examples/dev_sample.jsonl

{"dev_row": 0, "state": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type ?u\ninst : Structure L M\ninst : Structure L N\ninst : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nh : M ≅[L] N\n⊢ M ⊨ T ↔ N ⊨ T", "true_first_token": "rw", "true_tactic": "rw [Theory.model_iff_subset_completeTheory, Theory.model_iff_subset_completeTheory,\n  h.completeTheory_eq]"}
{"dev_row": 68, "state": "case codisjoint\nR : Type u_1\ninst : Ring R\nE : Type u_2\ninst : AddCommGroup E\ninst : Module R E\nF : Type ?u\ninst : AddCommGroup F\ninst : Module R F\nG : Type ?u\ninst : AddCommGroup G\ninst : Module R G\np q : Submodule R E\nS : Type ?u\ninst : Semiring S\nM : Type ?u\ninst : AddCommMonoid M\ninst : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\na : x ∈ ⊤\n⊢ ∃ y z, ↑y + ↑z = x", "true_first_token": "refine", "true_tactic": "refine' ⟨f x, ⟨x - f x, _⟩, add_sub_cancel'_right _ _⟩"}
{"dev_row": 136, "state": "case inr.mp\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\nhy : ∀ (i : ℕ), i < n → y i = x i\n⊢ n ≤ firstDiff y x", "true_first_token": "by", "true_tactic": "by_contra' H"}
{"dev_row": 214, "state": "E : Type u_1\nX : Type u_2\ninst : TopologicalSpace E\ninst : TopologicalSpace X\nf : E → X\ns : Set X\nhf : IsCoveringMapOn f s\n⊢ IsLocallyHomeomorphOn f (f ⁻¹' s)", "true_first_token": "refine", "true_tactic": "refine' IsLocallyHomeomorphOn.mk f (f ⁻¹' s) fun x hx => _"}
{"dev_row": 285, "state": "case neg.mp\nz : ℂ\nh₀ : ¬z = 0\nh : arg z = π\n⊢ (↑(↑abs z) * (cos ↑π + sin ↑π * I)).re < 0 ∧ (↑(↑abs z) * (cos ↑π + sin ↑π * I)).im = 0", "true_first_token": "simp", "true_tactic": "simp [h₀]"}
{"dev_row": 359, "state": "case this\nα : Type u\nβ : Type v\nX : Type ?u\ninst : PseudoEMetricSpace α\nx y z : α\nε ε₁ ε₂ : ℝ≥0∞\ns t : Set α\nhs : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ t, Set.Countable t ∧ (⋃ (x : α) (_ : x ∈ t), closedBall x ε) = univ\n⊢ ∀ (ε : ℝ≥0∞), ε > 0 → ∃ t, Set.Countable t ∧ univ ⊆ ⋃ (x : α) (_ : x ∈ t), closedBall x ε", "true_first_token": "simpa", "true_tactic": "simpa only [univ_subset_iff] using hs"}
{"dev_row": 432, "state": "V : Type u_1\nP : Type u_2\ninst : NormedAddCommGroup V\ninst : InnerProductSpace ℝ V\ninst : MetricSpace P\ninst : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ∠ p₁ p₂ p₃ = π / 2\nh0 : p₁ = p₂ ∨ p₃ ≠ p₂\n⊢ dist p₃ p₂ / Real.cos (∠ p₂ p₃ p₁) = dist p₁ p₃", "true_first_token": "rw", "true_tactic": "rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←\n  inner_neg_left, neg_vsub_eq_vsub_rev] at h"}
{"dev_row": 503, "state": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst : Field R\np q : R[X]\nhp0 : IsUnit (leadingCoeff p)\n⊢ Monic (↑normalize p)", "true_first_token": "rw", "true_tactic": "rw [Monic, leadingCoeff_normalize, normalize_eq_one]"}
{"dev_row": 578, "state": "V : Type u_1\nP : Type u_2\ninst : NormedAddCommGroup V\ninst : InnerProductSpace ℝ V\ninst : MetricSpace P\ninst : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst : FiniteDimensional ℝ { x // x ∈ direction s }\nhd : finrank ℝ { x // x ∈ direction s } = 2\nc₁ c₂ p₁ p₂ p : P\nhc₁s : c₁ ∈ s\nhc₂s : c₂ ∈ s\nhp₁s : p₁ ∈ s\nhp₂s : p₂ ∈ s\nhps : p ∈ s\nr₁ r₂ : ℝ\nhc : c₁ ≠ c₂\nhp : p₁ ≠ p₂\nhp₁c₁ : dist p₁ c₁ = r₁\nhp₂c₁ : dist p₂ c₁ = r₁\nhpc₁ : dist p c₁ = r₁\nhp₁c₂ : dist p₁ c₂ = r₂\nhp₂c₂ : dist p₂ c₂ = r₂\nhpc₂ : dist p c₂ = r₂\n⊢ p = p₁ ∨ p = p₂", "true_first_token": "have", "true_tactic": "have ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0 :=\n  inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hp₂c₁.symm) (hp₁c₂.trans hp₂c₂.symm)"}
{"dev_row": 648, "state": "ι : Type u_1\nα : Type u_2\nβ : Type ?u\nr : α → α → Prop\ns : Finset ι\nf : ι → Set α\n⊢ Set.PartiallyWellOrderedOn (sup ∅ f) r ↔ ∀ (i : ι), i ∈ ∅ → Set.PartiallyWellOrderedOn (f i) r", "true_first_token": "simp", "true_tactic": "simp"}
{"dev_row": 721, "state": "α : Type ?u\nβ : Type ?u\nγ : Type ?u\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nα : Type u_1\nr : α → α → Prop\nwo : IsWellOrder α r\nh : ∀ (a : Ordinal), a < type r → succ a < type r\nx : α\n⊢ ∃ y, r x y", "true_first_token": "use", "true_tactic": "use enum r (succ (typein r x)) (h _ (typein_lt_type r x))"}
{"dev_row": 793, "state": "R : Type u_1\ninst : CommRing R\nn : ℕ\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : ↑(pderiv true) x = 1\npderiv_true_y : ↑(pderiv true) y = 0\ne : Bool → R[X] := fun i => bif i then X else 1 - X\n⊢ ↑(MvPolynomial.aeval e) (↑(pderiv true) (↑(pderiv true) ((x + y) ^ n))) * X ^ 2 = (n * (n - 1)) • X ^ 2", "true_first_token": "simp", "true_tactic": "simp only [pderiv_one, pderiv_mul, (pderiv _).leibniz_pow, (pderiv _).map_coe_nat,\n  (pderiv true).map_add, pderiv_true_x, pderiv_true_y, Algebra.id.smul_eq_mul, add_zero,\n  mul_one, Derivation.map_smul_of_tower, map_nsmul, map_pow, map_add, Bool.cond_true,\n  Bool.cond_false, MvPolynomial.aeval_X, add_sub_cancel'_right, one_pow, smul_smul,\n  smul_one_mul]"}
{"dev_row": 865, "state": "case neg.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst : TopologicalSpace α\ninst : TopologicalSpace β\ninst : LinearOrder α\ninst : LinearOrder β\ninst : OrderTopology α\ninst : OrderTopology β\nt : Set α\nx : α\ninst : IsCountablyGenerated (𝓝 x)\nhtx : IsLUB t x\nht : Set.Nonempty t\nh : ¬x ∈ t\nu : ℕ → α\nhu : StrictMono u ∧ (∀ (n : ℕ), u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t\n⊢ ∃ u, Monotone u ∧ (∀ (n : ℕ), u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t", "true_first_token": "exact", "true_tactic": "exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩"}
{"dev_row": 934, "state": "case neg\nR : Type u_1\ninst : AddMonoidWithOne R\nn k : ℕ\nhk : ∀ (m : ℕ), m < succ k → Nat.binCast m = ↑m\nh : ¬(k + 1) % 2 = 0\n⊢ ↑(succ k / 2 + succ k / 2) + 1 = ↑(succ k % 2 + 2 * (succ k / 2))", "true_first_token": "have", "true_tactic": "have h1 := Or.resolve_left (Nat.mod_two_eq_zero_or_one (succ k)) h"}
{"dev_row": 1010, "state": "case neg\nα : Type u_1\nE : Type u_2\nF : Type ?u\nG : Type ?u\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst : NormedAddCommGroup E\ninst : NormedAddCommGroup F\ninst : NormedAddCommGroup G\np q : ℝ≥0∞\nhpq : p ≤ q\nf : α → E\nhf : AEStronglyMeasurable f μ\nhp0 : p ≠ 0\n⊢ snorm f p μ ≤ snorm f q μ * ↑↑μ Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q)", "true_first_token": "have", "true_tactic": "have hp0_lt : 0 < p := lt_of_le_of_ne (zero_le _) hp0.symm"}
{"dev_row": 1080, "state": "case inl\nn : Type u_2\ninst : Fintype n\ninst : DecidableEq n\nR : Type u_1\ninst : Field R\nA : Matrix n n R\nhAps : Splits (RingHom.id R) (charpoly A)\nh : IsEmpty n\n⊢ trace A = Multiset.sum (roots (charpoly A))", "true_first_token": "rw", "true_tactic": "rw [Matrix.trace, Fintype.sum_empty, Matrix.charpoly,\n  det_eq_one_of_card_eq_zero (Fintype.card_eq_zero_iff.2 h), Polynomial.roots_one,\n  Multiset.empty_eq_zero, Multiset.sum_zero]"}