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[ { "state_after": "α : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ singularPart s μ + withDensityᵥ μ (rnDeriv s μ) =\n toSignedMeasure (toJordanDecomposition s).posPart - toSignedMeasure (toJordanDecomposition s).negPart", "state_before": "α : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ singularPart s μ + withDensityᵥ μ (rnDeriv s μ) = s", "tactic": "conv_rhs =>\n rw [← toSignedMeasure_toJordanDecomposition s, JordanDecomposition.toSignedMeasure]" }, { "state_after": "α : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ toSignedMeasure\n (Measure.singularPart (toJordanDecomposition s).posPart μ +\n withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x) -\n toSignedMeasure\n ((withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x) +\n Measure.singularPart (toJordanDecomposition s).negPart μ) =\n toSignedMeasure (toJordanDecomposition s).posPart - toSignedMeasure (toJordanDecomposition s).negPart\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hfm\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hfm\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤", "state_before": "α : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ singularPart s μ + withDensityᵥ μ (rnDeriv s μ) =\n toSignedMeasure (toJordanDecomposition s).posPart - toSignedMeasure (toJordanDecomposition s).negPart", "tactic": "rw [singularPart, rnDeriv,\n withDensityᵥ_sub' (integrable_toReal_of_lintegral_ne_top _ _)\n (integrable_toReal_of_lintegral_ne_top _ _),\n withDensityᵥ_toReal, withDensityᵥ_toReal, sub_eq_add_neg, sub_eq_add_neg,\n add_comm (s.toJordanDecomposition.posPart.singularPart μ).toSignedMeasure, ← add_assoc,\n add_assoc (-(s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure),\n ← toSignedMeasure_add, add_comm, ← add_assoc, ← neg_add, ← toSignedMeasure_add, add_comm,\n ← sub_eq_add_neg]" }, { "state_after": "case h.e'_3.h.e'_5.h.e'_3\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (toJordanDecomposition s).posPart =\n Measure.singularPart (toJordanDecomposition s).posPart μ +\n withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x\n\ncase h.e'_3.h.e'_6.h.e'_3\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (toJordanDecomposition s).negPart =\n (withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x) +\n Measure.singularPart (toJordanDecomposition s).negPart μ\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hfm\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hfm\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤", "state_before": "α : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ toSignedMeasure\n (Measure.singularPart (toJordanDecomposition s).posPart μ +\n withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x) -\n toSignedMeasure\n ((withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x) +\n Measure.singularPart (toJordanDecomposition s).negPart μ) =\n toSignedMeasure (toJordanDecomposition s).posPart - toSignedMeasure (toJordanDecomposition s).negPart\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hfm\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hfm\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤", "tactic": "convert rfl" }, { "state_after": "no goals", "state_before": "case hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hfm\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\ncase hfm\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x\n\ncase hf\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).posPart μ x ∂μ) ≠ ⊤\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x\n\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤", "tactic": "all_goals\n first\n | exact (lintegral_rnDeriv_lt_top _ _).ne\n | measurability" }, { "state_after": "no goals", "state_before": "case h.e'_3.h.e'_5.h.e'_3\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (toJordanDecomposition s).posPart =\n Measure.singularPart (toJordanDecomposition s).posPart μ +\n withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).posPart μ x", "tactic": "exact s.toJordanDecomposition.posPart.haveLebesgueDecomposition_add μ" }, { "state_after": "case h.e'_3.h.e'_6.h.e'_3\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (toJordanDecomposition s).negPart =\n Measure.singularPart (toJordanDecomposition s).negPart μ +\n withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x", "state_before": "case h.e'_3.h.e'_6.h.e'_3\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (toJordanDecomposition s).negPart =\n (withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x) +\n Measure.singularPart (toJordanDecomposition s).negPart μ", "tactic": "rw [add_comm]" }, { "state_after": "no goals", "state_before": "case h.e'_3.h.e'_6.h.e'_3\nα : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (toJordanDecomposition s).negPart =\n Measure.singularPart (toJordanDecomposition s).negPart μ +\n withDensity μ fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x", "tactic": "exact s.toJordanDecomposition.negPart.haveLebesgueDecomposition_add μ" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤", "tactic": "first\n| exact (lintegral_rnDeriv_lt_top _ _).ne\n| measurability" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ (∫⁻ (x : α), Measure.rnDeriv (toJordanDecomposition s).negPart μ x ∂μ) ≠ ⊤", "tactic": "exact (lintegral_rnDeriv_lt_top _ _).ne" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.432299\nm : MeasurableSpace α\nμ ν : Measure α\ns t : SignedMeasure α\ninst✝ : HaveLebesgueDecomposition s μ\n⊢ AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart μ x", "tactic": "measurability" } ]

[ 953, 20 ]

[ 933, 1 ]

[]

[ 750, 29 ]

[ 748, 1 ]

[ { "state_after": "case h.h\nα : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\nA : Matrix m' (Fin (Nat.succ n)) α\ni : Fin n\nj : m'\n⊢ vecTail (↑of.symm Aᵀ) i j = (vecTail ∘ A)ᵀ i j", "state_before": "α : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\nA : Matrix m' (Fin (Nat.succ n)) α\n⊢ vecTail (↑of.symm Aᵀ) = (vecTail ∘ A)ᵀ", "tactic": "ext (i j)" }, { "state_after": "no goals", "state_before": "case h.h\nα : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\nA : Matrix m' (Fin (Nat.succ n)) α\ni : Fin n\nj : m'\n⊢ vecTail (↑of.symm Aᵀ) i j = (vecTail ∘ A)ᵀ i j", "tactic": "rfl" } ]

[ 220, 6 ]

[ 218, 1 ]

[]

[ 500, 12 ]

[ 499, 11 ]

[ { "state_after": "E : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "state_before": "E : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "tactic": "have hc : IsCompact (closure U) := hb.isCompact_closure" }, { "state_after": "E : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\n⊢ ∃ w, w ∈ closure U ∧ IsMaxOn (norm ∘ f) (closure U) w\n\ncase intro.intro\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhwU : w ∈ closure U\nhle : IsMaxOn (norm ∘ f) (closure U) w\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "state_before": "E : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "tactic": "obtain ⟨w, hwU, hle⟩ : ∃ w ∈ closure U, IsMaxOn (norm ∘ f) (closure U) w" }, { "state_after": "case intro.intro\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhwU : w ∈ closure U\nhle : IsMaxOn (norm ∘ f) (closure U) w\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "state_before": "E : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\n⊢ ∃ w, w ∈ closure U ∧ IsMaxOn (norm ∘ f) (closure U) w\n\ncase intro.intro\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhwU : w ∈ closure U\nhle : IsMaxOn (norm ∘ f) (closure U) w\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "tactic": "exact hc.exists_forall_ge hne.closure hd.continuousOn.norm" }, { "state_after": "case intro.intro\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhwU : w ∈ interior U ∨ w ∈ frontier U\nhle : IsMaxOn (norm ∘ f) (closure U) w\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "state_before": "case intro.intro\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhwU : w ∈ closure U\nhle : IsMaxOn (norm ∘ f) (closure U) w\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "tactic": "rw [closure_eq_interior_union_frontier, mem_union] at hwU" }, { "state_after": "case intro.intro.inl\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ interior U\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z\n\ncase intro.intro.inr\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ frontier U\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "state_before": "case intro.intro\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhwU : w ∈ interior U ∨ w ∈ frontier U\nhle : IsMaxOn (norm ∘ f) (closure U) w\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "tactic": "cases' hwU with hwU hwU" }, { "state_after": "case intro.intro.inr\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ frontier U\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z\n\ncase intro.intro.inl\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ interior U\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "state_before": "case intro.intro.inl\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ interior U\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z\n\ncase intro.intro.inr\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ frontier U\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "tactic": "rotate_left" }, { "state_after": "case intro.intro.inl\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ interior U\nthis : interior U ≠ univ\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "state_before": "case intro.intro.inl\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ interior U\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "tactic": "have : interior U ≠ univ := ne_top_of_le_ne_top hc.ne_univ interior_subset_closure" }, { "state_after": "case intro.intro.inl.intro.intro\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ interior U\nthis : interior U ≠ univ\nz : E\nhzU : z ∈ frontier (interior U)\nhzw : infDist w (interior Uᶜ) = dist w z\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "state_before": "case intro.intro.inl\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ interior U\nthis : interior U ≠ univ\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "tactic": "rcases exists_mem_frontier_infDist_compl_eq_dist hwU this with ⟨z, hzU, hzw⟩" }, { "state_after": "case intro.intro.inl.intro.intro\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ interior U\nthis : interior U ≠ univ\nz : E\nhzU : z ∈ frontier (interior U)\nhzw : infDist w (interior Uᶜ) = dist w z\nx : E\nhx : x ∈ closure U\n⊢ (norm ∘ f) w = (norm ∘ f) z", "state_before": "case intro.intro.inl.intro.intro\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ interior U\nthis : interior U ≠ univ\nz : E\nhzU : z ∈ frontier (interior U)\nhzw : infDist w (interior Uᶜ) = dist w z\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "tactic": "refine' ⟨z, frontier_interior_subset hzU, fun x hx => (hle hx).out.trans_eq _⟩" }, { "state_after": "case intro.intro.inl.intro.intro\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ interior U\nthis : interior U ≠ univ\nz : E\nhzU : z ∈ frontier (interior U)\nhzw : infDist w (interior Uᶜ) = dist w z\nx : E\nhx : x ∈ closure U\n⊢ ball w (dist z w) ⊆ U", "state_before": "case intro.intro.inl.intro.intro\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ interior U\nthis : interior U ≠ univ\nz : E\nhzU : z ∈ frontier (interior U)\nhzw : infDist w (interior Uᶜ) = dist w z\nx : E\nhx : x ∈ closure U\n⊢ (norm ∘ f) w = (norm ∘ f) z", "tactic": "refine' (norm_eq_norm_of_isMaxOn_of_ball_subset hd (hle.on_subset subset_closure) _).symm" }, { "state_after": "case intro.intro.inl.intro.intro\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ interior U\nthis : interior U ≠ univ\nz : E\nhzU : z ∈ frontier (interior U)\nhzw : infDist w (interior Uᶜ) = dist w z\nx : E\nhx : x ∈ closure U\n⊢ ball w (infDist w (interior Uᶜ)) ⊆ U", "state_before": "case intro.intro.inl.intro.intro\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ interior U\nthis : interior U ≠ univ\nz : E\nhzU : z ∈ frontier (interior U)\nhzw : infDist w (interior Uᶜ) = dist w z\nx : E\nhx : x ∈ closure U\n⊢ ball w (dist z w) ⊆ U", "tactic": "rw [dist_comm, ← hzw]" }, { "state_after": "no goals", "state_before": "case intro.intro.inl.intro.intro\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ interior U\nthis : interior U ≠ univ\nz : E\nhzU : z ∈ frontier (interior U)\nhzw : infDist w (interior Uᶜ) = dist w z\nx : E\nhx : x ∈ closure U\n⊢ ball w (infDist w (interior Uᶜ)) ⊆ U", "tactic": "exact ball_infDist_compl_subset.trans interior_subset" }, { "state_after": "no goals", "state_before": "case intro.intro.inr\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Metric.Bounded U\nhne : Set.Nonempty U\nhd : DiffContOnCl ℂ f U\nhc : IsCompact (closure U)\nw : E\nhle : IsMaxOn (norm ∘ f) (closure U) w\nhwU : w ∈ frontier U\n⊢ ∃ z, z ∈ frontier U ∧ IsMaxOn (norm ∘ f) (closure U) z", "tactic": "exact ⟨w, hwU, hle⟩" } ]

[ 385, 56 ]

[ 372, 1 ]

[]

[ 251, 21 ]

[ 250, 1 ]

[]

[ 107, 40 ]

[ 105, 1 ]

[ { "state_after": "case mk.mk\nι : Type u_1\nI J : Box ι\nx y l₁ u₁ : ι → ℝ\nh₁ : ∀ (i : ι), l₁ i < u₁ i\nl₂ u₂ : ι → ℝ\nh₂ : ∀ (i : ι), l₂ i < u₂ i\nh : ↑{ lower := l₁, upper := u₁, lower_lt_upper := h₁ } = ↑{ lower := l₂, upper := u₂, lower_lt_upper := h₂ }\n⊢ { lower := l₁, upper := u₁, lower_lt_upper := h₁ } = { lower := l₂, upper := u₂, lower_lt_upper := h₂ }", "state_before": "ι : Type u_1\nI J : Box ι\nx y : ι → ℝ\n⊢ Injective toSet", "tactic": "rintro ⟨l₁, u₁, h₁⟩ ⟨l₂, u₂, h₂⟩ h" }, { "state_after": "case mk.mk\nι : Type u_1\nI J : Box ι\nx y l₁ u₁ : ι → ℝ\nh₁ : ∀ (i : ι), l₁ i < u₁ i\nl₂ u₂ : ι → ℝ\nh₂ : ∀ (i : ι), l₂ i < u₂ i\nh : (l₂ ≤ l₁ ∧ u₁ ≤ u₂) ∧ l₁ ≤ l₂ ∧ u₂ ≤ u₁\n⊢ { lower := l₁, upper := u₁, lower_lt_upper := h₁ } = { lower := l₂, upper := u₂, lower_lt_upper := h₂ }", "state_before": "case mk.mk\nι : Type u_1\nI J : Box ι\nx y l₁ u₁ : ι → ℝ\nh₁ : ∀ (i : ι), l₁ i < u₁ i\nl₂ u₂ : ι → ℝ\nh₂ : ∀ (i : ι), l₂ i < u₂ i\nh : ↑{ lower := l₁, upper := u₁, lower_lt_upper := h₁ } = ↑{ lower := l₂, upper := u₂, lower_lt_upper := h₂ }\n⊢ { lower := l₁, upper := u₁, lower_lt_upper := h₁ } = { lower := l₂, upper := u₂, lower_lt_upper := h₂ }", "tactic": "simp only [Subset.antisymm_iff, coe_subset_coe, le_iff_bounds] at h" }, { "state_after": "case mk.mk.e_lower\nι : Type u_1\nI J : Box ι\nx y l₁ u₁ : ι → ℝ\nh₁ : ∀ (i : ι), l₁ i < u₁ i\nl₂ u₂ : ι → ℝ\nh₂ : ∀ (i : ι), l₂ i < u₂ i\nh : (l₂ ≤ l₁ ∧ u₁ ≤ u₂) ∧ l₁ ≤ l₂ ∧ u₂ ≤ u₁\n⊢ l₁ = l₂\n\ncase mk.mk.e_upper\nι : Type u_1\nI J : Box ι\nx y l₁ u₁ : ι → ℝ\nh₁ : ∀ (i : ι), l₁ i < u₁ i\nl₂ u₂ : ι → ℝ\nh₂ : ∀ (i : ι), l₂ i < u₂ i\nh : (l₂ ≤ l₁ ∧ u₁ ≤ u₂) ∧ l₁ ≤ l₂ ∧ u₂ ≤ u₁\n⊢ u₁ = u₂", "state_before": "case mk.mk\nι : Type u_1\nI J : Box ι\nx y l₁ u₁ : ι → ℝ\nh₁ : ∀ (i : ι), l₁ i < u₁ i\nl₂ u₂ : ι → ℝ\nh₂ : ∀ (i : ι), l₂ i < u₂ i\nh : (l₂ ≤ l₁ ∧ u₁ ≤ u₂) ∧ l₁ ≤ l₂ ∧ u₂ ≤ u₁\n⊢ { lower := l₁, upper := u₁, lower_lt_upper := h₁ } = { lower := l₂, upper := u₂, lower_lt_upper := h₂ }", "tactic": "congr" }, { "state_after": "no goals", "state_before": "case mk.mk.e_lower\nι : Type u_1\nI J : Box ι\nx y l₁ u₁ : ι → ℝ\nh₁ : ∀ (i : ι), l₁ i < u₁ i\nl₂ u₂ : ι → ℝ\nh₂ : ∀ (i : ι), l₂ i < u₂ i\nh : (l₂ ≤ l₁ ∧ u₁ ≤ u₂) ∧ l₁ ≤ l₂ ∧ u₂ ≤ u₁\n⊢ l₁ = l₂\n\ncase mk.mk.e_upper\nι : Type u_1\nI J : Box ι\nx y l₁ u₁ : ι → ℝ\nh₁ : ∀ (i : ι), l₁ i < u₁ i\nl₂ u₂ : ι → ℝ\nh₂ : ∀ (i : ι), l₂ i < u₂ i\nh : (l₂ ≤ l₁ ∧ u₁ ≤ u₂) ∧ l₁ ≤ l₂ ∧ u₂ ≤ u₁\n⊢ u₁ = u₂", "tactic": "exacts [le_antisymm h.2.1 h.1.1, le_antisymm h.1.2 h.2.2]" } ]

[ 181, 60 ]

[ 177, 1 ]

[]

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[ 1838, 1 ]

[ { "state_after": "R : Type u_1\ninst✝² : CommRing R\np n : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (Nat.Prime p)\n⊢ (-1) ^ p ^ n + 1 = 0", "state_before": "R : Type u_1\ninst✝² : CommRing R\np n : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (Nat.Prime p)\n⊢ (-1) ^ p ^ n = -1", "tactic": "rw [eq_neg_iff_add_eq_zero]" }, { "state_after": "R : Type u_1\ninst✝² : CommRing R\np n : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (Nat.Prime p)\n⊢ (-1) ^ p ^ n + 1 ^ p ^ n = 0", "state_before": "R : Type u_1\ninst✝² : CommRing R\np n : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (Nat.Prime p)\n⊢ (-1) ^ p ^ n + 1 = 0", "tactic": "nth_rw 2 [← one_pow (p ^ n)]" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝² : CommRing R\np n : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (Nat.Prime p)\n⊢ (-1) ^ p ^ n + 1 ^ p ^ n = 0", "tactic": "rw [← add_pow_char_pow, add_left_neg, zero_pow (pow_pos (Fact.out (p := Nat.Prime p)).pos _)]" } ]

[ 322, 96 ]

[ 318, 1 ]

[ { "state_after": "no goals", "state_before": "C : Type u₁\ninst✝⁶ : Category C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝² : PreservesZeroMorphisms F\nX Y : C\ninst✝¹ : HasBinaryBiproduct X Y\ninst✝ : PreservesBinaryBiproduct X Y F\nW : C\nf : X ⟶ W\ng : Y ⟶ W\n⊢ (mapBiprod F X Y).inv ≫ F.map (desc f g) = desc (F.map f) (F.map g)", "tactic": "apply biprod.hom_ext' <;> simp [mapBiprod, ← F.map_comp]" } ]

[ 463, 59 ]

[ 461, 1 ]

[ { "state_after": "no goals", "state_before": "𝕜 : Type u_2\n𝕜₂ : Type ?u.411250\n𝕜₃ : Type ?u.411253\nE : Type u_1\nEₗ : Type ?u.411259\nF : Type ?u.411262\nFₗ : Type ?u.411265\nG : Type ?u.411268\nGₗ : Type ?u.411271\n𝓕 : Type ?u.411274\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ x : E\n⊢ ‖↑(id 𝕜 E) x‖ ≤ 1 * ‖x‖", "tactic": "simp" } ]

[ 287, 50 ]

[ 286, 1 ]

[]

[ 232, 33 ]

[ 231, 1 ]

[]

[ 1265, 42 ]

[ 1264, 1 ]

[]

[ 579, 6 ]

[ 578, 1 ]

[]

[ 76, 6 ]

[ 75, 1 ]

[ { "state_after": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\n⊢ (forget D).map (colimMap (diagramPullback J P f) ≫ colimit.pre (diagram J P Y) (Functor.op (pullback J f)))\n ((forget D).map (colimit.ι (diagram J P X) S.op) (↑(Meq.equiv P S).symm x)) =\n (forget D).map (colimit.ι (diagram J P Y) (Cover.pullback S f).op)\n (↑(Meq.equiv P (Cover.pullback S f)).symm (Meq.pullback x f))", "state_before": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\n⊢ (forget D).map ((plusObj J P).map f.op) (mk x) = mk (Meq.pullback x f)", "tactic": "dsimp [mk, plusObj]" }, { "state_after": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\n⊢ (forget D).map ((diagramPullback J P f).app S.op ≫ colimit.ι (diagram J P Y) ((Functor.op (pullback J f)).obj S.op))\n (↑(Meq.equiv P S).symm x) =\n (forget D).map (colimit.ι (diagram J P Y) (Cover.pullback S f).op)\n (↑(Meq.equiv P (Cover.pullback S f)).symm (Meq.pullback x f))", "state_before": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\n⊢ (forget D).map (colimMap (diagramPullback J P f) ≫ colimit.pre (diagram J P Y) (Functor.op (pullback J f)))\n ((forget D).map (colimit.ι (diagram J P X) S.op) (↑(Meq.equiv P S).symm x)) =\n (forget D).map (colimit.ι (diagram J P Y) (Cover.pullback S f).op)\n (↑(Meq.equiv P (Cover.pullback S f)).symm (Meq.pullback x f))", "tactic": "simp only [← comp_apply, colimit.ι_pre, ι_colimMap_assoc]" }, { "state_after": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\n⊢ (forget D).map (colimit.ι (diagram J P Y) ((Functor.op (pullback J f)).obj S.op))\n ((forget D).map ((diagramPullback J P f).app S.op) (↑(Meq.equiv P S).symm x)) =\n (forget D).map (colimit.ι (diagram J P Y) (Cover.pullback S f).op)\n (↑(Meq.equiv P (Cover.pullback S f)).symm (Meq.pullback x f))", "state_before": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\n⊢ (forget D).map ((diagramPullback J P f).app S.op ≫ colimit.ι (diagram J P Y) ((Functor.op (pullback J f)).obj S.op))\n (↑(Meq.equiv P S).symm x) =\n (forget D).map (colimit.ι (diagram J P Y) (Cover.pullback S f).op)\n (↑(Meq.equiv P (Cover.pullback S f)).symm (Meq.pullback x f))", "tactic": "simp_rw [comp_apply]" }, { "state_after": "case h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\n⊢ (forget D).map ((diagramPullback J P f).app S.op) (↑(Meq.equiv P S).symm x) =\n ↑(Meq.equiv P (Cover.pullback S f)).symm (Meq.pullback x f)", "state_before": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\n⊢ (forget D).map (colimit.ι (diagram J P Y) ((Functor.op (pullback J f)).obj S.op))\n ((forget D).map ((diagramPullback J P f).app S.op) (↑(Meq.equiv P S).symm x)) =\n (forget D).map (colimit.ι (diagram J P Y) (Cover.pullback S f).op)\n (↑(Meq.equiv P (Cover.pullback S f)).symm (Meq.pullback x f))", "tactic": "apply congr_arg" }, { "state_after": "case h.a\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\n⊢ ↑(Meq.equiv P ((Functor.op (pullback J f)).obj S.op).unop)\n ((forget D).map ((diagramPullback J P f).app S.op) (↑(Meq.equiv P S).symm x)) =\n ↑(Meq.equiv P ((Functor.op (pullback J f)).obj S.op).unop)\n (↑(Meq.equiv P (Cover.pullback S f)).symm (Meq.pullback x f))", "state_before": "case h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\n⊢ (forget D).map ((diagramPullback J P f).app S.op) (↑(Meq.equiv P S).symm x) =\n ↑(Meq.equiv P (Cover.pullback S f)).symm (Meq.pullback x f)", "tactic": "apply (Meq.equiv P _).injective" }, { "state_after": "case h.a\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\n⊢ ↑(Meq.equiv P ((Functor.op (pullback J f)).obj S.op).unop)\n ((forget D).map ((diagramPullback J P f).app S.op) (↑(Meq.equiv P S).symm x)) =\n Meq.pullback x f", "state_before": "case h.a\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\n⊢ ↑(Meq.equiv P ((Functor.op (pullback J f)).obj S.op).unop)\n ((forget D).map ((diagramPullback J P f).app S.op) (↑(Meq.equiv P S).symm x)) =\n ↑(Meq.equiv P ((Functor.op (pullback J f)).obj S.op).unop)\n (↑(Meq.equiv P (Cover.pullback S f)).symm (Meq.pullback x f))", "tactic": "erw [Equiv.apply_symm_apply]" }, { "state_after": "case h.a.h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\ni : Cover.Arrow ((Functor.op (pullback J f)).obj S.op).unop\n⊢ ↑(↑(Meq.equiv P ((Functor.op (pullback J f)).obj S.op).unop)\n ((forget D).map ((diagramPullback J P f).app S.op) (↑(Meq.equiv P S).symm x)))\n i =\n ↑(Meq.pullback x f) i", "state_before": "case h.a\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\n⊢ ↑(Meq.equiv P ((Functor.op (pullback J f)).obj S.op).unop)\n ((forget D).map ((diagramPullback J P f).app S.op) (↑(Meq.equiv P S).symm x)) =\n Meq.pullback x f", "tactic": "ext i" }, { "state_after": "case h.a.h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\ni : Cover.Arrow ((Functor.op (pullback J f)).obj S.op).unop\n⊢ (forget D).map\n (Multiequalizer.lift (Cover.index ((Functor.op (pullback J f)).obj S.op).unop P) ((diagram J P X).obj S.op)\n (fun I => Multiequalizer.ι (Cover.index S.op.unop P) (Cover.Arrow.base I))\n (_ :\n ∀ (I : (Cover.index ((Functor.op (pullback J f)).obj S.op).unop P).R),\n Multiequalizer.ι (Cover.index S.op.unop P)\n (MulticospanIndex.fstTo (Cover.index S.op.unop P) (Cover.Relation.base I)) ≫\n MulticospanIndex.fst (Cover.index S.op.unop P) (Cover.Relation.base I) =\n Multiequalizer.ι (Cover.index S.op.unop P)\n (MulticospanIndex.sndTo (Cover.index S.op.unop P) (Cover.Relation.base I)) ≫\n MulticospanIndex.snd (Cover.index S.op.unop P) (Cover.Relation.base I)) ≫\n Multiequalizer.ι (Cover.index ((Functor.op (pullback J f)).obj S.op).unop P) i)\n (↑(Meq.equiv P S).symm x) =\n ↑x { Y := i.Y, f := i.f ≫ f, hf := (_ : (Cover.sieve ((pullback J f).obj S)).arrows i.f) }", "state_before": "case h.a.h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\ni : Cover.Arrow ((Functor.op (pullback J f)).obj S.op).unop\n⊢ ↑(↑(Meq.equiv P ((Functor.op (pullback J f)).obj S.op).unop)\n ((forget D).map ((diagramPullback J P f).app S.op) (↑(Meq.equiv P S).symm x)))\n i =\n ↑(Meq.pullback x f) i", "tactic": "simp only [diagramPullback_app, Meq.pullback_apply, Meq.equiv_apply, ← comp_apply]" }, { "state_after": "case h.a.h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\ni : Cover.Arrow ((Functor.op (pullback J f)).obj S.op).unop\n⊢ ↑x (Cover.Arrow.base i) = ↑x { Y := i.Y, f := i.f ≫ f, hf := (_ : (Cover.sieve ((pullback J f).obj S)).arrows i.f) }", "state_before": "case h.a.h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\ni : Cover.Arrow ((Functor.op (pullback J f)).obj S.op).unop\n⊢ (forget D).map\n (Multiequalizer.lift (Cover.index ((Functor.op (pullback J f)).obj S.op).unop P) ((diagram J P X).obj S.op)\n (fun I => Multiequalizer.ι (Cover.index S.op.unop P) (Cover.Arrow.base I))\n (_ :\n ∀ (I : (Cover.index ((Functor.op (pullback J f)).obj S.op).unop P).R),\n Multiequalizer.ι (Cover.index S.op.unop P)\n (MulticospanIndex.fstTo (Cover.index S.op.unop P) (Cover.Relation.base I)) ≫\n MulticospanIndex.fst (Cover.index S.op.unop P) (Cover.Relation.base I) =\n Multiequalizer.ι (Cover.index S.op.unop P)\n (MulticospanIndex.sndTo (Cover.index S.op.unop P) (Cover.Relation.base I)) ≫\n MulticospanIndex.snd (Cover.index S.op.unop P) (Cover.Relation.base I)) ≫\n Multiequalizer.ι (Cover.index ((Functor.op (pullback J f)).obj S.op).unop P) i)\n (↑(Meq.equiv P S).symm x) =\n ↑x { Y := i.Y, f := i.f ≫ f, hf := (_ : (Cover.sieve ((pullback J f).obj S)).arrows i.f) }", "tactic": "erw [Multiequalizer.lift_ι, Meq.equiv_symm_eq_apply]" }, { "state_after": "case h.a.h.mk\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\nY✝ : C\nf✝ : Y✝ ⟶ Y\nhf✝ : (Cover.sieve ((Functor.op (pullback J f)).obj S.op).unop).arrows f✝\n⊢ ↑x (Cover.Arrow.base { Y := Y✝, f := f✝, hf := hf✝ }) =\n ↑x\n { Y := { Y := Y✝, f := f✝, hf := hf✝ }.Y, f := { Y := Y✝, f := f✝, hf := hf✝ }.f ≫ f,\n hf := (_ : (Cover.sieve ((pullback J f).obj S)).arrows { Y := Y✝, f := f✝, hf := hf✝ }.f) }", "state_before": "case h.a.h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\ni : Cover.Arrow ((Functor.op (pullback J f)).obj S.op).unop\n⊢ ↑x (Cover.Arrow.base i) = ↑x { Y := i.Y, f := i.f ≫ f, hf := (_ : (Cover.sieve ((pullback J f).obj S)).arrows i.f) }", "tactic": "cases i" }, { "state_after": "no goals", "state_before": "case h.a.h.mk\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : Meq P S\nf : Y ⟶ X\nY✝ : C\nf✝ : Y✝ ⟶ Y\nhf✝ : (Cover.sieve ((Functor.op (pullback J f)).obj S.op).unop).arrows f✝\n⊢ ↑x (Cover.Arrow.base { Y := Y✝, f := f✝, hf := hf✝ }) =\n ↑x\n { Y := { Y := Y✝, f := f✝, hf := hf✝ }.Y, f := { Y := Y✝, f := f✝, hf := hf✝ }.f ≫ f,\n hf := (_ : (Cover.sieve ((pullback J f).obj S)).arrows { Y := Y✝, f := f✝, hf := hf✝ }.f) }", "tactic": "rfl" } ]

[ 177, 15 ]

[ 166, 1 ]

[]

[ 133, 79 ]

[ 132, 1 ]

[ { "state_after": "R : Type u\nS : Type v\nF : Type u_1\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : RingHomClass F R S\nf : F\nI✝ : Ideal R\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsMaximal I\nne_top : ¬map f I = ⊤\nJ : Ideal S\nhJ : map f I < J\n⊢ J = ⊤", "state_before": "R : Type u\nS : Type v\nF : Type u_1\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : RingHomClass F R S\nf : F\nI✝ : Ideal R\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsMaximal I\n⊢ map f I = ⊤ ∨ IsMaximal (map f I)", "tactic": "refine' or_iff_not_imp_left.2 fun ne_top => ⟨⟨fun h => ne_top h, fun J hJ => _⟩⟩" }, { "state_after": "case refine'_1\nR : Type u\nS : Type v\nF : Type u_1\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : RingHomClass F R S\nf : F\nI✝ : Ideal R\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsMaximal I\nne_top : ¬map f I = ⊤\nJ : Ideal S\nhJ : map f I < J\n⊢ I ≤ comap f J\n\ncase refine'_2\nR : Type u\nS : Type v\nF : Type u_1\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : RingHomClass F R S\nf : F\nI✝ : Ideal R\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsMaximal I\nne_top : ¬map f I = ⊤\nJ : Ideal S\nhJ : map f I < J\n⊢ I ≠ comap f J", "state_before": "R : Type u\nS : Type v\nF : Type u_1\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : RingHomClass F R S\nf : F\nI✝ : Ideal R\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsMaximal I\nne_top : ¬map f I = ⊤\nJ : Ideal S\nhJ : map f I < J\n⊢ J = ⊤", "tactic": "refine'\n (relIsoOfSurjective f hf).injective\n (Subtype.ext_iff.2 (Eq.trans (H.1.2 (comap f J) (lt_of_le_of_ne _ _)) comap_top.symm))" }, { "state_after": "no goals", "state_before": "case refine'_1\nR : Type u\nS : Type v\nF : Type u_1\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : RingHomClass F R S\nf : F\nI✝ : Ideal R\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsMaximal I\nne_top : ¬map f I = ⊤\nJ : Ideal S\nhJ : map f I < J\n⊢ I ≤ comap f J", "tactic": "exact map_le_iff_le_comap.1 (le_of_lt hJ)" }, { "state_after": "no goals", "state_before": "case refine'_2\nR : Type u\nS : Type v\nF : Type u_1\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : RingHomClass F R S\nf : F\nI✝ : Ideal R\nhf : Function.Surjective ↑f\nI : Ideal R\nH : IsMaximal I\nne_top : ¬map f I = ⊤\nJ : Ideal S\nhJ : map f I < J\n⊢ I ≠ comap f J", "tactic": "exact fun h => hJ.right (le_map_of_comap_le_of_surjective f hf (le_of_eq h.symm))" } ]

[ 1684, 88 ]

[ 1677, 1 ]

[ { "state_after": "U : Type u_1\ninst✝ : Quiver U\nu v v' w : U\np : Path u v\np' : Path u v'\ne : v ⟶ w\ne' : v' ⟶ w\nh : cons p e = cons p' e'\n⊢ HEq p p'", "state_before": "U : Type u_1\ninst✝ : Quiver U\nu v v' w : U\np : Path u v\np' : Path u v'\ne : v ⟶ w\ne' : v' ⟶ w\nh : cons p e = cons p' e'\n⊢ Path.cast (_ : u = u) (_ : v = v') p = p'", "tactic": "rw [Path.cast_eq_iff_heq]" }, { "state_after": "no goals", "state_before": "U : Type u_1\ninst✝ : Quiver U\nu v v' w : U\np : Path u v\np' : Path u v'\ne : v ⟶ w\ne' : v' ⟶ w\nh : cons p e = cons p' e'\n⊢ HEq p p'", "tactic": "exact heq_of_cons_eq_cons h" } ]

[ 142, 30 ]

[ 139, 1 ]

[]

[ 847, 31 ]

[ 844, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Type ?u.475243\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : Option α → β → γ\na : α\nas : List α\nb : β\nbs : List β\n⊢ map₂Right f [] bs = map (f none) bs", "tactic": "cases bs <;> rfl" } ]

[ 4136, 88 ]

[ 4136, 1 ]

[ { "state_after": "case pos\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : p = 0\n⊢ snorm f_lim p μ ≤ liminf (fun n => snorm (f n) p μ) atTop\n\ncase neg\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : ¬p = 0\n⊢ snorm f_lim p μ ≤ liminf (fun n => snorm (f n) p μ) atTop", "state_before": "α : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\n⊢ snorm f_lim p μ ≤ liminf (fun n => snorm (f n) p μ) atTop", "tactic": "by_cases hp0 : p = 0" }, { "state_after": "case neg\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : p ≠ 0\n⊢ snorm f_lim p μ ≤ liminf (fun n => snorm (f n) p μ) atTop", "state_before": "case neg\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : ¬p = 0\n⊢ snorm f_lim p μ ≤ liminf (fun n => snorm (f n) p μ) atTop", "tactic": "rw [← Ne.def] at hp0" }, { "state_after": "case pos\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : p ≠ 0\nhp_top : p = ⊤\n⊢ snorm f_lim p μ ≤ liminf (fun n => snorm (f n) p μ) atTop\n\ncase neg\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : p ≠ 0\nhp_top : ¬p = ⊤\n⊢ snorm f_lim p μ ≤ liminf (fun n => snorm (f n) p μ) atTop", "state_before": "case neg\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : p ≠ 0\n⊢ snorm f_lim p μ ≤ liminf (fun n => snorm (f n) p μ) atTop", "tactic": "by_cases hp_top : p = ∞" }, { "state_after": "case neg\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : p ≠ 0\nhp_top : ¬p = ⊤\n⊢ snorm' f_lim (ENNReal.toReal p) μ ≤ liminf (fun n => snorm' (f n) (ENNReal.toReal p) μ) atTop", "state_before": "case neg\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : p ≠ 0\nhp_top : ¬p = ⊤\n⊢ snorm f_lim p μ ≤ liminf (fun n => snorm (f n) p μ) atTop", "tactic": "simp_rw [snorm_eq_snorm' hp0 hp_top]" }, { "state_after": "case neg\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : p ≠ 0\nhp_top : ¬p = ⊤\nhp_pos : 0 < ENNReal.toReal p\n⊢ snorm' f_lim (ENNReal.toReal p) μ ≤ liminf (fun n => snorm' (f n) (ENNReal.toReal p) μ) atTop", "state_before": "case neg\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : p ≠ 0\nhp_top : ¬p = ⊤\n⊢ snorm' f_lim (ENNReal.toReal p) μ ≤ liminf (fun n => snorm' (f n) (ENNReal.toReal p) μ) atTop", "tactic": "have hp_pos : 0 < p.toReal := ENNReal.toReal_pos hp0 hp_top" }, { "state_after": "no goals", "state_before": "case neg\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : p ≠ 0\nhp_top : ¬p = ⊤\nhp_pos : 0 < ENNReal.toReal p\n⊢ snorm' f_lim (ENNReal.toReal p) μ ≤ liminf (fun n => snorm' (f n) (ENNReal.toReal p) μ) atTop", "tactic": "exact snorm'_lim_le_liminf_snorm' hp_pos hf h_lim" }, { "state_after": "no goals", "state_before": "case pos\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : p = 0\n⊢ snorm f_lim p μ ≤ liminf (fun n => snorm (f n) p μ) atTop", "tactic": "simp [hp0]" }, { "state_after": "case pos\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : p ≠ 0\nhp_top : p = ⊤\n⊢ snorm f_lim ⊤ μ ≤ liminf (fun n => snorm (f n) ⊤ μ) atTop", "state_before": "case pos\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : p ≠ 0\nhp_top : p = ⊤\n⊢ snorm f_lim p μ ≤ liminf (fun n => snorm (f n) p μ) atTop", "tactic": "simp_rw [hp_top]" }, { "state_after": "no goals", "state_before": "case pos\nα : Type u_2\nE✝ : Type ?u.8118264\nF : Type ?u.8118267\nG : Type ?u.8118270\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nhp0 : p ≠ 0\nhp_top : p = ⊤\n⊢ snorm f_lim ⊤ μ ≤ liminf (fun n => snorm (f n) ⊤ μ) atTop", "tactic": "exact snorm_exponent_top_lim_le_liminf_snorm_exponent_top h_lim" } ]

[ 1227, 52 ]

[ 1215, 1 ]

[ { "state_after": "μ : YoungDiagram\ni j : ℕ\n⊢ (i, j) ∈ μ ↔ ∀ (m : ℕ), m ≤ j → ¬¬(i, m) ∈ μ", "state_before": "μ : YoungDiagram\ni j : ℕ\n⊢ (i, j) ∈ μ ↔ j < rowLen μ i", "tactic": "rw [rowLen, Nat.lt_find_iff]" }, { "state_after": "μ : YoungDiagram\ni j : ℕ\n⊢ (i, j) ∈ μ ↔ ∀ (m : ℕ), m ≤ j → (i, m) ∈ μ", "state_before": "μ : YoungDiagram\ni j : ℕ\n⊢ (i, j) ∈ μ ↔ ∀ (m : ℕ), m ≤ j → ¬¬(i, m) ∈ μ", "tactic": "push_neg" }, { "state_after": "no goals", "state_before": "μ : YoungDiagram\ni j : ℕ\n⊢ (i, j) ∈ μ ↔ ∀ (m : ℕ), m ≤ j → (i, m) ∈ μ", "tactic": "exact ⟨fun h _ hmj => μ.up_left_mem (by rfl) hmj h, fun h => h _ (by rfl)⟩" }, { "state_after": "no goals", "state_before": "μ : YoungDiagram\ni j : ℕ\nh : (i, j) ∈ μ\nx✝ : ℕ\nhmj : x✝ ≤ j\n⊢ i ≤ i", "tactic": "rfl" }, { "state_after": "no goals", "state_before": "μ : YoungDiagram\ni j : ℕ\nh : ∀ (m : ℕ), m ≤ j → (i, m) ∈ μ\n⊢ j ≤ j", "tactic": "rfl" } ]

[ 314, 77 ]

[ 311, 1 ]

[ { "state_after": "α✝ : Type u_1\na : α✝\nl : List α✝\np : α✝ → Bool\nal : a ∈ l\npa : p a = true\nw✝ : α✝\nl₁ l₂ : List α✝\nleft✝¹ : ∀ (b : α✝), b ∈ l₁ → ¬p b = true\nleft✝ : p w✝ = true\ne₁ : l = l₁ ++ w✝ :: l₂\ne₂ : eraseP p l = l₁ ++ l₂\n⊢ length (eraseP p l) = pred (length l)", "state_before": "α✝ : Type u_1\na : α✝\nl : List α✝\np : α✝ → Bool\nal : a ∈ l\npa : p a = true\n⊢ length (eraseP p l) = pred (length l)", "tactic": "let ⟨_, l₁, l₂, _, _, e₁, e₂⟩ := exists_of_eraseP al pa" }, { "state_after": "α✝ : Type u_1\na : α✝\nl : List α✝\np : α✝ → Bool\nal : a ∈ l\npa : p a = true\nw✝ : α✝\nl₁ l₂ : List α✝\nleft✝¹ : ∀ (b : α✝), b ∈ l₁ → ¬p b = true\nleft✝ : p w✝ = true\ne₁ : l = l₁ ++ w✝ :: l₂\ne₂ : eraseP p l = l₁ ++ l₂\n⊢ length (l₁ ++ l₂) = pred (length l)", "state_before": "α✝ : Type u_1\na : α✝\nl : List α✝\np : α✝ → Bool\nal : a ∈ l\npa : p a = true\nw✝ : α✝\nl₁ l₂ : List α✝\nleft✝¹ : ∀ (b : α✝), b ∈ l₁ → ¬p b = true\nleft✝ : p w✝ = true\ne₁ : l = l₁ ++ w✝ :: l₂\ne₂ : eraseP p l = l₁ ++ l₂\n⊢ length (eraseP p l) = pred (length l)", "tactic": "rw [e₂]" }, { "state_after": "α✝ : Type u_1\na : α✝\nl : List α✝\np : α✝ → Bool\nal : a ∈ l\npa : p a = true\nw✝ : α✝\nl₁ l₂ : List α✝\nleft✝¹ : ∀ (b : α✝), b ∈ l₁ → ¬p b = true\nleft✝ : p w✝ = true\ne₁ : l = l₁ ++ w✝ :: l₂\ne₂ : eraseP p l = l₁ ++ l₂\n⊢ length l₁ + length l₂ = pred (length l₁ + succ (length l₂))", "state_before": "α✝ : Type u_1\na : α✝\nl : List α✝\np : α✝ → Bool\nal : a ∈ l\npa : p a = true\nw✝ : α✝\nl₁ l₂ : List α✝\nleft✝¹ : ∀ (b : α✝), b ∈ l₁ → ¬p b = true\nleft✝ : p w✝ = true\ne₁ : l = l₁ ++ w✝ :: l₂\ne₂ : eraseP p l = l₁ ++ l₂\n⊢ length (l₁ ++ l₂) = pred (length l)", "tactic": "simp [length_append, e₁]" }, { "state_after": "no goals", "state_before": "α✝ : Type u_1\na : α✝\nl : List α✝\np : α✝ → Bool\nal : a ∈ l\npa : p a = true\nw✝ : α✝\nl₁ l₂ : List α✝\nleft✝¹ : ∀ (b : α✝), b ∈ l₁ → ¬p b = true\nleft✝ : p w✝ = true\ne₁ : l = l₁ ++ w✝ :: l₂\ne₂ : eraseP p l = l₁ ++ l₂\n⊢ length l₁ + length l₂ = pred (length l₁ + succ (length l₂))", "tactic": "rfl" } ]

[ 967, 41 ]

[ 964, 9 ]

[]

[ 606, 38 ]

[ 605, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.8994\nγ : Type ?u.8997\nι : Sort u_2\nι' : Sort ?u.9003\np : ι → Prop\ns : ι → Set α\nh : IsBasis p s\nU : Set α\n⊢ U ∈ IsBasis.filter h ↔ ∃ i, p i ∧ s i ⊆ U", "tactic": "simp only [IsBasis.filter, FilterBasis.mem_filter_iff, mem_filterBasis_iff,\n exists_exists_and_eq_and]" } ]

[ 221, 30 ]

[ 218, 11 ]

[]

[ 501, 49 ]

[ 499, 1 ]

[ { "state_after": "α : Type u_1\ninst✝ : OrderedCommGroup α\ns t : Set α\na : α\n⊢ (⋃ (a : α) (_ : a ∈ s), a • ↑(upperClosure t)) = ⋃ (i : α) (_ : i ∈ s), ↑(i • upperClosure t)", "state_before": "α : Type u_1\ninst✝ : OrderedCommGroup α\ns t : Set α\na : α\n⊢ s * ↑(upperClosure t) = ↑(upperClosure (s * t))", "tactic": "simp_rw [← smul_eq_mul, ← Set.iUnion_smul_set, upperClosure_iUnion, upperClosure_smul,\n UpperSet.coe_iInf₂]" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : OrderedCommGroup α\ns t : Set α\na : α\n⊢ (⋃ (a : α) (_ : a ∈ s), a • ↑(upperClosure t)) = ⋃ (i : α) (_ : i ∈ s), ↑(i • upperClosure t)", "tactic": "rfl" } ]

[ 291, 6 ]

[ 288, 1 ]

[]

[ 222, 28 ]

[ 220, 11 ]

[]

[ 45, 6 ]

[ 44, 1 ]

[]

[ 1362, 73 ]

[ 1361, 1 ]

[ { "state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.154319\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set (β × β)\nf : α → α → β\nhf :\n Tendsto\n ((fun x => (f x.fst.fst x.fst.snd, f x.snd.fst x.snd.snd)) ∘ fun p =>\n ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd))\n (𝓤 α ×ˢ 𝓤 α) (𝓤 β)\nhs : s ∈ 𝓤 β\n⊢ ∃ u, u ∈ 𝓤 α ∧ ∀ (a b c : α), (a, b) ∈ u → (f a c, f b c) ∈ s", "state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.154319\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set (β × β)\nf : α → α → β\nhf : UniformContinuous fun p => f p.fst p.snd\nhs : s ∈ 𝓤 β\n⊢ ∃ u, u ∈ 𝓤 α ∧ ∀ (a b c : α), (a, b) ∈ u → (f a c, f b c) ∈ s", "tactic": "rw [UniformContinuous, uniformity_prod_eq_prod, tendsto_map'_iff] at hf" }, { "state_after": "case intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.154319\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set (β × β)\nf : α → α → β\nhf :\n Tendsto\n ((fun x => (f x.fst.fst x.fst.snd, f x.snd.fst x.snd.snd)) ∘ fun p =>\n ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd))\n (𝓤 α ×ˢ 𝓤 α) (𝓤 β)\nhs : s ∈ 𝓤 β\nu : Set (α × α)\nhu : u ∈ 𝓤 α\nv : Set (α × α)\nhv : v ∈ 𝓤 α\nhuvt :\n u ×ˢ v ⊆\n ((fun x => (f x.fst.fst x.fst.snd, f x.snd.fst x.snd.snd)) ∘ fun p =>\n ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) ⁻¹'\n s\n⊢ ∃ u, u ∈ 𝓤 α ∧ ∀ (a b c : α), (a, b) ∈ u → (f a c, f b c) ∈ s", "state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.154319\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set (β × β)\nf : α → α → β\nhf :\n Tendsto\n ((fun x => (f x.fst.fst x.fst.snd, f x.snd.fst x.snd.snd)) ∘ fun p =>\n ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd))\n (𝓤 α ×ˢ 𝓤 α) (𝓤 β)\nhs : s ∈ 𝓤 β\n⊢ ∃ u, u ∈ 𝓤 α ∧ ∀ (a b c : α), (a, b) ∈ u → (f a c, f b c) ∈ s", "tactic": "rcases mem_prod_iff.1 (mem_map.1 <| hf hs) with ⟨u, hu, v, hv, huvt⟩" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.154319\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set (β × β)\nf : α → α → β\nhf :\n Tendsto\n ((fun x => (f x.fst.fst x.fst.snd, f x.snd.fst x.snd.snd)) ∘ fun p =>\n ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd))\n (𝓤 α ×ˢ 𝓤 α) (𝓤 β)\nhs : s ∈ 𝓤 β\nu : Set (α × α)\nhu : u ∈ 𝓤 α\nv : Set (α × α)\nhv : v ∈ 𝓤 α\nhuvt :\n u ×ˢ v ⊆\n ((fun x => (f x.fst.fst x.fst.snd, f x.snd.fst x.snd.snd)) ∘ fun p =>\n ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) ⁻¹'\n s\n⊢ ∃ u, u ∈ 𝓤 α ∧ ∀ (a b c : α), (a, b) ∈ u → (f a c, f b c) ∈ s", "tactic": "exact ⟨u, hu, fun a b c hab => @huvt ((_, _), (_, _)) ⟨hab, refl_mem_uniformity hv⟩⟩" } ]

[ 1591, 87 ]

[ 1586, 1 ]

[ { "state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\n⊢ ∀ᶠ (n : ι × α) in p ×ˢ p', (f n.snd, F n.fst n.snd) ∈ u", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\n⊢ TendstoUniformlyOnFilter F f p p'", "tactic": "intro u hu" }, { "state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\n⊢ ∀ᶠ (n : ι × α) in p ×ˢ p', (f n.snd, F n.fst n.snd) ∈ u", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\n⊢ ∀ᶠ (n : ι × α) in p ×ˢ p', (f n.snd, F n.fst n.snd) ∈ u", "tactic": "rcases comp_symm_of_uniformity hu with ⟨t, ht, htsymm, htmem⟩" }, { "state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\n⊢ ∀ᶠ (n : ι × α) in p ×ˢ p', (f n.snd, F n.fst n.snd) ∈ u", "state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\n⊢ ∀ᶠ (n : ι × α) in p ×ˢ p', (f n.snd, F n.fst n.snd) ∈ u", "tactic": "have hmc : ∀ᶠ x in (p ×ˢ p) ×ˢ p', Tendsto (fun n : ι => F n x.snd) p (𝓝 (f x.snd)) := by\n rw [eventually_prod_iff]\n refine' ⟨fun _ => True, by simp, _, hF', by simp⟩" }, { "state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\n⊢ ∀ᶠ (y : α × ι) in p' ×ˢ p, (f (Prod.swap y).snd, F (Prod.swap y).fst (Prod.swap y).snd) ∈ u", "state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\n⊢ ∀ᶠ (n : ι × α) in p ×ˢ p', (f n.snd, F n.fst n.snd) ∈ u", "tactic": "rw [Filter.eventually_swap_iff]" }, { "state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\n⊢ ∀ᶠ (y : α × ι) in p' ×ˢ p, (f (Prod.swap y).snd, F (Prod.swap y).fst (Prod.swap y).snd) ∈ u", "state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\n⊢ ∀ᶠ (y : α × ι) in p' ×ˢ p, (f (Prod.swap y).snd, F (Prod.swap y).fst (Prod.swap y).snd) ∈ u", "tactic": "have := tendsto_prodAssoc.eventually (tendsto_prod_swap.eventually ((hF t ht).and hmc))" }, { "state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\n⊢ ∀ (x : α × ι),\n (∀ᶠ (y : ι) in p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) (x, y))).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) (x, y))).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) (x, y))).fst.snd\n (Prod.swap (↑(Equiv.prodAssoc α ι ι) (x, y))).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) (x, y))).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) (x, y))).snd))) →\n (f (Prod.swap x).snd, F (Prod.swap x).fst (Prod.swap x).snd) ∈ u", "state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\n⊢ ∀ᶠ (y : α × ι) in p' ×ˢ p, (f (Prod.swap y).snd, F (Prod.swap y).fst (Prod.swap y).snd) ∈ u", "tactic": "apply this.curry.mono" }, { "state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\n⊢ ∀ (a : α) (b : ι), (∀ᶠ (x : ι) in p, (F b a, F x a) ∈ t) → Tendsto (fun n => F n a) p (𝓝 (f a)) → (f a, F b a) ∈ u", "state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\n⊢ ∀ (x : α × ι),\n (∀ᶠ (y : ι) in p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) (x, y))).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) (x, y))).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) (x, y))).fst.snd\n (Prod.swap (↑(Equiv.prodAssoc α ι ι) (x, y))).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) (x, y))).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) (x, y))).snd))) →\n (f (Prod.swap x).snd, F (Prod.swap x).fst (Prod.swap x).snd) ∈ u", "tactic": "simp only [Equiv.prodAssoc_apply, eventually_and, eventually_const, Prod.snd_swap, Prod.fst_swap,\n and_imp, Prod.forall]" }, { "state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\nx : α\nn : ι\nhx : ∀ᶠ (x_1 : ι) in p, (F n x, F x_1 x) ∈ t\nhm' : Tendsto (fun n => F n x) p (𝓝 (f x))\n⊢ (f x, F n x) ∈ u", "state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\n⊢ ∀ (a : α) (b : ι), (∀ᶠ (x : ι) in p, (F b a, F x a) ∈ t) → Tendsto (fun n => F n a) p (𝓝 (f a)) → (f a, F b a) ∈ u", "tactic": "intro x n hx hm'" }, { "state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\nx : α\nn : ι\nhx : ∀ᶠ (x_1 : ι) in p, (F n x, F x_1 x) ∈ t\nhm' : Tendsto (fun n => F n x) p (𝓝 (f x))\n⊢ ∃ z, (f x, z) ∈ t ∧ (z, F n x) ∈ t", "state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\nx : α\nn : ι\nhx : ∀ᶠ (x_1 : ι) in p, (F n x, F x_1 x) ∈ t\nhm' : Tendsto (fun n => F n x) p (𝓝 (f x))\n⊢ (f x, F n x) ∈ u", "tactic": "refine' Set.mem_of_mem_of_subset (mem_compRel.mpr _) htmem" }, { "state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\nx : α\nn : ι\nhx : ∀ᶠ (x_1 : ι) in p, (F n x, F x_1 x) ∈ t\nhm' : Tendsto (fun x_1 => (f x, F x_1 x)) p (𝓤 β)\n⊢ ∃ z, (f x, z) ∈ t ∧ (z, F n x) ∈ t", "state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\nx : α\nn : ι\nhx : ∀ᶠ (x_1 : ι) in p, (F n x, F x_1 x) ∈ t\nhm' : Tendsto (fun n => F n x) p (𝓝 (f x))\n⊢ ∃ z, (f x, z) ∈ t ∧ (z, F n x) ∈ t", "tactic": "rw [Uniform.tendsto_nhds_right] at hm'" }, { "state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis✝ :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\nx : α\nn : ι\nhx : ∀ᶠ (x_1 : ι) in p, (F n x, F x_1 x) ∈ t\nhm' : Tendsto (fun x_1 => (f x, F x_1 x)) p (𝓤 β)\nthis : ∀ᶠ (x_1 : ι) in p, (F n x, F x_1 x) ∈ t ∧ (fun x_2 => (f x, F x_2 x)) x_1 ∈ t\n⊢ ∃ z, (f x, z) ∈ t ∧ (z, F n x) ∈ t", "state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\nx : α\nn : ι\nhx : ∀ᶠ (x_1 : ι) in p, (F n x, F x_1 x) ∈ t\nhm' : Tendsto (fun x_1 => (f x, F x_1 x)) p (𝓤 β)\n⊢ ∃ z, (f x, z) ∈ t ∧ (z, F n x) ∈ t", "tactic": "have := hx.and (hm' ht)" }, { "state_after": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis✝ :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\nx : α\nn : ι\nhx : ∀ᶠ (x_1 : ι) in p, (F n x, F x_1 x) ∈ t\nhm' : Tendsto (fun x_1 => (f x, F x_1 x)) p (𝓤 β)\nthis : ∀ᶠ (x_1 : ι) in p, (F n x, F x_1 x) ∈ t ∧ (fun x_2 => (f x, F x_2 x)) x_1 ∈ t\nm : ι\nhm : (F n x, F m x) ∈ t ∧ (fun x_1 => (f x, F x_1 x)) m ∈ t\n⊢ ∃ z, (f x, z) ∈ t ∧ (z, F n x) ∈ t", "state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis✝ :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\nx : α\nn : ι\nhx : ∀ᶠ (x_1 : ι) in p, (F n x, F x_1 x) ∈ t\nhm' : Tendsto (fun x_1 => (f x, F x_1 x)) p (𝓤 β)\nthis : ∀ᶠ (x_1 : ι) in p, (F n x, F x_1 x) ∈ t ∧ (fun x_2 => (f x, F x_2 x)) x_1 ∈ t\n⊢ ∃ z, (f x, z) ∈ t ∧ (z, F n x) ∈ t", "tactic": "obtain ⟨m, hm⟩ := this.exists" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\nhmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))\nthis✝ :\n ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p,\n (F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.fst (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd,\n F (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).fst.snd (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) ∈\n t ∧\n Tendsto (fun n => F n (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd) p\n (𝓝 (f (Prod.swap (↑(Equiv.prodAssoc α ι ι) x)).snd))\nx : α\nn : ι\nhx : ∀ᶠ (x_1 : ι) in p, (F n x, F x_1 x) ∈ t\nhm' : Tendsto (fun x_1 => (f x, F x_1 x)) p (𝓤 β)\nthis : ∀ᶠ (x_1 : ι) in p, (F n x, F x_1 x) ∈ t ∧ (fun x_2 => (f x, F x_2 x)) x_1 ∈ t\nm : ι\nhm : (F n x, F m x) ∈ t ∧ (fun x_1 => (f x, F x_1 x)) m ∈ t\n⊢ ∃ z, (f x, z) ∈ t ∧ (z, F n x) ∈ t", "tactic": "exact ⟨F m x, ⟨hm.2, htsymm hm.1⟩⟩" }, { "state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\n⊢ ∃ pa,\n (∀ᶠ (x : ι × ι) in p ×ˢ p, pa x) ∧\n ∃ pb,\n (∀ᶠ (y : α) in p', pb y) ∧\n ∀ {x : ι × ι}, pa x → ∀ {y : α}, pb y → Tendsto (fun n => F n (x, y).snd) p (𝓝 (f (x, y).snd))", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\n⊢ ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.snd) p (𝓝 (f x.snd))", "tactic": "rw [eventually_prod_iff]" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\n⊢ ∃ pa,\n (∀ᶠ (x : ι × ι) in p ×ˢ p, pa x) ∧\n ∃ pb,\n (∀ᶠ (y : α) in p', pb y) ∧\n ∀ {x : ι × ι}, pa x → ∀ {y : α}, pb y → Tendsto (fun n => F n (x, y).snd) p (𝓝 (f (x, y).snd))", "tactic": "refine' ⟨fun _ => True, by simp, _, hF', by simp⟩" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\n⊢ ∀ᶠ (x : ι × ι) in p ×ˢ p, (fun x => True) x", "tactic": "simp" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : NeBot p\nhF : UniformCauchySeqOnFilter F p p'\nhF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t\nhtmem : t ○ t ⊆ u\n⊢ ∀ {x : ι × ι},\n (fun x => True) x →\n ∀ {y : α}, Tendsto (fun n => F n y) p (𝓝 (f y)) → Tendsto (fun n => F n (x, y).snd) p (𝓝 (f (x, y).snd))", "tactic": "simp" } ]

[ 467, 37 ]

[ 441, 1 ]

[]

[ 175, 30 ]

[ 174, 1 ]

[]

[ 199, 53 ]

[ 198, 1 ]

[]

[ 69, 35 ]

[ 66, 1 ]

[]

[ 1358, 78 ]

[ 1356, 1 ]

[]

[ 249, 7 ]

[ 248, 1 ]

[]

[ 170, 6 ]

[ 168, 1 ]

[]

[ 402, 6 ]

[ 401, 1 ]

[ { "state_after": "case mp\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective (map f)\nx y : α\nhxy : f x = f y\n⊢ x = y\n\ncase mpr\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective f\nx y : List α\nhxy : map f x = map f y\n⊢ x = y", "state_before": "ι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\n⊢ Injective (map f) ↔ Injective f", "tactic": "constructor <;> intro h x y hxy" }, { "state_after": "case mp\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective (map f)\nx y : α\nhxy : f x = f y\n⊢ [x] = [y]", "state_before": "case mp\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective (map f)\nx y : α\nhxy : f x = f y\n⊢ x = y", "tactic": "suffices [x] = [y] by simpa using this" }, { "state_after": "case mp.a\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective (map f)\nx y : α\nhxy : f x = f y\n⊢ map f [x] = map f [y]", "state_before": "case mp\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective (map f)\nx y : α\nhxy : f x = f y\n⊢ [x] = [y]", "tactic": "apply h" }, { "state_after": "no goals", "state_before": "case mp.a\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective (map f)\nx y : α\nhxy : f x = f y\n⊢ map f [x] = map f [y]", "tactic": "simp [hxy]" }, { "state_after": "no goals", "state_before": "ι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective (map f)\nx y : α\nhxy : f x = f y\nthis : [x] = [y]\n⊢ x = y", "tactic": "simpa using this" }, { "state_after": "case mpr.nil\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective f\nx✝ y : List α\nhxy✝ : map f x✝ = map f y\nx : List α\nhxy : map f x = map f []\n⊢ x = []\n\ncase mpr.cons\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective f\nx✝ y : List α\nhxy✝ : map f x✝ = map f y\nyh : α\nyt : List α\ny_ih : ∀ ⦃x : List α⦄, map f x = map f yt → x = yt\nx : List α\nhxy : map f x = map f (yh :: yt)\n⊢ x = yh :: yt", "state_before": "case mpr\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective f\nx y : List α\nhxy : map f x = map f y\n⊢ x = y", "tactic": "induction' y with yh yt y_ih generalizing x" }, { "state_after": "case mpr.cons.nil\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective f\nx y : List α\nhxy✝ : map f x = map f y\nyh : α\nyt : List α\ny_ih : ∀ ⦃x : List α⦄, map f x = map f yt → x = yt\nhxy : map f [] = map f (yh :: yt)\n⊢ [] = yh :: yt\n\ncase mpr.cons.cons\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective f\nx y : List α\nhxy✝ : map f x = map f y\nyh : α\nyt : List α\ny_ih : ∀ ⦃x : List α⦄, map f x = map f yt → x = yt\nhead✝ : α\ntail✝ : List α\nhxy : map f (head✝ :: tail✝) = map f (yh :: yt)\n⊢ head✝ :: tail✝ = yh :: yt", "state_before": "case mpr.cons\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective f\nx✝ y : List α\nhxy✝ : map f x✝ = map f y\nyh : α\nyt : List α\ny_ih : ∀ ⦃x : List α⦄, map f x = map f yt → x = yt\nx : List α\nhxy : map f x = map f (yh :: yt)\n⊢ x = yh :: yt", "tactic": "cases x" }, { "state_after": "no goals", "state_before": "case mpr.nil\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective f\nx✝ y : List α\nhxy✝ : map f x✝ = map f y\nx : List α\nhxy : map f x = map f []\n⊢ x = []", "tactic": "simpa using hxy" }, { "state_after": "no goals", "state_before": "case mpr.cons.nil\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective f\nx y : List α\nhxy✝ : map f x = map f y\nyh : α\nyt : List α\ny_ih : ∀ ⦃x : List α⦄, map f x = map f yt → x = yt\nhxy : map f [] = map f (yh :: yt)\n⊢ [] = yh :: yt", "tactic": "simp at hxy" }, { "state_after": "case mpr.cons.cons\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective f\nx y : List α\nhxy✝ : map f x = map f y\nyh : α\nyt : List α\ny_ih : ∀ ⦃x : List α⦄, map f x = map f yt → x = yt\nhead✝ : α\ntail✝ : List α\nhxy : f head✝ = f yh ∧ map f tail✝ = map f yt\n⊢ head✝ :: tail✝ = yh :: yt", "state_before": "case mpr.cons.cons\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective f\nx y : List α\nhxy✝ : map f x = map f y\nyh : α\nyt : List α\ny_ih : ∀ ⦃x : List α⦄, map f x = map f yt → x = yt\nhead✝ : α\ntail✝ : List α\nhxy : map f (head✝ :: tail✝) = map f (yh :: yt)\n⊢ head✝ :: tail✝ = yh :: yt", "tactic": "simp only [map, cons.injEq] at hxy" }, { "state_after": "no goals", "state_before": "case mpr.cons.cons\nι : Type ?u.135734\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : α → β\nh : Injective f\nx y : List α\nhxy✝ : map f x = map f y\nyh : α\nyt : List α\ny_ih : ∀ ⦃x : List α⦄, map f x = map f yt → x = yt\nhead✝ : α\ntail✝ : List α\nhxy : f head✝ = f yh ∧ map f tail✝ = map f yt\n⊢ head✝ :: tail✝ = yh :: yt", "tactic": "simp [y_ih hxy.2, h hxy.1]" } ]

[ 1842, 33 ]

[ 1832, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nl : List α\na : α\nn : ℕ\n⊢ rotate' (a :: l) (succ n) = rotate' (l ++ [a]) n", "tactic": "simp [rotate']" } ]

[ 58, 81 ]

[ 57, 1 ]

[ { "state_after": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX✝ Y✝ X Y : C\nf : X ⟶ Y\n⊢ ∀ {F₁ F₂ : C ⥤ D}, (F₁ ≅ F₂) → (IsIso (F₁.map f) ↔ IsIso (F₂.map f))", "state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX✝ Y✝ : C\nF₁ F₂ : C ⥤ D\ne : F₁ ≅ F₂\nX Y : C\nf : X ⟶ Y\n⊢ IsIso (F₁.map f) ↔ IsIso (F₂.map f)", "tactic": "revert F₁ F₂" }, { "state_after": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX✝ Y✝ X Y : C\nf : X ⟶ Y\n⊢ ∀ {F₁ F₂ : C ⥤ D}, (F₁ ≅ F₂) → IsIso (F₁.map f) → IsIso (F₂.map f)", "state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX✝ Y✝ X Y : C\nf : X ⟶ Y\n⊢ ∀ {F₁ F₂ : C ⥤ D}, (F₁ ≅ F₂) → (IsIso (F₁.map f) ↔ IsIso (F₂.map f))", "tactic": "suffices ∀ {F₁ F₂ : C ⥤ D} (_ : F₁ ≅ F₂) (_ : IsIso (F₁.map f)), IsIso (F₂.map f) by\n exact fun F₁ F₂ e => ⟨this e, this e.symm⟩" }, { "state_after": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX✝ Y✝ X Y : C\nf : X ⟶ Y\nF₁ F₂ : C ⥤ D\ne : F₁ ≅ F₂\nhf : IsIso (F₁.map f)\n⊢ IsIso (F₂.map f)", "state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX✝ Y✝ X Y : C\nf : X ⟶ Y\n⊢ ∀ {F₁ F₂ : C ⥤ D}, (F₁ ≅ F₂) → IsIso (F₁.map f) → IsIso (F₂.map f)", "tactic": "intro F₁ F₂ e hf" }, { "state_after": "case refine'_1\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX✝ Y✝ X Y : C\nf : X ⟶ Y\nF₁ F₂ : C ⥤ D\ne : F₁ ≅ F₂\nhf : IsIso (F₁.map f)\n⊢ F₂.map f ≫ e.inv.app Y ≫ inv (F₁.map f) ≫ e.hom.app X = 𝟙 (F₂.obj X)\n\ncase refine'_2\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX✝ Y✝ X Y : C\nf : X ⟶ Y\nF₁ F₂ : C ⥤ D\ne : F₁ ≅ F₂\nhf : IsIso (F₁.map f)\n⊢ (e.inv.app Y ≫ inv (F₁.map f) ≫ e.hom.app X) ≫ F₂.map f = 𝟙 (F₂.obj Y)", "state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX✝ Y✝ X Y : C\nf : X ⟶ Y\nF₁ F₂ : C ⥤ D\ne : F₁ ≅ F₂\nhf : IsIso (F₁.map f)\n⊢ IsIso (F₂.map f)", "tactic": "refine' IsIso.mk ⟨e.inv.app Y ≫ inv (F₁.map f) ≫ e.hom.app X, _, _⟩" }, { "state_after": "no goals", "state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX✝ Y✝ X Y : C\nf : X ⟶ Y\nthis : ∀ {F₁ F₂ : C ⥤ D}, (F₁ ≅ F₂) → IsIso (F₁.map f) → IsIso (F₂.map f)\n⊢ ∀ {F₁ F₂ : C ⥤ D}, (F₁ ≅ F₂) → (IsIso (F₁.map f) ↔ IsIso (F₂.map f))", "tactic": "exact fun F₁ F₂ e => ⟨this e, this e.symm⟩" }, { "state_after": "no goals", "state_before": "case refine'_1\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX✝ Y✝ X Y : C\nf : X ⟶ Y\nF₁ F₂ : C ⥤ D\ne : F₁ ≅ F₂\nhf : IsIso (F₁.map f)\n⊢ F₂.map f ≫ e.inv.app Y ≫ inv (F₁.map f) ≫ e.hom.app X = 𝟙 (F₂.obj X)", "tactic": "simp only [NatTrans.naturality_assoc, IsIso.hom_inv_id_assoc, Iso.inv_hom_id_app]" }, { "state_after": "no goals", "state_before": "case refine'_2\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX✝ Y✝ X Y : C\nf : X ⟶ Y\nF₁ F₂ : C ⥤ D\ne : F₁ ≅ F₂\nhf : IsIso (F₁.map f)\n⊢ (e.inv.app Y ≫ inv (F₁.map f) ≫ e.hom.app X) ≫ F₂.map f = 𝟙 (F₂.obj Y)", "tactic": "simp only [assoc, ← e.hom.naturality, IsIso.inv_hom_id_assoc, Iso.inv_hom_id_app]" } ]

[ 267, 86 ]

[ 259, 1 ]

[]

[ 973, 32 ]

[ 972, 1 ]

[]

[ 178, 49 ]

[ 177, 1 ]

[]

[ 271, 27 ]

[ 270, 1 ]

[]

[ 1510, 25 ]

[ 1509, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.97279\nγ : Type ?u.97282\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\no₁ o₂ : Ordinal\nh₁ : o₁ < type r\nh₂ : o₂ < type r\n⊢ r (enum r o₁ h₁) (enum r o₂ h₂) ↔ o₁ < o₂", "tactic": "rw [← typein_lt_typein r, typein_enum, typein_enum]" } ]

[ 539, 54 ]

[ 537, 1 ]

[ { "state_after": "case refl\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX : C\n⊢ F.map (eqToHom (_ : X = X)) = eqToHom (_ : F.obj X = F.obj X)", "state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y : C\np : X = Y\n⊢ F.map (eqToHom p) = eqToHom (_ : F.obj X = F.obj Y)", "tactic": "cases p" }, { "state_after": "no goals", "state_before": "case refl\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX : C\n⊢ F.map (eqToHom (_ : X = X)) = eqToHom (_ : F.obj X = F.obj X)", "tactic": "simp" } ]

[ 274, 72 ]

[ 273, 1 ]

[ { "state_after": "no goals", "state_before": "a b c : Prop\n⊢ a ∧ b ∧ c ↔ b ∧ c ∧ a", "tactic": "simp only [and_left_comm, and_comm]" } ]

[ 187, 38 ]

[ 186, 1 ]

[]

[ 106, 6 ]

[ 105, 1 ]

[]

[ 274, 16 ]

[ 273, 1 ]

[ { "state_after": "no goals", "state_before": "p : ℝ≥0∞\n𝕜 : Type ?u.509175\n𝕜' : Type ?u.509178\nι : Type u_2\nα : ι → Type ?u.509186\nβ✝ : ι → Type ?u.509191\ninst✝¹⁰ : Fintype ι\ninst✝⁹ : Fact (1 ≤ p)\ninst✝⁸ : NormedField 𝕜\ninst✝⁷ : NormedField 𝕜'\ninst✝⁶ : (i : ι) → SeminormedAddCommGroup (β✝ i)\ninst✝⁵ : (i : ι) → NormedSpace 𝕜 (β✝ i)\nc : 𝕜\nx y : PiLp p β✝\nx' y' : (i : ι) → β✝ i\ni : ι\nι' : Type ?u.509387\ninst✝⁴ : Fintype ι'\nE : Type ?u.509393\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nβ : Type u_1\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : Nonempty ι\nb : β\n⊢ ↑(↑(Fintype.card ι) ^ ENNReal.toReal (1 / p) * ‖b‖₊) = ↑(↑(Fintype.card ι) ^ ENNReal.toReal (1 / p)) * ‖b‖", "tactic": "simp" } ]

[ 927, 77 ]

[ 924, 1 ]

[ { "state_after": "R : Type u_2\nM : Type u_1\nr : R\nx y : M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np' : Submodule R M\n⊢ Nonempty (Unique (M ⧸ ⊤))", "state_before": "R : Type u_2\nM : Type u_1\nr : R\nx y : M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' : Submodule R M\n⊢ p = ⊤ → Nonempty (Unique (M ⧸ p))", "tactic": "rintro rfl" }, { "state_after": "no goals", "state_before": "R : Type u_2\nM : Type u_1\nr : R\nx y : M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np' : Submodule R M\n⊢ Nonempty (Unique (M ⧸ ⊤))", "tactic": "exact ⟨QuotientTop.unique⟩" } ]

[ 294, 32 ]

[ 291, 1 ]

[ { "state_after": "no goals", "state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.1513482\ninst✝¹ : Semiring k\ninst✝ : Group G\nf g : MonoidAlgebra k G\nx : G\n⊢ ↑(f * g) x = sum f fun a b => ↑(single a b * g) x", "tactic": "rw [← Finsupp.sum_apply, ← Finsupp.sum_mul g f, f.sum_single]" }, { "state_after": "no goals", "state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.1513482\ninst✝¹ : Semiring k\ninst✝ : Group G\nf g : MonoidAlgebra k G\nx : G\n⊢ (sum f fun a b => ↑(single a b * g) x) = sum f fun a b => b * ↑g (a⁻¹ * x)", "tactic": "simp only [single_mul_apply, Finsupp.sum]" } ]

[ 1066, 58 ]

[ 1061, 1 ]

[]

[ 1858, 41 ]

[ 1857, 1 ]

[ { "state_after": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ (if 0 = 0 then 1 else 0) = 1", "state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ coeff 1 0 = 1", "tactic": "rw [← monomial_zero_one, coeff_monomial]" }, { "state_after": "no goals", "state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ (if 0 = 0 then 1 else 0) = 1", "tactic": "simp" } ]

[ 686, 7 ]

[ 684, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.5460\nf✝ fa : α → α\nfb : β → β\nx✝ y : α\nm n✝ : ℕ\nf : α → α\nn : ℕ\nhn : 0 < n\nx : α\nhx : x ∈ ptsOfPeriod f n\n⊢ f ((f^[Nat.pred n]) x) = x", "tactic": "rw [← comp_apply (f := f), comp_iterate_pred_of_pos f hn, hx.eq]" } ]

[ 214, 73 ]

[ 209, 1 ]

[ { "state_after": "case refine'_1\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_freq : ∃ᶠ (n : ι) in l, p n\n⊢ ∃ x, Tendsto x atTop l ∧ ∃ᶠ (n : ℕ) in atTop, p (x n)\n\ncase refine'_2\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_exists_freq : ∃ x, Tendsto x atTop l ∧ ∃ᶠ (n : ℕ) in atTop, p (x n)\n⊢ ∃ᶠ (n : ι) in l, p n", "state_before": "ι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\n⊢ (∃ᶠ (n : ι) in l, p n) ↔ ∃ x, Tendsto x atTop l ∧ ∃ᶠ (n : ℕ) in atTop, p (x n)", "tactic": "refine' ⟨fun h_freq => _, fun h_exists_freq => _⟩" }, { "state_after": "case refine'_1\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_freq : ∃ᶠ (n : ι) in l, p n\nthis : NeBot (l ⊓ 𝓟 {x | p x})\n⊢ ∃ x, Tendsto x atTop l ∧ ∃ᶠ (n : ℕ) in atTop, p (x n)", "state_before": "case refine'_1\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_freq : ∃ᶠ (n : ι) in l, p n\n⊢ ∃ x, Tendsto x atTop l ∧ ∃ᶠ (n : ℕ) in atTop, p (x n)", "tactic": "have : NeBot (l ⊓ 𝓟 { x : ι | p x }) := by simpa [neBot_iff, inf_principal_eq_bot]" }, { "state_after": "case refine'_1.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_freq : ∃ᶠ (n : ι) in l, p n\nthis : NeBot (l ⊓ 𝓟 {x | p x})\nx : ℕ → ι\nhx : Tendsto x atTop (l ⊓ 𝓟 {x | p x})\n⊢ ∃ x, Tendsto x atTop l ∧ ∃ᶠ (n : ℕ) in atTop, p (x n)", "state_before": "case refine'_1\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_freq : ∃ᶠ (n : ι) in l, p n\nthis : NeBot (l ⊓ 𝓟 {x | p x})\n⊢ ∃ x, Tendsto x atTop l ∧ ∃ᶠ (n : ℕ) in atTop, p (x n)", "tactic": "obtain ⟨x, hx⟩ := exists_seq_tendsto (l ⊓ 𝓟 { x : ι | p x })" }, { "state_after": "case refine'_1.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_freq : ∃ᶠ (n : ι) in l, p n\nthis : NeBot (l ⊓ 𝓟 {x | p x})\nx : ℕ → ι\nhx : Tendsto x atTop l ∧ Tendsto x atTop (𝓟 {x | p x})\n⊢ ∃ x, Tendsto x atTop l ∧ ∃ᶠ (n : ℕ) in atTop, p (x n)", "state_before": "case refine'_1.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_freq : ∃ᶠ (n : ι) in l, p n\nthis : NeBot (l ⊓ 𝓟 {x | p x})\nx : ℕ → ι\nhx : Tendsto x atTop (l ⊓ 𝓟 {x | p x})\n⊢ ∃ x, Tendsto x atTop l ∧ ∃ᶠ (n : ℕ) in atTop, p (x n)", "tactic": "rw [tendsto_inf] at hx" }, { "state_after": "case refine'_1.intro.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_freq : ∃ᶠ (n : ι) in l, p n\nthis : NeBot (l ⊓ 𝓟 {x | p x})\nx : ℕ → ι\nhx_l : Tendsto x atTop l\nhx_p : Tendsto x atTop (𝓟 {x | p x})\n⊢ ∃ x, Tendsto x atTop l ∧ ∃ᶠ (n : ℕ) in atTop, p (x n)", "state_before": "case refine'_1.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_freq : ∃ᶠ (n : ι) in l, p n\nthis : NeBot (l ⊓ 𝓟 {x | p x})\nx : ℕ → ι\nhx : Tendsto x atTop l ∧ Tendsto x atTop (𝓟 {x | p x})\n⊢ ∃ x, Tendsto x atTop l ∧ ∃ᶠ (n : ℕ) in atTop, p (x n)", "tactic": "cases' hx with hx_l hx_p" }, { "state_after": "case refine'_1.intro.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_freq : ∃ᶠ (n : ι) in l, p n\nthis : NeBot (l ⊓ 𝓟 {x | p x})\nx : ℕ → ι\nhx_l : Tendsto x atTop l\nhx_p : Tendsto x atTop (𝓟 {x | p x})\n⊢ ∃ᶠ (n : ℕ) in atTop, p (x n)", "state_before": "case refine'_1.intro.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_freq : ∃ᶠ (n : ι) in l, p n\nthis : NeBot (l ⊓ 𝓟 {x | p x})\nx : ℕ → ι\nhx_l : Tendsto x atTop l\nhx_p : Tendsto x atTop (𝓟 {x | p x})\n⊢ ∃ x, Tendsto x atTop l ∧ ∃ᶠ (n : ℕ) in atTop, p (x n)", "tactic": "refine' ⟨x, hx_l, _⟩" }, { "state_after": "case refine'_1.intro.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_freq : ∃ᶠ (n : ι) in l, p n\nthis : NeBot (l ⊓ 𝓟 {x | p x})\nx : ℕ → ι\nhx_l : Tendsto x atTop l\nhx_p : ∀ᶠ (a : ℕ) in atTop, x a ∈ {x | p x}\n⊢ ∃ᶠ (n : ℕ) in atTop, p (x n)", "state_before": "case refine'_1.intro.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_freq : ∃ᶠ (n : ι) in l, p n\nthis : NeBot (l ⊓ 𝓟 {x | p x})\nx : ℕ → ι\nhx_l : Tendsto x atTop l\nhx_p : Tendsto x atTop (𝓟 {x | p x})\n⊢ ∃ᶠ (n : ℕ) in atTop, p (x n)", "tactic": "rw [tendsto_principal] at hx_p" }, { "state_after": "no goals", "state_before": "case refine'_1.intro.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_freq : ∃ᶠ (n : ι) in l, p n\nthis : NeBot (l ⊓ 𝓟 {x | p x})\nx : ℕ → ι\nhx_l : Tendsto x atTop l\nhx_p : ∀ᶠ (a : ℕ) in atTop, x a ∈ {x | p x}\n⊢ ∃ᶠ (n : ℕ) in atTop, p (x n)", "tactic": "exact hx_p.frequently" }, { "state_after": "no goals", "state_before": "ι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_freq : ∃ᶠ (n : ι) in l, p n\n⊢ NeBot (l ⊓ 𝓟 {x | p x})", "tactic": "simpa [neBot_iff, inf_principal_eq_bot]" }, { "state_after": "case refine'_2.intro.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nx : ℕ → ι\nhx_tendsto : Tendsto x atTop l\nhx_freq : ∃ᶠ (n : ℕ) in atTop, p (x n)\n⊢ ∃ᶠ (n : ι) in l, p n", "state_before": "case refine'_2\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nh_exists_freq : ∃ x, Tendsto x atTop l ∧ ∃ᶠ (n : ℕ) in atTop, p (x n)\n⊢ ∃ᶠ (n : ι) in l, p n", "tactic": "obtain ⟨x, hx_tendsto, hx_freq⟩ := h_exists_freq" }, { "state_after": "case refine'_2.intro.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nx : ℕ → ι\nhx_tendsto : Tendsto x atTop l\nhx_freq : ¬{x_1 | ¬p (x x_1)} ∈ atTop\n⊢ ¬{x | ¬p x} ∈ l", "state_before": "case refine'_2.intro.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nx : ℕ → ι\nhx_tendsto : Tendsto x atTop l\nhx_freq : ∃ᶠ (n : ℕ) in atTop, p (x n)\n⊢ ∃ᶠ (n : ι) in l, p n", "tactic": "simp_rw [Filter.Frequently, Filter.Eventually] at hx_freq⊢" }, { "state_after": "case refine'_2.intro.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nx : ℕ → ι\nhx_tendsto : Tendsto x atTop l\nhx_freq : ¬{x_1 | ¬p (x x_1)} ∈ atTop\nthis : {n | ¬p (x n)} = {n | x n ∈ {y | ¬p y}}\n⊢ ¬{x | ¬p x} ∈ l", "state_before": "case refine'_2.intro.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nx : ℕ → ι\nhx_tendsto : Tendsto x atTop l\nhx_freq : ¬{x_1 | ¬p (x x_1)} ∈ atTop\n⊢ ¬{x | ¬p x} ∈ l", "tactic": "have : { n : ℕ | ¬p (x n) } = { n | x n ∈ { y | ¬p y } } := rfl" }, { "state_after": "case refine'_2.intro.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nx : ℕ → ι\nhx_tendsto : Tendsto x atTop l\nhx_freq : ¬{y | ¬p y} ∈ map (fun n => x n) atTop\nthis : {n | ¬p (x n)} = {n | x n ∈ {y | ¬p y}}\n⊢ ¬{x | ¬p x} ∈ l", "state_before": "case refine'_2.intro.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nx : ℕ → ι\nhx_tendsto : Tendsto x atTop l\nhx_freq : ¬{x_1 | ¬p (x x_1)} ∈ atTop\nthis : {n | ¬p (x n)} = {n | x n ∈ {y | ¬p y}}\n⊢ ¬{x | ¬p x} ∈ l", "tactic": "rw [this, ← mem_map'] at hx_freq" }, { "state_after": "no goals", "state_before": "case refine'_2.intro.intro\nι✝ : Type ?u.368461\nι' : Type ?u.368464\nα : Type ?u.368467\nβ : Type ?u.368470\nγ : Type ?u.368473\nι : Type u_1\nl : Filter ι\np : ι → Prop\nhl : IsCountablyGenerated l\nx : ℕ → ι\nhx_tendsto : Tendsto x atTop l\nhx_freq : ¬{y | ¬p y} ∈ map (fun n => x n) atTop\nthis : {n | ¬p (x n)} = {n | x n ∈ {y | ¬p y}}\n⊢ ¬{x | ¬p x} ∈ l", "tactic": "exact mt (@hx_tendsto _) hx_freq" } ]

[ 1861, 37 ]

[ 1846, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.68398\nγ : Type ?u.68401\nι : Sort u_2\nι' : Sort ?u.68407\nι₂ : Sort ?u.68410\nκ : ι → Sort u_3\nκ₁ : ι → Sort ?u.68420\nκ₂ : ι → Sort ?u.68425\nκ' : ι' → Sort ?u.68430\ns : (i : ι) → κ i → Set α\n⊢ (⋂ (i : ι) (j : κ i), s i j)ᶜ = ⋃ (i : ι) (j : κ i), s i jᶜ", "tactic": "simp_rw [compl_iInter]" } ]

[ 508, 25 ]

[ 507, 1 ]

[ { "state_after": "case intro\nα : Type u_1\nβ : Type ?u.326633\nι : Type ?u.326636\nG : Type ?u.326639\nM : Type u_2\nN : Type ?u.326645\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\nt : Set α\nht : Set.Finite t\ns : Finset α\n⊢ ((∏ᶠ (i : α) (_ : i ∈ ↑s ∪ t), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑s ∩ t), f i) =\n (∏ᶠ (i : α) (_ : i ∈ ↑s), f i) * ∏ᶠ (i : α) (_ : i ∈ t), f i", "state_before": "α : Type u_1\nβ : Type ?u.326633\nι : Type ?u.326636\nG : Type ?u.326639\nM : Type u_2\nN : Type ?u.326645\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Set α\nhs : Set.Finite s\nht : Set.Finite t\n⊢ ((∏ᶠ (i : α) (_ : i ∈ s ∪ t), f i) * ∏ᶠ (i : α) (_ : i ∈ s ∩ t), f i) =\n (∏ᶠ (i : α) (_ : i ∈ s), f i) * ∏ᶠ (i : α) (_ : i ∈ t), f i", "tactic": "lift s to Finset α using hs" }, { "state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.326633\nι : Type ?u.326636\nG : Type ?u.326639\nM : Type u_2\nN : Type ?u.326645\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Finset α\n⊢ ((∏ᶠ (i : α) (_ : i ∈ ↑s ∪ ↑t), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑s ∩ ↑t), f i) =\n (∏ᶠ (i : α) (_ : i ∈ ↑s), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑t), f i", "state_before": "case intro\nα : Type u_1\nβ : Type ?u.326633\nι : Type ?u.326636\nG : Type ?u.326639\nM : Type u_2\nN : Type ?u.326645\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\nt : Set α\nht : Set.Finite t\ns : Finset α\n⊢ ((∏ᶠ (i : α) (_ : i ∈ ↑s ∪ t), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑s ∩ t), f i) =\n (∏ᶠ (i : α) (_ : i ∈ ↑s), f i) * ∏ᶠ (i : α) (_ : i ∈ t), f i", "tactic": "lift t to Finset α using ht" }, { "state_after": "no goals", "state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.326633\nι : Type ?u.326636\nG : Type ?u.326639\nM : Type u_2\nN : Type ?u.326645\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Finset α\n⊢ ((∏ᶠ (i : α) (_ : i ∈ ↑s ∪ ↑t), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑s ∩ ↑t), f i) =\n (∏ᶠ (i : α) (_ : i ∈ ↑s), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑t), f i", "tactic": "classical\n rw [← Finset.coe_union, ← Finset.coe_inter]\n simp only [finprod_mem_coe_finset, Finset.prod_union_inter]" }, { "state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.326633\nι : Type ?u.326636\nG : Type ?u.326639\nM : Type u_2\nN : Type ?u.326645\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Finset α\n⊢ ((∏ᶠ (i : α) (_ : i ∈ ↑(s ∪ t)), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑(s ∩ t)), f i) =\n (∏ᶠ (i : α) (_ : i ∈ ↑s), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑t), f i", "state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.326633\nι : Type ?u.326636\nG : Type ?u.326639\nM : Type u_2\nN : Type ?u.326645\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Finset α\n⊢ ((∏ᶠ (i : α) (_ : i ∈ ↑s ∪ ↑t), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑s ∩ ↑t), f i) =\n (∏ᶠ (i : α) (_ : i ∈ ↑s), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑t), f i", "tactic": "rw [← Finset.coe_union, ← Finset.coe_inter]" }, { "state_after": "no goals", "state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.326633\nι : Type ?u.326636\nG : Type ?u.326639\nM : Type u_2\nN : Type ?u.326645\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Finset α\n⊢ ((∏ᶠ (i : α) (_ : i ∈ ↑(s ∪ t)), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑(s ∩ t)), f i) =\n (∏ᶠ (i : α) (_ : i ∈ ↑s), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑t), f i", "tactic": "simp only [finprod_mem_coe_finset, Finset.prod_union_inter]" } ]

[ 772, 64 ]

[ 767, 1 ]

[ { "state_after": "case inl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : s ×ˢ t = ∅\n⊢ s ×ˢ t ⊆ s₁ ×ˢ t₁ ↔ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅\n\ncase inr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\n⊢ s ×ˢ t ⊆ s₁ ×ˢ t₁ ↔ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\n⊢ s ×ˢ t ⊆ s₁ ×ˢ t₁ ↔ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅", "tactic": "cases' (s ×ˢ t).eq_empty_or_nonempty with h h" }, { "state_after": "case inr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\n⊢ s ×ˢ t ⊆ s₁ ×ˢ t₁ ↔ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅", "state_before": "case inr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\n⊢ s ×ˢ t ⊆ s₁ ×ˢ t₁ ↔ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅", "tactic": "have st : s.Nonempty ∧ t.Nonempty := by rwa [prod_nonempty_iff] at h" }, { "state_after": "case inr.refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\nH : s ×ˢ t ⊆ s₁ ×ˢ t₁\n⊢ s ⊆ s₁\n\ncase inr.refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\nH : s ×ˢ t ⊆ s₁ ×ˢ t₁\n⊢ t ⊆ t₁\n\ncase inr.refine'_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\n⊢ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅ → s ×ˢ t ⊆ s₁ ×ˢ t₁", "state_before": "case inr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\n⊢ s ×ˢ t ⊆ s₁ ×ˢ t₁ ↔ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅", "tactic": "refine' ⟨fun H => Or.inl ⟨_, _⟩, _⟩" }, { "state_after": "no goals", "state_before": "case inl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : s ×ˢ t = ∅\n⊢ s ×ˢ t ⊆ s₁ ×ˢ t₁ ↔ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅", "tactic": "simp [h, prod_eq_empty_iff.1 h]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\n⊢ Set.Nonempty s ∧ Set.Nonempty t", "tactic": "rwa [prod_nonempty_iff] at h" }, { "state_after": "case inr.refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\nH : s ×ˢ t ⊆ s₁ ×ˢ t₁\nthis : Prod.fst '' s ×ˢ t ⊆ Prod.fst '' s₁ ×ˢ t₁\n⊢ s ⊆ s₁", "state_before": "case inr.refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\nH : s ×ˢ t ⊆ s₁ ×ˢ t₁\n⊢ s ⊆ s₁", "tactic": "have := image_subset (Prod.fst : α × β → α) H" }, { "state_after": "no goals", "state_before": "case inr.refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\nH : s ×ˢ t ⊆ s₁ ×ˢ t₁\nthis : Prod.fst '' s ×ˢ t ⊆ Prod.fst '' s₁ ×ˢ t₁\n⊢ s ⊆ s₁", "tactic": "rwa [fst_image_prod _ st.2, fst_image_prod _ (h.mono H).snd] at this" }, { "state_after": "case inr.refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\nH : s ×ˢ t ⊆ s₁ ×ˢ t₁\nthis : Prod.snd '' s ×ˢ t ⊆ Prod.snd '' s₁ ×ˢ t₁\n⊢ t ⊆ t₁", "state_before": "case inr.refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\nH : s ×ˢ t ⊆ s₁ ×ˢ t₁\n⊢ t ⊆ t₁", "tactic": "have := image_subset (Prod.snd : α × β → β) H" }, { "state_after": "no goals", "state_before": "case inr.refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\nH : s ×ˢ t ⊆ s₁ ×ˢ t₁\nthis : Prod.snd '' s ×ˢ t ⊆ Prod.snd '' s₁ ×ˢ t₁\n⊢ t ⊆ t₁", "tactic": "rwa [snd_image_prod st.1, snd_image_prod (h.mono H).fst] at this" }, { "state_after": "case inr.refine'_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\nH : s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅\n⊢ s ×ˢ t ⊆ s₁ ×ˢ t₁", "state_before": "case inr.refine'_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\n⊢ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅ → s ×ˢ t ⊆ s₁ ×ˢ t₁", "tactic": "intro H" }, { "state_after": "case inr.refine'_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\nH : s ⊆ s₁ ∧ t ⊆ t₁\n⊢ s ×ˢ t ⊆ s₁ ×ˢ t₁", "state_before": "case inr.refine'_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\nH : s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅\n⊢ s ×ˢ t ⊆ s₁ ×ˢ t₁", "tactic": "simp only [st.1.ne_empty, st.2.ne_empty, or_false_iff] at H" }, { "state_after": "no goals", "state_before": "case inr.refine'_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.87049\nδ : Type ?u.87052\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nh : Set.Nonempty (s ×ˢ t)\nst : Set.Nonempty s ∧ Set.Nonempty t\nH : s ⊆ s₁ ∧ t ⊆ t₁\n⊢ s ×ˢ t ⊆ s₁ ×ˢ t₁", "tactic": "exact prod_mono H.1 H.2" } ]

[ 394, 28 ]

[ 383, 1 ]

[ { "state_after": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → G β\nh : β → γ\nx : Multiset α\n⊢ ∀ (a : List α),\n Functor.map h <$> traverse g (Quotient.mk (isSetoid α) a) =\n traverse (Functor.map h ∘ g) (Quotient.mk (isSetoid α) a)", "state_before": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → G β\nh : β → γ\nx : Multiset α\n⊢ Functor.map h <$> traverse g x = traverse (Functor.map h ∘ g) x", "tactic": "refine' Quotient.inductionOn x _" }, { "state_after": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → G β\nh : β → γ\nx : Multiset α\na✝ : List α\n⊢ Functor.map h <$> traverse g (Quotient.mk (isSetoid α) a✝) =\n traverse (Functor.map h ∘ g) (Quotient.mk (isSetoid α) a✝)", "state_before": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → G β\nh : β → γ\nx : Multiset α\n⊢ ∀ (a : List α),\n Functor.map h <$> traverse g (Quotient.mk (isSetoid α) a) =\n traverse (Functor.map h ∘ g) (Quotient.mk (isSetoid α) a)", "tactic": "intro" }, { "state_after": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → G β\nh : β → γ\nx : Multiset α\na✝ : List α\n⊢ (Coe.coe ∘ Functor.map h) <$> Traversable.traverse g a✝ = Coe.coe <$> Traversable.traverse (Functor.map h ∘ g) a✝", "state_before": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → G β\nh : β → γ\nx : Multiset α\na✝ : List α\n⊢ Functor.map h <$> traverse g (Quotient.mk (isSetoid α) a✝) =\n traverse (Functor.map h ∘ g) (Quotient.mk (isSetoid α) a✝)", "tactic": "simp only [traverse, quot_mk_to_coe, lift_coe, Function.comp_apply, Functor.map_map, map_comp_coe]" }, { "state_after": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → G β\nh : β → γ\nx : Multiset α\na✝ : List α\n⊢ Coe.coe <$> Functor.map h <$> Traversable.traverse g a✝ =\n Coe.coe <$> (Functor.map (Functor.map h) ∘ Traversable.traverse g) a✝", "state_before": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → G β\nh : β → γ\nx : Multiset α\na✝ : List α\n⊢ (Coe.coe ∘ Functor.map h) <$> Traversable.traverse g a✝ = Coe.coe <$> Traversable.traverse (Functor.map h ∘ g) a✝", "tactic": "rw [LawfulFunctor.comp_map, Traversable.map_traverse']" }, { "state_after": "no goals", "state_before": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → G β\nh : β → γ\nx : Multiset α\na✝ : List α\n⊢ Coe.coe <$> Functor.map h <$> Traversable.traverse g a✝ =\n Coe.coe <$> (Functor.map (Functor.map h) ∘ Traversable.traverse g) a✝", "tactic": "rfl" } ]

[ 127, 6 ]

[ 120, 1 ]

[ { "state_after": "no goals", "state_before": "p : ℝ≥0∞\n𝕜 : Type ?u.505702\n𝕜' : Type ?u.505705\nι : Type u_2\nα : ι → Type ?u.505713\nβ✝ : ι → Type ?u.505718\ninst✝⁹ : Fintype ι\ninst✝⁸ : Fact (1 ≤ p)\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedField 𝕜'\ninst✝⁵ : (i : ι) → SeminormedAddCommGroup (β✝ i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (β✝ i)\nc : 𝕜\nx y : PiLp p β✝\nx' y' : (i : ι) → β✝ i\ni : ι\nι' : Type ?u.505914\ninst✝³ : Fintype ι'\nE : Type ?u.505920\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nβ : Type u_1\ninst✝ : SeminormedAddCommGroup β\nhp : p ≠ ⊤\nb : β\n⊢ ↑(↑(Fintype.card ι) ^ ENNReal.toReal (1 / p) * ‖b‖₊) = ↑(↑(Fintype.card ι) ^ ENNReal.toReal (1 / p)) * ‖b‖", "tactic": "simp" } ]

[ 917, 79 ]

[ 914, 1 ]

[ { "state_after": "no goals", "state_before": "⊢ ¬(Set.Nonempty ∅ ∧ BddAbove ∅)", "tactic": "simp" } ]

[ 75, 20 ]

[ 74, 1 ]

[ { "state_after": "case zero\nx : ZMod Nat.zero\n⊢ valMinAbs x = 0 ↔ x = 0\n\ncase succ\nn : ℕ\nx : ZMod (Nat.succ n)\n⊢ valMinAbs x = 0 ↔ x = 0", "state_before": "n : ℕ\nx : ZMod n\n⊢ valMinAbs x = 0 ↔ x = 0", "tactic": "cases' n with n" }, { "state_after": "case succ\nn : ℕ\nx : ZMod (Nat.succ n)\n⊢ valMinAbs x = valMinAbs 0 ↔ x = 0", "state_before": "case succ\nn : ℕ\nx : ZMod (Nat.succ n)\n⊢ valMinAbs x = 0 ↔ x = 0", "tactic": "rw [← valMinAbs_zero n.succ]" }, { "state_after": "no goals", "state_before": "case succ\nn : ℕ\nx : ZMod (Nat.succ n)\n⊢ valMinAbs x = valMinAbs 0 ↔ x = 0", "tactic": "apply injective_valMinAbs.eq_iff" }, { "state_after": "no goals", "state_before": "case zero\nx : ZMod Nat.zero\n⊢ valMinAbs x = 0 ↔ x = 0", "tactic": "simp" } ]

[ 1001, 35 ]

[ 997, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type u_1\nβ₂ : Type ?u.114350\nγ : Type ?u.114353\nι : Sort ?u.114356\nι' : Sort ?u.114359\nκ : ι → Sort ?u.114364\nκ' : ι' → Sort ?u.114369\ninst✝ : CompleteLattice α\nf✝ g s✝ t✝ : ι → α\na b : α\nf : β → α\ns t : Set β\n⊢ (⨆ (x : β) (_ : x ∈ s ∪ t), f x) = (⨆ (x : β) (_ : x ∈ s), f x) ⊔ ⨆ (x : β) (_ : x ∈ t), f x", "tactic": "simp_rw [mem_union, iSup_or, iSup_sup_eq]" } ]

[ 1415, 47 ]

[ 1414, 1 ]

[]

[ 120, 65 ]

[ 118, 1 ]

[]

[ 46, 25 ]

[ 45, 1 ]

[]

[ 213, 51 ]

[ 212, 1 ]

[]

[ 3979, 24 ]

[ 3978, 1 ]

[ { "state_after": "A : Type ?u.179006\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ∑ k in range (succ n),\n ↑(Nat.choose ((k, n - k).fst + (k, n - k).snd) (k, n - k).snd) / (↑(k, n - k).snd + 1) *\n bernoulli' (k, n - k).fst =\n ∑ k in range (succ n), ↑(Nat.choose n (n - k)) / (↑n - ↑k + 1) * bernoulli' k", "state_before": "A : Type ?u.179006\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ∑ k in antidiagonal n, ↑(Nat.choose (k.fst + k.snd) k.snd) / (↑k.snd + 1) * bernoulli' k.fst = 1", "tactic": "refine' ((sum_antidiagonal_eq_sum_range_succ_mk _ n).trans _).trans (bernoulli'_spec n)" }, { "state_after": "A : Type ?u.179006\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn x : ℕ\nhx : x ∈ range (succ n)\n⊢ ↑(Nat.choose ((x, n - x).fst + (x, n - x).snd) (x, n - x).snd) / (↑(x, n - x).snd + 1) * bernoulli' (x, n - x).fst =\n ↑(Nat.choose n (n - x)) / (↑n - ↑x + 1) * bernoulli' x", "state_before": "A : Type ?u.179006\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ∑ k in range (succ n),\n ↑(Nat.choose ((k, n - k).fst + (k, n - k).snd) (k, n - k).snd) / (↑(k, n - k).snd + 1) *\n bernoulli' (k, n - k).fst =\n ∑ k in range (succ n), ↑(Nat.choose n (n - k)) / (↑n - ↑k + 1) * bernoulli' k", "tactic": "refine' sum_congr rfl fun x hx => _" }, { "state_after": "no goals", "state_before": "A : Type ?u.179006\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn x : ℕ\nhx : x ∈ range (succ n)\n⊢ ↑(Nat.choose ((x, n - x).fst + (x, n - x).snd) (x, n - x).snd) / (↑(x, n - x).snd + 1) * bernoulli' (x, n - x).fst =\n ↑(Nat.choose n (n - x)) / (↑n - ↑x + 1) * bernoulli' x", "tactic": "simp only [add_tsub_cancel_of_le, mem_range_succ_iff.mp hx, cast_sub]" } ]

[ 97, 72 ]

[ 93, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Type ?u.81246\nα : Type u_1\nβ : Type ?u.81252\nr : α → α → Prop\nr' : β → β → Prop\nf : α → β\ns t : Set α\na : α\ninst✝ : IsRefl α r\n⊢ PartiallyWellOrderedOn (insert a s) r ↔ PartiallyWellOrderedOn s r", "tactic": "simp only [← singleton_union, partiallyWellOrderedOn_union,\n partiallyWellOrderedOn_singleton, true_and_iff]" } ]

[ 318, 52 ]

[ 315, 1 ]

[ { "state_after": "no goals", "state_before": "z : ℂ\n⊢ (z * I).im = z.re", "tactic": "simp" } ]

[ 342, 56 ]

[ 342, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nβ₂ : Type ?u.211513\nγ : Type ?u.211516\nι : Sort u_3\nι' : Sort ?u.211522\nκ : ι → Sort ?u.211527\nκ' : ι' → Sort ?u.211532\ninst✝¹ : InfSet α\ninst✝ : InfSet β\nf : ι → α × β\n⊢ swap (iInf f) = ⨅ (i : ι), swap (f i)", "tactic": "simp_rw [iInf, swap_sInf, ←range_comp, Function.comp]" } ]

[ 1866, 56 ]

[ 1865, 1 ]

[ { "state_after": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvs : Set.InjOn v ↑s\nhi : i ∈ s\nhj : j ∈ s\nhij : i ≠ j\n⊢ {i, j} ⊆ s", "state_before": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvs : Set.InjOn v ↑s\nhi : i ∈ s\nhj : j ∈ s\nhij : i ≠ j\n⊢ ↑(interpolate s v) r =\n ↑(interpolate (Finset.erase s j) v) r * basisDivisor (v i) (v j) +\n ↑(interpolate (Finset.erase s i) v) r * basisDivisor (v j) (v i)", "tactic": "rw [interpolate_eq_sum_interpolate_insert_sdiff _ hvs ⟨i, mem_insert_self i {j}⟩ _,\n sum_insert (not_mem_singleton.mpr hij), sum_singleton, basis_pair_left hij,\n basis_pair_right hij, sdiff_insert_insert_of_mem_of_not_mem hi (not_mem_singleton.mpr hij),\n sdiff_singleton_eq_erase, pair_comm,\n sdiff_insert_insert_of_mem_of_not_mem hj (not_mem_singleton.mpr hij.symm),\n sdiff_singleton_eq_erase]" }, { "state_after": "no goals", "state_before": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvs : Set.InjOn v ↑s\nhi : i ∈ s\nhj : j ∈ s\nhij : i ≠ j\n⊢ {i, j} ⊆ s", "tactic": "exact insert_subset.mpr ⟨hi, singleton_subset_iff.mpr hj⟩" } ]

[ 468, 62 ]

[ 457, 1 ]

[]

[ 142, 61 ]

[ 141, 1 ]

[ { "state_after": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nha : a ≠ 0\n⊢ Finset.max {n} = ↑n", "state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nha : a ≠ 0\n⊢ degree (↑(monomial n) a) = ↑n", "tactic": "rw [degree, support_monomial n ha]" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nha : a ≠ 0\n⊢ Finset.max {n} = ↑n", "tactic": "rfl" } ]

[ 290, 42 ]

[ 289, 1 ]

[ { "state_after": "case h.h\nα : Type u_4\nM : Type u_1\nN : Type ?u.445605\nP : Type ?u.445608\nR : Type u_2\nS : Type ?u.445614\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : AddCommMonoid N\ninst✝⁷ : Module R N\ninst✝⁶ : AddCommMonoid P\ninst✝⁵ : Module R P\nα' : Type u_3\nM' : Type ?u.445709\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M'\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M'\ninst✝ : Module R M\nv : α → M\nv' : α' → M'\nf : α' → α\na✝ : α'\n⊢ ↑(comp (Finsupp.total α' M R (v ∘ f)) (lsingle a✝)) 1 =\n ↑(comp (comp (Finsupp.total α M R v) (lmapDomain R R f)) (lsingle a✝)) 1", "state_before": "α : Type u_4\nM : Type u_1\nN : Type ?u.445605\nP : Type ?u.445608\nR : Type u_2\nS : Type ?u.445614\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : AddCommMonoid N\ninst✝⁷ : Module R N\ninst✝⁶ : AddCommMonoid P\ninst✝⁵ : Module R P\nα' : Type u_3\nM' : Type ?u.445709\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M'\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M'\ninst✝ : Module R M\nv : α → M\nv' : α' → M'\nf : α' → α\n⊢ Finsupp.total α' M R (v ∘ f) = comp (Finsupp.total α M R v) (lmapDomain R R f)", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case h.h\nα : Type u_4\nM : Type u_1\nN : Type ?u.445605\nP : Type ?u.445608\nR : Type u_2\nS : Type ?u.445614\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : AddCommMonoid N\ninst✝⁷ : Module R N\ninst✝⁶ : AddCommMonoid P\ninst✝⁵ : Module R P\nα' : Type u_3\nM' : Type ?u.445709\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M'\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M'\ninst✝ : Module R M\nv : α → M\nv' : α' → M'\nf : α' → α\na✝ : α'\n⊢ ↑(comp (Finsupp.total α' M R (v ∘ f)) (lsingle a✝)) 1 =\n ↑(comp (comp (Finsupp.total α M R v) (lmapDomain R R f)) (lsingle a✝)) 1", "tactic": "simp [total_apply]" } ]

[ 746, 21 ]

[ 743, 1 ]

[ { "state_after": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX : limit (F ⋙ objects)\nj : J\n⊢ limit.π (homDiagram X X) j (eqToHom (_ : X = X)) = eqToHom (_ : limit.π (F ⋙ objects) j X = limit.π (F ⋙ objects) j X)", "state_before": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX Y : limit (F ⋙ objects)\nj : J\nh : X = Y\n⊢ limit.π (homDiagram X Y) j (eqToHom h) = eqToHom (_ : limit.π (F ⋙ objects) j X = limit.π (F ⋙ objects) j Y)", "tactic": "subst h" }, { "state_after": "no goals", "state_before": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX : limit (F ⋙ objects)\nj : J\n⊢ limit.π (homDiagram X X) j (eqToHom (_ : X = X)) = eqToHom (_ : limit.π (F ⋙ objects) j X = limit.π (F ⋙ objects) j X)", "tactic": "simp" } ]

[ 135, 7 ]

[ 130, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nH : X = 0\n⊢ False", "tactic": "simpa only [coeff_one_X, one_ne_zero, map_zero] using congr_arg (coeff R 1) H" } ]

[ 1447, 80 ]

[ 1446, 1 ]

[]

[ 410, 58 ]

[ 408, 1 ]

[ { "state_after": "case a.mk\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.70206\ns s' : Finset α\nt t' : Finset β\na : α\nb✝ b : β\nx : α\ny : β\n⊢ (x, y) ∈ s ×ˢ {b} ↔ (x, y) ∈ map { toFun := fun i => (i, b), inj' := (_ : Function.Injective fun a => (a, b)) } s", "state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.70206\ns s' : Finset α\nt t' : Finset β\na : α\nb✝ b : β\n⊢ s ×ˢ {b} = map { toFun := fun i => (i, b), inj' := (_ : Function.Injective fun a => (a, b)) } s", "tactic": "ext ⟨x, y⟩" }, { "state_after": "no goals", "state_before": "case a.mk\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.70206\ns s' : Finset α\nt t' : Finset β\na : α\nb✝ b : β\nx : α\ny : β\n⊢ (x, y) ∈ s ×ˢ {b} ↔ (x, y) ∈ map { toFun := fun i => (i, b), inj' := (_ : Function.Injective fun a => (a, b)) } s", "tactic": "simp [and_left_comm, eq_comm]" } ]

[ 226, 32 ]

[ 224, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nf : α → β → γ\na : Option α\nb : Option β\nc : Option γ\ng : β → α → γ\nh_comm : ∀ (a : α) (b : β), f a b = g b a\n⊢ map₂ f a b = map₂ g b a", "tactic": "cases a <;> cases b <;> simp [h_comm]" } ]

[ 131, 43 ]

[ 130, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\na✝ b c d : R\nn✝ m : ℕ\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\nn : ℕ\nhn : 0 < n\na : R\nh : X ^ n + ↑C a = 1\n⊢ n = 0", "tactic": "simpa only [natDegree_X_pow_add_C, natDegree_one] using congr_arg natDegree h" } ]

[ 1408, 93 ]

[ 1407, 1 ]

[ { "state_after": "case refine_1\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\n⊢ invFun R A (↑(toFunAlgHom R A) 0) = 0\n\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\n⊢ ∀ (x : A) (y : R[X]), invFun R A (↑(toFunAlgHom R A) (x ⊗ₜ[R] y)) = x ⊗ₜ[R] y\n\ncase refine_3\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\n⊢ ∀ (x y : A ⊗[R] R[X]),\n invFun R A (↑(toFunAlgHom R A) x) = x →\n invFun R A (↑(toFunAlgHom R A) y) = y → invFun R A (↑(toFunAlgHom R A) (x + y)) = x + y", "state_before": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\n⊢ invFun R A (↑(toFunAlgHom R A) x) = x", "tactic": "refine TensorProduct.induction_on x ?_ ?_ ?_" }, { "state_after": "no goals", "state_before": "case refine_1\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\n⊢ invFun R A (↑(toFunAlgHom R A) 0) = 0", "tactic": "simp [invFun]" }, { "state_after": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\na : A\np : R[X]\n⊢ invFun R A (↑(toFunAlgHom R A) (a ⊗ₜ[R] p)) = a ⊗ₜ[R] p", "state_before": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\n⊢ ∀ (x : A) (y : R[X]), invFun R A (↑(toFunAlgHom R A) (x ⊗ₜ[R] y)) = x ⊗ₜ[R] y", "tactic": "intro a p" }, { "state_after": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\na : A\np : R[X]\n⊢ eval₂ (↑includeLeft) (1 ⊗ₜ[R] X) (↑(toFunAlgHom R A) (a ⊗ₜ[R] p)) = a ⊗ₜ[R] p", "state_before": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\na : A\np : R[X]\n⊢ invFun R A (↑(toFunAlgHom R A) (a ⊗ₜ[R] p)) = a ⊗ₜ[R] p", "tactic": "dsimp only [invFun]" }, { "state_after": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\na : A\np : R[X]\n⊢ (sum p fun n a_1 => eval₂ (↑includeLeft) (1 ⊗ₜ[R] X) (↑(monomial n) (a * ↑(algebraMap R A) a_1))) = a ⊗ₜ[R] p", "state_before": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\na : A\np : R[X]\n⊢ eval₂ (↑includeLeft) (1 ⊗ₜ[R] X) (↑(toFunAlgHom R A) (a ⊗ₜ[R] p)) = a ⊗ₜ[R] p", "tactic": "rw [toFunAlgHom_apply_tmul, eval₂_sum]" }, { "state_after": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\na : A\np : R[X]\n⊢ (a ⊗ₜ[R] Finset.sum (support p) fun a => coeff p a • X ^ a) = a ⊗ₜ[R] p", "state_before": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\na : A\np : R[X]\n⊢ (sum p fun n a_1 => eval₂ (↑includeLeft) (1 ⊗ₜ[R] X) (↑(monomial n) (a * ↑(algebraMap R A) a_1))) = a ⊗ₜ[R] p", "tactic": "simp_rw [eval₂_monomial, AlgHom.coe_toRingHom, Algebra.TensorProduct.tmul_pow, one_pow,\n Algebra.TensorProduct.includeLeft_apply, Algebra.TensorProduct.tmul_mul_tmul, mul_one,\n one_mul, ← Algebra.commutes, ← Algebra.smul_def, smul_tmul, sum_def, ← tmul_sum]" }, { "state_after": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\na : A\np : R[X]\n⊢ (a ⊗ₜ[R] Finset.sum (support p) fun a => coeff p a • X ^ a) = a ⊗ₜ[R] sum p fun n a => ↑C a * X ^ n", "state_before": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\na : A\np : R[X]\n⊢ (a ⊗ₜ[R] Finset.sum (support p) fun a => coeff p a • X ^ a) = a ⊗ₜ[R] p", "tactic": "conv_rhs => rw [← sum_C_mul_X_pow_eq p]" }, { "state_after": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\na : A\np : R[X]\n⊢ (a ⊗ₜ[R] Finset.sum (support p) fun x => ↑(algebraMap R R[X]) (coeff p x) * X ^ x) =\n a ⊗ₜ[R] sum p fun n a => ↑C a * X ^ n", "state_before": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\na : A\np : R[X]\n⊢ (a ⊗ₜ[R] Finset.sum (support p) fun a => coeff p a • X ^ a) = a ⊗ₜ[R] sum p fun n a => ↑C a * X ^ n", "tactic": "simp only [Algebra.smul_def]" }, { "state_after": "no goals", "state_before": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\na : A\np : R[X]\n⊢ (a ⊗ₜ[R] Finset.sum (support p) fun x => ↑(algebraMap R R[X]) (coeff p x) * X ^ x) =\n a ⊗ₜ[R] sum p fun n a => ↑C a * X ^ n", "tactic": "rfl" }, { "state_after": "case refine_3\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx p q : A ⊗[R] R[X]\nhp : invFun R A (↑(toFunAlgHom R A) p) = p\nhq : invFun R A (↑(toFunAlgHom R A) q) = q\n⊢ invFun R A (↑(toFunAlgHom R A) (p + q)) = p + q", "state_before": "case refine_3\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\n⊢ ∀ (x y : A ⊗[R] R[X]),\n invFun R A (↑(toFunAlgHom R A) x) = x →\n invFun R A (↑(toFunAlgHom R A) y) = y → invFun R A (↑(toFunAlgHom R A) (x + y)) = x + y", "tactic": "intro p q hp hq" }, { "state_after": "no goals", "state_before": "case refine_3\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx p q : A ⊗[R] R[X]\nhp : invFun R A (↑(toFunAlgHom R A) p) = p\nhq : invFun R A (↑(toFunAlgHom R A) q) = q\n⊢ invFun R A (↑(toFunAlgHom R A) (p + q)) = p + q", "tactic": "simp only [AlgHom.map_add, invFun_add, hp, hq]" } ]

[ 167, 51 ]

[ 154, 1 ]

[]

[ 4150, 34 ]

[ 4148, 1 ]

[ { "state_after": "no goals", "state_before": "F : Type ?u.242140\nα : Type u_1\nβ : Type ?u.242146\nγ : Type ?u.242149\nδ : Type ?u.242152\nε : Type ?u.242155\ninst✝ : Monoid α\nf g : Filter α\ns : Set α\na : α\nm n✝ n : ℕ\nhn : n ≠ 0\n⊢ ⊥ ^ n = ⊥", "tactic": "rw [← tsub_add_cancel_of_le (Nat.succ_le_of_lt <| Nat.pos_of_ne_zero hn), pow_succ, bot_mul]" } ]

[ 686, 95 ]

[ 685, 1 ]

[]

[ 116, 6 ]

[ 115, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ eval₂ f g (p + q) = eval₂ f g p + eval₂ f g q", "tactic": "classical exact Finsupp.sum_add_index (by simp [f.map_zero]) (by simp [add_mul, f.map_add])" }, { "state_after": "no goals", "state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ eval₂ f g (p + q) = eval₂ f g p + eval₂ f g q", "tactic": "exact Finsupp.sum_add_index (by simp [f.map_zero]) (by simp [add_mul, f.map_add])" }, { "state_after": "no goals", "state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ (a : σ →₀ ℕ), a ∈ p.support ∪ q.support → (↑f 0 * Finsupp.prod a fun n e => g n ^ e) = 0", "tactic": "simp [f.map_zero]" }, { "state_after": "no goals", "state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ (a : σ →₀ ℕ),\n a ∈ p.support ∪ q.support →\n ∀ (b₁ b₂ : R),\n (↑f (b₁ + b₂) * Finsupp.prod a fun n e => g n ^ e) =\n (↑f b₁ * Finsupp.prod a fun n e => g n ^ e) + ↑f b₂ * Finsupp.prod a fun n e => g n ^ e", "tactic": "simp [add_mul, f.map_add]" } ]

[ 957, 94 ]

[ 956, 1 ]

[]

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[]

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[]

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[ 1641, 1 ]

[]

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[]

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[]

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[]

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[]

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