\n⊢ z ∈ closure (ball c r)", "state_before": "case h\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\n⊢ ‖dslope f c z‖ ≤ R₂ / r", "tactic": "refine' norm_le_of_forall_mem_frontier_norm_le bounded_ball hd _ _" }, { "state_after": "E : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\nthis : ∀ᶠ (r : ℝ) in 𝓝[Iio R₁] R₁, ‖dslope f c z‖ ≤ R₂ / r\n⊢ Tendsto (fun c => R₂ / c) (𝓝[Iio R₁] R₁) (𝓝 (R₂ / R₁))", "state_before": "E : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\nthis : ∀ᶠ (r : ℝ) in 𝓝[Iio R₁] R₁, ‖dslope f c z‖ ≤ R₂ / r\n⊢ ‖dslope f c z‖ ≤ R₂ / R₁", "tactic": "refine' ge_of_tendsto _ this" }, { "state_after": "no goals", "state_before": "E : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\nthis : ∀ᶠ (r : ℝ) in 𝓝[Iio R₁] R₁, ‖dslope f c z‖ ≤ R₂ / r\n⊢ Tendsto (fun c => R₂ / c) (𝓝[Iio R₁] R₁) (𝓝 (R₂ / R₁))", "tactic": "exact (tendsto_const_nhds.div tendsto_id hR₁.ne').mono_left nhdsWithin_le_nhds" }, { "state_after": "case hd\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\n⊢ DifferentiableOn ℂ (dslope f c) (closure (ball c r))", "state_before": "case hd\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\n⊢ DiffContOnCl ℂ (dslope f c) (ball c r)", "tactic": "refine' DifferentiableOn.diffContOnCl _" }, { "state_after": "case hd\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\n⊢ DifferentiableOn ℂ (dslope f c) (closedBall c r)", "state_before": "case hd\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\n⊢ DifferentiableOn ℂ (dslope f c) (closure (ball c r))", "tactic": "rw [closure_ball c hr₀.ne']" }, { "state_after": "no goals", "state_before": "case hd\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\n⊢ DifferentiableOn ℂ (dslope f c) (closedBall c r)", "tactic": "exact ((differentiableOn_dslope <| ball_mem_nhds _ hR₁).mpr hd).mono\n (closedBall_subset_ball hr.2)" }, { "state_after": "case h.refine'_1\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\n⊢ ∀ (z : ℂ), z ∈ sphere c r → ‖dslope f c z‖ ≤ R₂ / r", "state_before": "case h.refine'_1\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\n⊢ ∀ (z : ℂ), z ∈ frontier (ball c r) → ‖dslope f c z‖ ≤ R₂ / r", "tactic": "rw [frontier_ball c hr₀.ne']" }, { "state_after": "case h.refine'_1\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z✝ z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz✝ : dist z✝ c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z✝ c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\nz : ℂ\nhz : z ∈ sphere c r\n⊢ ‖dslope f c z‖ ≤ R₂ / r", "state_before": "case h.refine'_1\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\n⊢ ∀ (z : ℂ), z ∈ sphere c r → ‖dslope f c z‖ ≤ R₂ / r", "tactic": "intro z hz" }, { "state_after": "case h.refine'_1\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z✝ z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz✝ : dist z✝ c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z✝ c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\nz : ℂ\nhz : z ∈ sphere c r\nhz' : z ≠ c\n⊢ ‖dslope f c z‖ ≤ R₂ / r", "state_before": "case h.refine'_1\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z✝ z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz✝ : dist z✝ c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z✝ c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\nz : ℂ\nhz : z ∈ sphere c r\n⊢ ‖dslope f c z‖ ≤ R₂ / r", "tactic": "have hz' : z ≠ c := ne_of_mem_sphere hz hr₀.ne'" }, { "state_after": "case h.refine'_1\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z✝ z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz✝ : dist z✝ c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z✝ c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\nz : ℂ\nhz : z ∈ sphere c r\nhz' : z ≠ c\n⊢ dist (f z) (f c) ≤ R₂", "state_before": "case h.refine'_1\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z✝ z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz✝ : dist z✝ c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z✝ c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\nz : ℂ\nhz : z ∈ sphere c r\nhz' : z ≠ c\n⊢ ‖dslope f c z‖ ≤ R₂ / r", "tactic": "rw [dslope_of_ne _ hz', slope_def_module, norm_smul, norm_inv, mem_sphere_iff_norm.1 hz, ←\n div_eq_inv_mul, div_le_div_right hr₀, ← dist_eq_norm]" }, { "state_after": "no goals", "state_before": "case h.refine'_1\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z✝ z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz✝ : dist z✝ c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z✝ c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\nz : ℂ\nhz : z ∈ sphere c r\nhz' : z ≠ c\n⊢ dist (f z) (f c) ≤ R₂", "tactic": "exact le_of_lt (h_maps (mem_ball.2 (by rw [mem_sphere.1 hz]; exact hr.2)))" }, { "state_after": "E : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z✝ z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz✝ : dist z✝ c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z✝ c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\nz : ℂ\nhz : z ∈ sphere c r\nhz' : z ≠ c\n⊢ r < R₁", "state_before": "E : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z✝ z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz✝ : dist z✝ c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z✝ c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\nz : ℂ\nhz : z ∈ sphere c r\nhz' : z ≠ c\n⊢ dist z c < R₁", "tactic": "rw [mem_sphere.1 hz]" }, { "state_after": "no goals", "state_before": "E : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z✝ z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz✝ : dist z✝ c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z✝ c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\nz : ℂ\nhz : z ∈ sphere c r\nhz' : z ≠ c\n⊢ r < R₁", "tactic": "exact hr.2" }, { "state_after": "case h.refine'_2\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\n⊢ dist z c ≤ r", "state_before": "case h.refine'_2\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\n⊢ z ∈ closure (ball c r)", "tactic": "rw [closure_ball c hr₀.ne', mem_closedBall]" }, { "state_after": "no goals", "state_before": "case h.refine'_2\nE : Type ?u.115\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf✝ : ℂ → E\nc z z₀ : ℂ\nf : ℂ → ℂ\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : dist z c < R₁\nhR₁ : 0 < R₁\nr : ℝ\nhr : r ∈ Ioo (dist z c) R₁\nhr₀ : 0 < r\nhd : DiffContOnCl ℂ (dslope f c) (ball c r)\n⊢ dist z c ≤ r", "tactic": "exact hr.1.le" } ]
[ 90, 18 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 67, 1 ]
Mathlib/LinearAlgebra/Span.lean
LinearMap.eqOn_span
[ { "state_after": "no goals", "state_before": "R : Type u_2\nR₂ : Type u_3\nK : Type ?u.336789\nM : Type u_1\nM₂ : Type u_4\nV : Type ?u.336798\nS : Type ?u.336801\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Semiring R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\nσ₁₂ : R →+* R₂\ns : Set M\nf g : M →ₛₗ[σ₁₂] M₂\nH : Set.EqOn (↑f) (↑g) s\nx : M\nh : x ∈ span R s\n⊢ ↑f x = ↑g x", "tactic": "refine' span_induction h H _ _ _ <;> simp (config := { contextual := true })" } ]
[ 950, 97 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 949, 1 ]
Mathlib/Topology/MetricSpace/Basic.lean
IsCompact.bounded
[]
[ 2403, 27 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 2401, 1 ]
Mathlib/Algebra/BigOperators/Finprod.lean
finprod_mem_insert'
[ { "state_after": "α : Type u_1\nβ : Type ?u.342023\nι : Type ?u.342026\nG : Type ?u.342029\nM : Type u_2\nN : Type ?u.342035\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns t : Set α\nf : α → M\nh : ¬a ∈ s\nhs : Set.Finite (s ∩ mulSupport f)\n⊢ Disjoint {a} s\n\nα : Type u_1\nβ : Type ?u.342023\nι : Type ?u.342026\nG : Type ?u.342029\nM : Type u_2\nN : Type ?u.342035\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns t : Set α\nf : α → M\nh : ¬a ∈ s\nhs : Set.Finite (s ∩ mulSupport f)\n⊢ Set.Finite ({a} ∩ mulSupport f)", "state_before": "α : Type u_1\nβ : Type ?u.342023\nι : Type ?u.342026\nG : Type ?u.342029\nM : Type u_2\nN : Type ?u.342035\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns t : Set α\nf : α → M\nh : ¬a ∈ s\nhs : Set.Finite (s ∩ mulSupport f)\n⊢ (∏ᶠ (i : α) (_ : i ∈ insert a s), f i) = f a * ∏ᶠ (i : α) (_ : i ∈ s), f i", "tactic": "rw [insert_eq, finprod_mem_union' _ _ hs, finprod_mem_singleton]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.342023\nι : Type ?u.342026\nG : Type ?u.342029\nM : Type u_2\nN : Type ?u.342035\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns t : Set α\nf : α → M\nh : ¬a ∈ s\nhs : Set.Finite (s ∩ mulSupport f)\n⊢ Disjoint {a} s", "tactic": "rwa [disjoint_singleton_left]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.342023\nι : Type ?u.342026\nG : Type ?u.342029\nM : Type u_2\nN : Type ?u.342035\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns t : Set α\nf : α → M\nh : ¬a ∈ s\nhs : Set.Finite (s ∩ mulSupport f)\n⊢ Set.Finite ({a} ∩ mulSupport f)", "tactic": "exact (finite_singleton a).inter_of_left _" } ]
[ 854, 47 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 850, 1 ]
Mathlib/Data/Set/Intervals/Instances.lean
Set.Icc.nonneg
[]
[ 109, 8 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 108, 1 ]
Mathlib/Order/LiminfLimsup.lean
Filter.limsInf_le_limsInf
[]
[ 509, 25 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 505, 1 ]
Mathlib/Algebra/Module/Submodule/Basic.lean
Submodule.sub_mem_iff_right
[ { "state_after": "no goals", "state_before": "G : Type u''\nS : Type u'\nR : Type u\nM : Type v\nι : Type w\ninst✝¹ : Ring R\ninst✝ : AddCommGroup M\nmodule_M : Module R M\np p' : Submodule R M\nr : R\nx y : M\nhx : x ∈ p\n⊢ x - y ∈ p ↔ y ∈ p", "tactic": "rw [sub_eq_add_neg, p.add_mem_iff_right hx, p.neg_mem_iff]" } ]
[ 587, 61 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 586, 1 ]
Mathlib/Topology/MetricSpace/Metrizable.lean
Embedding.metrizableSpace
[]
[ 132, 33 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 129, 1 ]
Mathlib/Data/Set/Pointwise/Basic.lean
Set.preimage_mul_left_one
[ { "state_after": "no goals", "state_before": "F : Type ?u.130016\nα : Type u_1\nβ : Type ?u.130022\nγ : Type ?u.130025\ninst✝ : Group α\ns t : Set α\na b : α\n⊢ (fun x x_1 => x * x_1) a ⁻¹' 1 = {a⁻¹}", "tactic": "rw [← image_mul_left', image_one, mul_one]" } ]
[ 1234, 45 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1233, 1 ]
Mathlib/RingTheory/IntegralClosure.lean
IsIntegralClosure.noZeroSMulDivisors
[ { "state_after": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : Algebra A B\ninst✝³ : IsIntegralClosure A R B\ninst✝² : Algebra R A\ninst✝¹ : IsScalarTower R A B\ninst✝ : NoZeroSMulDivisors R B\nx✝¹ : R\nx✝ : A\n⊢ ↑(algebraMap A B) (x✝¹ • x✝) = x✝¹ • ↑(algebraMap A B) x✝", "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : Algebra A B\ninst✝³ : IsIntegralClosure A R B\ninst✝² : Algebra R A\ninst✝¹ : IsScalarTower R A B\ninst✝ : NoZeroSMulDivisors R B\n⊢ NoZeroSMulDivisors R A", "tactic": "refine'\n Function.Injective.noZeroSMulDivisors _ (IsIntegralClosure.algebraMap_injective A R B)\n (map_zero _) fun _ _ => _" }, { "state_after": "no goals", "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : Algebra A B\ninst✝³ : IsIntegralClosure A R B\ninst✝² : Algebra R A\ninst✝¹ : IsScalarTower R A B\ninst✝ : NoZeroSMulDivisors R B\nx✝¹ : R\nx✝ : A\n⊢ ↑(algebraMap A B) (x✝¹ • x✝) = x✝¹ • ↑(algebraMap A B) x✝", "tactic": "simp only [Algebra.algebraMap_eq_smul_one, IsScalarTower.smul_assoc]" } ]
[ 856, 71 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 851, 1 ]
Mathlib/Algebra/Category/ModuleCat/Images.lean
ModuleCat.imageIsoRange_inv_image_ι
[]
[ 115, 27 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 113, 1 ]
Mathlib/Tactic/NormNum/Prime.lean
Mathlib.Meta.NormNum.isNat_not_prime
[]
[ 182, 94 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 182, 1 ]
Mathlib/Algebra/Category/ModuleCat/Kernels.lean
ModuleCat.cokernel_π_ext
[ { "state_after": "R : Type u\ninst✝ : Ring R\nG H : ModuleCat R\nf✝ : G ⟶ H\nM N : ModuleCat R\nf : M ⟶ N\ny : ↑N\nm : ↑M\n⊢ ↑(cokernel.π f) (y + ↑f m) = ↑(cokernel.π f) y", "state_before": "R : Type u\ninst✝ : Ring R\nG H : ModuleCat R\nf✝ : G ⟶ H\nM N : ModuleCat R\nf : M ⟶ N\nx y : ↑N\nm : ↑M\nw : x = y + ↑f m\n⊢ ↑(cokernel.π f) x = ↑(cokernel.π f) y", "tactic": "subst w" }, { "state_after": "no goals", "state_before": "R : Type u\ninst✝ : Ring R\nG H : ModuleCat R\nf✝ : G ⟶ H\nM N : ModuleCat R\nf : M ⟶ N\ny : ↑N\nm : ↑M\n⊢ ↑(cokernel.π f) (y + ↑f m) = ↑(cokernel.π f) y", "tactic": "simpa only [map_add, add_right_eq_self] using cokernel.condition_apply f m" } ]
[ 143, 77 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 140, 1 ]
Mathlib/LinearAlgebra/Ray.lean
SameRay.zero_left
[]
[ 58, 13 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 57, 1 ]
Mathlib/Order/Compare.lean
Ordering.compares_iff_of_compares_impl
[ { "state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\n⊢ Compares o a b", "state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\n⊢ Compares o a b ↔ Compares o a' b'", "tactic": "refine' ⟨h, fun ho => _⟩" }, { "state_after": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\nhab : a < b\n⊢ Compares o a b\n\ncase inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\nhab : a = b ∨ b < a\n⊢ Compares o a b", "state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\n⊢ Compares o a b", "tactic": "cases' lt_trichotomy a b with hab hab" }, { "state_after": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\nhab✝ : a < b\nhab : Compares lt a b\n⊢ Compares o a b", "state_before": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\nhab : a < b\n⊢ Compares o a b", "tactic": "have hab : Compares Ordering.lt a b := hab" }, { "state_after": "no goals", "state_before": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\nhab✝ : a < b\nhab : Compares lt a b\n⊢ Compares o a b", "tactic": "rwa [ho.inj (h hab)]" }, { "state_after": "case inr.inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\nhab : a = b\n⊢ Compares o a b\n\ncase inr.inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\nhab : b < a\n⊢ Compares o a b", "state_before": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\nhab : a = b ∨ b < a\n⊢ Compares o a b", "tactic": "cases' hab with hab hab" }, { "state_after": "case inr.inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\nhab✝ : a = b\nhab : Compares eq a b\n⊢ Compares o a b", "state_before": "case inr.inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\nhab : a = b\n⊢ Compares o a b", "tactic": "have hab : Compares Ordering.eq a b := hab" }, { "state_after": "no goals", "state_before": "case inr.inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\nhab✝ : a = b\nhab : Compares eq a b\n⊢ Compares o a b", "tactic": "rwa [ho.inj (h hab)]" }, { "state_after": "case inr.inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\nhab✝ : b < a\nhab : Compares gt a b\n⊢ Compares o a b", "state_before": "case inr.inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\nhab : b < a\n⊢ Compares o a b", "tactic": "have hab : Compares Ordering.gt a b := hab" }, { "state_after": "no goals", "state_before": "case inr.inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, Compares o a b → Compares o a' b'\no : Ordering\nho : Compares o a' b'\nhab✝ : b < a\nhab : Compares gt a b\n⊢ Compares o a b", "tactic": "rwa [ho.inj (h hab)]" } ]
[ 141, 27 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 131, 1 ]
Mathlib/Analysis/Calculus/Taylor.lean
taylorWithin_succ
[ { "state_after": "𝕜 : Type ?u.66235\nE : Type u_1\nF : Type ?u.66241\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nn : ℕ\ns : Set ℝ\nx₀ : ℝ\n⊢ ∑ k in Finset.range (n + 1 + 1),\n ↑(PolynomialModule.comp (Polynomial.X - ↑Polynomial.C x₀))\n (↑(PolynomialModule.single ℝ k) (taylorCoeffWithin f k s x₀)) =\n ∑ k in Finset.range (n + 1),\n ↑(PolynomialModule.comp (Polynomial.X - ↑Polynomial.C x₀))\n (↑(PolynomialModule.single ℝ k) (taylorCoeffWithin f k s x₀)) +\n ↑(PolynomialModule.comp (Polynomial.X - ↑Polynomial.C x₀))\n (↑(PolynomialModule.single ℝ (n + 1)) (taylorCoeffWithin f (n + 1) s x₀))", "state_before": "𝕜 : Type ?u.66235\nE : Type u_1\nF : Type ?u.66241\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nn : ℕ\ns : Set ℝ\nx₀ : ℝ\n⊢ taylorWithin f (n + 1) s x₀ =\n taylorWithin f n s x₀ +\n ↑(PolynomialModule.comp (Polynomial.X - ↑Polynomial.C x₀))\n (↑(PolynomialModule.single ℝ (n + 1)) (taylorCoeffWithin f (n + 1) s x₀))", "tactic": "dsimp only [taylorWithin]" }, { "state_after": "no goals", "state_before": "𝕜 : Type ?u.66235\nE : Type u_1\nF : Type ?u.66241\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nn : ℕ\ns : Set ℝ\nx₀ : ℝ\n⊢ ∑ k in Finset.range (n + 1 + 1),\n ↑(PolynomialModule.comp (Polynomial.X - ↑Polynomial.C x₀))\n (↑(PolynomialModule.single ℝ k) (taylorCoeffWithin f k s x₀)) =\n ∑ k in Finset.range (n + 1),\n ↑(PolynomialModule.comp (Polynomial.X - ↑Polynomial.C x₀))\n (↑(PolynomialModule.single ℝ k) (taylorCoeffWithin f k s x₀)) +\n ↑(PolynomialModule.comp (Polynomial.X - ↑Polynomial.C x₀))\n (↑(PolynomialModule.single ℝ (n + 1)) (taylorCoeffWithin f (n + 1) s x₀))", "tactic": "rw [Finset.sum_range_succ]" } ]
[ 82, 29 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 77, 1 ]
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean
AffineEquiv.surjective
[]
[ 232, 23 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 231, 11 ]
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
Real.Angle.coe_zero
[]
[ 83, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 82, 1 ]
Mathlib/Order/Filter/Pointwise.lean
Filter.bot_smul
[]
[ 977, 16 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 976, 1 ]
Mathlib/Data/Set/Basic.lean
Set.nonempty_iff_univ_nonempty
[]
[ 532, 47 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 531, 1 ]
Mathlib/Order/GaloisConnection.lean
GaloisConnection.l_comm_iff_u_comm
[]
[ 371, 71 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 365, 1 ]
Mathlib/Data/Set/Pointwise/SMul.lean
Set.Nonempty.smul
[]
[ 135, 18 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 134, 1 ]
Mathlib/Data/Seq/WSeq.lean
Stream'.WSeq.dropn_ofSeq
[ { "state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nn : ℕ\n⊢ tail (drop (↑s) n) = ↑(Seq.tail (Seq.drop s n))", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nn : ℕ\n⊢ drop (↑s) (n + 1) = ↑(Seq.drop s (n + 1))", "tactic": "simp only [drop, Nat.add_eq, add_zero, Seq.drop]" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nn : ℕ\n⊢ tail (drop (↑s) n) = ↑(Seq.tail (Seq.drop s n))", "tactic": "rw [dropn_ofSeq s n, tail_ofSeq]" } ]
[ 1376, 37 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1372, 1 ]
Mathlib/SetTheory/ZFC/Basic.lean
ZFSet.subset_iff
[]
[ 787, 26 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 782, 1 ]
Mathlib/MeasureTheory/Function/L1Space.lean
MeasureTheory.integrable_smul_measure
[ { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.848051\nδ : Type ?u.848054\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nc : ℝ≥0∞\nh₁ : c ≠ 0\nh₂ : c ≠ ⊤\nh : Integrable f\n⊢ Integrable f", "tactic": "simpa only [smul_smul, ENNReal.inv_mul_cancel h₁ h₂, one_smul] using\n h.smul_measure (ENNReal.inv_ne_top.2 h₁)" } ]
[ 574, 32 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 569, 1 ]
Mathlib/Algebra/Order/Monoid/Lemmas.lean
mul_lt_of_lt_of_le_one
[]
[ 573, 17 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 567, 1 ]
Mathlib/RingTheory/Subring/Basic.lean
Subring.prod_mono_left
[]
[ 1117, 27 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1116, 1 ]
Mathlib/RingTheory/Multiplicity.lean
multiplicity.pow'
[ { "state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a : α\nhp : Prime p\nha : Finite p a\nk : ℕ\n⊢ Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha", "state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a : α\nhp : Prime p\nha : Finite p a\n⊢ ∀ {k : ℕ}, Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha", "tactic": "intro k" }, { "state_after": "case zero\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a : α\nhp : Prime p\nha : Finite p a\n⊢ Part.get (multiplicity p (a ^ zero)) (_ : Finite p (a ^ zero)) = zero * Part.get (multiplicity p a) ha\n\ncase succ\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a : α\nhp : Prime p\nha : Finite p a\nk : ℕ\nhk : Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha\n⊢ Part.get (multiplicity p (a ^ succ k)) (_ : Finite p (a ^ succ k)) = succ k * Part.get (multiplicity p a) ha", "state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a : α\nhp : Prime p\nha : Finite p a\nk : ℕ\n⊢ Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha", "tactic": "induction' k with k hk" }, { "state_after": "no goals", "state_before": "case zero\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a : α\nhp : Prime p\nha : Finite p a\n⊢ Part.get (multiplicity p (a ^ zero)) (_ : Finite p (a ^ zero)) = zero * Part.get (multiplicity p a) ha", "tactic": "simp [one_right hp.not_unit]" }, { "state_after": "case succ\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a : α\nhp : Prime p\nha : Finite p a\nk : ℕ\nhk : Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha\nthis : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k)\n⊢ Part.get (multiplicity p (a ^ succ k)) (_ : Finite p (a ^ succ k)) = succ k * Part.get (multiplicity p a) ha", "state_before": "case succ\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a : α\nhp : Prime p\nha : Finite p a\nk : ℕ\nhk : Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha\n⊢ Part.get (multiplicity p (a ^ succ k)) (_ : Finite p (a ^ succ k)) = succ k * Part.get (multiplicity p a) ha", "tactic": "have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ]" }, { "state_after": "no goals", "state_before": "case succ\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a : α\nhp : Prime p\nha : Finite p a\nk : ℕ\nhk : Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha\nthis : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k)\n⊢ Part.get (multiplicity p (a ^ succ k)) (_ : Finite p (a ^ succ k)) = succ k * Part.get (multiplicity p a) ha", "tactic": "rw [succ_eq_add_one, get_eq_get_of_eq _ _ this,\n multiplicity.mul' hp, hk, add_mul, one_mul, add_comm]" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a : α\nhp : Prime p\nha : Finite p a\nk : ℕ\nhk : Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha\n⊢ multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k)", "tactic": "rw [_root_.pow_succ]" } ]
[ 606, 60 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 599, 11 ]
Mathlib/Order/Antisymmetrization.lean
AntisymmRel.image
[]
[ 140, 19 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 138, 1 ]
Mathlib/LinearAlgebra/Basic.lean
LinearMap.iterate_injective
[ { "state_after": "R : Type u_2\nR₁ : Type ?u.187673\nR₂ : Type ?u.187676\nR₃ : Type ?u.187679\nR₄ : Type ?u.187682\nS : Type ?u.187685\nK : Type ?u.187688\nK₂ : Type ?u.187691\nM : Type u_1\nM' : Type ?u.187697\nM₁ : Type ?u.187700\nM₂ : Type ?u.187703\nM₃ : Type ?u.187706\nM₄ : Type ?u.187709\nN : Type ?u.187712\nN₂ : Type ?u.187715\nι : Type ?u.187718\nV : Type ?u.187721\nV₂ : Type ?u.187724\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\nf' : M →ₗ[R] M\nh : Injective ↑f'\nn : ℕ\n⊢ Injective ↑(comp (f' ^ n) f')", "state_before": "R : Type u_2\nR₁ : Type ?u.187673\nR₂ : Type ?u.187676\nR₃ : Type ?u.187679\nR₄ : Type ?u.187682\nS : Type ?u.187685\nK : Type ?u.187688\nK₂ : Type ?u.187691\nM : Type u_1\nM' : Type ?u.187697\nM₁ : Type ?u.187700\nM₂ : Type ?u.187703\nM₃ : Type ?u.187706\nM₄ : Type ?u.187709\nN : Type ?u.187712\nN₂ : Type ?u.187715\nι : Type ?u.187718\nV : Type ?u.187721\nV₂ : Type ?u.187724\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\nf' : M →ₗ[R] M\nh : Injective ↑f'\nn : ℕ\n⊢ Injective ↑(f' ^ (n + 1))", "tactic": "rw [iterate_succ]" }, { "state_after": "no goals", "state_before": "R : Type u_2\nR₁ : Type ?u.187673\nR₂ : Type ?u.187676\nR₃ : Type ?u.187679\nR₄ : Type ?u.187682\nS : Type ?u.187685\nK : Type ?u.187688\nK₂ : Type ?u.187691\nM : Type u_1\nM' : Type ?u.187697\nM₁ : Type ?u.187700\nM₂ : Type ?u.187703\nM₃ : Type ?u.187706\nM₄ : Type ?u.187709\nN : Type ?u.187712\nN₂ : Type ?u.187715\nι : Type ?u.187718\nV : Type ?u.187721\nV₂ : Type ?u.187724\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\nf' : M →ₗ[R] M\nh : Injective ↑f'\nn : ℕ\n⊢ Injective ↑(comp (f' ^ n) f')", "tactic": "exact (iterate_injective h n).comp h" } ]
[ 385, 41 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 381, 1 ]
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
Complex.cos_sub_two_pi
[]
[ 1220, 24 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1219, 1 ]
Mathlib/CategoryTheory/Category/Preorder.lean
CategoryTheory.Equivalence.toOrderIso_apply
[]
[ 188, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 187, 1 ]
Mathlib/SetTheory/Ordinal/Exponential.lean
Ordinal.opow_lt_opow_left_of_succ
[ { "state_after": "a b c : Ordinal\nab : a < b\n⊢ a ^ c * a < b ^ c * b", "state_before": "a b c : Ordinal\nab : a < b\n⊢ a ^ succ c < b ^ succ c", "tactic": "rw [opow_succ, opow_succ]" }, { "state_after": "no goals", "state_before": "a b c : Ordinal\nab : a < b\n⊢ a ^ c * a < b ^ c * b", "tactic": "exact\n (mul_le_mul_right' (opow_le_opow_left c ab.le) a).trans_lt\n (mul_lt_mul_of_pos_left ab (opow_pos c ((Ordinal.zero_le a).trans_lt ab)))" } ]
[ 184, 81 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 180, 1 ]
Mathlib/Data/List/Cycle.lean
Cycle.nodup_coe_iff
[]
[ 663, 10 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 662, 1 ]
Mathlib/Computability/Primrec.lean
Primrec.list_concat
[]
[ 1106, 55 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1105, 1 ]
Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
ContinuousWithinAt.rpow
[]
[ 298, 15 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 295, 8 ]
Mathlib/Data/Set/Intervals/OrdConnected.lean
Set.ordConnected_empty
[]
[ 183, 24 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 182, 1 ]
Mathlib/RingTheory/UniqueFactorizationDomain.lean
UniqueFactorizationMonoid.factors_unique
[]
[ 252, 55 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 249, 1 ]
Mathlib/RingTheory/Valuation/Basic.lean
AddValuation.comap_comp
[]
[ 757, 29 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 755, 1 ]
Mathlib/Data/Real/Irrational.lean
Irrational.of_mul_nat
[]
[ 374, 17 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 373, 1 ]
Mathlib/Algebra/Order/Floor.lean
Int.fract_sub_int
[ { "state_after": "F : Type ?u.147946\nα : Type u_1\nβ : Type ?u.147952\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nm : ℤ\n⊢ a - ↑m - ↑⌊a - ↑m⌋ = fract a", "state_before": "F : Type ?u.147946\nα : Type u_1\nβ : Type ?u.147952\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nm : ℤ\n⊢ fract (a - ↑m) = fract a", "tactic": "rw [fract]" }, { "state_after": "no goals", "state_before": "F : Type ?u.147946\nα : Type u_1\nβ : Type ?u.147952\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nm : ℤ\n⊢ a - ↑m - ↑⌊a - ↑m⌋ = fract a", "tactic": "simp" } ]
[ 835, 7 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 833, 1 ]