Datasets:

tasksource

/

leandojo

like 5

[ { "state_after": "no goals", "state_before": "α : Type u\ninst✝ : Monoid α\na✝ b✝ c✝ u a : αˣ\nb c : α\nh : ↑a * b = ↑a * c\n⊢ b = c", "tactic": "simpa only [inv_mul_cancel_left] using congr_arg (fun x : α => ↑(a⁻¹ : αˣ) * x) h" } ]

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[ { "state_after": "case refine'_1\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nhe : exponent G ≠ 0\n⊢ Set.Finite (Set.range orderOf)\n\ncase refine'_2\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nhe : Set.Finite (Set.range orderOf)\n⊢ exponent G ≠ 0", "state_before": "G : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\n⊢ exponent G ≠ 0 ↔ Set.Finite (Set.range orderOf)", "tactic": "refine' ⟨fun he => _, fun he => _⟩" }, { "state_after": "case refine'_1\nG : Type u\ninst✝ : Monoid G\nh✝ : ∀ (g : G), 0 < orderOf g\nhe : exponent G ≠ 0\nh : ¬Set.Finite (Set.range orderOf)\n⊢ False", "state_before": "case refine'_1\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nhe : exponent G ≠ 0\n⊢ Set.Finite (Set.range orderOf)", "tactic": "by_contra h" }, { "state_after": "case refine'_1.intro.intro.intro\nG : Type u\ninst✝ : Monoid G\nh✝ : ∀ (g : G), 0 < orderOf g\nhe : exponent G ≠ 0\nh : ¬Set.Finite (Set.range orderOf)\nt : G\nhet : exponent G < orderOf t\n⊢ False", "state_before": "case refine'_1\nG : Type u\ninst✝ : Monoid G\nh✝ : ∀ (g : G), 0 < orderOf g\nhe : exponent G ≠ 0\nh : ¬Set.Finite (Set.range orderOf)\n⊢ False", "tactic": "obtain ⟨m, ⟨t, rfl⟩, het⟩ := Set.Infinite.exists_gt h (exponent G)" }, { "state_after": "no goals", "state_before": "case refine'_1.intro.intro.intro\nG : Type u\ninst✝ : Monoid G\nh✝ : ∀ (g : G), 0 < orderOf g\nhe : exponent G ≠ 0\nh : ¬Set.Finite (Set.range orderOf)\nt : G\nhet : exponent G < orderOf t\n⊢ False", "tactic": "exact pow_ne_one_of_lt_orderOf' he het (pow_exponent_eq_one t)" }, { "state_after": "case refine'_2.intro\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\n⊢ exponent G ≠ 0", "state_before": "case refine'_2\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nhe : Set.Finite (Set.range orderOf)\n⊢ exponent G ≠ 0", "tactic": "lift Set.range (orderOf (G := G)) to Finset ℕ using he with t ht" }, { "state_after": "case refine'_2.intro\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\n⊢ exponent G ≠ 0", "state_before": "case refine'_2.intro\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\n⊢ exponent G ≠ 0", "tactic": "have htpos : 0 < t.prod id := by\n refine' Finset.prod_pos fun a ha => _\n rw [← Finset.mem_coe, ht] at ha\n obtain ⟨k, rfl⟩ := ha\n exact h k" }, { "state_after": "case refine'_2.intro\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\n⊢ exponent G ∣ Finset.prod t id", "state_before": "case refine'_2.intro\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\n⊢ exponent G ≠ 0", "tactic": "suffices exponent G ∣ t.prod id by\n intro h\n rw [h, zero_dvd_iff] at this\n exact htpos.ne' this" }, { "state_after": "case refine'_2.intro\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\ng : G\n⊢ g ^ Finset.prod t id = 1", "state_before": "case refine'_2.intro\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\n⊢ exponent G ∣ Finset.prod t id", "tactic": "refine' exponent_dvd_of_forall_pow_eq_one _ _ fun g => _" }, { "state_after": "case refine'_2.intro\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\ng : G\n⊢ orderOf g ∣ Finset.prod t id", "state_before": "case refine'_2.intro\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\ng : G\n⊢ g ^ Finset.prod t id = 1", "tactic": "rw [pow_eq_mod_orderOf, Nat.mod_eq_zero_of_dvd, pow_zero g]" }, { "state_after": "case refine'_2.intro.ha\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\ng : G\n⊢ Function.minimalPeriod (fun x => g * x) 1 ∈ t", "state_before": "case refine'_2.intro\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\ng : G\n⊢ orderOf g ∣ Finset.prod t id", "tactic": "apply Finset.dvd_prod_of_mem" }, { "state_after": "case refine'_2.intro.ha\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\ng : G\n⊢ Function.minimalPeriod (fun x => g * x) 1 ∈ Set.range orderOf", "state_before": "case refine'_2.intro.ha\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\ng : G\n⊢ Function.minimalPeriod (fun x => g * x) 1 ∈ t", "tactic": "rw [← Finset.mem_coe, ht]" }, { "state_after": "no goals", "state_before": "case refine'_2.intro.ha\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\ng : G\n⊢ Function.minimalPeriod (fun x => g * x) 1 ∈ Set.range orderOf", "tactic": "exact Set.mem_range_self g" }, { "state_after": "G : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\na : ℕ\nha : a ∈ t\n⊢ 0 < id a", "state_before": "G : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\n⊢ 0 < Finset.prod t id", "tactic": "refine' Finset.prod_pos fun a ha => _" }, { "state_after": "G : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\na : ℕ\nha : a ∈ Set.range orderOf\n⊢ 0 < id a", "state_before": "G : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\na : ℕ\nha : a ∈ t\n⊢ 0 < id a", "tactic": "rw [← Finset.mem_coe, ht] at ha" }, { "state_after": "case intro\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nk : G\n⊢ 0 < id (orderOf k)", "state_before": "G : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\na : ℕ\nha : a ∈ Set.range orderOf\n⊢ 0 < id a", "tactic": "obtain ⟨k, rfl⟩ := ha" }, { "state_after": "no goals", "state_before": "case intro\nG : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nk : G\n⊢ 0 < id (orderOf k)", "tactic": "exact h k" }, { "state_after": "G : Type u\ninst✝ : Monoid G\nh✝ : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\nthis : exponent G ∣ Finset.prod t id\nh : exponent G = 0\n⊢ False", "state_before": "G : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\nthis : exponent G ∣ Finset.prod t id\n⊢ exponent G ≠ 0", "tactic": "intro h" }, { "state_after": "G : Type u\ninst✝ : Monoid G\nh✝ : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\nthis : Finset.prod t id = 0\nh : exponent G = 0\n⊢ False", "state_before": "G : Type u\ninst✝ : Monoid G\nh✝ : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\nthis : exponent G ∣ Finset.prod t id\nh : exponent G = 0\n⊢ False", "tactic": "rw [h, zero_dvd_iff] at this" }, { "state_after": "no goals", "state_before": "G : Type u\ninst✝ : Monoid G\nh✝ : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite✝ exponent_ne_zero_iff_range_orderOf_finite :\n (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ Set.Finite ↑t)\nhtpos : 0 < Finset.prod t id\nthis : Finset.prod t id = 0\nh : exponent G = 0\n⊢ False", "tactic": "exact htpos.ne' this" } ]

[ 244, 31 ]

[ 224, 1 ]

[]

[ 598, 38 ]

[ 597, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.495766\nγ : Type ?u.495769\nδ : Type ?u.495772\nι : Type ?u.495775\nR : Type ?u.495778\nR' : Type ?u.495781\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nμa μa' : Measure α\nμb μb' : Measure β\nμc : Measure γ\nf✝ : α → β\nf : α → α\nhf : QuasiMeasurePreserving f\nhs : f ⁻¹' s =ᵐ[μ] s\nn : ℕ\n⊢ (preimage f^[n]) s =ᵐ[μ] s", "state_before": "α : Type u_1\nβ : Type ?u.495766\nγ : Type ?u.495769\nδ : Type ?u.495772\nι : Type ?u.495775\nR : Type ?u.495778\nR' : Type ?u.495781\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nμa μa' : Measure α\nμb μb' : Measure β\nμc : Measure γ\nf✝ : α → β\nf : α → α\nhf : QuasiMeasurePreserving f\nhs : f ⁻¹' s =ᵐ[μ] s\n⊢ ∀ (n : ℕ), (preimage f^[n]) s =ᵐ[μ] s", "tactic": "intro n" }, { "state_after": "case zero\nα : Type u_1\nβ : Type ?u.495766\nγ : Type ?u.495769\nδ : Type ?u.495772\nι : Type ?u.495775\nR : Type ?u.495778\nR' : Type ?u.495781\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nμa μa' : Measure α\nμb μb' : Measure β\nμc : Measure γ\nf✝ : α → β\nf : α → α\nhf : QuasiMeasurePreserving f\nhs : f ⁻¹' s =ᵐ[μ] s\n⊢ (preimage f^[Nat.zero]) s =ᵐ[μ] s\n\ncase succ\nα : Type u_1\nβ : Type ?u.495766\nγ : Type ?u.495769\nδ : Type ?u.495772\nι : Type ?u.495775\nR : Type ?u.495778\nR' : Type ?u.495781\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nμa μa' : Measure α\nμb μb' : Measure β\nμc : Measure γ\nf✝ : α → β\nf : α → α\nhf : QuasiMeasurePreserving f\nhs : f ⁻¹' s =ᵐ[μ] s\nn : ℕ\nih : (preimage f^[n]) s =ᵐ[μ] s\n⊢ (preimage f^[Nat.succ n]) s =ᵐ[μ] s", "state_before": "α : Type u_1\nβ : Type ?u.495766\nγ : Type ?u.495769\nδ : Type ?u.495772\nι : Type ?u.495775\nR : Type ?u.495778\nR' : Type ?u.495781\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nμa μa' : Measure α\nμb μb' : Measure β\nμc : Measure γ\nf✝ : α → β\nf : α → α\nhf : QuasiMeasurePreserving f\nhs : f ⁻¹' s =ᵐ[μ] s\nn : ℕ\n⊢ (preimage f^[n]) s =ᵐ[μ] s", "tactic": "induction' n with n ih" }, { "state_after": "no goals", "state_before": "case succ\nα : Type u_1\nβ : Type ?u.495766\nγ : Type ?u.495769\nδ : Type ?u.495772\nι : Type ?u.495775\nR : Type ?u.495778\nR' : Type ?u.495781\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nμa μa' : Measure α\nμb μb' : Measure β\nμc : Measure γ\nf✝ : α → β\nf : α → α\nhf : QuasiMeasurePreserving f\nhs : f ⁻¹' s =ᵐ[μ] s\nn : ℕ\nih : (preimage f^[n]) s =ᵐ[μ] s\n⊢ (preimage f^[Nat.succ n]) s =ᵐ[μ] s", "tactic": "simpa only [iterate_succ', comp_apply] using ae_eq_trans (hf.ae_eq ih) hs" }, { "state_after": "no goals", "state_before": "case zero\nα : Type u_1\nβ : Type ?u.495766\nγ : Type ?u.495769\nδ : Type ?u.495772\nι : Type ?u.495775\nR : Type ?u.495778\nR' : Type ?u.495781\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nμa μa' : Measure α\nμb μb' : Measure β\nμc : Measure γ\nf✝ : α → β\nf : α → α\nhf : QuasiMeasurePreserving f\nhs : f ⁻¹' s =ᵐ[μ] s\n⊢ (preimage f^[Nat.zero]) s =ᵐ[μ] s", "tactic": "rfl" } ]

[ 2557, 89 ]

[ 2549, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u\ninst✝² : CommRing R\nS : Type u\ninst✝¹ : CommRing S\nP : Type u\ninst✝ : CommRing P\nU V : Opens ↑(PrimeSpectrum.Top R)\nhUV : U = V\np : ↑(PrimeSpectrum.Top R)\nhpV : p ∈ V.carrier\n⊢ p ∈ ↑(PrimeSpectrum.comap (RingHom.id R)) ⁻¹' U.carrier", "tactic": "rwa [hUV, PrimeSpectrum.comap_id]" }, { "state_after": "no goals", "state_before": "R : Type u\ninst✝² : CommRing R\nS : Type u\ninst✝¹ : CommRing S\nP : Type u\ninst✝ : CommRing P\nU V : Opens ↑(PrimeSpectrum.Top R)\nhUV : U = V\n⊢ (structureSheaf R).val.obj U.op = (structureSheaf R).val.obj V.op", "tactic": "rw [hUV]" }, { "state_after": "no goals", "state_before": "R : Type u\ninst✝² : CommRing R\nS : Type u\ninst✝¹ : CommRing S\nP : Type u\ninst✝ : CommRing P\nU V : Opens ↑(PrimeSpectrum.Top R)\nhUV : U = V\n⊢ comap (RingHom.id R) U V\n (_ : ∀ (p : ↑(PrimeSpectrum.Top R)), p ∈ V.carrier → p ∈ ↑(PrimeSpectrum.comap (RingHom.id R)) ⁻¹' U.carrier) =\n eqToHom (_ : (structureSheaf R).val.obj U.op = (structureSheaf R).val.obj V.op)", "tactic": "erw [comap_id_eq_map U V (eqToHom hUV.symm), eqToHom_op, eqToHom_map]" } ]

[ 1190, 75 ]

[ 1187, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nm : Set (OuterMeasure α)\ns : Set α\nhm : Set.Nonempty m\n⊢ ↑(restrict s) (sInf m) = sInf (↑(restrict s) '' m)", "tactic": "simp only [sInf_eq_iInf, restrict_biInf, hm, iInf_image]" } ]

[ 1289, 59 ]

[ 1287, 1 ]

[ { "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 : C\napp' : (X : C) → F.obj X ≅ G.obj X\nnaturality : ∀ {X Y : C} (f : X ⟶ Y), F.map f ≫ (app' Y).hom = (app' X).hom ≫ G.map f\nX : C\n⊢ (ofComponents app').app X = app' X", "tactic": "aesop" } ]

[ 237, 62 ]

[ 236, 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\nc : γ\n⊢ c ∈ map₂ f a b ↔ ∃ a' b', a' ∈ a ∧ b' ∈ b ∧ f a' b' = c", "tactic": "simp [map₂]" } ]

[ 81, 17 ]

[ 80, 1 ]

[]

[ 501, 57 ]

[ 499, 1 ]

[]

[ 124, 6 ]

[ 123, 1 ]

[]

[ 850, 6 ]

[ 849, 1 ]

[]

[ 857, 62 ]

[ 856, 1 ]

[]

[ 254, 6 ]

[ 253, 1 ]

[ { "state_after": "no goals", "state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nS✝ R : Sieve X\nJ : GrothendieckTopology C\nS : Sieve X\nf : Y ⟶ X\n⊢ Covers ⊥ S f ↔ S.arrows f", "tactic": "rw [covers_iff, bot_covering, ← Sieve.pullback_eq_top_iff_mem]" } ]

[ 339, 65 ]

[ 338, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.125652\nb₁ b₂ : β\nl : List α\nh : b₁ ∈ map (const α b₂) ↑l\n⊢ b₁ ∈ replicate (?m.125825 h) b₂", "tactic": "rwa [map_const] at h" } ]

[ 1304, 49 ]

[ 1302, 1 ]

[]

[ 75, 57 ]

[ 71, 1 ]

[ { "state_after": "ι : Type v\ndec_ι : DecidableEq ι\nβ : ι → Type w\ninst✝ : (i : ι) → AddCommMonoid (β i)\nC : (⨁ (i : ι), β i) → Prop\nx : ⨁ (i : ι), β i\nH_zero : C 0\nH_basic : ∀ (i : ι) (x : β i), C (↑(of β i) x)\nH_plus : ∀ (x y : ⨁ (i : ι), β i), C x → C y → C (x + y)\n⊢ ∀ (i : ι) (b : (fun i => β i) i) (f : Π₀ (i : ι), (fun i => β i) i),\n ↑f i = 0 → b ≠ 0 → C f → C (Dfinsupp.single i b + f)", "state_before": "ι : Type v\ndec_ι : DecidableEq ι\nβ : ι → Type w\ninst✝ : (i : ι) → AddCommMonoid (β i)\nC : (⨁ (i : ι), β i) → Prop\nx : ⨁ (i : ι), β i\nH_zero : C 0\nH_basic : ∀ (i : ι) (x : β i), C (↑(of β i) x)\nH_plus : ∀ (x y : ⨁ (i : ι), β i), C x → C y → C (x + y)\n⊢ C x", "tactic": "apply Dfinsupp.induction x H_zero" }, { "state_after": "ι : Type v\ndec_ι : DecidableEq ι\nβ : ι → Type w\ninst✝ : (i : ι) → AddCommMonoid (β i)\nC : (⨁ (i : ι), β i) → Prop\nx : ⨁ (i : ι), β i\nH_zero : C 0\nH_basic : ∀ (i : ι) (x : β i), C (↑(of β i) x)\nH_plus : ∀ (x y : ⨁ (i : ι), β i), C x → C y → C (x + y)\ni : ι\nb : β i\nf : Π₀ (i : ι), (fun i => β i) i\nh1 : ↑f i = 0\nh2 : b ≠ 0\nih : C f\n⊢ C (Dfinsupp.single i b + f)", "state_before": "ι : Type v\ndec_ι : DecidableEq ι\nβ : ι → Type w\ninst✝ : (i : ι) → AddCommMonoid (β i)\nC : (⨁ (i : ι), β i) → Prop\nx : ⨁ (i : ι), β i\nH_zero : C 0\nH_basic : ∀ (i : ι) (x : β i), C (↑(of β i) x)\nH_plus : ∀ (x y : ⨁ (i : ι), β i), C x → C y → C (x + y)\n⊢ ∀ (i : ι) (b : (fun i => β i) i) (f : Π₀ (i : ι), (fun i => β i) i),\n ↑f i = 0 → b ≠ 0 → C f → C (Dfinsupp.single i b + f)", "tactic": "intro i b f h1 h2 ih" }, { "state_after": "no goals", "state_before": "ι : Type v\ndec_ι : DecidableEq ι\nβ : ι → Type w\ninst✝ : (i : ι) → AddCommMonoid (β i)\nC : (⨁ (i : ι), β i) → Prop\nx : ⨁ (i : ι), β i\nH_zero : C 0\nH_basic : ∀ (i : ι) (x : β i), C (↑(of β i) x)\nH_plus : ∀ (x y : ⨁ (i : ι), β i), C x → C y → C (x + y)\ni : ι\nb : β i\nf : Π₀ (i : ι), (fun i => β i) i\nh1 : ↑f i = 0\nh2 : b ≠ 0\nih : C f\n⊢ C (Dfinsupp.single i b + f)", "tactic": "solve_by_elim" } ]

[ 170, 16 ]

[ 165, 11 ]

[]

[ 652, 6 ]

[ 651, 1 ]

[]

[ 147, 6 ]

[ 145, 1 ]

[]

[ 612, 10 ]

[ 611, 1 ]

[]

[ 245, 6 ]

[ 244, 1 ]

[]

[ 814, 49 ]

[ 812, 1 ]

[]

[ 97, 12 ]

[ 96, 1 ]

[ { "state_after": "case pos\nα : Type u_1\nf : α → ℝ\nhf : Summable f\n⊢ (∑' (a : α), ↑(f a)) = ↑(∑' (a : α), f a)\n\ncase neg\nα : Type u_1\nf : α → ℝ\nhf : ¬Summable f\n⊢ (∑' (a : α), ↑(f a)) = ↑(∑' (a : α), f a)", "state_before": "α : Type u_1\nf : α → ℝ\n⊢ (∑' (a : α), ↑(f a)) = ↑(∑' (a : α), f a)", "tactic": "by_cases hf : Summable f" }, { "state_after": "no goals", "state_before": "case pos\nα : Type u_1\nf : α → ℝ\nhf : Summable f\n⊢ (∑' (a : α), ↑(f a)) = ↑(∑' (a : α), f a)", "tactic": "exact (hasSum_coe.mpr hf.hasSum).tsum_eq" }, { "state_after": "no goals", "state_before": "case neg\nα : Type u_1\nf : α → ℝ\nhf : ¬Summable f\n⊢ (∑' (a : α), ↑(f a)) = ↑(∑' (a : α), f a)", "tactic": "simp [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable (summable_coe.not.mpr hf)]" } ]

[ 258, 99 ]

[ 255, 1 ]

[]

[ 422, 34 ]

[ 421, 1 ]

[]

[ 139, 10 ]

[ 138, 1 ]

[ { "state_after": "no goals", "state_before": "p q : ℕ\nhp : Prime p\nh : ↑(Nat.factorization p) q ≠ 0\n⊢ p = q", "tactic": "simpa [hp.factorization, single_apply] using h" } ]

[ 299, 63 ]

[ 298, 1 ]

[]

[ 1426, 32 ]

[ 1425, 1 ]

[ { "state_after": "no goals", "state_before": "⊢ Prime 3", "tactic": "decide" } ]

[ 175, 43 ]

[ 175, 1 ]

[ { "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\n⊢ f /ₘ g = q ∧ f %ₘ g = r", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n⊢ f /ₘ g = q ∧ f %ₘ g = r", "tactic": "nontriviality R" }, { "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\n⊢ f /ₘ g = q ∧ f %ₘ g = r", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\n⊢ f /ₘ g = q ∧ f %ₘ g = r", "tactic": "have h₁ : r - f %ₘ g = -g * (q - f /ₘ g) :=\n eq_of_sub_eq_zero\n (by\n rw [← sub_eq_zero_of_eq (h.1.trans (modByMonic_add_div f hg).symm)]\n simp [mul_add, mul_comm, sub_eq_add_neg, add_comm, add_left_comm, add_assoc])" }, { "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\nh₂ : degree (r - f %ₘ g) = degree (g * (q - f /ₘ g))\n⊢ f /ₘ g = q ∧ f %ₘ g = r", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\n⊢ f /ₘ g = q ∧ f %ₘ g = r", "tactic": "have h₂ : degree (r - f %ₘ g) = degree (g * (q - f /ₘ g)) := by simp [h₁]" }, { "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\nh₂ : degree (r - f %ₘ g) = degree (g * (q - f /ₘ g))\nh₄ : degree (r - f %ₘ g) < degree g\n⊢ f /ₘ g = q ∧ f %ₘ g = r", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\nh₂ : degree (r - f %ₘ g) = degree (g * (q - f /ₘ g))\n⊢ f /ₘ g = q ∧ f %ₘ g = r", "tactic": "have h₄ : degree (r - f %ₘ g) < degree g :=\n calc\n degree (r - f %ₘ g) ≤ max (degree r) (degree (f %ₘ g)) := degree_sub_le _ _\n _ < degree g := max_lt_iff.2 ⟨h.2, degree_modByMonic_lt _ hg⟩" }, { "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\nh₂ : degree (r - f %ₘ g) = degree (g * (q - f /ₘ g))\nh₄ : degree (r - f %ₘ g) < degree g\nh₅ : q - f /ₘ g = 0\n⊢ f /ₘ g = q ∧ f %ₘ g = r", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\nh₂ : degree (r - f %ₘ g) = degree (g * (q - f /ₘ g))\nh₄ : degree (r - f %ₘ g) < degree g\n⊢ f /ₘ g = q ∧ f %ₘ g = r", "tactic": "have h₅ : q - f /ₘ g = 0 :=\n _root_.by_contradiction fun hqf =>\n not_le_of_gt h₄ <|\n calc\n degree g ≤ degree g + degree (q - f /ₘ g) := by\n erw [degree_eq_natDegree hg.ne_zero, degree_eq_natDegree hqf, WithBot.coe_le_coe]\n exact Nat.le_add_right _ _\n _ = degree (r - f %ₘ g) := by rw [h₂, degree_mul']; simpa [Monic.def.1 hg]" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\nh₂ : degree (r - f %ₘ g) = degree (g * (q - f /ₘ g))\nh₄ : degree (r - f %ₘ g) < degree g\nh₅ : q - f /ₘ g = 0\n⊢ f /ₘ g = q ∧ f %ₘ g = r", "tactic": "exact ⟨Eq.symm <| eq_of_sub_eq_zero h₅, Eq.symm <| eq_of_sub_eq_zero <| by simpa [h₅] using h₁⟩" }, { "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\n⊢ r - f %ₘ g - -g * (q - f /ₘ g) = r + g * q - (f %ₘ g + g * (f /ₘ g))", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\n⊢ r - f %ₘ g - -g * (q - f /ₘ g) = 0", "tactic": "rw [← sub_eq_zero_of_eq (h.1.trans (modByMonic_add_div f hg).symm)]" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\n⊢ r - f %ₘ g - -g * (q - f /ₘ g) = r + g * q - (f %ₘ g + g * (f /ₘ g))", "tactic": "simp [mul_add, mul_comm, sub_eq_add_neg, add_comm, add_left_comm, add_assoc]" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\n⊢ degree (r - f %ₘ g) = degree (g * (q - f /ₘ g))", "tactic": "simp [h₁]" }, { "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\nh₂ : degree (r - f %ₘ g) = degree (g * (q - f /ₘ g))\nh₄ : degree (r - f %ₘ g) < degree g\nhqf : ¬q - f /ₘ g = 0\n⊢ ↑(natDegree g) ≤ (fun x x_1 => x + x_1) ↑(natDegree g) ↑(natDegree (q - f /ₘ g))", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\nh₂ : degree (r - f %ₘ g) = degree (g * (q - f /ₘ g))\nh₄ : degree (r - f %ₘ g) < degree g\nhqf : ¬q - f /ₘ g = 0\n⊢ degree g ≤ degree g + degree (q - f /ₘ g)", "tactic": "erw [degree_eq_natDegree hg.ne_zero, degree_eq_natDegree hqf, WithBot.coe_le_coe]" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\nh₂ : degree (r - f %ₘ g) = degree (g * (q - f /ₘ g))\nh₄ : degree (r - f %ₘ g) < degree g\nhqf : ¬q - f /ₘ g = 0\n⊢ ↑(natDegree g) ≤ (fun x x_1 => x + x_1) ↑(natDegree g) ↑(natDegree (q - f /ₘ g))", "tactic": "exact Nat.le_add_right _ _" }, { "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\nh₂ : degree (r - f %ₘ g) = degree (g * (q - f /ₘ g))\nh₄ : degree (r - f %ₘ g) < degree g\nhqf : ¬q - f /ₘ g = 0\n⊢ leadingCoeff g * leadingCoeff (q - f /ₘ g) ≠ 0", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\nh₂ : degree (r - f %ₘ g) = degree (g * (q - f /ₘ g))\nh₄ : degree (r - f %ₘ g) < degree g\nhqf : ¬q - f /ₘ g = 0\n⊢ degree g + degree (q - f /ₘ g) = degree (r - f %ₘ g)", "tactic": "rw [h₂, degree_mul']" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\nh₂ : degree (r - f %ₘ g) = degree (g * (q - f /ₘ g))\nh₄ : degree (r - f %ₘ g) < degree g\nhqf : ¬q - f /ₘ g = 0\n⊢ leadingCoeff g * leadingCoeff (q - f /ₘ g) ≠ 0", "tactic": "simpa [Monic.def.1 hg]" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q✝ : R[X]\nf g : R[X]\nq r : R[X]\nhg : Monic g\nh : r + g * q = f ∧ degree r < degree g\n✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\nh₂ : degree (r - f %ₘ g) = degree (g * (q - f /ₘ g))\nh₄ : degree (r - f %ₘ g) < degree g\nh₅ : q - f /ₘ g = 0\n⊢ r - f %ₘ g = 0", "tactic": "simpa [h₅] using h₁" } ]

[ 357, 98 ]

[ 336, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nf : α → α\nn✝ n : ℕ\nihn : id^[n] = id\n⊢ id^[Nat.succ n] = id", "tactic": "rw [iterate_succ, ihn, comp.left_id]" } ]

[ 64, 70 ]

[ 63, 1 ]

[]

[ 506, 35 ]

[ 504, 1 ]

[]

[ 221, 27 ]

[ 220, 1 ]

[]

[ 970, 56 ]

[ 968, 1 ]

[]

[ 122, 73 ]

[ 119, 1 ]

[]

[ 293, 12 ]

[ 292, 1 ]

[]

[ 1682, 89 ]

[ 1681, 1 ]

[]

[ 819, 55 ]

[ 818, 1 ]

[]

[ 184, 6 ]

[ 183, 1 ]

[]

[ 1266, 93 ]

[ 1266, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u_1\nR₁ : Type ?u.1083452\nR₂ : Type u_2\nR₃ : Type ?u.1083458\nR₄ : Type ?u.1083461\nS : Type ?u.1083464\nK : Type ?u.1083467\nK₂ : Type ?u.1083470\nM : Type u_4\nM' : Type ?u.1083476\nM₁ : Type ?u.1083479\nM₂ : Type u_3\nM₃ : Type ?u.1083485\nM₄ : Type ?u.1083488\nN : Type ?u.1083491\nN₂ : Type ?u.1083494\nι : Type ?u.1083497\nV : Type ?u.1083500\nV₂ : Type ?u.1083503\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : Semiring R₃\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : Module R M\ninst✝³ : Module R₂ M₂\ninst✝² : Module R₃ M₃\nσ₂₁ : R₂ →+* R\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝¹ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nF : Type u_5\nsc : SemilinearMapClass F τ₁₂ M M₂\ninst✝ : RingHomSurjective τ₁₂\nf : F\n⊢ range f = ⊤ ↔ Surjective ↑f", "tactic": "rw [SetLike.ext'_iff, range_coe, top_coe, Set.range_iff_surjective]" } ]

[ 1240, 70 ]

[ 1239, 1 ]

[]

[ 33, 82 ]

[ 32, 1 ]

[ { "state_after": "case a\nF : Type ?u.100208\nα : Type ?u.100211\nβ : Type ?u.100214\nR : Type u_1\nS : Type u_2\nS' : Type ?u.100223\ninst✝² : NonAssocSemiring R\ninst✝¹ : NonAssocSemiring S\ninst✝ : NonAssocSemiring S'\ne : R ≃+* S\nx✝ : S\n⊢ ↑(RingHom.comp (toRingHom e) (toRingHom (RingEquiv.symm e))) x✝ = ↑(RingHom.id S) x✝", "state_before": "F : Type ?u.100208\nα : Type ?u.100211\nβ : Type ?u.100214\nR : Type u_1\nS : Type u_2\nS' : Type ?u.100223\ninst✝² : NonAssocSemiring R\ninst✝¹ : NonAssocSemiring S\ninst✝ : NonAssocSemiring S'\ne : R ≃+* S\n⊢ RingHom.comp (toRingHom e) (toRingHom (RingEquiv.symm e)) = RingHom.id S", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case a\nF : Type ?u.100208\nα : Type ?u.100211\nβ : Type ?u.100214\nR : Type u_1\nS : Type u_2\nS' : Type ?u.100223\ninst✝² : NonAssocSemiring R\ninst✝¹ : NonAssocSemiring S\ninst✝ : NonAssocSemiring S'\ne : R ≃+* S\nx✝ : S\n⊢ ↑(RingHom.comp (toRingHom e) (toRingHom (RingEquiv.symm e))) x✝ = ↑(RingHom.id S) x✝", "tactic": "simp" } ]

[ 769, 7 ]

[ 766, 1 ]

[ { "state_after": "G : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\n⊢ ∀ {g' : G}, g' ^ n ^ j = 1 → g' • (fun x => x ^ n)^[j] ⁻¹' s ⊆ (fun x => x ^ n)^[j] ⁻¹' s", "state_before": "G : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\n⊢ ∀ {g' : G}, g' ^ n ^ j = 1 → g' • s ⊆ s", "tactic": "rw [(IsFixedPt.preimage_iterate hs j : zpowGroupHom n^[j] ⁻¹' s = s).symm]" }, { "state_after": "case intro.intro\nG : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\ng' : G\nhg' : g' ^ n ^ j = 1\ny : G\nhy : y ∈ (fun x => x ^ n)^[j] ⁻¹' s\n⊢ (fun x => g' • x) y ∈ (fun x => x ^ n)^[j] ⁻¹' s", "state_before": "G : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\n⊢ ∀ {g' : G}, g' ^ n ^ j = 1 → g' • (fun x => x ^ n)^[j] ⁻¹' s ⊆ (fun x => x ^ n)^[j] ⁻¹' s", "tactic": "rintro g' hg' - ⟨y, hy, rfl⟩" }, { "state_after": "case intro.intro\nG : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\ng' : G\nhg' : g' ^ n ^ j = 1\ny : G\nhy : y ∈ (fun x => x ^ n)^[j] ⁻¹' s\n⊢ (↑(zpowGroupHom n)^[j]) (g' * y) ∈ s", "state_before": "case intro.intro\nG : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\ng' : G\nhg' : g' ^ n ^ j = 1\ny : G\nhy : y ∈ (fun x => x ^ n)^[j] ⁻¹' s\n⊢ (fun x => g' • x) y ∈ (fun x => x ^ n)^[j] ⁻¹' s", "tactic": "change (zpowGroupHom n^[j]) (g' * y) ∈ s" }, { "state_after": "case hg'\nG : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\ng' : G\nhg' : g' ^ n ^ j = 1\ny : G\nhy : y ∈ (fun x => x ^ n)^[j] ⁻¹' s\n⊢ (↑(zpowGroupHom n)^[j]) g' = 1\n\ncase intro.intro\nG : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\ng' y : G\nhy : y ∈ (fun x => x ^ n)^[j] ⁻¹' s\nhg' : (↑(zpowGroupHom n)^[j]) g' = 1\n⊢ (↑(zpowGroupHom n)^[j]) (g' * y) ∈ s", "state_before": "case intro.intro\nG : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\ng' : G\nhg' : g' ^ n ^ j = 1\ny : G\nhy : y ∈ (fun x => x ^ n)^[j] ⁻¹' s\n⊢ (↑(zpowGroupHom n)^[j]) (g' * y) ∈ s", "tactic": "replace hg' : (zpowGroupHom n^[j]) g' = 1" }, { "state_after": "no goals", "state_before": "case intro.intro\nG : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\ng' y : G\nhy : y ∈ (fun x => x ^ n)^[j] ⁻¹' s\nhg' : (↑(zpowGroupHom n)^[j]) g' = 1\n⊢ (↑(zpowGroupHom n)^[j]) (g' * y) ∈ s", "tactic": "rwa [MonoidHom.iterate_map_mul, hg', one_mul]" }, { "state_after": "G : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\nthis : ∀ {g' : G}, g' ^ n ^ j = 1 → g' • s ⊆ s\n⊢ s ≤ g • s", "state_before": "G : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\nthis : ∀ {g' : G}, g' ^ n ^ j = 1 → g' • s ⊆ s\n⊢ g • s = s", "tactic": "refine' le_antisymm (this hg) _" }, { "state_after": "G : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\nthis : ∀ {g' : G}, g' ^ n ^ j = 1 → g' • s ⊆ s\n⊢ g • g⁻¹ • s ≤ g • s", "state_before": "G : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\nthis : ∀ {g' : G}, g' ^ n ^ j = 1 → g' • s ⊆ s\n⊢ s ≤ g • s", "tactic": "conv_lhs => rw [← smul_inv_smul g s]" }, { "state_after": "case hg\nG : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\nthis : ∀ {g' : G}, g' ^ n ^ j = 1 → g' • s ⊆ s\n⊢ g⁻¹ ^ n ^ j = 1\n\nG : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nthis : ∀ {g' : G}, g' ^ n ^ j = 1 → g' • s ⊆ s\nhg : g⁻¹ ^ n ^ j = 1\n⊢ g • g⁻¹ • s ≤ g • s", "state_before": "G : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\nthis : ∀ {g' : G}, g' ^ n ^ j = 1 → g' • s ⊆ s\n⊢ g • g⁻¹ • s ≤ g • s", "tactic": "replace hg : g⁻¹ ^ n ^ j = 1" }, { "state_after": "no goals", "state_before": "G : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nthis : ∀ {g' : G}, g' ^ n ^ j = 1 → g' • s ⊆ s\nhg : g⁻¹ ^ n ^ j = 1\n⊢ g • g⁻¹ • s ≤ g • s", "tactic": "simpa only [le_eq_subset, set_smul_subset_set_smul_iff] using this hg" }, { "state_after": "no goals", "state_before": "case hg\nG : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\nthis : ∀ {g' : G}, g' ^ n ^ j = 1 → g' • s ⊆ s\n⊢ g⁻¹ ^ n ^ j = 1", "tactic": "rw [inv_zpow, hg, inv_one]" }, { "state_after": "no goals", "state_before": "case hg'\nG : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x => x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nhg : g ^ n ^ j = 1\ng' : G\nhg' : g' ^ n ^ j = 1\ny : G\nhy : y ∈ (fun x => x ^ n)^[j] ⁻¹' s\n⊢ (↑(zpowGroupHom n)^[j]) g' = 1", "tactic": "simpa [zpowGroupHom]" } ]

[ 47, 48 ]

[ 34, 1 ]

[ { "state_after": "α : Type u_1\nl : List α\nhl : Nodup l\n⊢ (∃ x y _h, x ∈ ↑l ∧ y ∈ ↑l) ↔ 2 ≤ List.length l", "state_before": "α : Type u_1\nl : List α\nhl : Nodup l\n⊢ Nontrivial ↑l ↔ 2 ≤ List.length l", "tactic": "rw [Nontrivial]" }, { "state_after": "case nil\nα : Type u_1\nhl : Nodup []\n⊢ (∃ x y _h, x ∈ ↑[] ∧ y ∈ ↑[]) ↔ 2 ≤ List.length []\n\ncase cons.nil\nα : Type u_1\nhd : α\nhl : Nodup [hd]\n⊢ (∃ x y _h, x ∈ ↑[hd] ∧ y ∈ ↑[hd]) ↔ 2 ≤ List.length [hd]\n\ncase cons.cons\nα : Type u_1\nhd hd' : α\ntl : List α\nhl : Nodup (hd :: hd' :: tl)\n⊢ (∃ x y _h, x ∈ ↑(hd :: hd' :: tl) ∧ y ∈ ↑(hd :: hd' :: tl)) ↔ 2 ≤ List.length (hd :: hd' :: tl)", "state_before": "α : Type u_1\nl : List α\nhl : Nodup l\n⊢ (∃ x y _h, x ∈ ↑l ∧ y ∈ ↑l) ↔ 2 ≤ List.length l", "tactic": "rcases l with (_ | ⟨hd, _ | ⟨hd', tl⟩⟩)" }, { "state_after": "no goals", "state_before": "case nil\nα : Type u_1\nhl : Nodup []\n⊢ (∃ x y _h, x ∈ ↑[] ∧ y ∈ ↑[]) ↔ 2 ≤ List.length []", "tactic": "simp" }, { "state_after": "no goals", "state_before": "case cons.nil\nα : Type u_1\nhd : α\nhl : Nodup [hd]\n⊢ (∃ x y _h, x ∈ ↑[hd] ∧ y ∈ ↑[hd]) ↔ 2 ≤ List.length [hd]", "tactic": "simp" }, { "state_after": "case cons.cons\nα : Type u_1\nhd hd' : α\ntl : List α\nhl : Nodup (hd :: hd' :: tl)\n⊢ ∃ x y, ¬x = y ∧ (x = hd ∨ x = hd' ∨ x ∈ tl) ∧ (y = hd ∨ y = hd' ∨ y ∈ tl)", "state_before": "case cons.cons\nα : Type u_1\nhd hd' : α\ntl : List α\nhl : Nodup (hd :: hd' :: tl)\n⊢ (∃ x y _h, x ∈ ↑(hd :: hd' :: tl) ∧ y ∈ ↑(hd :: hd' :: tl)) ↔ 2 ≤ List.length (hd :: hd' :: tl)", "tactic": "simp only [mem_cons, exists_prop, mem_coe_iff, List.length, Ne.def, Nat.succ_le_succ_iff,\n zero_le, iff_true_iff]" }, { "state_after": "case cons.cons\nα : Type u_1\nhd hd' : α\ntl : List α\nhl : Nodup (hd :: hd' :: tl)\n⊢ ¬hd = hd'", "state_before": "case cons.cons\nα : Type u_1\nhd hd' : α\ntl : List α\nhl : Nodup (hd :: hd' :: tl)\n⊢ ∃ x y, ¬x = y ∧ (x = hd ∨ x = hd' ∨ x ∈ tl) ∧ (y = hd ∨ y = hd' ∨ y ∈ tl)", "tactic": "refine' ⟨hd, hd', _, by simp⟩" }, { "state_after": "case cons.cons\nα : Type u_1\nhd hd' : α\ntl : List α\nhl : (¬hd = hd' ∧ ¬hd ∈ tl) ∧ ¬hd' ∈ tl ∧ Nodup tl\n⊢ ¬hd = hd'", "state_before": "case cons.cons\nα : Type u_1\nhd hd' : α\ntl : List α\nhl : Nodup (hd :: hd' :: tl)\n⊢ ¬hd = hd'", "tactic": "simp only [not_or, mem_cons, nodup_cons] at hl" }, { "state_after": "no goals", "state_before": "case cons.cons\nα : Type u_1\nhd hd' : α\ntl : List α\nhl : (¬hd = hd' ∧ ¬hd ∈ tl) ∧ ¬hd' ∈ tl ∧ Nodup tl\n⊢ ¬hd = hd'", "tactic": "exact hl.left.left" }, { "state_after": "no goals", "state_before": "α : Type u_1\nhd hd' : α\ntl : List α\nhl : Nodup (hd :: hd' :: tl)\n⊢ (hd = hd ∨ hd = hd' ∨ hd ∈ tl) ∧ (hd' = hd ∨ hd' = hd' ∨ hd' ∈ tl)", "tactic": "simp" } ]

[ 633, 23 ]

[ 623, 1 ]

[]

[ 400, 23 ]

[ 399, 11 ]

[]

[ 69, 55 ]

[ 68, 1 ]

[]

[ 144, 26 ]

[ 142, 1 ]

[ { "state_after": "case intro\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type ?u.32484\nδ : Type ?u.32487\nr r₁ r₂ : α → α → Prop\ns s₁ s₂ : β → β → Prop\na a₁ a₂ : α\nb✝¹ b₁ b₂ : β\nx y : α ⊕ β\naca : ∀ (a : α), Acc (Lex r s) (inl a)\nb✝ b : β\nh✝ : ∀ (y : β), s y b → Acc s y\nIH : ∀ (y : β), s y b → Acc (Lex r s) (inr y)\n⊢ Acc (Lex r s) (inr b)", "state_before": "α : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type ?u.32484\nδ : Type ?u.32487\nr r₁ r₂ : α → α → Prop\ns s₁ s₂ : β → β → Prop\na a₁ a₂ : α\nb✝ b₁ b₂ : β\nx y : α ⊕ β\naca : ∀ (a : α), Acc (Lex r s) (inl a)\nb : β\nacb : Acc s b\n⊢ Acc (Lex r s) (inr b)", "tactic": "induction' acb with b _ IH" }, { "state_after": "case intro.h\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type ?u.32484\nδ : Type ?u.32487\nr r₁ r₂ : α → α → Prop\ns s₁ s₂ : β → β → Prop\na a₁ a₂ : α\nb✝¹ b₁ b₂ : β\nx y : α ⊕ β\naca : ∀ (a : α), Acc (Lex r s) (inl a)\nb✝ b : β\nh✝ : ∀ (y : β), s y b → Acc s y\nIH : ∀ (y : β), s y b → Acc (Lex r s) (inr y)\n⊢ ∀ (y : α ⊕ β), Lex r s y (inr b) → Acc (Lex r s) y", "state_before": "case intro\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type ?u.32484\nδ : Type ?u.32487\nr r₁ r₂ : α → α → Prop\ns s₁ s₂ : β → β → Prop\na a₁ a₂ : α\nb✝¹ b₁ b₂ : β\nx y : α ⊕ β\naca : ∀ (a : α), Acc (Lex r s) (inl a)\nb✝ b : β\nh✝ : ∀ (y : β), s y b → Acc s y\nIH : ∀ (y : β), s y b → Acc (Lex r s) (inr y)\n⊢ Acc (Lex r s) (inr b)", "tactic": "constructor" }, { "state_after": "case intro.h\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type ?u.32484\nδ : Type ?u.32487\nr r₁ r₂ : α → α → Prop\ns s₁ s₂ : β → β → Prop\na a₁ a₂ : α\nb✝¹ b₁ b₂ : β\nx y✝ : α ⊕ β\naca : ∀ (a : α), Acc (Lex r s) (inl a)\nb✝ b : β\nh✝ : ∀ (y : β), s y b → Acc s y\nIH : ∀ (y : β), s y b → Acc (Lex r s) (inr y)\ny : α ⊕ β\nh : Lex r s y (inr b)\n⊢ Acc (Lex r s) y", "state_before": "case intro.h\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type ?u.32484\nδ : Type ?u.32487\nr r₁ r₂ : α → α → Prop\ns s₁ s₂ : β → β → Prop\na a₁ a₂ : α\nb✝¹ b₁ b₂ : β\nx y : α ⊕ β\naca : ∀ (a : α), Acc (Lex r s) (inl a)\nb✝ b : β\nh✝ : ∀ (y : β), s y b → Acc s y\nIH : ∀ (y : β), s y b → Acc (Lex r s) (inr y)\n⊢ ∀ (y : α ⊕ β), Lex r s y (inr b) → Acc (Lex r s) y", "tactic": "intro y h" }, { "state_after": "case intro.h.inr\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type ?u.32484\nδ : Type ?u.32487\nr r₁ r₂ : α → α → Prop\ns s₁ s₂ : β → β → Prop\na a₁ a₂ : α\nb✝¹ b₁ b₂ : β\nx y : α ⊕ β\naca : ∀ (a : α), Acc (Lex r s) (inl a)\nb✝ b : β\nh✝ : ∀ (y : β), s y b → Acc s y\nIH : ∀ (y : β), s y b → Acc (Lex r s) (inr y)\nb' : β\nh' : s b' b\n⊢ Acc (Lex r s) (inr b')\n\ncase intro.h.sep\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type ?u.32484\nδ : Type ?u.32487\nr r₁ r₂ : α → α → Prop\ns s₁ s₂ : β → β → Prop\na✝ a₁ a₂ : α\nb✝¹ b₁ b₂ : β\nx y : α ⊕ β\naca : ∀ (a : α), Acc (Lex r s) (inl a)\nb✝ b : β\nh✝ : ∀ (y : β), s y b → Acc s y\nIH : ∀ (y : β), s y b → Acc (Lex r s) (inr y)\na : α\n⊢ Acc (Lex r s) (inl a)", "state_before": "case intro.h\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type ?u.32484\nδ : Type ?u.32487\nr r₁ r₂ : α → α → Prop\ns s₁ s₂ : β → β → Prop\na a₁ a₂ : α\nb✝¹ b₁ b₂ : β\nx y✝ : α ⊕ β\naca : ∀ (a : α), Acc (Lex r s) (inl a)\nb✝ b : β\nh✝ : ∀ (y : β), s y b → Acc s y\nIH : ∀ (y : β), s y b → Acc (Lex r s) (inr y)\ny : α ⊕ β\nh : Lex r s y (inr b)\n⊢ Acc (Lex r s) y", "tactic": "cases' h with _ _ _ b' _ h' a" }, { "state_after": "no goals", "state_before": "case intro.h.inr\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type ?u.32484\nδ : Type ?u.32487\nr r₁ r₂ : α → α → Prop\ns s₁ s₂ : β → β → Prop\na a₁ a₂ : α\nb✝¹ b₁ b₂ : β\nx y : α ⊕ β\naca : ∀ (a : α), Acc (Lex r s) (inl a)\nb✝ b : β\nh✝ : ∀ (y : β), s y b → Acc s y\nIH : ∀ (y : β), s y b → Acc (Lex r s) (inr y)\nb' : β\nh' : s b' b\n⊢ Acc (Lex r s) (inr b')", "tactic": "exact IH _ h'" }, { "state_after": "no goals", "state_before": "case intro.h.sep\nα : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type ?u.32484\nδ : Type ?u.32487\nr r₁ r₂ : α → α → Prop\ns s₁ s₂ : β → β → Prop\na✝ a₁ a₂ : α\nb✝¹ b₁ b₂ : β\nx y : α ⊕ β\naca : ∀ (a : α), Acc (Lex r s) (inl a)\nb✝ b : β\nh✝ : ∀ (y : β), s y b → Acc s y\nIH : ∀ (y : β), s y b → Acc (Lex r s) (inr y)\na : α\n⊢ Acc (Lex r s) (inl a)", "tactic": "exact aca _" } ]

[ 542, 16 ]

[ 535, 1 ]

[ { "state_after": "m n k : Nat\ne : n = m * k\n⊢ 0 < n → m ≤ n", "state_before": "m n : Nat\nh : 0 < n\nk : Nat\ne : n = m * k\n⊢ m ≤ n", "tactic": "revert h" }, { "state_after": "m n k : Nat\ne : n = m * k\n⊢ 0 < m * k → m ≤ m * k", "state_before": "m n k : Nat\ne : n = m * k\n⊢ 0 < n → m ≤ n", "tactic": "rw [e]" }, { "state_after": "no goals", "state_before": "m n k : Nat\ne : n = m * k\n⊢ 0 < m * k → m ≤ m * k", "tactic": "match k with\n| 0 => intro hn; simp at hn\n| pk+1 =>\n intro\n have := Nat.mul_le_mul_left m (succ_pos pk)\n rwa [Nat.mul_one] at this" }, { "state_after": "m n k : Nat\ne : n = m * 0\nhn : 0 < m * 0\n⊢ m ≤ m * 0", "state_before": "m n k : Nat\ne : n = m * 0\n⊢ 0 < m * 0 → m ≤ m * 0", "tactic": "intro hn" }, { "state_after": "no goals", "state_before": "m n k : Nat\ne : n = m * 0\nhn : 0 < m * 0\n⊢ m ≤ m * 0", "tactic": "simp at hn" }, { "state_after": "m n k pk : Nat\ne : n = m * (pk + 1)\nh✝ : 0 < m * (pk + 1)\n⊢ m ≤ m * (pk + 1)", "state_before": "m n k pk : Nat\ne : n = m * (pk + 1)\n⊢ 0 < m * (pk + 1) → m ≤ m * (pk + 1)", "tactic": "intro" }, { "state_after": "m n k pk : Nat\ne : n = m * (pk + 1)\nh✝ : 0 < m * (pk + 1)\nthis : m * succ 0 ≤ m * succ pk\n⊢ m ≤ m * (pk + 1)", "state_before": "m n k pk : Nat\ne : n = m * (pk + 1)\nh✝ : 0 < m * (pk + 1)\n⊢ m ≤ m * (pk + 1)", "tactic": "have := Nat.mul_le_mul_left m (succ_pos pk)" }, { "state_after": "no goals", "state_before": "m n k pk : Nat\ne : n = m * (pk + 1)\nh✝ : 0 < m * (pk + 1)\nthis : m * succ 0 ≤ m * succ pk\n⊢ m ≤ m * (pk + 1)", "tactic": "rwa [Nat.mul_one] at this" } ]

[ 705, 32 ]

[ 696, 1 ]

[]

[ 139, 34 ]

[ 138, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nn : ℤ\nx : α\n⊢ ↑(f ^ n) x ∈ support f ↔ x ∈ support f", "tactic": "simp only [mem_support, ne_eq, apply_zpow_apply_eq_iff]" } ]

[ 382, 58 ]

[ 381, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Encodable β\ninst✝ : CompleteLattice α\nf : β → α\n⊢ (⨆ (j : β) (i : ℕ) (_ : j ∈ decode₂ β i), f j) = ⨆ (b : β), f b", "state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Encodable β\ninst✝ : CompleteLattice α\nf : β → α\n⊢ (⨆ (i : ℕ) (b : β) (_ : b ∈ decode₂ β i), f b) = ⨆ (b : β), f b", "tactic": "rw [iSup_comm]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Encodable β\ninst✝ : CompleteLattice α\nf : β → α\n⊢ (⨆ (j : β) (i : ℕ) (_ : j ∈ decode₂ β i), f j) = ⨆ (b : β), f b", "tactic": "simp only [mem_decode₂, iSup_iSup_eq_right]" } ]

[ 35, 46 ]

[ 32, 1 ]

[ { "state_after": "ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nf : BumpCovering ι X s\ni : ι\nx : X\nt : Finset ι\nht : ∀ (j : ι), WellOrderingRel j i → ↑(toFun s f j) x ≠ 0 → j ∈ t\nj : ι\nhj : 1 - ↑(toFun s f j) x ≠ 1\n⊢ WellOrderingRel j i ↔ j ∈ Finset.filter (fun j => WellOrderingRel j i) t", "state_before": "ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nf : BumpCovering ι X s\ni : ι\nx : X\nt : Finset ι\nht : ∀ (j : ι), WellOrderingRel j i → ↑(toFun s f j) x ≠ 0 → j ∈ t\n⊢ toPouFun f i x = ↑(toFun s f i) x * ∏ j in Finset.filter (fun j => WellOrderingRel j i) t, (1 - ↑(toFun s f j) x)", "tactic": "refine' congr_arg _ (finprod_cond_eq_prod_of_cond_iff _ fun {j} hj => _)" }, { "state_after": "ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nf : BumpCovering ι X s\ni : ι\nx : X\nt : Finset ι\nht : ∀ (j : ι), WellOrderingRel j i → ↑(toFun s f j) x ≠ 0 → j ∈ t\nj : ι\nhj : ¬↑(toFun s f j) x = 0\n⊢ WellOrderingRel j i ↔ j ∈ Finset.filter (fun j => WellOrderingRel j i) t", "state_before": "ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nf : BumpCovering ι X s\ni : ι\nx : X\nt : Finset ι\nht : ∀ (j : ι), WellOrderingRel j i → ↑(toFun s f j) x ≠ 0 → j ∈ t\nj : ι\nhj : 1 - ↑(toFun s f j) x ≠ 1\n⊢ WellOrderingRel j i ↔ j ∈ Finset.filter (fun j => WellOrderingRel j i) t", "tactic": "rw [Ne.def, sub_eq_self] at hj" }, { "state_after": "ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nf : BumpCovering ι X s\ni : ι\nx : X\nt : Finset ι\nht : ∀ (j : ι), WellOrderingRel j i → ↑(toFun s f j) x ≠ 0 → j ∈ t\nj : ι\nhj : ¬↑(toFun s f j) x = 0\n⊢ WellOrderingRel j i → j ∈ t", "state_before": "ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nf : BumpCovering ι X s\ni : ι\nx : X\nt : Finset ι\nht : ∀ (j : ι), WellOrderingRel j i → ↑(toFun s f j) x ≠ 0 → j ∈ t\nj : ι\nhj : ¬↑(toFun s f j) x = 0\n⊢ WellOrderingRel j i ↔ j ∈ Finset.filter (fun j => WellOrderingRel j i) t", "tactic": "rw [Finset.mem_filter, Iff.comm, and_iff_right_iff_imp]" }, { "state_after": "no goals", "state_before": "ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nf : BumpCovering ι X s\ni : ι\nx : X\nt : Finset ι\nht : ∀ (j : ι), WellOrderingRel j i → ↑(toFun s f j) x ≠ 0 → j ∈ t\nj : ι\nhj : ¬↑(toFun s f j) x = 0\n⊢ WellOrderingRel j i → j ∈ t", "tactic": "exact flip (ht j) hj" } ]

[ 395, 23 ]

[ 389, 1 ]

[]

[ 94, 88 ]

[ 94, 1 ]

[]

[ 92, 49 ]

[ 90, 1 ]

[]

[ 204, 19 ]

[ 203, 1 ]

[]

[ 241, 6 ]

[ 240, 1 ]

[]

[ 446, 6 ]

[ 445, 1 ]

[]

[ 100, 10 ]

[ 98, 1 ]

[]

[ 203, 37 ]

[ 201, 1 ]

[ { "state_after": "case intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₁ l₂ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ l₂\nl : List α\np : l ~ l₁\ns : l <+ l₂\n⊢ a :: l₁ <+~ l₂", "state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₁ l₂ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ l₂\ns : l₁ <+~ l₂\n⊢ a :: l₁ <+~ l₂", "tactic": "rcases s with ⟨l, p, s⟩" }, { "state_after": "case intro.intro.slnil\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l l₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ []\np : [] ~ l₁\n⊢ a :: l₁ <+~ []\n\ncase intro.intro.cons\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\na : α\nl₂ l l₁✝ l₂✝ : List α\na✝¹ : α\na✝ : l₁✝ <+ l₂✝\na_ih✝ : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ l₂✝ → l₁✝ ~ l₁ → a :: l₁ <+~ l₂✝\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ a✝¹ :: l₂✝\np : l₁✝ ~ l₁\n⊢ a :: l₁ <+~ a✝¹ :: l₂✝\n\ncase intro.intro.cons₂\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\na : α\nl₂ l l₁✝ l₂✝ : List α\na✝¹ : α\na✝ : l₁✝ <+ l₂✝\na_ih✝ : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ l₂✝ → l₁✝ ~ l₁ → a :: l₁ <+~ l₂✝\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ a✝¹ :: l₂✝\np : a✝¹ :: l₁✝ ~ l₁\n⊢ a :: l₁ <+~ a✝¹ :: l₂✝", "state_before": "case intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₁ l₂ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ l₂\nl : List α\np : l ~ l₁\ns : l <+ l₂\n⊢ a :: l₁ <+~ l₂", "tactic": "induction s generalizing l₁" }, { "state_after": "case intro.intro.cons\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\na : α\nl₂ l l₁✝ l₂✝ : List α\na✝¹ : α\na✝ : l₁✝ <+ l₂✝\na_ih✝ : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ l₂✝ → l₁✝ ~ l₁ → a :: l₁ <+~ l₂✝\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ a✝¹ :: l₂✝\np : l₁✝ ~ l₁\n⊢ a :: l₁ <+~ a✝¹ :: l₂✝\n\ncase intro.intro.cons₂\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\na : α\nl₂ l l₁✝ l₂✝ : List α\na✝¹ : α\na✝ : l₁✝ <+ l₂✝\na_ih✝ : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ l₂✝ → l₁✝ ~ l₁ → a :: l₁ <+~ l₂✝\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ a✝¹ :: l₂✝\np : a✝¹ :: l₁✝ ~ l₁\n⊢ a :: l₁ <+~ a✝¹ :: l₂✝", "state_before": "case intro.intro.slnil\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l l₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ []\np : [] ~ l₁\n⊢ a :: l₁ <+~ []\n\ncase intro.intro.cons\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\na : α\nl₂ l l₁✝ l₂✝ : List α\na✝¹ : α\na✝ : l₁✝ <+ l₂✝\na_ih✝ : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ l₂✝ → l₁✝ ~ l₁ → a :: l₁ <+~ l₂✝\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ a✝¹ :: l₂✝\np : l₁✝ ~ l₁\n⊢ a :: l₁ <+~ a✝¹ :: l₂✝\n\ncase intro.intro.cons₂\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\na : α\nl₂ l l₁✝ l₂✝ : List α\na✝¹ : α\na✝ : l₁✝ <+ l₂✝\na_ih✝ : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ l₂✝ → l₁✝ ~ l₁ → a :: l₁ <+~ l₂✝\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ a✝¹ :: l₂✝\np : a✝¹ :: l₁✝ ~ l₁\n⊢ a :: l₁ <+~ a✝¹ :: l₂✝", "tactic": "case slnil => cases h₂" }, { "state_after": "no goals", "state_before": "case intro.intro.cons₂\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\na : α\nl₂ l l₁✝ l₂✝ : List α\na✝¹ : α\na✝ : l₁✝ <+ l₂✝\na_ih✝ : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ l₂✝ → l₁✝ ~ l₁ → a :: l₁ <+~ l₂✝\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ a✝¹ :: l₂✝\np : a✝¹ :: l₁✝ ~ l₁\n⊢ a :: l₁ <+~ a✝¹ :: l₂✝", "tactic": "case cons₂ r₁ r₂ b _ ih =>\n have bm : b ∈ l₁ := p.subset <| mem_cons_self _ _\n have am : a ∈ r₂ := by\n simp only [find?, mem_cons] at h₂\n exact h₂.resolve_left fun e => h₁ <| e.symm ▸ bm\n rcases mem_split bm with ⟨t₁, t₂, rfl⟩\n have st : t₁ ++ t₂ <+ t₁ ++ b :: t₂ := by simp\n rcases ih (d₁.sublist st) (mt (fun x => st.subset x) h₁) am\n (Perm.cons_inv <| p.trans perm_middle) with\n ⟨t, p', s'⟩\n exact\n ⟨b :: t, (p'.cons b).trans <| (swap _ _ _).trans (perm_middle.symm.cons a), s'.cons₂ _⟩" }, { "state_after": "no goals", "state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l l₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ []\np : [] ~ l₁\n⊢ a :: l₁ <+~ []", "tactic": "cases h₂" }, { "state_after": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\ns' : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\np : r₁ ~ l₁\nh₂ : a = b ∨ a ∈ r₂\n⊢ a :: l₁ <+~ b :: r₂", "state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\ns' : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ b :: r₂\np : r₁ ~ l₁\n⊢ a :: l₁ <+~ b :: r₂", "tactic": "simp at h₂" }, { "state_after": "case inl\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\ns' : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\np : r₁ ~ l₁\ne : a = b\n⊢ a :: l₁ <+~ b :: r₂\n\ncase inr\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\ns' : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\np : r₁ ~ l₁\nm : a ∈ r₂\n⊢ a :: l₁ <+~ b :: r₂", "state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\ns' : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\np : r₁ ~ l₁\nh₂ : a = b ∨ a ∈ r₂\n⊢ a :: l₁ <+~ b :: r₂", "tactic": "cases' h₂ with e m" }, { "state_after": "case inl\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\ns' : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\np : r₁ ~ l₁\n⊢ a :: l₁ <+~ a :: r₂", "state_before": "case inl\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\ns' : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\np : r₁ ~ l₁\ne : a = b\n⊢ a :: l₁ <+~ b :: r₂", "tactic": "subst b" }, { "state_after": "no goals", "state_before": "case inl\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\ns' : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\np : r₁ ~ l₁\n⊢ a :: l₁ <+~ a :: r₂", "tactic": "exact ⟨a :: r₁, p.cons a, s'.cons₂ _⟩" }, { "state_after": "case inr.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\ns'✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\np : r₁ ~ l₁\nm : a ∈ r₂\nt : List α\np' : t ~ a :: l₁\ns' : t <+ r₂\n⊢ a :: l₁ <+~ b :: r₂", "state_before": "case inr\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\ns' : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\np : r₁ ~ l₁\nm : a ∈ r₂\n⊢ a :: l₁ <+~ b :: r₂", "tactic": "rcases ih d₁ h₁ m p with ⟨t, p', s'⟩" }, { "state_after": "no goals", "state_before": "case inr.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\ns'✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\np : r₁ ~ l₁\nm : a ∈ r₂\nt : List α\np' : t ~ a :: l₁\ns' : t <+ r₂\n⊢ a :: l₁ <+~ b :: r₂", "tactic": "exact ⟨t, p', s'.cons _⟩" }, { "state_after": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\na✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ b :: r₂\np : b :: r₁ ~ l₁\nbm : b ∈ l₁\n⊢ a :: l₁ <+~ b :: r₂", "state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\na✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ b :: r₂\np : b :: r₁ ~ l₁\n⊢ a :: l₁ <+~ b :: r₂", "tactic": "have bm : b ∈ l₁ := p.subset <| mem_cons_self _ _" }, { "state_after": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\na✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ b :: r₂\np : b :: r₁ ~ l₁\nbm : b ∈ l₁\nam : a ∈ r₂\n⊢ a :: l₁ <+~ b :: r₂", "state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\na✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ b :: r₂\np : b :: r₁ ~ l₁\nbm : b ∈ l₁\n⊢ a :: l₁ <+~ b :: r₂", "tactic": "have am : a ∈ r₂ := by\n simp only [find?, mem_cons] at h₂\n exact h₂.resolve_left fun e => h₁ <| e.symm ▸ bm" }, { "state_after": "case intro.intro\nα : Type uu\nβ : Type vv\nl₁ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\na✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nh₂ : a ∈ b :: r₂\nam : a ∈ r₂\nt₁ t₂ : List α\nd₁ : Nodup (t₁ ++ b :: t₂)\nh₁ : ¬a ∈ t₁ ++ b :: t₂\np : b :: r₁ ~ t₁ ++ b :: t₂\nbm : b ∈ t₁ ++ b :: t₂\n⊢ a :: (t₁ ++ b :: t₂) <+~ b :: r₂", "state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\na✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ b :: r₂\np : b :: r₁ ~ l₁\nbm : b ∈ l₁\nam : a ∈ r₂\n⊢ a :: l₁ <+~ b :: r₂", "tactic": "rcases mem_split bm with ⟨t₁, t₂, rfl⟩" }, { "state_after": "case intro.intro\nα : Type uu\nβ : Type vv\nl₁ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\na✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nh₂ : a ∈ b :: r₂\nam : a ∈ r₂\nt₁ t₂ : List α\nd₁ : Nodup (t₁ ++ b :: t₂)\nh₁ : ¬a ∈ t₁ ++ b :: t₂\np : b :: r₁ ~ t₁ ++ b :: t₂\nbm : b ∈ t₁ ++ b :: t₂\nst : t₁ ++ t₂ <+ t₁ ++ b :: t₂\n⊢ a :: (t₁ ++ b :: t₂) <+~ b :: r₂", "state_before": "case intro.intro\nα : Type uu\nβ : Type vv\nl₁ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\na✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nh₂ : a ∈ b :: r₂\nam : a ∈ r₂\nt₁ t₂ : List α\nd₁ : Nodup (t₁ ++ b :: t₂)\nh₁ : ¬a ∈ t₁ ++ b :: t₂\np : b :: r₁ ~ t₁ ++ b :: t₂\nbm : b ∈ t₁ ++ b :: t₂\n⊢ a :: (t₁ ++ b :: t₂) <+~ b :: r₂", "tactic": "have st : t₁ ++ t₂ <+ t₁ ++ b :: t₂ := by simp" }, { "state_after": "case intro.intro.intro.intro\nα : Type uu\nβ : Type vv\nl₁ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\na✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nh₂ : a ∈ b :: r₂\nam : a ∈ r₂\nt₁ t₂ : List α\nd₁ : Nodup (t₁ ++ b :: t₂)\nh₁ : ¬a ∈ t₁ ++ b :: t₂\np : b :: r₁ ~ t₁ ++ b :: t₂\nbm : b ∈ t₁ ++ b :: t₂\nst : t₁ ++ t₂ <+ t₁ ++ b :: t₂\nt : List α\np' : t ~ a :: (t₁ ++ t₂)\ns' : t <+ r₂\n⊢ a :: (t₁ ++ b :: t₂) <+~ b :: r₂", "state_before": "case intro.intro\nα : Type uu\nβ : Type vv\nl₁ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\na✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nh₂ : a ∈ b :: r₂\nam : a ∈ r₂\nt₁ t₂ : List α\nd₁ : Nodup (t₁ ++ b :: t₂)\nh₁ : ¬a ∈ t₁ ++ b :: t₂\np : b :: r₁ ~ t₁ ++ b :: t₂\nbm : b ∈ t₁ ++ b :: t₂\nst : t₁ ++ t₂ <+ t₁ ++ b :: t₂\n⊢ a :: (t₁ ++ b :: t₂) <+~ b :: r₂", "tactic": "rcases ih (d₁.sublist st) (mt (fun x => st.subset x) h₁) am\n (Perm.cons_inv <| p.trans perm_middle) with\n ⟨t, p', s'⟩" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.intro\nα : Type uu\nβ : Type vv\nl₁ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\na✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nh₂ : a ∈ b :: r₂\nam : a ∈ r₂\nt₁ t₂ : List α\nd₁ : Nodup (t₁ ++ b :: t₂)\nh₁ : ¬a ∈ t₁ ++ b :: t₂\np : b :: r₁ ~ t₁ ++ b :: t₂\nbm : b ∈ t₁ ++ b :: t₂\nst : t₁ ++ t₂ <+ t₁ ++ b :: t₂\nt : List α\np' : t ~ a :: (t₁ ++ t₂)\ns' : t <+ r₂\n⊢ a :: (t₁ ++ b :: t₂) <+~ b :: r₂", "tactic": "exact\n ⟨b :: t, (p'.cons b).trans <| (swap _ _ _).trans (perm_middle.symm.cons a), s'.cons₂ _⟩" }, { "state_after": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\na✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\np : b :: r₁ ~ l₁\nbm : b ∈ l₁\nh₂ : a = b ∨ a ∈ r₂\n⊢ a ∈ r₂", "state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\na✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\nh₂ : a ∈ b :: r₂\np : b :: r₁ ~ l₁\nbm : b ∈ l₁\n⊢ a ∈ r₂", "tactic": "simp only [find?, mem_cons] at h₂" }, { "state_after": "no goals", "state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\na✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nl₁ : List α\nd₁ : Nodup l₁\nh₁ : ¬a ∈ l₁\np : b :: r₁ ~ l₁\nbm : b ∈ l₁\nh₂ : a = b ∨ a ∈ r₂\n⊢ a ∈ r₂", "tactic": "exact h₂.resolve_left fun e => h₁ <| e.symm ▸ bm" }, { "state_after": "no goals", "state_before": "α : Type uu\nβ : Type vv\nl₁ l₂✝ : List α\na : α\nl₂ l r₁ r₂ : List α\nb : α\na✝ : r₁ <+ r₂\nih : ∀ {l₁ : List α}, Nodup l₁ → ¬a ∈ l₁ → a ∈ r₂ → r₁ ~ l₁ → a :: l₁ <+~ r₂\nh₂ : a ∈ b :: r₂\nam : a ∈ r₂\nt₁ t₂ : List α\nd₁ : Nodup (t₁ ++ b :: t₂)\nh₁ : ¬a ∈ t₁ ++ b :: t₂\np : b :: r₁ ~ t₁ ++ b :: t₂\nbm : b ∈ t₁ ++ b :: t₂\n⊢ t₁ ++ t₂ <+ t₁ ++ b :: t₂", "tactic": "simp" } ]

[ 708, 94 ]

[ 685, 1 ]

[ { "state_after": "α : Type u_2\nβ : Type ?u.176434\nγ : Type ?u.176437\nι : Type ?u.176440\nM : Type u_1\nM' : Type ?u.176446\nN : Type ?u.176449\nP : Type ?u.176452\nG : Type ?u.176455\nH : Type ?u.176458\nR : Type ?u.176461\nS : Type ?u.176464\ninst✝¹ : Zero M\ninst✝ : DecidableEq M\ns : Finset α\nf : α → M\nhf : ∀ (a : α), f a ≠ 0 → a ∈ s\n⊢ filter (fun x => ¬f x = 0) s = filter (fun a => ¬f a = 0) s", "state_before": "α : Type u_2\nβ : Type ?u.176434\nγ : Type ?u.176437\nι : Type ?u.176440\nM : Type u_1\nM' : Type ?u.176446\nN : Type ?u.176449\nP : Type ?u.176452\nG : Type ?u.176455\nH : Type ?u.176458\nR : Type ?u.176461\nS : Type ?u.176464\ninst✝¹ : Zero M\ninst✝ : DecidableEq M\ns : Finset α\nf : α → M\nhf : ∀ (a : α), f a ≠ 0 → a ∈ s\n⊢ (onFinset s f hf).support = filter (fun a => f a ≠ 0) s", "tactic": "dsimp [onFinset]" }, { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type ?u.176434\nγ : Type ?u.176437\nι : Type ?u.176440\nM : Type u_1\nM' : Type ?u.176446\nN : Type ?u.176449\nP : Type ?u.176452\nG : Type ?u.176455\nH : Type ?u.176458\nR : Type ?u.176461\nS : Type ?u.176464\ninst✝¹ : Zero M\ninst✝ : DecidableEq M\ns : Finset α\nf : α → M\nhf : ∀ (a : α), f a ≠ 0 → a ∈ s\n⊢ filter (fun x => ¬f x = 0) s = filter (fun a => ¬f a = 0) s", "tactic": "congr" } ]

[ 719, 26 ]

[ 716, 1 ]

[ { "state_after": "no goals", "state_before": "S : Type ?u.120411\nT : Type ?u.120414\nR : Type u_1\ninst✝ : CommRing R\nc₁ c₂ r x y z : R\na b c : ℍ[R,c₁,c₂]\n⊢ ↑(x * y) = ↑x * ↑y", "tactic": "ext <;> simp" } ]

[ 479, 72 ]

[ 479, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u\nS : Type ?u.133744\nA : Type v\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Ring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\nB : Type u_1\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\na : A\nh : IsAlgebraic R a\np : R[X]\nhp : p ≠ 0\nha : ↑(aeval a) p = 0\n⊢ ↑(aeval (↑f a)) p = 0", "tactic": "rw [aeval_algHom, f.comp_apply, ha, map_zero]" } ]

[ 152, 60 ]

[ 149, 1 ]

[ { "state_after": "α : Type u_1\nsize✝ : ℕ\nl : Ordnode α\nx✝ : α\nr : Ordnode α\nb : BalancedSz (size l) (size r)\nbl : Balanced l\nbr : Balanced r\n⊢ BalancedSz (size r) (size l)", "state_before": "α : Type u_1\nsize✝ : ℕ\nl : Ordnode α\nx✝ : α\nr : Ordnode α\nb : BalancedSz (size l) (size r)\nbl : Balanced l\nbr : Balanced r\n⊢ BalancedSz (size (Ordnode.dual r)) (size (Ordnode.dual l))", "tactic": "rw [size_dual, size_dual]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nsize✝ : ℕ\nl : Ordnode α\nx✝ : α\nr : Ordnode α\nb : BalancedSz (size l) (size r)\nbl : Balanced l\nbr : Balanced r\n⊢ BalancedSz (size r) (size l)", "tactic": "exact b.symm" } ]

[ 222, 96 ]

[ 220, 1 ]

[]

[ 177, 45 ]

[ 176, 1 ]

[]

[ 2111, 6 ]

[ 2110, 1 ]

[]

[ 36, 6 ]

[ 34, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns✝ t✝ : Multiset α\na : α\nf : α → γ\ng : β → γ\ns : Multiset α\nt : Multiset β\nhs : Nodup s\nht : Nodup t\ni : (a : α) → a ∈ s → β\nhi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t\nh : ∀ (a : α) (ha : a ∈ s), f a = g (i a ha)\ni_inj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂\ni_surj : ∀ (b : β), b ∈ t → ∃ a ha, b = i a ha\nx : β\n⊢ x ∈ t ↔ ∃ a h, x = i a (_ : ↑{ val := a, property := (_ : a ∈ s) } ∈ s)", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns✝ t✝ : Multiset α\na : α\nf : α → γ\ng : β → γ\ns : Multiset α\nt : Multiset β\nhs : Nodup s\nht : Nodup t\ni : (a : α) → a ∈ s → β\nhi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t\nh : ∀ (a : α) (ha : a ∈ s), f a = g (i a ha)\ni_inj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂\ni_surj : ∀ (b : β), b ∈ t → ∃ a ha, b = i a ha\nx : β\n⊢ x ∈ t ↔ x ∈ map (fun x => i ↑x (_ : ↑x ∈ s)) (attach s)", "tactic": "simp only [mem_map, true_and_iff, Subtype.exists, eq_comm, mem_attach]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns✝ t✝ : Multiset α\na : α\nf : α → γ\ng : β → γ\ns : Multiset α\nt : Multiset β\nhs : Nodup s\nht : Nodup t\ni : (a : α) → a ∈ s → β\nhi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t\nh : ∀ (a : α) (ha : a ∈ s), f a = g (i a ha)\ni_inj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂\ni_surj : ∀ (b : β), b ∈ t → ∃ a ha, b = i a ha\nx : β\n⊢ x ∈ t ↔ ∃ a h, x = i a (_ : ↑{ val := a, property := (_ : a ∈ s) } ∈ s)", "tactic": "exact ⟨i_surj _, fun ⟨y, hy⟩ => hy.snd.symm ▸ hi _ _⟩" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns✝ t✝ : Multiset α\na : α\nf : α → γ\ng : β → γ\ns : Multiset α\nt : Multiset β\nhs : Nodup s\nht : Nodup t\ni : (a : α) → a ∈ s → β\nhi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t\nh : ∀ (a : α) (ha : a ∈ s), f a = g (i a ha)\ni_inj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂\ni_surj : ∀ (b : β), b ∈ t → ∃ a ha, b = i a ha\nthis : t = map (fun x => i ↑x (_ : ↑x ∈ s)) (attach s)\n⊢ map f s = pmap (fun x x_1 => f x) s (_ : ∀ (x : α), x ∈ s → x ∈ s)", "tactic": "rw [pmap_eq_map]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns✝ t✝ : Multiset α\na : α\nf : α → γ\ng : β → γ\ns : Multiset α\nt : Multiset β\nhs : Nodup s\nht : Nodup t\ni : (a : α) → a ∈ s → β\nhi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t\nh : ∀ (a : α) (ha : a ∈ s), f a = g (i a ha)\ni_inj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂\ni_surj : ∀ (b : β), b ∈ t → ∃ a ha, b = i a ha\nthis : t = map (fun x => i ↑x (_ : ↑x ∈ s)) (attach s)\n⊢ pmap (fun x x_1 => f x) s (_ : ∀ (x : α), x ∈ s → x ∈ s) = map (fun x => f ↑x) (attach s)", "tactic": "rw [pmap_eq_map_attach]" }, { "state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns✝ t✝ : Multiset α\na : α\nf : α → γ\ng : β → γ\ns : Multiset α\nt : Multiset β\nhs : Nodup s\nht : Nodup t\ni : (a : α) → a ∈ s → β\nhi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t\nh : ∀ (a : α) (ha : a ∈ s), f a = g (i a ha)\ni_inj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂\ni_surj : ∀ (b : β), b ∈ t → ∃ a ha, b = i a ha\nthis : t = map (fun x => i ↑x (_ : ↑x ∈ s)) (attach s)\n⊢ map (fun x => f ↑x) (attach s) = map (g ∘ fun x => i ↑x (_ : ↑x ∈ s)) (attach s)", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns✝ t✝ : Multiset α\na : α\nf : α → γ\ng : β → γ\ns : Multiset α\nt : Multiset β\nhs : Nodup s\nht : Nodup t\ni : (a : α) → a ∈ s → β\nhi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t\nh : ∀ (a : α) (ha : a ∈ s), f a = g (i a ha)\ni_inj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂\ni_surj : ∀ (b : β), b ∈ t → ∃ a ha, b = i a ha\nthis : t = map (fun x => i ↑x (_ : ↑x ∈ s)) (attach s)\n⊢ map (fun x => f ↑x) (attach s) = map g t", "tactic": "rw [this, Multiset.map_map]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns✝ t✝ : Multiset α\na : α\nf : α → γ\ng : β → γ\ns : Multiset α\nt : Multiset β\nhs : Nodup s\nht : Nodup t\ni : (a : α) → a ∈ s → β\nhi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t\nh : ∀ (a : α) (ha : a ∈ s), f a = g (i a ha)\ni_inj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂\ni_surj : ∀ (b : β), b ∈ t → ∃ a ha, b = i a ha\nthis : t = map (fun x => i ↑x (_ : ↑x ∈ s)) (attach s)\n⊢ map (fun x => f ↑x) (attach s) = map (g ∘ fun x => i ↑x (_ : ↑x ∈ s)) (attach s)", "tactic": "exact map_congr rfl fun x _ => h _ _" } ]

[ 273, 88 ]

[ 258, 1 ]

[ { "state_after": "no goals", "state_before": "α : Sort u\nβ : Sort v\nγ : Sort w\nα' : Type u_2\nβ' : Type u_1\ne : α' ≃ β'\n⊢ ↑(permCongr e) (Equiv.refl α') = Equiv.refl β'", "tactic": "simp [permCongr_def]" } ]

[ 436, 23 ]

[ 435, 9 ]

[ { "state_after": "no goals", "state_before": "n : ℕ\nα : Type u_1\nβ : Type ?u.60181\nγ : Type ?u.60184\nv : Vector α n\ni j : Fin n\na : α\n⊢ get (set v i a) j = if i = j then a else get v j", "tactic": "split_ifs <;> try simp [*] <;> try rw [get_set_of_ne] ; assumption" }, { "state_after": "no goals", "state_before": "case inr\nn : ℕ\nα : Type u_1\nβ : Type ?u.60181\nγ : Type ?u.60184\nv : Vector α n\ni j : Fin n\na : α\nh✝ : ¬i = j\n⊢ get (set v i a) j = get v j", "tactic": "simp [*] <;> try rw [get_set_of_ne] ; assumption" }, { "state_after": "case inr.h\nn : ℕ\nα : Type u_1\nβ : Type ?u.60181\nγ : Type ?u.60184\nv : Vector α n\ni j : Fin n\na : α\nh✝ : ¬i = j\n⊢ i ≠ j", "state_before": "case inr\nn : ℕ\nα : Type u_1\nβ : Type ?u.60181\nγ : Type ?u.60184\nv : Vector α n\ni j : Fin n\na : α\nh✝ : ¬i = j\n⊢ get (set v i a) j = get v j", "tactic": "rw [get_set_of_ne]" }, { "state_after": "no goals", "state_before": "case inr.h\nn : ℕ\nα : Type u_1\nβ : Type ?u.60181\nγ : Type ?u.60184\nv : Vector α n\ni j : Fin n\na : α\nh✝ : ¬i = j\n⊢ i ≠ j", "tactic": "assumption" } ]

[ 609, 69 ]

[ 607, 1 ]

[ { "state_after": "case pos\nα : Type ?u.98739\nβ : Type ?u.98742\nι : Type ?u.98745\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\nh0 : r₁ = 0\n⊢ (fun n => ↑n ^ k * r₁ ^ n) =o[atTop] fun n => r₂ ^ n\n\ncase neg\nα : Type ?u.98739\nβ : Type ?u.98742\nι : Type ?u.98745\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\nh0 : ¬r₁ = 0\n⊢ (fun n => ↑n ^ k * r₁ ^ n) =o[atTop] fun n => r₂ ^ n", "state_before": "α : Type ?u.98739\nβ : Type ?u.98742\nι : Type ?u.98745\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\n⊢ (fun n => ↑n ^ k * r₁ ^ n) =o[atTop] fun n => r₂ ^ n", "tactic": "by_cases h0 : r₁ = 0" }, { "state_after": "case neg\nα : Type ?u.98739\nβ : Type ?u.98742\nι : Type ?u.98745\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\nh0 : 0 < ‖r₁‖\n⊢ (fun n => ↑n ^ k * r₁ ^ n) =o[atTop] fun n => r₂ ^ n", "state_before": "case neg\nα : Type ?u.98739\nβ : Type ?u.98742\nι : Type ?u.98745\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\nh0 : ¬r₁ = 0\n⊢ (fun n => ↑n ^ k * r₁ ^ n) =o[atTop] fun n => r₂ ^ n", "tactic": "rw [← Ne.def, ← norm_pos_iff] at h0" }, { "state_after": "case neg\nα : Type ?u.98739\nβ : Type ?u.98742\nι : Type ?u.98745\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\nh0 : 0 < ‖r₁‖\nA : (fun n => ↑n ^ k) =o[atTop] fun n => (r₂ / ‖r₁‖) ^ n\n⊢ (fun n => ↑n ^ k * r₁ ^ n) =o[atTop] fun n => r₂ ^ n", "state_before": "case neg\nα : Type ?u.98739\nβ : Type ?u.98742\nι : Type ?u.98745\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\nh0 : 0 < ‖r₁‖\n⊢ (fun n => ↑n ^ k * r₁ ^ n) =o[atTop] fun n => r₂ ^ n", "tactic": "have A : (fun n ↦ (n : R) ^ k : ℕ → R) =o[atTop] fun n ↦ (r₂ / ‖r₁‖) ^ n :=\n isLittleO_pow_const_const_pow_of_one_lt k ((one_lt_div h0).2 h)" }, { "state_after": "case neg\nα : Type ?u.98739\nβ : Type ?u.98742\nι : Type ?u.98745\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\nh0 : 0 < ‖r₁‖\nA : (fun n => ↑n ^ k) =o[atTop] fun n => (r₂ / ‖r₁‖) ^ n\n⊢ (fun n => r₁ ^ n) =O[atTop] fun n => ‖r₁‖ ^ n", "state_before": "case neg\nα : Type ?u.98739\nβ : Type ?u.98742\nι : Type ?u.98745\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\nh0 : 0 < ‖r₁‖\nA : (fun n => ↑n ^ k) =o[atTop] fun n => (r₂ / ‖r₁‖) ^ n\n⊢ (fun n => ↑n ^ k * r₁ ^ n) =o[atTop] fun n => r₂ ^ n", "tactic": "suffices (fun n ↦ r₁ ^ n) =O[atTop] fun n ↦ ‖r₁‖ ^ n by\n simpa [div_mul_cancel _ (pow_pos h0 _).ne'] using A.mul_isBigO this" }, { "state_after": "no goals", "state_before": "case neg\nα : Type ?u.98739\nβ : Type ?u.98742\nι : Type ?u.98745\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\nh0 : 0 < ‖r₁‖\nA : (fun n => ↑n ^ k) =o[atTop] fun n => (r₂ / ‖r₁‖) ^ n\n⊢ (fun n => r₁ ^ n) =O[atTop] fun n => ‖r₁‖ ^ n", "tactic": "exact IsBigO.of_bound 1 (by simpa using eventually_norm_pow_le r₁)" }, { "state_after": "case pos\nα : Type ?u.98739\nβ : Type ?u.98742\nι : Type ?u.98745\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\nh0 : r₁ = 0\nn : ℕ\nhn : n ≥ 1\n⊢ n ∈ {x | (fun x => (fun _x => 0) x = (fun n => ↑n ^ k * r₁ ^ n) x) x}", "state_before": "case pos\nα : Type ?u.98739\nβ : Type ?u.98742\nι : Type ?u.98745\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\nh0 : r₁ = 0\n⊢ (fun n => ↑n ^ k * r₁ ^ n) =o[atTop] fun n => r₂ ^ n", "tactic": "refine' (isLittleO_zero _ _).congr' (mem_atTop_sets.2 <| ⟨1, fun n hn ↦ _⟩) EventuallyEq.rfl" }, { "state_after": "no goals", "state_before": "case pos\nα : Type ?u.98739\nβ : Type ?u.98742\nι : Type ?u.98745\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\nh0 : r₁ = 0\nn : ℕ\nhn : n ≥ 1\n⊢ n ∈ {x | (fun x => (fun _x => 0) x = (fun n => ↑n ^ k * r₁ ^ n) x) x}", "tactic": "simp [zero_pow (zero_lt_one.trans_le hn), h0]" }, { "state_after": "no goals", "state_before": "α : Type ?u.98739\nβ : Type ?u.98742\nι : Type ?u.98745\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\nh0 : 0 < ‖r₁‖\nA : (fun n => ↑n ^ k) =o[atTop] fun n => (r₂ / ‖r₁‖) ^ n\nthis : (fun n => r₁ ^ n) =O[atTop] fun n => ‖r₁‖ ^ n\n⊢ (fun n => ↑n ^ k * r₁ ^ n) =o[atTop] fun n => r₂ ^ n", "tactic": "simpa [div_mul_cancel _ (pow_pos h0 _).ne'] using A.mul_isBigO this" }, { "state_after": "no goals", "state_before": "α : Type ?u.98739\nβ : Type ?u.98742\nι : Type ?u.98745\nR : Type u_1\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\nh0 : 0 < ‖r₁‖\nA : (fun n => ↑n ^ k) =o[atTop] fun n => (r₂ / ‖r₁‖) ^ n\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖r₁ ^ x‖ ≤ 1 * ‖‖r₁‖ ^ x‖", "tactic": "simpa using eventually_norm_pow_le r₁" } ]

[ 227, 69 ]

[ 216, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.162120\nβ₂ : Type ?u.162123\nγ : Type ?u.162126\nι : Sort ?u.162129\nι' : Sort ?u.162132\nκ : ι → Sort ?u.162137\nκ' : ι' → Sort ?u.162142\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na b : α\nf : Bool → α\n⊢ (⨆ (b : Bool), f b) = f true ⊔ f false", "tactic": "rw [iSup, Bool.range_eq, sSup_pair, sup_comm]" } ]

[ 1515, 48 ]

[ 1514, 1 ]

[ { "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s\n⊢ direction (affineSpan k (↑s ∪ ↑(affineSpan k {p2}))) = direction s ⊔ Submodule.span k {p2 -ᵥ p1}", "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s\n⊢ direction (affineSpan k (insert p2 ↑s)) = Submodule.span k {p2 -ᵥ p1} ⊔ direction s", "tactic": "rw [sup_comm, ← Set.union_singleton, ← coe_affineSpan_singleton k V p2]" }, { "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s\n⊢ direction (s ⊔ affineSpan k {p2}) = direction s ⊔ Submodule.span k {p2 -ᵥ p1}", "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s\n⊢ direction (affineSpan k (↑s ∪ ↑(affineSpan k {p2}))) = direction s ⊔ Submodule.span k {p2 -ᵥ p1}", "tactic": "change (s ⊔ affineSpan k {p2}).direction = _" }, { "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s\n⊢ direction s ⊔ vectorSpan k {p2} ⊔ Submodule.span k {p2 -ᵥ p1} = direction s ⊔ Submodule.span k {p2 -ᵥ p1}", "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s\n⊢ direction (s ⊔ affineSpan k {p2}) = direction s ⊔ Submodule.span k {p2 -ᵥ p1}", "tactic": "rw [direction_sup hp1 (mem_affineSpan k (Set.mem_singleton _)), direction_affineSpan]" }, { "state_after": "no goals", "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s\n⊢ direction s ⊔ vectorSpan k {p2} ⊔ Submodule.span k {p2 -ᵥ p1} = direction s ⊔ Submodule.span k {p2 -ᵥ p1}", "tactic": "simp" } ]

[ 1452, 7 ]

[ 1446, 1 ]

[]

[ 573, 38 ]

[ 571, 1 ]

[]

[ 443, 47 ]

[ 442, 1 ]

[]

[ 559, 61 ]

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[]

[ 155, 6 ]

[ 154, 1 ]

[ { "state_after": "no goals", "state_before": "x : ℝ\nhx₁ : -π < x\nhx₂ : x < π\nh : x = 0\n⊢ sin x = 0", "tactic": "simp [h]" } ]

[ 506, 26 ]

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[]

[ 282, 54 ]

[ 280, 1 ]

[ { "state_after": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.282975\nP : Type u_3\nP' : Type ?u.282981\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhyz : WOppSide s y z\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\n⊢ WOppSide s x z", "state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.282975\nP : Type u_3\nP' : Type ?u.282981\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhxy : WSameSide s x y\nhyz : WOppSide s y z\nhy : ¬y ∈ s\n⊢ WOppSide s x z", "tactic": "rcases hxy with ⟨p₁, hp₁, p₂, hp₂, hxy⟩" }, { "state_after": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.282975\nP : Type u_3\nP' : Type ?u.282981\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhyz : ∃ p₂_1, p₂_1 ∈ s ∧ SameRay R (y -ᵥ p₂) (p₂_1 -ᵥ z)\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\n⊢ WOppSide s x z", "state_before": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.282975\nP : Type u_3\nP' : Type ?u.282981\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhyz : WOppSide s y z\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\n⊢ WOppSide s x z", "tactic": "rw [wOppSide_iff_exists_left hp₂, or_iff_right hy] at hyz" }, { "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.282975\nP : Type u_3\nP' : Type ?u.282981\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (p₃ -ᵥ z)\n⊢ WOppSide s x z", "state_before": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.282975\nP : Type u_3\nP' : Type ?u.282981\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhyz : ∃ p₂_1, p₂_1 ∈ s ∧ SameRay R (y -ᵥ p₂) (p₂_1 -ᵥ z)\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\n⊢ WOppSide s x z", "tactic": "rcases hyz with ⟨p₃, hp₃, hyz⟩" }, { "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.282975\nP : Type u_3\nP' : Type ?u.282981\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (p₃ -ᵥ z)\n⊢ y -ᵥ p₂ = 0 → x -ᵥ p₁ = 0 ∨ p₃ -ᵥ z = 0", "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.282975\nP : Type u_3\nP' : Type ?u.282981\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (p₃ -ᵥ z)\n⊢ WOppSide s x z", "tactic": "refine' ⟨p₁, hp₁, p₃, hp₃, hxy.trans hyz _⟩" }, { "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.282975\nP : Type u_3\nP' : Type ?u.282981\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (p₃ -ᵥ z)\nh : y -ᵥ p₂ = 0\n⊢ False", "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.282975\nP : Type u_3\nP' : Type ?u.282981\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (p₃ -ᵥ z)\n⊢ y -ᵥ p₂ = 0 → x -ᵥ p₁ = 0 ∨ p₃ -ᵥ z = 0", "tactic": "refine' fun h => False.elim _" }, { "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.282975\nP : Type u_3\nP' : Type ?u.282981\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (p₃ -ᵥ z)\nh : y = p₂\n⊢ False", "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.282975\nP : Type u_3\nP' : Type ?u.282981\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (p₃ -ᵥ z)\nh : y -ᵥ p₂ = 0\n⊢ False", "tactic": "rw [vsub_eq_zero_iff_eq] at h" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.282975\nP : Type u_3\nP' : Type ?u.282981\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (p₃ -ᵥ z)\nh : y = p₂\n⊢ False", "tactic": "exact hy (h.symm ▸ hp₂)" } ]

[ 549, 26 ]

[ 541, 1 ]

[]

[ 160, 29 ]

[ 159, 1 ]

[ { "state_after": "no goals", "state_before": "α✝ : Type ?u.230952\nE✝ : Type ?u.230955\nF : Type ?u.230958\nG : Type ?u.230961\nm m0 : MeasurableSpace α✝\np✝ : ℝ≥0∞\nq : ℝ\nμ✝ ν : Measure α✝\ninst✝⁴ : NormedAddCommGroup E✝\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedAddCommGroup G\nα : Type u_1\nE : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : α → E\nhfp : Memℒp f p\n⊢ snorm (↑(AEEqFun.mk f (_ : AEStronglyMeasurable f μ))) p μ < ⊤", "tactic": "simp [hfp.2]" } ]

[ 96, 58 ]

[ 94, 1 ]

[]

[ 403, 89 ]

[ 400, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type ?u.167857\nS : Type ?u.167860\n⊢ Module.rank ℝ ℂ = 2", "tactic": "simp [← finrank_eq_rank, finrank_real_complex]" } ]

[ 191, 101 ]

[ 191, 1 ]

[ { "state_after": "case refine'_1\nα : Type u_2\nβ✝ : Type ?u.2792\nγ : Type u_4\nι✝ : Sort ?u.2798\nf✝ f₁ f₂ : Filter α\ng✝ g₁ g₂ : Set α → Filter β✝\nι : Sort u_1\np : ι → Prop\ns✝ : ι → Set α\nf : Filter α\nhf : HasBasis f p s✝\nβ : ι → Type u_3\npg : (i : ι) → β i → Prop\nsg : (i : ι) → β i → Set γ\ng : Set α → Filter γ\nhg : ∀ (i : ι), HasBasis (g (s✝ i)) (pg i) (sg i)\ngm : Monotone g\ns : Set γ\n⊢ DirectedOn ((fun s => g s) ⁻¹'o fun x x_1 => x ≥ x_1) f.sets\n\ncase refine'_2\nα : Type u_2\nβ✝ : Type ?u.2792\nγ : Type u_4\nι✝ : Sort ?u.2798\nf✝ f₁ f₂ : Filter α\ng✝ g₁ g₂ : Set α → Filter β✝\nι : Sort u_1\np : ι → Prop\ns✝ : ι → Set α\nf : Filter α\nhf : HasBasis f p s✝\nβ : ι → Type u_3\npg : (i : ι) → β i → Prop\nsg : (i : ι) → β i → Set γ\ng : Set α → Filter γ\nhg : ∀ (i : ι), HasBasis (g (s✝ i)) (pg i) (sg i)\ngm : Monotone g\ns : Set γ\n⊢ (∃ i, i ∈ f.sets ∧ s ∈ g i) ↔ ∃ i, p i ∧ ∃ x, pg i x ∧ sg i x ⊆ s", "state_before": "α : Type u_2\nβ✝ : Type ?u.2792\nγ : Type u_4\nι✝ : Sort ?u.2798\nf✝ f₁ f₂ : Filter α\ng✝ g₁ g₂ : Set α → Filter β✝\nι : Sort u_1\np : ι → Prop\ns✝ : ι → Set α\nf : Filter α\nhf : HasBasis f p s✝\nβ : ι → Type u_3\npg : (i : ι) → β i → Prop\nsg : (i : ι) → β i → Set γ\ng : Set α → Filter γ\nhg : ∀ (i : ι), HasBasis (g (s✝ i)) (pg i) (sg i)\ngm : Monotone g\ns : Set γ\n⊢ s ∈ Filter.lift f g ↔ ∃ i, p i ∧ ∃ x, pg i x ∧ sg i x ⊆ s", "tactic": "refine' (mem_biInf_of_directed _ ⟨univ, univ_sets _⟩).trans _" }, { "state_after": "case refine'_1\nα : Type u_2\nβ✝ : Type ?u.2792\nγ : Type u_4\nι✝ : Sort ?u.2798\nf✝ f₁ f₂ : Filter α\ng✝ g₁ g₂ : Set α → Filter β✝\nι : Sort u_1\np : ι → Prop\ns✝ : ι → Set α\nf : Filter α\nhf : HasBasis f p s✝\nβ : ι → Type u_3\npg : (i : ι) → β i → Prop\nsg : (i : ι) → β i → Set γ\ng : Set α → Filter γ\nhg : ∀ (i : ι), HasBasis (g (s✝ i)) (pg i) (sg i)\ngm : Monotone g\ns : Set γ\nt₁ : Set α\nht₁ : t₁ ∈ f.sets\nt₂ : Set α\nht₂ : t₂ ∈ f.sets\n⊢ ∃ z, z ∈ f.sets ∧ ((fun s => g s) ⁻¹'o fun x x_1 => x ≥ x_1) t₁ z ∧ ((fun s => g s) ⁻¹'o fun x x_1 => x ≥ x_1) t₂ z", "state_before": "case refine'_1\nα : Type u_2\nβ✝ : Type ?u.2792\nγ : Type u_4\nι✝ : Sort ?u.2798\nf✝ f₁ f₂ : Filter α\ng✝ g₁ g₂ : Set α → Filter β✝\nι : Sort u_1\np : ι → Prop\ns✝ : ι → Set α\nf : Filter α\nhf : HasBasis f p s✝\nβ : ι → Type u_3\npg : (i : ι) → β i → Prop\nsg : (i : ι) → β i → Set γ\ng : Set α → Filter γ\nhg : ∀ (i : ι), HasBasis (g (s✝ i)) (pg i) (sg i)\ngm : Monotone g\ns : Set γ\n⊢ DirectedOn ((fun s => g s) ⁻¹'o fun x x_1 => x ≥ x_1) f.sets", "tactic": "intro t₁ ht₁ t₂ ht₂" }, { "state_after": "no goals", "state_before": "case refine'_1\nα : Type u_2\nβ✝ : Type ?u.2792\nγ : Type u_4\nι✝ : Sort ?u.2798\nf✝ f₁ f₂ : Filter α\ng✝ g₁ g₂ : Set α → Filter β✝\nι : Sort u_1\np : ι → Prop\ns✝ : ι → Set α\nf : Filter α\nhf : HasBasis f p s✝\nβ : ι → Type u_3\npg : (i : ι) → β i → Prop\nsg : (i : ι) → β i → Set γ\ng : Set α → Filter γ\nhg : ∀ (i : ι), HasBasis (g (s✝ i)) (pg i) (sg i)\ngm : Monotone g\ns : Set γ\nt₁ : Set α\nht₁ : t₁ ∈ f.sets\nt₂ : Set α\nht₂ : t₂ ∈ f.sets\n⊢ ∃ z, z ∈ f.sets ∧ ((fun s => g s) ⁻¹'o fun x x_1 => x ≥ x_1) t₁ z ∧ ((fun s => g s) ⁻¹'o fun x x_1 => x ≥ x_1) t₂ z", "tactic": "exact ⟨t₁ ∩ t₂, inter_mem ht₁ ht₂, gm <| inter_subset_left _ _, gm <| inter_subset_right _ _⟩" }, { "state_after": "case refine'_2\nα : Type u_2\nβ✝ : Type ?u.2792\nγ : Type u_4\nι✝ : Sort ?u.2798\nf✝ f₁ f₂ : Filter α\ng✝ g₁ g₂ : Set α → Filter β✝\nι : Sort u_1\np : ι → Prop\ns✝ : ι → Set α\nf : Filter α\nhf : HasBasis f p s✝\nβ : ι → Type u_3\npg : (i : ι) → β i → Prop\nsg : (i : ι) → β i → Set γ\ng : Set α → Filter γ\nhg : ∀ (i : ι), HasBasis (g (s✝ i)) (pg i) (sg i)\ngm : Monotone g\ns : Set γ\n⊢ (∃ i, i ∈ f.sets ∧ s ∈ g i) ↔ ∃ i, p i ∧ s ∈ g (s✝ i)", "state_before": "case refine'_2\nα : Type u_2\nβ✝ : Type ?u.2792\nγ : Type u_4\nι✝ : Sort ?u.2798\nf✝ f₁ f₂ : Filter α\ng✝ g₁ g₂ : Set α → Filter β✝\nι : Sort u_1\np : ι → Prop\ns✝ : ι → Set α\nf : Filter α\nhf : HasBasis f p s✝\nβ : ι → Type u_3\npg : (i : ι) → β i → Prop\nsg : (i : ι) → β i → Set γ\ng : Set α → Filter γ\nhg : ∀ (i : ι), HasBasis (g (s✝ i)) (pg i) (sg i)\ngm : Monotone g\ns : Set γ\n⊢ (∃ i, i ∈ f.sets ∧ s ∈ g i) ↔ ∃ i, p i ∧ ∃ x, pg i x ∧ sg i x ⊆ s", "tactic": "simp only [← (hg _).mem_iff]" }, { "state_after": "no goals", "state_before": "case refine'_2\nα : Type u_2\nβ✝ : Type ?u.2792\nγ : Type u_4\nι✝ : Sort ?u.2798\nf✝ f₁ f₂ : Filter α\ng✝ g₁ g₂ : Set α → Filter β✝\nι : Sort u_1\np : ι → Prop\ns✝ : ι → Set α\nf : Filter α\nhf : HasBasis f p s✝\nβ : ι → Type u_3\npg : (i : ι) → β i → Prop\nsg : (i : ι) → β i → Set γ\ng : Set α → Filter γ\nhg : ∀ (i : ι), HasBasis (g (s✝ i)) (pg i) (sg i)\ngm : Monotone g\ns : Set γ\n⊢ (∃ i, i ∈ f.sets ∧ s ∈ g i) ↔ ∃ i, p i ∧ s ∈ g (s✝ i)", "tactic": "exact hf.exists_iff fun t₁ t₂ ht H => gm ht H" } ]

[ 55, 50 ]

[ 47, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nδ : Type u_3\nf : α → β → δ\nl : AssocList α β\n⊢ toList (mapVal f l) =\n List.map\n (fun x =>\n match x with\n | (a, b) => (a, f a b))\n (toList l)", "tactic": "induction l <;> simp [*]" } ]

[ 96, 27 ]

[ 94, 9 ]

[]

[ 354, 6 ]

[ 352, 1 ]

[ { "state_after": "R : Type u_1\nR₁ : Type ?u.395006\nR₂ : Type ?u.395009\nR₃ : Type ?u.395012\nM : Type u_3\nM₁ : Type u_2\nM₂ : Type ?u.395021\nMₗ₁ : Type ?u.395024\nMₗ₁' : Type ?u.395027\nMₗ₂ : Type ?u.395030\nMₗ₂' : Type ?u.395033\nK : Type ?u.395036\nK₁ : Type ?u.395039\nK₂ : Type ?u.395042\nV : Type ?u.395045\nV₁ : Type ?u.395048\nV₂ : Type ?u.395051\nn : Type ?u.395054\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB F : M →ₗ[R] M →ₗ[R] R\ne : M₁ ≃ₗ[R] M\nf : Module.End R M\nhₗ : comp (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = compl₁₂ (comp F f) ↑e ↑e\n⊢ IsPairSelfAdjoint B F f ↔\n IsPairSelfAdjoint (compl₁₂ B ↑e ↑e) (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)", "state_before": "R : Type u_1\nR₁ : Type ?u.395006\nR₂ : Type ?u.395009\nR₃ : Type ?u.395012\nM : Type u_3\nM₁ : Type u_2\nM₂ : Type ?u.395021\nMₗ₁ : Type ?u.395024\nMₗ₁' : Type ?u.395027\nMₗ₂ : Type ?u.395030\nMₗ₂' : Type ?u.395033\nK : Type ?u.395036\nK₁ : Type ?u.395039\nK₂ : Type ?u.395042\nV : Type ?u.395045\nV₁ : Type ?u.395048\nV₂ : Type ?u.395051\nn : Type ?u.395054\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB F : M →ₗ[R] M →ₗ[R] R\ne : M₁ ≃ₗ[R] M\nf : Module.End R M\n⊢ IsPairSelfAdjoint B F f ↔\n IsPairSelfAdjoint (compl₁₂ B ↑e ↑e) (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)", "tactic": "have hₗ :\n (F.compl₁₂ (↑e : M₁ →ₗ[R] M) (↑e : M₁ →ₗ[R] M)).comp (e.symm.conj f) =\n (F.comp f).compl₁₂ (↑e : M₁ →ₗ[R] M) (↑e : M₁ →ₗ[R] M) := by\n ext\n simp only [LinearEquiv.symm_conj_apply, coe_comp, LinearEquiv.coe_coe, compl₁₂_apply,\n LinearEquiv.apply_symm_apply, Function.comp_apply]" }, { "state_after": "R : Type u_1\nR₁ : Type ?u.395006\nR₂ : Type ?u.395009\nR₃ : Type ?u.395012\nM : Type u_3\nM₁ : Type u_2\nM₂ : Type ?u.395021\nMₗ₁ : Type ?u.395024\nMₗ₁' : Type ?u.395027\nMₗ₂ : Type ?u.395030\nMₗ₂' : Type ?u.395033\nK : Type ?u.395036\nK₁ : Type ?u.395039\nK₂ : Type ?u.395042\nV : Type ?u.395045\nV₁ : Type ?u.395048\nV₂ : Type ?u.395051\nn : Type ?u.395054\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB F : M →ₗ[R] M →ₗ[R] R\ne : M₁ ≃ₗ[R] M\nf : Module.End R M\nhₗ : comp (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = compl₁₂ (comp F f) ↑e ↑e\nhᵣ : compl₂ (compl₁₂ B ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = compl₁₂ (compl₂ B f) ↑e ↑e\n⊢ IsPairSelfAdjoint B F f ↔\n IsPairSelfAdjoint (compl₁₂ B ↑e ↑e) (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)", "state_before": "R : Type u_1\nR₁ : Type ?u.395006\nR₂ : Type ?u.395009\nR₃ : Type ?u.395012\nM : Type u_3\nM₁ : Type u_2\nM₂ : Type ?u.395021\nMₗ₁ : Type ?u.395024\nMₗ₁' : Type ?u.395027\nMₗ₂ : Type ?u.395030\nMₗ₂' : Type ?u.395033\nK : Type ?u.395036\nK₁ : Type ?u.395039\nK₂ : Type ?u.395042\nV : Type ?u.395045\nV₁ : Type ?u.395048\nV₂ : Type ?u.395051\nn : Type ?u.395054\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB F : M →ₗ[R] M →ₗ[R] R\ne : M₁ ≃ₗ[R] M\nf : Module.End R M\nhₗ : comp (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = compl₁₂ (comp F f) ↑e ↑e\n⊢ IsPairSelfAdjoint B F f ↔\n IsPairSelfAdjoint (compl₁₂ B ↑e ↑e) (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)", "tactic": "have hᵣ :\n (B.compl₁₂ (↑e : M₁ →ₗ[R] M) (↑e : M₁ →ₗ[R] M)).compl₂ (e.symm.conj f) =\n (B.compl₂ f).compl₁₂ (↑e : M₁ →ₗ[R] M) (↑e : M₁ →ₗ[R] M) := by\n ext\n simp only [LinearEquiv.symm_conj_apply, compl₂_apply, coe_comp, LinearEquiv.coe_coe,\n compl₁₂_apply, LinearEquiv.apply_symm_apply, Function.comp_apply]" }, { "state_after": "R : Type u_1\nR₁ : Type ?u.395006\nR₂ : Type ?u.395009\nR₃ : Type ?u.395012\nM : Type u_3\nM₁ : Type u_2\nM₂ : Type ?u.395021\nMₗ₁ : Type ?u.395024\nMₗ₁' : Type ?u.395027\nMₗ₂ : Type ?u.395030\nMₗ₂' : Type ?u.395033\nK : Type ?u.395036\nK₁ : Type ?u.395039\nK₂ : Type ?u.395042\nV : Type ?u.395045\nV₁ : Type ?u.395048\nV₂ : Type ?u.395051\nn : Type ?u.395054\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB F : M →ₗ[R] M →ₗ[R] R\ne : M₁ ≃ₗ[R] M\nf : Module.End R M\nhₗ : comp (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = compl₁₂ (comp F f) ↑e ↑e\nhᵣ : compl₂ (compl₁₂ B ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = compl₁₂ (compl₂ B f) ↑e ↑e\nhe : Function.Surjective ↑↑e\n⊢ IsPairSelfAdjoint B F f ↔\n IsPairSelfAdjoint (compl₁₂ B ↑e ↑e) (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)", "state_before": "R : Type u_1\nR₁ : Type ?u.395006\nR₂ : Type ?u.395009\nR₃ : Type ?u.395012\nM : Type u_3\nM₁ : Type u_2\nM₂ : Type ?u.395021\nMₗ₁ : Type ?u.395024\nMₗ₁' : Type ?u.395027\nMₗ₂ : Type ?u.395030\nMₗ₂' : Type ?u.395033\nK : Type ?u.395036\nK₁ : Type ?u.395039\nK₂ : Type ?u.395042\nV : Type ?u.395045\nV₁ : Type ?u.395048\nV₂ : Type ?u.395051\nn : Type ?u.395054\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB F : M →ₗ[R] M →ₗ[R] R\ne : M₁ ≃ₗ[R] M\nf : Module.End R M\nhₗ : comp (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = compl₁₂ (comp F f) ↑e ↑e\nhᵣ : compl₂ (compl₁₂ B ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = compl₁₂ (compl₂ B f) ↑e ↑e\n⊢ IsPairSelfAdjoint B F f ↔\n IsPairSelfAdjoint (compl₁₂ B ↑e ↑e) (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)", "tactic": "have he : Function.Surjective (⇑(↑e : M₁ →ₗ[R] M) : M₁ → M) := e.surjective" }, { "state_after": "no goals", "state_before": "R : Type u_1\nR₁ : Type ?u.395006\nR₂ : Type ?u.395009\nR₃ : Type ?u.395012\nM : Type u_3\nM₁ : Type u_2\nM₂ : Type ?u.395021\nMₗ₁ : Type ?u.395024\nMₗ₁' : Type ?u.395027\nMₗ₂ : Type ?u.395030\nMₗ₂' : Type ?u.395033\nK : Type ?u.395036\nK₁ : Type ?u.395039\nK₂ : Type ?u.395042\nV : Type ?u.395045\nV₁ : Type ?u.395048\nV₂ : Type ?u.395051\nn : Type ?u.395054\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB F : M →ₗ[R] M →ₗ[R] R\ne : M₁ ≃ₗ[R] M\nf : Module.End R M\nhₗ : comp (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = compl₁₂ (comp F f) ↑e ↑e\nhᵣ : compl₂ (compl₁₂ B ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = compl₁₂ (compl₂ B f) ↑e ↑e\nhe : Function.Surjective ↑↑e\n⊢ IsPairSelfAdjoint B F f ↔\n IsPairSelfAdjoint (compl₁₂ B ↑e ↑e) (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)", "tactic": "simp_rw [IsPairSelfAdjoint, isAdjointPair_iff_comp_eq_compl₂, hₗ, hᵣ, compl₁₂_inj he he]" }, { "state_after": "case h.h\nR : Type u_1\nR₁ : Type ?u.395006\nR₂ : Type ?u.395009\nR₃ : Type ?u.395012\nM : Type u_3\nM₁ : Type u_2\nM₂ : Type ?u.395021\nMₗ₁ : Type ?u.395024\nMₗ₁' : Type ?u.395027\nMₗ₂ : Type ?u.395030\nMₗ₂' : Type ?u.395033\nK : Type ?u.395036\nK₁ : Type ?u.395039\nK₂ : Type ?u.395042\nV : Type ?u.395045\nV₁ : Type ?u.395048\nV₂ : Type ?u.395051\nn : Type ?u.395054\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB F : M →ₗ[R] M →ₗ[R] R\ne : M₁ ≃ₗ[R] M\nf : Module.End R M\nx✝¹ x✝ : M₁\n⊢ ↑(↑(comp (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)) x✝¹) x✝ =\n ↑(↑(compl₁₂ (comp F f) ↑e ↑e) x✝¹) x✝", "state_before": "R : Type u_1\nR₁ : Type ?u.395006\nR₂ : Type ?u.395009\nR₃ : Type ?u.395012\nM : Type u_3\nM₁ : Type u_2\nM₂ : Type ?u.395021\nMₗ₁ : Type ?u.395024\nMₗ₁' : Type ?u.395027\nMₗ₂ : Type ?u.395030\nMₗ₂' : Type ?u.395033\nK : Type ?u.395036\nK₁ : Type ?u.395039\nK₂ : Type ?u.395042\nV : Type ?u.395045\nV₁ : Type ?u.395048\nV₂ : Type ?u.395051\nn : Type ?u.395054\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB F : M →ₗ[R] M →ₗ[R] R\ne : M₁ ≃ₗ[R] M\nf : Module.End R M\n⊢ comp (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = compl₁₂ (comp F f) ↑e ↑e", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case h.h\nR : Type u_1\nR₁ : Type ?u.395006\nR₂ : Type ?u.395009\nR₃ : Type ?u.395012\nM : Type u_3\nM₁ : Type u_2\nM₂ : Type ?u.395021\nMₗ₁ : Type ?u.395024\nMₗ₁' : Type ?u.395027\nMₗ₂ : Type ?u.395030\nMₗ₂' : Type ?u.395033\nK : Type ?u.395036\nK₁ : Type ?u.395039\nK₂ : Type ?u.395042\nV : Type ?u.395045\nV₁ : Type ?u.395048\nV₂ : Type ?u.395051\nn : Type ?u.395054\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB F : M →ₗ[R] M →ₗ[R] R\ne : M₁ ≃ₗ[R] M\nf : Module.End R M\nx✝¹ x✝ : M₁\n⊢ ↑(↑(comp (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)) x✝¹) x✝ =\n ↑(↑(compl₁₂ (comp F f) ↑e ↑e) x✝¹) x✝", "tactic": "simp only [LinearEquiv.symm_conj_apply, coe_comp, LinearEquiv.coe_coe, compl₁₂_apply,\n LinearEquiv.apply_symm_apply, Function.comp_apply]" }, { "state_after": "case h.h\nR : Type u_1\nR₁ : Type ?u.395006\nR₂ : Type ?u.395009\nR₃ : Type ?u.395012\nM : Type u_3\nM₁ : Type u_2\nM₂ : Type ?u.395021\nMₗ₁ : Type ?u.395024\nMₗ₁' : Type ?u.395027\nMₗ₂ : Type ?u.395030\nMₗ₂' : Type ?u.395033\nK : Type ?u.395036\nK₁ : Type ?u.395039\nK₂ : Type ?u.395042\nV : Type ?u.395045\nV₁ : Type ?u.395048\nV₂ : Type ?u.395051\nn : Type ?u.395054\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB F : M →ₗ[R] M →ₗ[R] R\ne : M₁ ≃ₗ[R] M\nf : Module.End R M\nhₗ : comp (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = compl₁₂ (comp F f) ↑e ↑e\nx✝¹ x✝ : M₁\n⊢ ↑(↑(compl₂ (compl₁₂ B ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)) x✝¹) x✝ =\n ↑(↑(compl₁₂ (compl₂ B f) ↑e ↑e) x✝¹) x✝", "state_before": "R : Type u_1\nR₁ : Type ?u.395006\nR₂ : Type ?u.395009\nR₃ : Type ?u.395012\nM : Type u_3\nM₁ : Type u_2\nM₂ : Type ?u.395021\nMₗ₁ : Type ?u.395024\nMₗ₁' : Type ?u.395027\nMₗ₂ : Type ?u.395030\nMₗ₂' : Type ?u.395033\nK : Type ?u.395036\nK₁ : Type ?u.395039\nK₂ : Type ?u.395042\nV : Type ?u.395045\nV₁ : Type ?u.395048\nV₂ : Type ?u.395051\nn : Type ?u.395054\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB F : M →ₗ[R] M →ₗ[R] R\ne : M₁ ≃ₗ[R] M\nf : Module.End R M\nhₗ : comp (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = compl₁₂ (comp F f) ↑e ↑e\n⊢ compl₂ (compl₁₂ B ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = compl₁₂ (compl₂ B f) ↑e ↑e", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case h.h\nR : Type u_1\nR₁ : Type ?u.395006\nR₂ : Type ?u.395009\nR₃ : Type ?u.395012\nM : Type u_3\nM₁ : Type u_2\nM₂ : Type ?u.395021\nMₗ₁ : Type ?u.395024\nMₗ₁' : Type ?u.395027\nMₗ₂ : Type ?u.395030\nMₗ₂' : Type ?u.395033\nK : Type ?u.395036\nK₁ : Type ?u.395039\nK₂ : Type ?u.395042\nV : Type ?u.395045\nV₁ : Type ?u.395048\nV₂ : Type ?u.395051\nn : Type ?u.395054\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB F : M →ₗ[R] M →ₗ[R] R\ne : M₁ ≃ₗ[R] M\nf : Module.End R M\nhₗ : comp (compl₁₂ F ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f) = compl₁₂ (comp F f) ↑e ↑e\nx✝¹ x✝ : M₁\n⊢ ↑(↑(compl₂ (compl₁₂ B ↑e ↑e) (↑(LinearEquiv.conj (LinearEquiv.symm e)) f)) x✝¹) x✝ =\n ↑(↑(compl₁₂ (compl₂ B f) ↑e ↑e) x✝¹) x✝", "tactic": "simp only [LinearEquiv.symm_conj_apply, compl₂_apply, coe_comp, LinearEquiv.coe_coe,\n compl₁₂_apply, LinearEquiv.apply_symm_apply, Function.comp_apply]" } ]

[ 585, 91 ]

[ 569, 1 ]

[]

[ 597, 11 ]

[ 596, 1 ]

[]

[ 135, 60 ]

[ 135, 1 ]

[ { "state_after": "R : Type u\nι : Type ?u.281942\ninst✝ : CommSemiring R\nI J K L : Ideal R\nn : ℕ\nh : I ⊔ J = ⊤\n⊢ (∏ _x in Finset.range n, I) ⊔ J = ⊤", "state_before": "R : Type u\nι : Type ?u.281942\ninst✝ : CommSemiring R\nI J K L : Ideal R\nn : ℕ\nh : I ⊔ J = ⊤\n⊢ I ^ n ⊔ J = ⊤", "tactic": "rw [← Finset.card_range n, ← Finset.prod_const]" }, { "state_after": "no goals", "state_before": "R : Type u\nι : Type ?u.281942\ninst✝ : CommSemiring R\nI J K L : Ideal R\nn : ℕ\nh : I ⊔ J = ⊤\n⊢ (∏ _x in Finset.range n, I) ⊔ J = ⊤", "tactic": "exact prod_sup_eq_top fun _ _ => h" } ]

[ 734, 37 ]

[ 732, 1 ]

[ { "state_after": "case nil\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ map g' (permutationsAux2 t ts r [] f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g []) f').snd\n\ncase cons\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ map g' (permutationsAux2 t ts r (ys_hd :: tail✝) f).snd =\n (permutationsAux2 (g t) (map g ts) (map g' r) (map g (ys_hd :: tail✝)) f').snd", "state_before": "α✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts ys : List α\nr : List β\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ map g' (permutationsAux2 t ts r ys f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g ys) f').snd", "tactic": "induction' ys with ys_hd _ ys_ih generalizing f f'" }, { "state_after": "case cons\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ map g' (permutationsAux2 t ts r (ys_hd :: tail✝) f).snd =\n (permutationsAux2 (g t) (map g ts) (map g' r) (map g (ys_hd :: tail✝)) f').snd", "state_before": "case nil\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ map g' (permutationsAux2 t ts r [] f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g []) f').snd\n\ncase cons\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ map g' (permutationsAux2 t ts r (ys_hd :: tail✝) f).snd =\n (permutationsAux2 (g t) (map g ts) (map g' r) (map g (ys_hd :: tail✝)) f').snd", "tactic": ". simp" }, { "state_after": "no goals", "state_before": "case cons\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ map g' (permutationsAux2 t ts r (ys_hd :: tail✝) f).snd =\n (permutationsAux2 (g t) (map g ts) (map g' r) (map g (ys_hd :: tail✝)) f').snd", "tactic": ". simp only [map, permutationsAux2_snd_cons, cons_append, cons.injEq]\n rw [ys_ih, permutationsAux2_fst]\n refine' ⟨_, rfl⟩\n . simp only [← map_cons, ← map_append]; apply H\n . intro a; apply H" }, { "state_after": "no goals", "state_before": "case nil\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ map g' (permutationsAux2 t ts r [] f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g []) f').snd", "tactic": "simp" }, { "state_after": "case cons\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ g' (f (t :: ys_hd :: (permutationsAux2 t ts r tail✝ fun x => f (ys_hd :: x)).fst)) =\n f' (g t :: g ys_hd :: (map g tail✝ ++ map g ts)) ∧\n map g' (permutationsAux2 t ts r tail✝ fun x => f (ys_hd :: x)).snd =\n (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) fun x => f' (g ys_hd :: x)).snd", "state_before": "case cons\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ map g' (permutationsAux2 t ts r (ys_hd :: tail✝) f).snd =\n (permutationsAux2 (g t) (map g ts) (map g' r) (map g (ys_hd :: tail✝)) f').snd", "tactic": "simp only [map, permutationsAux2_snd_cons, cons_append, cons.injEq]" }, { "state_after": "case cons\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ g' (f (t :: ys_hd :: (tail✝ ++ ts))) = f' (g t :: g ys_hd :: (map g tail✝ ++ map g ts)) ∧\n (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) ?cons.f').snd =\n (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) fun x => f' (g ys_hd :: x)).snd\n\ncase cons.f'\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ List α' → β'\n\ncase cons.f'\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ List α' → β'\n\ncase cons.H\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ ∀ (a : List α), g' (f (ys_hd :: a)) = ?cons.f' (map g a)", "state_before": "case cons\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ g' (f (t :: ys_hd :: (permutationsAux2 t ts r tail✝ fun x => f (ys_hd :: x)).fst)) =\n f' (g t :: g ys_hd :: (map g tail✝ ++ map g ts)) ∧\n map g' (permutationsAux2 t ts r tail✝ fun x => f (ys_hd :: x)).snd =\n (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) fun x => f' (g ys_hd :: x)).snd", "tactic": "rw [ys_ih, permutationsAux2_fst]" }, { "state_after": "case cons\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ g' (f (t :: ys_hd :: (tail✝ ++ ts))) = f' (g t :: g ys_hd :: (map g tail✝ ++ map g ts))\n\ncase cons.H\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ ∀ (a : List α), g' (f (ys_hd :: a)) = f' (g ys_hd :: map g a)", "state_before": "case cons\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ g' (f (t :: ys_hd :: (tail✝ ++ ts))) = f' (g t :: g ys_hd :: (map g tail✝ ++ map g ts)) ∧\n (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) ?cons.f').snd =\n (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) fun x => f' (g ys_hd :: x)).snd\n\ncase cons.f'\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ List α' → β'\n\ncase cons.f'\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ List α' → β'\n\ncase cons.H\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ ∀ (a : List α), g' (f (ys_hd :: a)) = ?cons.f' (map g a)", "tactic": "refine' ⟨_, rfl⟩" }, { "state_after": "case cons.H\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ ∀ (a : List α), g' (f (ys_hd :: a)) = f' (g ys_hd :: map g a)", "state_before": "case cons\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ g' (f (t :: ys_hd :: (tail✝ ++ ts))) = f' (g t :: g ys_hd :: (map g tail✝ ++ map g ts))\n\ncase cons.H\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ ∀ (a : List α), g' (f (ys_hd :: a)) = f' (g ys_hd :: map g a)", "tactic": ". simp only [← map_cons, ← map_append]; apply H" }, { "state_after": "no goals", "state_before": "case cons.H\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ ∀ (a : List α), g' (f (ys_hd :: a)) = f' (g ys_hd :: map g a)", "tactic": ". intro a; apply H" }, { "state_after": "case cons\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ g' (f (t :: ys_hd :: (tail✝ ++ ts))) = f' (map g (t :: ys_hd :: (tail✝ ++ ts)))", "state_before": "case cons\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ g' (f (t :: ys_hd :: (tail✝ ++ ts))) = f' (g t :: g ys_hd :: (map g tail✝ ++ map g ts))", "tactic": "simp only [← map_cons, ← map_append]" }, { "state_after": "no goals", "state_before": "case cons\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ g' (f (t :: ys_hd :: (tail✝ ++ ts))) = f' (map g (t :: ys_hd :: (tail✝ ++ ts)))", "tactic": "apply H" }, { "state_after": "case cons.H\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\na : List α\n⊢ g' (f (ys_hd :: a)) = f' (g ys_hd :: map g a)", "state_before": "case cons.H\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\n⊢ ∀ (a : List α), g' (f (ys_hd :: a)) = f' (g ys_hd :: map g a)", "tactic": "intro a" }, { "state_after": "no goals", "state_before": "case cons.H\nα✝ : Type ?u.3849\nβ✝ : Type ?u.3852\nα : Type u_1\nβ : Type u_2\nα' : Type u_3\nβ' : Type u_4\ng : α → α'\ng' : β → β'\nt : α\nts : List α\nr : List β\nf✝ : List α → β\nf'✝ : List α' → β'\nH✝ : ∀ (a : List α), g' (f✝ a) = f'✝ (map g a)\nys_hd : α\ntail✝ : List α\nys_ih :\n ∀ (f : List α → β) (f' : List α' → β'),\n (∀ (a : List α), g' (f a) = f' (map g a)) →\n map g' (permutationsAux2 t ts r tail✝ f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail✝) f').snd\nf : List α → β\nf' : List α' → β'\nH : ∀ (a : List α), g' (f a) = f' (map g a)\na : List α\n⊢ g' (f (ys_hd :: a)) = f' (g ys_hd :: map g a)", "tactic": "apply H" } ]

[ 101, 23 ]

[ 91, 1 ]