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[ { "state_after": "case disjoint\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\n⊢ Disjoint p (ker f)\n\ncase codisjoint\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\n⊢ Codisjoint p (ker f)", "state_before": "R : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\n⊢ IsCompl p (ker f)", "tactic": "constructor" }, { "state_after": "case disjoint\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\n⊢ p ⊓ ker f ≤ ⊥", "state_before": "case disjoint\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\n⊢ Disjoint p (ker f)", "tactic": "rw [disjoint_iff_inf_le]" }, { "state_after": "case disjoint.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\nhpx : x ∈ ↑p\nhfx : x ∈ ↑(ker f)\n⊢ x ∈ ⊥", "state_before": "case disjoint\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\n⊢ p ⊓ ker f ≤ ⊥", "tactic": "rintro x ⟨hpx, hfx⟩" }, { "state_after": "case disjoint.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\nhpx : x ∈ ↑p\nhfx : x = 0\n⊢ x ∈ ⊥", "state_before": "case disjoint.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\nhpx : x ∈ ↑p\nhfx : x ∈ ↑(ker f)\n⊢ x ∈ ⊥", "tactic": "erw [SetLike.mem_coe, mem_ker, hf ⟨x, hpx⟩, mk_eq_zero] at hfx" }, { "state_after": "no goals", "state_before": "case disjoint.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\nhpx : x ∈ ↑p\nhfx : x = 0\n⊢ x ∈ ⊥", "tactic": "simp only [hfx, SetLike.mem_coe, zero_mem]" }, { "state_after": "case codisjoint\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\n⊢ ⊤ ≤ p ⊔ ker f", "state_before": "case codisjoint\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\n⊢ Codisjoint p (ker f)", "tactic": "rw [codisjoint_iff_le_sup]" }, { "state_after": "case codisjoint\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\na✝ : x ∈ ⊤\n⊢ x ∈ p ⊔ ker f", "state_before": "case codisjoint\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\n⊢ ⊤ ≤ p ⊔ ker f", "tactic": "intro x _" }, { "state_after": "case codisjoint\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\na✝ : x ∈ ⊤\n⊢ ∃ y z, ↑y + ↑z = x", "state_before": "case codisjoint\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\na✝ : x ∈ ⊤\n⊢ x ∈ p ⊔ ker f", "tactic": "rw [mem_sup']" }, { "state_after": "case codisjoint\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\na✝ : x ∈ ⊤\n⊢ x - ↑(↑f x) ∈ ker f", "state_before": "case codisjoint\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\na✝ : x ∈ ⊤\n⊢ ∃ y z, ↑y + ↑z = x", "tactic": "refine' ⟨f x, ⟨x - f x, _⟩, add_sub_cancel'_right _ _⟩" }, { "state_after": "no goals", "state_before": "case codisjoint\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.23402\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.23918\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.24881\ninst✝² : Semiring S\nM : Type ?u.24887\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : E →ₗ[R] { x // x ∈ p }\nhf : ∀ (x : { x // x ∈ p }), ↑f ↑x = x\nx : E\na✝ : x ∈ ⊤\n⊢ x - ↑(↑f x) ∈ ker f", "tactic": "rw [mem_ker, LinearMap.map_sub, hf, sub_self]" } ]

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[ { "state_after": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\n⊢ ¬f ∈ z.asHomogeneousIdeal ↔ ∃ u, (u ∈ Set.range fun i => basicOpen 𝒜 (↑(GradedAlgebra.proj 𝒜 i) f)) ∧ z ∈ u", "state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\n⊢ z ∈ ↑(basicOpen 𝒜 f) ↔ z ∈ ↑(⨆ (i : ℕ), basicOpen 𝒜 (↑(GradedAlgebra.proj 𝒜 i) f))", "tactic": "erw [mem_coe_basicOpen, TopologicalSpace.Opens.mem_sSup]" }, { "state_after": "case mp\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nhz : ¬f ∈ z.asHomogeneousIdeal\n⊢ ∃ u, (u ∈ Set.range fun i => basicOpen 𝒜 (↑(GradedAlgebra.proj 𝒜 i) f)) ∧ z ∈ u\n\ncase mpr\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nhz : ∃ u, (u ∈ Set.range fun i => basicOpen 𝒜 (↑(GradedAlgebra.proj 𝒜 i) f)) ∧ z ∈ u\n⊢ ¬f ∈ z.asHomogeneousIdeal", "state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\n⊢ ¬f ∈ z.asHomogeneousIdeal ↔ ∃ u, (u ∈ Set.range fun i => basicOpen 𝒜 (↑(GradedAlgebra.proj 𝒜 i) f)) ∧ z ∈ u", "tactic": "constructor <;> intro hz" }, { "state_after": "case mp.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nhz : ¬f ∈ z.asHomogeneousIdeal\ni : ℕ\nhi : ¬↑(GradedAlgebra.proj 𝒜 i) f ∈ z.asHomogeneousIdeal\n⊢ ∃ u, (u ∈ Set.range fun i => basicOpen 𝒜 (↑(GradedAlgebra.proj 𝒜 i) f)) ∧ z ∈ u", "state_before": "case mp\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nhz : ¬f ∈ z.asHomogeneousIdeal\n⊢ ∃ u, (u ∈ Set.range fun i => basicOpen 𝒜 (↑(GradedAlgebra.proj 𝒜 i) f)) ∧ z ∈ u", "tactic": "rcases show ∃ i, GradedAlgebra.proj 𝒜 i f ∉ z.asHomogeneousIdeal by\n contrapose! hz with H\n classical\n rw [← DirectSum.sum_support_decompose 𝒜 f]\n apply Ideal.sum_mem _ fun i _ => H i with ⟨i, hi⟩" }, { "state_after": "no goals", "state_before": "case mp.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nhz : ¬f ∈ z.asHomogeneousIdeal\ni : ℕ\nhi : ¬↑(GradedAlgebra.proj 𝒜 i) f ∈ z.asHomogeneousIdeal\n⊢ ∃ u, (u ∈ Set.range fun i => basicOpen 𝒜 (↑(GradedAlgebra.proj 𝒜 i) f)) ∧ z ∈ u", "tactic": "exact ⟨basicOpen 𝒜 (GradedAlgebra.proj 𝒜 i f), ⟨i, rfl⟩, by rwa [mem_basicOpen]⟩" }, { "state_after": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nH : ∀ (i : ℕ), ↑(GradedAlgebra.proj 𝒜 i) f ∈ z.asHomogeneousIdeal\n⊢ f ∈ z.asHomogeneousIdeal", "state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nhz : ¬f ∈ z.asHomogeneousIdeal\n⊢ ∃ i, ¬↑(GradedAlgebra.proj 𝒜 i) f ∈ z.asHomogeneousIdeal", "tactic": "contrapose! hz with H" }, { "state_after": "no goals", "state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nH : ∀ (i : ℕ), ↑(GradedAlgebra.proj 𝒜 i) f ∈ z.asHomogeneousIdeal\n⊢ f ∈ z.asHomogeneousIdeal", "tactic": "classical\nrw [← DirectSum.sum_support_decompose 𝒜 f]\napply Ideal.sum_mem _ fun i _ => H i" }, { "state_after": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nH : ∀ (i : ℕ), ↑(GradedAlgebra.proj 𝒜 i) f ∈ z.asHomogeneousIdeal\n⊢ ∑ i in Dfinsupp.support (↑(decompose 𝒜) f), ↑(↑(↑(decompose 𝒜) f) i) ∈ z.asHomogeneousIdeal", "state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nH : ∀ (i : ℕ), ↑(GradedAlgebra.proj 𝒜 i) f ∈ z.asHomogeneousIdeal\n⊢ f ∈ z.asHomogeneousIdeal", "tactic": "rw [← DirectSum.sum_support_decompose 𝒜 f]" }, { "state_after": "no goals", "state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nH : ∀ (i : ℕ), ↑(GradedAlgebra.proj 𝒜 i) f ∈ z.asHomogeneousIdeal\n⊢ ∑ i in Dfinsupp.support (↑(decompose 𝒜) f), ↑(↑(↑(decompose 𝒜) f) i) ∈ z.asHomogeneousIdeal", "tactic": "apply Ideal.sum_mem _ fun i _ => H i" }, { "state_after": "no goals", "state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nhz : ¬f ∈ z.asHomogeneousIdeal\ni : ℕ\nhi : ¬↑(GradedAlgebra.proj 𝒜 i) f ∈ z.asHomogeneousIdeal\n⊢ z ∈ basicOpen 𝒜 (↑(GradedAlgebra.proj 𝒜 i) f)", "tactic": "rwa [mem_basicOpen]" }, { "state_after": "case mpr.intro.intro.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\ni : ℕ\nhz : z ∈ (fun i => basicOpen 𝒜 (↑(GradedAlgebra.proj 𝒜 i) f)) i\n⊢ ¬f ∈ z.asHomogeneousIdeal", "state_before": "case mpr\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nhz : ∃ u, (u ∈ Set.range fun i => basicOpen 𝒜 (↑(GradedAlgebra.proj 𝒜 i) f)) ∧ z ∈ u\n⊢ ¬f ∈ z.asHomogeneousIdeal", "tactic": "obtain ⟨_, ⟨i, rfl⟩, hz⟩ := hz" }, { "state_after": "no goals", "state_before": "case mpr.intro.intro.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\ni : ℕ\nhz : z ∈ (fun i => basicOpen 𝒜 (↑(GradedAlgebra.proj 𝒜 i) f)) i\n⊢ ¬f ∈ z.asHomogeneousIdeal", "tactic": "exact fun rid => hz (z.1.2 i rid)" } ]

[ 454, 42 ]

[ 441, 1 ]

[ { "state_after": "case mp\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\n⊢ IsCompactElement k → ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\n\ncase mpr\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\n⊢ (∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s) → IsCompactElement k", "state_before": "ι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\n⊢ IsCompactElement k ↔ ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s", "tactic": "constructor" }, { "state_after": "case mp\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\n⊢ ∃ t, k ≤ Finset.sup t s", "state_before": "case mp\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\n⊢ IsCompactElement k → ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s", "tactic": "intro H ι s hs" }, { "state_after": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\n⊢ ∃ t, k ≤ Finset.sup t s", "state_before": "case mp\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\n⊢ ∃ t, k ≤ Finset.sup t s", "tactic": "obtain ⟨t, ht, ht'⟩ := H (Set.range s) hs" }, { "state_after": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nthis : ∀ (x : { x // x ∈ t }), ∃ i, s i = ↑x\n⊢ ∃ t, k ≤ Finset.sup t s", "state_before": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\n⊢ ∃ t, k ≤ Finset.sup t s", "tactic": "have : ∀ x : t, ∃ i, s i = x := fun x => ht x.prop" }, { "state_after": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\n⊢ ∃ t, k ≤ Finset.sup t s", "state_before": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nthis : ∀ (x : { x // x ∈ t }), ∃ i, s i = ↑x\n⊢ ∃ t, k ≤ Finset.sup t s", "tactic": "choose f hf using this" }, { "state_after": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\n⊢ Finset.sup t id ≤ Finset.sup (Finset.image f Finset.univ) s", "state_before": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\n⊢ ∃ t, k ≤ Finset.sup t s", "tactic": "refine' ⟨Finset.univ.image f, ht'.trans _⟩" }, { "state_after": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\n⊢ ∀ (b : α), b ∈ t → id b ≤ Finset.sup (Finset.image f Finset.univ) s", "state_before": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\n⊢ Finset.sup t id ≤ Finset.sup (Finset.image f Finset.univ) s", "tactic": "rw [Finset.sup_le_iff]" }, { "state_after": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\nb : α\nhb : b ∈ t\n⊢ id b ≤ Finset.sup (Finset.image f Finset.univ) s", "state_before": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\n⊢ ∀ (b : α), b ∈ t → id b ≤ Finset.sup (Finset.image f Finset.univ) s", "tactic": "intro b hb" }, { "state_after": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\nb : α\nhb : b ∈ t\n⊢ s (f { val := b, property := hb }) ≤ Finset.sup (Finset.image f Finset.univ) s", "state_before": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\nb : α\nhb : b ∈ t\n⊢ id b ≤ Finset.sup (Finset.image f Finset.univ) s", "tactic": "rw [← show s (f ⟨b, hb⟩) = id b from hf _]" }, { "state_after": "no goals", "state_before": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\nb : α\nhb : b ∈ t\n⊢ s (f { val := b, property := hb }) ≤ Finset.sup (Finset.image f Finset.univ) s", "tactic": "exact Finset.le_sup (Finset.mem_image_of_mem f <| Finset.mem_univ (Subtype.mk b hb))" }, { "state_after": "case mpr\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\n⊢ ∃ t, ↑t ⊆ s ∧ k ≤ Finset.sup t id", "state_before": "case mpr\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\n⊢ (∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s) → IsCompactElement k", "tactic": "intro H s hs" }, { "state_after": "case mpr.intro\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\nt : Finset ↑s\nht : k ≤ Finset.sup t Subtype.val\n⊢ ∃ t, ↑t ⊆ s ∧ k ≤ Finset.sup t id", "state_before": "case mpr\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\n⊢ ∃ t, ↑t ⊆ s ∧ k ≤ Finset.sup t id", "tactic": "obtain ⟨t, ht⟩ :=\n H s Subtype.val\n (by\n delta iSup\n rwa [Subtype.range_coe])" }, { "state_after": "case mpr.intro\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\nt : Finset ↑s\nht : k ≤ Finset.sup t Subtype.val\n⊢ Finset.sup t Subtype.val ≤ Finset.sup (Finset.image Subtype.val t) id", "state_before": "case mpr.intro\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\nt : Finset ↑s\nht : k ≤ Finset.sup t Subtype.val\n⊢ ∃ t, ↑t ⊆ s ∧ k ≤ Finset.sup t id", "tactic": "refine' ⟨t.image Subtype.val, by simp, ht.trans _⟩" }, { "state_after": "case mpr.intro\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\nt : Finset ↑s\nht : k ≤ Finset.sup t Subtype.val\n⊢ ∀ (b : ↑s), b ∈ t → ↑b ≤ Finset.sup (Finset.image Subtype.val t) id", "state_before": "case mpr.intro\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\nt : Finset ↑s\nht : k ≤ Finset.sup t Subtype.val\n⊢ Finset.sup t Subtype.val ≤ Finset.sup (Finset.image Subtype.val t) id", "tactic": "rw [Finset.sup_le_iff]" }, { "state_after": "no goals", "state_before": "case mpr.intro\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\nt : Finset ↑s\nht : k ≤ Finset.sup t Subtype.val\n⊢ ∀ (b : ↑s), b ∈ t → ↑b ≤ Finset.sup (Finset.image Subtype.val t) id", "tactic": "exact fun x hx => @Finset.le_sup _ _ _ _ _ id _ (Finset.mem_image_of_mem Subtype.val hx)" }, { "state_after": "ι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\n⊢ k ≤ sSup (Set.range Subtype.val)", "state_before": "ι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\n⊢ k ≤ iSup Subtype.val", "tactic": "delta iSup" }, { "state_after": "no goals", "state_before": "ι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\n⊢ k ≤ sSup (Set.range Subtype.val)", "tactic": "rwa [Subtype.range_coe]" }, { "state_after": "no goals", "state_before": "ι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\nt : Finset ↑s\nht : k ≤ Finset.sup t Subtype.val\n⊢ ↑(Finset.image Subtype.val t) ⊆ s", "tactic": "simp" } ]

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[]

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[]

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[ { "state_after": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nh₁ : p ∣ q\nh₂ : natDegree q < natDegree p\nhc : ¬q = 0\n⊢ False", "state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nh₁ : p ∣ q\nh₂ : natDegree q < natDegree p\n⊢ q = 0", "tactic": "by_contra hc" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nh₁ : p ∣ q\nh₂ : natDegree q < natDegree p\nhc : ¬q = 0\n⊢ False", "tactic": "exact (lt_iff_not_ge _ _).mp h₂ (natDegree_le_of_dvd h₁ hc)" } ]

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[]

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[ { "state_after": "no goals", "state_before": "α : Type u\ninst✝ : Monoid α\na✝ b c u : αˣ\na : α\n⊢ ↑u * a = 1 ↔ ↑u⁻¹ = a", "tactic": "rw [← inv_mul_eq_one, inv_inv]" } ]

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[]

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[]

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[ { "state_after": "no goals", "state_before": "α : Type ?u.253194\nβ : Type ?u.253197\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nh0 : a ≠ 0\nhI : a ≠ ⊤\n⊢ a * (b / a) = b", "tactic": "rw [mul_comm, ENNReal.div_mul_cancel h0 hI]" } ]

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[]

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[ { "state_after": "no goals", "state_before": "b x y : Ordinal\nxy : x ≤ y\nhx : x = 0\n⊢ log b x ≤ log b y", "tactic": "simp only [hx, log_zero_right, Ordinal.zero_le]" }, { "state_after": "no goals", "state_before": "b x y : Ordinal\nxy : x ≤ y\nhx : ¬x = 0\nhb : ¬1 < b\n⊢ log b x ≤ log b y", "tactic": "simp only [log_of_not_one_lt_left hb, Ordinal.zero_le]" } ]

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[]

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[ { "state_after": "E : Type ?u.10318\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nz w : ℂ\n⊢ dist z w = Real.sqrt ((z.re - w.re) * (z.re - w.re) + (z.im - w.im) * (z.im - w.im))", "state_before": "E : Type ?u.10318\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nz w : ℂ\n⊢ dist z w = Real.sqrt ((z.re - w.re) ^ 2 + (z.im - w.im) ^ 2)", "tactic": "rw [sq, sq]" }, { "state_after": "no goals", "state_before": "E : Type ?u.10318\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nz w : ℂ\n⊢ dist z w = Real.sqrt ((z.re - w.re) * (z.re - w.re) + (z.im - w.im) * (z.im - w.im))", "tactic": "rfl" } ]

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[ { "state_after": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R M₂\ninst✝⁷ : AddCommMonoid M₃\ninst✝⁶ : Module R M₃\nφ : ι → Type i\ninst✝⁵ : (i : ι) → AddCommMonoid (φ i)\ninst✝⁴ : (i : ι) → Module R (φ i)\ninst✝³ : Finite ι\ninst✝² : DecidableEq ι\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ((i : ι) → φ i) →ₗ[R] M\nh : ∀ (i : ι), comp f (single i) = comp g (single i)\ni : ι\nx : φ i\n⊢ ↑f (Pi.single i x) = ↑g (Pi.single i x)", "state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R M₂\ninst✝⁷ : AddCommMonoid M₃\ninst✝⁶ : Module R M₃\nφ : ι → Type i\ninst✝⁵ : (i : ι) → AddCommMonoid (φ i)\ninst✝⁴ : (i : ι) → Module R (φ i)\ninst✝³ : Finite ι\ninst✝² : DecidableEq ι\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ((i : ι) → φ i) →ₗ[R] M\nh : ∀ (i : ι), comp f (single i) = comp g (single i)\n⊢ f = g", "tactic": "refine' pi_ext fun i x => _" }, { "state_after": "no goals", "state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R M₂\ninst✝⁷ : AddCommMonoid M₃\ninst✝⁶ : Module R M₃\nφ : ι → Type i\ninst✝⁵ : (i : ι) → AddCommMonoid (φ i)\ninst✝⁴ : (i : ι) → Module R (φ i)\ninst✝³ : Finite ι\ninst✝² : DecidableEq ι\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ((i : ι) → φ i) →ₗ[R] M\nh : ∀ (i : ι), comp f (single i) = comp g (single i)\ni : ι\nx : φ i\n⊢ ↑f (Pi.single i x) = ↑g (Pi.single i x)", "tactic": "convert LinearMap.congr_fun (h i) x" } ]

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[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nb : β\nf✝ g✝ : α → β\ninst✝ : NonUnitalNonAssocSemiring β\nn k : ℕ\nf g : ℕ → β\n⊢ (∑ i in range (n + 1), f i) * ∑ i in range (k + 1), g i =\n (∑ i in range n, f i) * ∑ i in range k, g i + f n * ∑ i in range k, g i + (∑ i in range n, f i) * g k + f n * g k", "tactic": "simp only [add_mul, mul_add, add_assoc, sum_range_succ]" } ]

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[]

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[ { "state_after": "case zero\nR : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\n⊢ (↑derivative^[Nat.zero]) 0 = 0\n\ncase succ\nR : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\nk : ℕ\nih : (↑derivative^[k]) 0 = 0\n⊢ (↑derivative^[Nat.succ k]) 0 = 0", "state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\nk : ℕ\n⊢ (↑derivative^[k]) 0 = 0", "tactic": "induction' k with k ih" }, { "state_after": "no goals", "state_before": "case zero\nR : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\n⊢ (↑derivative^[Nat.zero]) 0 = 0", "tactic": "simp" }, { "state_after": "no goals", "state_before": "case succ\nR : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\nk : ℕ\nih : (↑derivative^[k]) 0 = 0\n⊢ (↑derivative^[Nat.succ k]) 0 = 0", "tactic": "simp [ih]" } ]

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[ { "state_after": "K : Type u\nV : Type v\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW✝ W : Subspace K V\nx : Module.Dual K { x // x ∈ W }\n⊢ ↑1 x = x", "state_before": "K : Type u\nV : Type v\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW✝ W : Subspace K V\nx : Module.Dual K { x // x ∈ W }\n⊢ ↑(comp (dualRestrict W) (dualLift W)) x = x", "tactic": "rw [dualRestrict_comp_dualLift]" }, { "state_after": "no goals", "state_before": "K : Type u\nV : Type v\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW✝ W : Subspace K V\nx : Module.Dual K { x // x ∈ W }\n⊢ ↑1 x = x", "tactic": "rfl" } ]

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[ { "state_after": "R : Type ?u.1213100\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1213307\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J✝ I J J' : FractionalIdeal R₁⁰ K\nhJ' : J' ≠ 0\n⊢ I ≤ { val := ↑J / ↑J', property := (_ : IsFractional R₁⁰ (↑J / ↑J')) } ↔ I * J' ≤ J", "state_before": "R : Type ?u.1213100\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1213307\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J✝ I J J' : FractionalIdeal R₁⁰ K\nhJ' : J' ≠ 0\n⊢ I ≤ J / J' ↔ I * J' ≤ J", "tactic": "rw [div_nonzero hJ']" }, { "state_after": "R : Type ?u.1213100\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1213307\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J✝ I J J' : FractionalIdeal R₁⁰ K\nhJ' : J' ≠ 0\n⊢ I ≤ { val := ↑J / ↑J', property := (_ : IsFractional R₁⁰ (↑J / ↑J')) } ↔ ↑I * ↑J' ≤ ↑J", "state_before": "R : Type ?u.1213100\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1213307\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J✝ I J J' : FractionalIdeal R₁⁰ K\nhJ' : J' ≠ 0\n⊢ I ≤ { val := ↑J / ↑J', property := (_ : IsFractional R₁⁰ (↑J / ↑J')) } ↔ I * J' ≤ J", "tactic": "rw [← coe_le_coe (I := I * J') (J := J), coe_mul]" }, { "state_after": "no goals", "state_before": "R : Type ?u.1213100\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1213307\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J✝ I J J' : FractionalIdeal R₁⁰ K\nhJ' : J' ≠ 0\n⊢ I ≤ { val := ↑J / ↑J', property := (_ : IsFractional R₁⁰ (↑J / ↑J')) } ↔ ↑I * ↑J' ≤ ↑J", "tactic": "exact Submodule.le_div_iff_mul_le" } ]

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[]

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[ { "state_after": "x y : ℝ\nf g : CauSeq ℚ abs\n⊢ (match { cauchy := Quotient.mk equiv f }, { cauchy := Quotient.mk equiv g } with\n | { cauchy := x }, { cauchy := y } => Quotient.liftOn₂ x y (fun x x_1 => x < x_1) Real.definition.proof_1✝) ↔\n f < g", "state_before": "x y : ℝ\nf g : CauSeq ℚ abs\n⊢ Real.lt { cauchy := Quotient.mk equiv f } { cauchy := Quotient.mk equiv g } ↔ f < g", "tactic": "rw [lt_def]" }, { "state_after": "no goals", "state_before": "x y : ℝ\nf g : CauSeq ℚ abs\n⊢ (match { cauchy := Quotient.mk equiv f }, { cauchy := Quotient.mk equiv g } with\n | { cauchy := x }, { cauchy := y } => Quotient.liftOn₂ x y (fun x x_1 => x < x_1) Real.definition.proof_1✝) ↔\n f < g", "tactic": "rfl" } ]

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[ { "state_after": "no goals", "state_before": "K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\n⊢ TerminatedAt (of v) n ↔ IntFractPair.stream v (n + 1) = none", "tactic": "rw [of_terminatedAt_iff_intFractPair_seq1_terminatedAt, Stream'.Seq.TerminatedAt,\n IntFractPair.get?_seq1_eq_succ_get?_stream]" } ]

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[ 212, 1 ]

[ { "state_after": "q : ℚ\nx y : ℝ\nh : Irrational x\nm : ℤ\nhm : m ≠ 0\n⊢ Irrational (x / ↑↑m)", "state_before": "q : ℚ\nx y : ℝ\nh : Irrational x\nm : ℤ\nhm : m ≠ 0\n⊢ Irrational (x / ↑m)", "tactic": "rw [← cast_coe_int]" }, { "state_after": "q : ℚ\nx y : ℝ\nh : Irrational x\nm : ℤ\nhm : m ≠ 0\n⊢ ↑m ≠ 0", "state_before": "q : ℚ\nx y : ℝ\nh : Irrational x\nm : ℤ\nhm : m ≠ 0\n⊢ Irrational (x / ↑↑m)", "tactic": "refine' h.div_rat _" }, { "state_after": "no goals", "state_before": "q : ℚ\nx y : ℝ\nh : Irrational x\nm : ℤ\nhm : m ≠ 0\n⊢ ↑m ≠ 0", "tactic": "rwa [Int.cast_ne_zero]" } ]

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[]

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[ { "state_after": "case a.mk.h.mk\nR : Type ?u.5462\nα : Type u_1\nα' : Type ?u.5468\nβ : Type u_2\nβ' : Type ?u.5474\nγ : Type u_3\nγ' : Type ?u.5480\nl : Type ?u.5483\nm : Type u_4\nn : Type u_5\np : Type ?u.5492\nq : Type ?u.5495\nr : Type ?u.5498\nl' : Type ?u.5501\nm' : Type ?u.5504\nn' : Type ?u.5507\np' : Type ?u.5510\ninst✝⁴ : Zero α\ninst✝³ : Zero β\ninst✝² : Zero γ\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq n\nf : α → β → γ\nhf₁ : ∀ (b : β), f 0 b = 0\nhf₂ : ∀ (a : α), f a 0 = 0\na : m → α\nb : n → β\ni₁ : m\ni₂ : n\nj₁ : m\nj₂ : n\n⊢ kroneckerMap f (diagonal a) (diagonal b) (i₁, i₂) (j₁, j₂) =\n diagonal (fun mn => f (a mn.fst) (b mn.snd)) (i₁, i₂) (j₁, j₂)", "state_before": "R : Type ?u.5462\nα : Type u_1\nα' : Type ?u.5468\nβ : Type u_2\nβ' : Type ?u.5474\nγ : Type u_3\nγ' : Type ?u.5480\nl : Type ?u.5483\nm : Type u_4\nn : Type u_5\np : Type ?u.5492\nq : Type ?u.5495\nr : Type ?u.5498\nl' : Type ?u.5501\nm' : Type ?u.5504\nn' : Type ?u.5507\np' : Type ?u.5510\ninst✝⁴ : Zero α\ninst✝³ : Zero β\ninst✝² : Zero γ\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq n\nf : α → β → γ\nhf₁ : ∀ (b : β), f 0 b = 0\nhf₂ : ∀ (a : α), f a 0 = 0\na : m → α\nb : n → β\n⊢ kroneckerMap f (diagonal a) (diagonal b) = diagonal fun mn => f (a mn.fst) (b mn.snd)", "tactic": "ext (⟨i₁, i₂⟩⟨j₁, j₂⟩)" }, { "state_after": "no goals", "state_before": "case a.mk.h.mk\nR : Type ?u.5462\nα : Type u_1\nα' : Type ?u.5468\nβ : Type u_2\nβ' : Type ?u.5474\nγ : Type u_3\nγ' : Type ?u.5480\nl : Type ?u.5483\nm : Type u_4\nn : Type u_5\np : Type ?u.5492\nq : Type ?u.5495\nr : Type ?u.5498\nl' : Type ?u.5501\nm' : Type ?u.5504\nn' : Type ?u.5507\np' : Type ?u.5510\ninst✝⁴ : Zero α\ninst✝³ : Zero β\ninst✝² : Zero γ\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq n\nf : α → β → γ\nhf₁ : ∀ (b : β), f 0 b = 0\nhf₂ : ∀ (a : α), f a 0 = 0\na : m → α\nb : n → β\ni₁ : m\ni₂ : n\nj₁ : m\nj₂ : n\n⊢ kroneckerMap f (diagonal a) (diagonal b) (i₁, i₂) (j₁, j₂) =\n diagonal (fun mn => f (a mn.fst) (b mn.snd)) (i₁, i₂) (j₁, j₂)", "tactic": "simp [diagonal, apply_ite f, ite_and, ite_apply, apply_ite (f (a i₁)), hf₁, hf₂]" } ]

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[]

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[]

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[ { "state_after": "no goals", "state_before": "K : Type u\ninst✝³ : CommRing K\ninst✝² : IsDomain K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y : ℕ × K\nx✝ : ∃ z, (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd\nz : ℕ\nH : (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd\n⊢ (↑(frobenius K p)^[z]) ((↑(frobenius K p)^[y.fst]) x.snd) = (↑(frobenius K p)^[z]) ((↑(frobenius K p)^[x.fst]) y.snd)", "tactic": "simpa only [add_comm, iterate_add] using H" } ]

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[]

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[ { "state_after": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : p z\n⊢ ↑(↑ofSubtype (swap x y)) z = ↑(swap ↑x ↑y) z\n\ncase neg\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\n⊢ ↑(↑ofSubtype (swap x y)) z = ↑(swap ↑x ↑y) z", "state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\n⊢ ↑(↑ofSubtype (swap x y)) z = ↑(swap ↑x ↑y) z", "tactic": "by_cases hz : p z" }, { "state_after": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : p z\n⊢ ↑(↑(swap x y) { val := z, property := hz }) = if z = ↑x then ↑y else if z = ↑y then ↑x else z", "state_before": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : p z\n⊢ ↑(↑ofSubtype (swap x y)) z = ↑(swap ↑x ↑y) z", "tactic": "rw [swap_apply_def, ofSubtype_apply_of_mem _ hz]" }, { "state_after": "case pos.inl\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : p z\nhzx : z = ↑x\n⊢ ↑(↑(swap x y) { val := z, property := hz }) = ↑y\n\ncase pos.inr.inl\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : p z\nhzx : ¬z = ↑x\nhzy : z = ↑y\n⊢ ↑(↑(swap x y) { val := z, property := hz }) = ↑x\n\ncase pos.inr.inr\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : p z\nhzx : ¬z = ↑x\nhzy : ¬z = ↑y\n⊢ ↑(↑(swap x y) { val := z, property := hz }) = z", "state_before": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : p z\n⊢ ↑(↑(swap x y) { val := z, property := hz }) = if z = ↑x then ↑y else if z = ↑y then ↑x else z", "tactic": "split_ifs with hzx hzy" }, { "state_after": "no goals", "state_before": "case pos.inl\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : p z\nhzx : z = ↑x\n⊢ ↑(↑(swap x y) { val := z, property := hz }) = ↑y", "tactic": "simp_rw [hzx, Subtype.coe_eta, swap_apply_left]" }, { "state_after": "no goals", "state_before": "case pos.inr.inl\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : p z\nhzx : ¬z = ↑x\nhzy : z = ↑y\n⊢ ↑(↑(swap x y) { val := z, property := hz }) = ↑x", "tactic": "simp_rw [hzy, Subtype.coe_eta, swap_apply_right]" }, { "state_after": "no goals", "state_before": "case pos.inr.inr\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : p z\nhzx : ¬z = ↑x\nhzy : ¬z = ↑y\n⊢ ↑(↑(swap x y) { val := z, property := hz }) = z", "tactic": "rw [swap_apply_of_ne_of_ne] <;>\nsimp [Subtype.ext_iff, *]" }, { "state_after": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\n⊢ z ≠ ↑x\n\ncase neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\n⊢ z ≠ ↑y", "state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\n⊢ ↑(↑ofSubtype (swap x y)) z = ↑(swap ↑x ↑y) z", "tactic": "rw [ofSubtype_apply_of_not_mem _ hz, swap_apply_of_ne_of_ne]" }, { "state_after": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\nh : z = ↑x\n⊢ False\n\ncase neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\n⊢ z ≠ ↑y", "state_before": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\n⊢ z ≠ ↑x\n\ncase neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\n⊢ z ≠ ↑y", "tactic": "intro h" }, { "state_after": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\nh : z = ↑x\n⊢ p z\n\ncase neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\n⊢ z ≠ ↑y", "state_before": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\nh : z = ↑x\n⊢ False\n\ncase neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\n⊢ z ≠ ↑y", "tactic": "apply hz" }, { "state_after": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\nh : z = ↑x\n⊢ p ↑x\n\ncase neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\n⊢ z ≠ ↑y", "state_before": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\nh : z = ↑x\n⊢ p z\n\ncase neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\n⊢ z ≠ ↑y", "tactic": "rw [h]" }, { "state_after": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\n⊢ z ≠ ↑y", "state_before": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\nh : z = ↑x\n⊢ p ↑x\n\ncase neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\n⊢ z ≠ ↑y", "tactic": "exact Subtype.prop x" }, { "state_after": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\nh : z = ↑y\n⊢ False", "state_before": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\n⊢ z ≠ ↑y", "tactic": "intro h" }, { "state_after": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\nh : z = ↑y\n⊢ p z", "state_before": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\nh : z = ↑y\n⊢ False", "tactic": "apply hz" }, { "state_after": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\nh : z = ↑y\n⊢ p ↑y", "state_before": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\nh : z = ↑y\n⊢ p z", "tactic": "rw [h]" }, { "state_after": "no goals", "state_before": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\np : α → Prop\ninst✝ : DecidablePred p\nx y : Subtype p\nz : α\nhz : ¬p z\nh : z = ↑y\n⊢ p ↑y", "tactic": "exact Subtype.prop y" } ]

[ 232, 27 ]

[ 214, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝ : Ring R\n⊢ cyclotomic 0 R = 1", "tactic": "simp only [cyclotomic, dif_pos]" } ]

[ 308, 34 ]

[ 307, 1 ]

[ { "state_after": "case mk.mk\n𝕜 : Type u_5\nV : Type u_1\nV₁ : Type ?u.25742\nV₂ : Type u_2\nV₃ : Type ?u.25748\nV₄ : Type ?u.25751\nP₁ : Type ?u.25754\nP : Type u_3\nP₂ : Type u_4\nP₃ : Type ?u.25763\nP₄ : Type ?u.25766\ninst✝²⁰ : NormedField 𝕜\ninst✝¹⁹ : SeminormedAddCommGroup V\ninst✝¹⁸ : SeminormedAddCommGroup V₁\ninst✝¹⁷ : SeminormedAddCommGroup V₂\ninst✝¹⁶ : SeminormedAddCommGroup V₃\ninst✝¹⁵ : SeminormedAddCommGroup V₄\ninst✝¹⁴ : NormedSpace 𝕜 V\ninst✝¹³ : NormedSpace 𝕜 V₁\ninst✝¹² : NormedSpace 𝕜 V₂\ninst✝¹¹ : NormedSpace 𝕜 V₃\ninst✝¹⁰ : NormedSpace 𝕜 V₄\ninst✝⁹ : PseudoMetricSpace P\ninst✝⁸ : MetricSpace P₁\ninst✝⁷ : PseudoMetricSpace P₂\ninst✝⁶ : PseudoMetricSpace P₃\ninst✝⁵ : PseudoMetricSpace P₄\ninst✝⁴ : NormedAddTorsor V P\ninst✝³ : NormedAddTorsor V₁ P₁\ninst✝² : NormedAddTorsor V₂ P₂\ninst✝¹ : NormedAddTorsor V₃ P₃\ninst✝ : NormedAddTorsor V₄ P₄\nf✝ : P →ᵃⁱ[𝕜] P₂\nf : P →ᵃ[𝕜] P₂\nnorm_map✝¹ : ∀ (x : V), ‖↑f.linear x‖ = ‖x‖\nnorm_map✝ : ∀ (x : V), ‖↑{ toAffineMap := f, norm_map := norm_map✝¹ }.toAffineMap.linear x‖ = ‖x‖\n⊢ { toAffineMap := f, norm_map := norm_map✝¹ } =\n { toAffineMap := { toAffineMap := f, norm_map := norm_map✝¹ }.toAffineMap, norm_map := norm_map✝ }", "state_before": "𝕜 : Type u_5\nV : Type u_1\nV₁ : Type ?u.25742\nV₂ : Type u_2\nV₃ : Type ?u.25748\nV₄ : Type ?u.25751\nP₁ : Type ?u.25754\nP : Type u_3\nP₂ : Type u_4\nP₃ : Type ?u.25763\nP₄ : Type ?u.25766\ninst✝²⁰ : NormedField 𝕜\ninst✝¹⁹ : SeminormedAddCommGroup V\ninst✝¹⁸ : SeminormedAddCommGroup V₁\ninst✝¹⁷ : SeminormedAddCommGroup V₂\ninst✝¹⁶ : SeminormedAddCommGroup V₃\ninst✝¹⁵ : SeminormedAddCommGroup V₄\ninst✝¹⁴ : NormedSpace 𝕜 V\ninst✝¹³ : NormedSpace 𝕜 V₁\ninst✝¹² : NormedSpace 𝕜 V₂\ninst✝¹¹ : NormedSpace 𝕜 V₃\ninst✝¹⁰ : NormedSpace 𝕜 V₄\ninst✝⁹ : PseudoMetricSpace P\ninst✝⁸ : MetricSpace P₁\ninst✝⁷ : PseudoMetricSpace P₂\ninst✝⁶ : PseudoMetricSpace P₃\ninst✝⁵ : PseudoMetricSpace P₄\ninst✝⁴ : NormedAddTorsor V P\ninst✝³ : NormedAddTorsor V₁ P₁\ninst✝² : NormedAddTorsor V₂ P₂\ninst✝¹ : NormedAddTorsor V₃ P₃\ninst✝ : NormedAddTorsor V₄ P₄\nf : P →ᵃⁱ[𝕜] P₂\n⊢ Injective toAffineMap", "tactic": "rintro ⟨f, _⟩ ⟨g, _⟩ rfl" }, { "state_after": "no goals", "state_before": "case mk.mk\n𝕜 : Type u_5\nV : Type u_1\nV₁ : Type ?u.25742\nV₂ : Type u_2\nV₃ : Type ?u.25748\nV₄ : Type ?u.25751\nP₁ : Type ?u.25754\nP : Type u_3\nP₂ : Type u_4\nP₃ : Type ?u.25763\nP₄ : Type ?u.25766\ninst✝²⁰ : NormedField 𝕜\ninst✝¹⁹ : SeminormedAddCommGroup V\ninst✝¹⁸ : SeminormedAddCommGroup V₁\ninst✝¹⁷ : SeminormedAddCommGroup V₂\ninst✝¹⁶ : SeminormedAddCommGroup V₃\ninst✝¹⁵ : SeminormedAddCommGroup V₄\ninst✝¹⁴ : NormedSpace 𝕜 V\ninst✝¹³ : NormedSpace 𝕜 V₁\ninst✝¹² : NormedSpace 𝕜 V₂\ninst✝¹¹ : NormedSpace 𝕜 V₃\ninst✝¹⁰ : NormedSpace 𝕜 V₄\ninst✝⁹ : PseudoMetricSpace P\ninst✝⁸ : MetricSpace P₁\ninst✝⁷ : PseudoMetricSpace P₂\ninst✝⁶ : PseudoMetricSpace P₃\ninst✝⁵ : PseudoMetricSpace P₄\ninst✝⁴ : NormedAddTorsor V P\ninst✝³ : NormedAddTorsor V₁ P₁\ninst✝² : NormedAddTorsor V₂ P₂\ninst✝¹ : NormedAddTorsor V₃ P₃\ninst✝ : NormedAddTorsor V₄ P₄\nf✝ : P →ᵃⁱ[𝕜] P₂\nf : P →ᵃ[𝕜] P₂\nnorm_map✝¹ : ∀ (x : V), ‖↑f.linear x‖ = ‖x‖\nnorm_map✝ : ∀ (x : V), ‖↑{ toAffineMap := f, norm_map := norm_map✝¹ }.toAffineMap.linear x‖ = ‖x‖\n⊢ { toAffineMap := f, norm_map := norm_map✝¹ } =\n { toAffineMap := { toAffineMap := f, norm_map := norm_map✝¹ }.toAffineMap, norm_map := norm_map✝ }", "tactic": "rfl" } ]

[ 88, 6 ]

[ 86, 1 ]

[]

[ 188, 74 ]

[ 187, 1 ]

[]

[ 42, 68 ]

[ 41, 1 ]

[]

[ 1428, 47 ]

[ 1426, 1 ]

[ { "state_after": "p : ℕ\nG : Type u_1\ninst✝⁴ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type ?u.345348\ninst✝³ : MulAction G α\ninst✝² : Fintype α\ninst✝¹ : Nontrivial G\ninst✝ : Finite G\nthis : Nontrivial { x // x ∈ Subgroup.center G }\n⊢ ⊥ < Subgroup.center G", "state_before": "p : ℕ\nG : Type u_1\ninst✝⁴ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type ?u.345348\ninst✝³ : MulAction G α\ninst✝² : Fintype α\ninst✝¹ : Nontrivial G\ninst✝ : Finite G\n⊢ ⊥ < Subgroup.center G", "tactic": "haveI := center_nontrivial hG" }, { "state_after": "case intro\np : ℕ\nG : Type u_1\ninst✝⁴ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type ?u.345348\ninst✝³ : MulAction G α\ninst✝² : Fintype α\ninst✝¹ : Nontrivial G\ninst✝ : Finite G\nthis : Nontrivial { x // x ∈ Subgroup.center G }\nval✝ : Fintype G\n⊢ ⊥ < Subgroup.center G", "state_before": "p : ℕ\nG : Type u_1\ninst✝⁴ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type ?u.345348\ninst✝³ : MulAction G α\ninst✝² : Fintype α\ninst✝¹ : Nontrivial G\ninst✝ : Finite G\nthis : Nontrivial { x // x ∈ Subgroup.center G }\n⊢ ⊥ < Subgroup.center G", "tactic": "cases nonempty_fintype G" }, { "state_after": "no goals", "state_before": "case intro\np : ℕ\nG : Type u_1\ninst✝⁴ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type ?u.345348\ninst✝³ : MulAction G α\ninst✝² : Fintype α\ninst✝¹ : Nontrivial G\ninst✝ : Finite G\nthis : Nontrivial { x // x ∈ Subgroup.center G }\nval✝ : Fintype G\n⊢ ⊥ < Subgroup.center G", "tactic": "classical exact\n bot_lt_iff_ne_bot.mpr ((Subgroup.center G).one_lt_card_iff_ne_bot.mp Fintype.one_lt_card)" }, { "state_after": "no goals", "state_before": "case intro\np : ℕ\nG : Type u_1\ninst✝⁴ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type ?u.345348\ninst✝³ : MulAction G α\ninst✝² : Fintype α\ninst✝¹ : Nontrivial G\ninst✝ : Finite G\nthis : Nontrivial { x // x ∈ Subgroup.center G }\nval✝ : Fintype G\n⊢ ⊥ < Subgroup.center G", "tactic": "exact\nbot_lt_iff_ne_bot.mpr ((Subgroup.center G).one_lt_card_iff_ne_bot.mp Fintype.one_lt_card)" } ]

[ 264, 96 ]

[ 260, 1 ]

[]

[ 1557, 6 ]

[ 1556, 1 ]

[ { "state_after": "K : Type u_2\nL : Type u_1\nL' : Type ?u.277529\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nS F E : IntermediateField K L\n⊢ (∀ (x : L), x ∈ F.toSubalgebra ↔ x ∈ E.toSubalgebra) ↔ ∀ (x : L), x ∈ ↑F ↔ x ∈ ↑E", "state_before": "K : Type u_2\nL : Type u_1\nL' : Type ?u.277529\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nS F E : IntermediateField K L\n⊢ F.toSubalgebra = E.toSubalgebra ↔ F = E", "tactic": "rw [SetLike.ext_iff, SetLike.ext'_iff, Set.ext_iff]" }, { "state_after": "no goals", "state_before": "K : Type u_2\nL : Type u_1\nL' : Type ?u.277529\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nS F E : IntermediateField K L\n⊢ (∀ (x : L), x ∈ F.toSubalgebra ↔ x ∈ E.toSubalgebra) ↔ ∀ (x : L), x ∈ ↑F ↔ x ∈ ↑E", "tactic": "rfl" } ]

[ 699, 6 ]

[ 697, 1 ]

[]

[ 145, 18 ]

[ 144, 1 ]

[]

[ 207, 16 ]

[ 206, 11 ]

[ { "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✝ Z : C\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\nF G : C ⥤ D\nh : F = G\nX Y : C\nf : X ⟶ Y\n⊢ HEq (F.map f) (G.map f)", "tactic": "rw [h]" } ]

[ 260, 9 ]

[ 259, 1 ]

[]

[ 916, 41 ]

[ 915, 11 ]

[]

[ 263, 34 ]

[ 258, 11 ]

[]

[ 265, 41 ]

[ 264, 1 ]

[ { "state_after": "case ofNat\nα : Type u\nβ : Type v\ninst✝ : Group α\na b : α\na✝ : ℕ\n⊢ (a * b * a⁻¹) ^ Int.ofNat a✝ = a * b ^ Int.ofNat a✝ * a⁻¹\n\ncase negSucc\nα : Type u\nβ : Type v\ninst✝ : Group α\na b : α\na✝ : ℕ\n⊢ (a * b * a⁻¹) ^ Int.negSucc a✝ = a * b ^ Int.negSucc a✝ * a⁻¹", "state_before": "α : Type u\nβ : Type v\ninst✝ : Group α\ni : ℤ\na b : α\n⊢ (a * b * a⁻¹) ^ i = a * b ^ i * a⁻¹", "tactic": "induction' i" }, { "state_after": "case ofNat\nα : Type u\nβ : Type v\ninst✝ : Group α\na b : α\na✝ : ℕ\n⊢ (a * b * a⁻¹) ^ Int.ofNat a✝ = a * b ^ Int.ofNat a✝ * a⁻¹", "state_before": "case ofNat\nα : Type u\nβ : Type v\ninst✝ : Group α\na b : α\na✝ : ℕ\n⊢ (a * b * a⁻¹) ^ Int.ofNat a✝ = a * b ^ Int.ofNat a✝ * a⁻¹", "tactic": "change (a * b * a⁻¹) ^ (_ : ℤ) = a * b ^ (_ : ℤ) * a⁻¹" }, { "state_after": "no goals", "state_before": "case ofNat\nα : Type u\nβ : Type v\ninst✝ : Group α\na b : α\na✝ : ℕ\n⊢ (a * b * a⁻¹) ^ Int.ofNat a✝ = a * b ^ Int.ofNat a✝ * a⁻¹", "tactic": "simp [zpow_ofNat]" }, { "state_after": "case negSucc\nα : Type u\nβ : Type v\ninst✝ : Group α\na b : α\na✝ : ℕ\n⊢ a * ((b ^ (a✝ + 1))⁻¹ * a⁻¹) = a * (b ^ (a✝ + 1))⁻¹ * a⁻¹", "state_before": "case negSucc\nα : Type u\nβ : Type v\ninst✝ : Group α\na b : α\na✝ : ℕ\n⊢ (a * b * a⁻¹) ^ Int.negSucc a✝ = a * b ^ Int.negSucc a✝ * a⁻¹", "tactic": "simp [zpow_negSucc, conj_pow]" }, { "state_after": "no goals", "state_before": "case negSucc\nα : Type u\nβ : Type v\ninst✝ : Group α\na b : α\na✝ : ℕ\n⊢ a * ((b ^ (a✝ + 1))⁻¹ * a⁻¹) = a * (b ^ (a✝ + 1))⁻¹ * a⁻¹", "tactic": "rw [mul_assoc]" } ]

[ 123, 19 ]

[ 118, 1 ]

[]

[ 1286, 67 ]

[ 1284, 1 ]

[]

[ 37, 21 ]

[ 35, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.26992\nι : Type ?u.26995\nM : Type u_2\nN : Type ?u.27001\ninst✝³ : One M\ninst✝² : One N\ns✝ t : Set α\nf g : α → M\na✝ : α\nU : Set α\ns : Set M\na : M\ninst✝¹ : Decidable (a ∈ s)\ninst✝ : Decidable (1 ∈ s)\n⊢ Set.ite U (if a ∈ s then univ else ∅) (if 1 ∈ s then univ else ∅) = (if a ∈ s then U else ∅) ∪ if 1 ∈ s then Uᶜ else ∅", "state_before": "α : Type u_1\nβ : Type ?u.26992\nι : Type ?u.26995\nM : Type u_2\nN : Type ?u.27001\ninst✝³ : One M\ninst✝² : One N\ns✝ t : Set α\nf g : α → M\na✝ : α\nU : Set α\ns : Set M\na : M\ninst✝¹ : Decidable (a ∈ s)\ninst✝ : Decidable (1 ∈ s)\n⊢ (mulIndicator U fun x => a) ⁻¹' s = (if a ∈ s then U else ∅) ∪ if 1 ∈ s then Uᶜ else ∅", "tactic": "rw [mulIndicator_preimage, preimage_one, preimage_const]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.26992\nι : Type ?u.26995\nM : Type u_2\nN : Type ?u.27001\ninst✝³ : One M\ninst✝² : One N\ns✝ t : Set α\nf g : α → M\na✝ : α\nU : Set α\ns : Set M\na : M\ninst✝¹ : Decidable (a ∈ s)\ninst✝ : Decidable (1 ∈ s)\n⊢ Set.ite U (if a ∈ s then univ else ∅) (if 1 ∈ s then univ else ∅) = (if a ∈ s then U else ∅) ∪ if 1 ∈ s then Uᶜ else ∅", "tactic": "split_ifs <;> simp [← compl_eq_univ_diff]" } ]

[ 312, 44 ]

[ 308, 1 ]

[ { "state_after": "M : Type w\nA : Set M\nL : Language\ninst✝ : Structure L M\nα : Type u₁\nβ : Type ?u.1235\nB : Set M\ns : Set (α → M)\n⊢ (∃ φ, s = setOf (Formula.Realize φ)) ↔\n ∃ b, s = setOf (Formula.Realize (↑(LEquiv.onFormula (LEquiv.addEmptyConstants L ↑∅)).symm b))", "state_before": "M : Type w\nA : Set M\nL : Language\ninst✝ : Structure L M\nα : Type u₁\nβ : Type ?u.1235\nB : Set M\ns : Set (α → M)\n⊢ Definable ∅ L s ↔ ∃ φ, s = setOf (Formula.Realize φ)", "tactic": "rw [Definable, Equiv.exists_congr_left (LEquiv.addEmptyConstants L (∅ : Set M)).onFormula]" }, { "state_after": "no goals", "state_before": "M : Type w\nA : Set M\nL : Language\ninst✝ : Structure L M\nα : Type u₁\nβ : Type ?u.1235\nB : Set M\ns : Set (α → M)\n⊢ (∃ φ, s = setOf (Formula.Realize φ)) ↔\n ∃ b, s = setOf (Formula.Realize (↑(LEquiv.onFormula (LEquiv.addEmptyConstants L ↑∅)).symm b))", "tactic": "simp [-constantsOn]" } ]

[ 65, 22 ]

[ 62, 1 ]

[ { "state_after": "L : Language\nM : Type w\nN : Type ?u.683008\nP : Type ?u.683011\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\n⊢ withConstants (LowerAdjoint.toFun (closure L) (A ∪ s)) (_ : A ⊆ ↑(LowerAdjoint.toFun (closure L) (A ∪ s))) ≤\n LowerAdjoint.toFun (closure (L[[↑A]])) s", "state_before": "L : Language\nM : Type w\nN : Type ?u.683008\nP : Type ?u.683011\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\n⊢ LowerAdjoint.toFun (closure (L[[↑A]])) s =\n withConstants (LowerAdjoint.toFun (closure L) (A ∪ s)) (_ : A ⊆ ↑(LowerAdjoint.toFun (closure L) (A ∪ s)))", "tactic": "refine' closure_eq_of_le ((A.subset_union_right s).trans subset_closure) _" }, { "state_after": "L : Language\nM : Type w\nN : Type ?u.683008\nP : Type ?u.683011\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\n⊢ ↑(LHom.substructureReduct (lhomWithConstants L ↑A))\n (withConstants (LowerAdjoint.toFun (closure L) (A ∪ s)) (_ : A ⊆ ↑(LowerAdjoint.toFun (closure L) (A ∪ s)))) ≤\n ↑(LHom.substructureReduct (lhomWithConstants L ↑A)) (LowerAdjoint.toFun (closure (L[[↑A]])) s)", "state_before": "L : Language\nM : Type w\nN : Type ?u.683008\nP : Type ?u.683011\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\n⊢ withConstants (LowerAdjoint.toFun (closure L) (A ∪ s)) (_ : A ⊆ ↑(LowerAdjoint.toFun (closure L) (A ∪ s))) ≤\n LowerAdjoint.toFun (closure (L[[↑A]])) s", "tactic": "rw [← (L.lhomWithConstants A).substructureReduct.le_iff_le]" }, { "state_after": "L : Language\nM : Type w\nN : Type ?u.683008\nP : Type ?u.683011\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\n⊢ A ⊆ ↑(LowerAdjoint.toFun (closure (L[[↑A]])) s)", "state_before": "L : Language\nM : Type w\nN : Type ?u.683008\nP : Type ?u.683011\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\n⊢ ↑(LHom.substructureReduct (lhomWithConstants L ↑A))\n (withConstants (LowerAdjoint.toFun (closure L) (A ∪ s)) (_ : A ⊆ ↑(LowerAdjoint.toFun (closure L) (A ∪ s)))) ≤\n ↑(LHom.substructureReduct (lhomWithConstants L ↑A)) (LowerAdjoint.toFun (closure (L[[↑A]])) s)", "tactic": "simp only [subset_closure, reduct_withConstants, closure_le, LHom.coe_substructureReduct,\n Set.union_subset_iff, and_true_iff]" }, { "state_after": "no goals", "state_before": "L : Language\nM : Type w\nN : Type ?u.683008\nP : Type ?u.683011\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\n⊢ A ⊆ ↑(LowerAdjoint.toFun (closure (L[[↑A]])) s)", "tactic": "exact subset_closure_withConstants" } ]

[ 784, 39 ]

[ 777, 1 ]

[]

[ 260, 23 ]

[ 259, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.187524\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\na a₁ a₂ b b₁ b₂ c d : α\n⊢ Ioi a₁ ∩ Iio b₁ ∩ (Ioi a₂ ∩ Iio b₂) = Ioi a₁ ∩ Ioi a₂ ∩ (Iio b₁ ∩ Iio b₂)", "state_before": "α : Type u_1\nβ : Type ?u.187524\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\na a₁ a₂ b b₁ b₂ c d : α\n⊢ Ioo a₁ b₁ ∩ Ioo a₂ b₂ = Ioo (a₁ ⊔ a₂) (b₁ ⊓ b₂)", "tactic": "simp only [Ioi_inter_Iio.symm, Ioi_inter_Ioi.symm, Iio_inter_Iio.symm]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.187524\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\na a₁ a₂ b b₁ b₂ c d : α\n⊢ Ioi a₁ ∩ Iio b₁ ∩ (Ioi a₂ ∩ Iio b₂) = Ioi a₁ ∩ Ioi a₂ ∩ (Iio b₁ ∩ Iio b₂)", "tactic": "ac_rfl" } ]

[ 1792, 81 ]

[ 1791, 1 ]

[ { "state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : I = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1\n\ncase neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1", "state_before": "R : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\n⊢ ↑I * (↑I)⁻¹ = 1", "tactic": "by_cases hI1 : I = ⊤" }, { "state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : IsField A\n⊢ ↑I * (↑I)⁻¹ = 1\n\ncase neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\n⊢ ↑I * (↑I)⁻¹ = 1", "state_before": "case neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1", "tactic": "by_cases hNF : IsField A" }, { "state_after": "case neg.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\n⊢ ↑I * (↑I)⁻¹ = 1", "state_before": "case neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\n⊢ ↑I * (↑I)⁻¹ = 1", "tactic": "obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=\n le_one_iff_exists_coeIdeal.mp mul_one_div_le_one" }, { "state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : J = ⊥\n⊢ ↑I * (↑I)⁻¹ = 1\n\ncase neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\n⊢ ↑I * (↑I)⁻¹ = 1", "state_before": "case neg.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\n⊢ ↑I * (↑I)⁻¹ = 1", "tactic": "by_cases hJ0 : J = ⊥" }, { "state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : J = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1\n\ncase neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1", "state_before": "case neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\n⊢ ↑I * (↑I)⁻¹ = 1", "tactic": "by_cases hJ1 : J = ⊤" }, { "state_after": "case neg.intro.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\nx : K\nhx : x ∈ (↑J)⁻¹\nhx1 : ¬x ∈ 1\n⊢ ↑I * (↑I)⁻¹ = 1", "state_before": "case neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1", "tactic": "obtain ⟨x, hx, hx1⟩ :\n ∃ x : K, x ∈ (J : FractionalIdeal A⁰ K)⁻¹ ∧ x ∉ (1 : FractionalIdeal A⁰ K) :=\n exists_not_mem_one_of_ne_bot hNF hJ0 hJ1" }, { "state_after": "case neg.intro.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\nx : K\nhx : x ∈ (↑J)⁻¹\nh_abs : ↑I * (↑I)⁻¹ ≠ 1\n⊢ x ∈ 1", "state_before": "case neg.intro.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\nx : K\nhx : x ∈ (↑J)⁻¹\nhx1 : ¬x ∈ 1\n⊢ ↑I * (↑I)⁻¹ = 1", "tactic": "contrapose! hx1 with h_abs" }, { "state_after": "case neg.intro.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\nx : K\nhx : x ∈ (↑I * (↑I)⁻¹)⁻¹\nh_abs : ↑I * (↑I)⁻¹ ≠ 1\n⊢ x ∈ 1", "state_before": "case neg.intro.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\nx : K\nhx : x ∈ (↑J)⁻¹\nh_abs : ↑I * (↑I)⁻¹ ≠ 1\n⊢ x ∈ 1", "tactic": "rw [hJ] at hx" }, { "state_after": "no goals", "state_before": "case neg.intro.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\nx : K\nhx : x ∈ (↑I * (↑I)⁻¹)⁻¹\nh_abs : ↑I * (↑I)⁻¹ ≠ 1\n⊢ x ∈ 1", "tactic": "exact hI hx" }, { "state_after": "no goals", "state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : I = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1", "tactic": "rw [hI1, coeIdeal_top, one_mul, inv_one]" }, { "state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : IsField A\nthis : Field A := IsField.toField hNF\n⊢ ↑I * (↑I)⁻¹ = 1", "state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : IsField A\n⊢ ↑I * (↑I)⁻¹ = 1", "tactic": "letI := hNF.toField" }, { "state_after": "no goals", "state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : IsField A\nthis : Field A := IsField.toField hNF\n⊢ ↑I * (↑I)⁻¹ = 1", "tactic": "rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩" }, { "state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nhJ : ↑⊥ = ↑I * (↑I)⁻¹\n⊢ ↑I * (↑I)⁻¹ = 1", "state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : J = ⊥\n⊢ ↑I * (↑I)⁻¹ = 1", "tactic": "subst hJ0" }, { "state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nhJ : ↑⊥ = ↑I * (↑I)⁻¹\n⊢ I = ⊥", "state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nhJ : ↑⊥ = ↑I * (↑I)⁻¹\n⊢ ↑I * (↑I)⁻¹ = 1", "tactic": "refine' absurd _ hI0" }, { "state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nhJ : ↑⊥ = ↑I * (↑I)⁻¹\n⊢ ↑I ≤ ↑I * (↑I)⁻¹", "state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nhJ : ↑⊥ = ↑I * (↑I)⁻¹\n⊢ I = ⊥", "tactic": "rw [eq_bot_iff, ← coeIdeal_le_coeIdeal K, hJ]" }, { "state_after": "no goals", "state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nhJ : ↑⊥ = ↑I * (↑I)⁻¹\n⊢ ↑I ≤ ↑I * (↑I)⁻¹", "tactic": "exact coe_ideal_le_self_mul_inv K I" }, { "state_after": "no goals", "state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : J = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1", "tactic": "rw [← hJ, hJ1, coeIdeal_top]" } ]

[ 485, 14 ]

[ 462, 1 ]

[ { "state_after": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\n⊢ degree 1 ≤ 0", "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\n⊢ degree (p ^ 0) ≤ 0 • degree p", "tactic": "rw [pow_zero, zero_nsmul]" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\n⊢ degree 1 ≤ 0", "tactic": "exact degree_one_le" }, { "state_after": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\nn : ℕ\n⊢ degree (p * p ^ n) ≤ degree p + degree (p ^ n)", "state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\nn : ℕ\n⊢ degree (p ^ (n + 1)) ≤ degree p + degree (p ^ n)", "tactic": "rw [pow_succ]" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\nn : ℕ\n⊢ degree (p * p ^ n) ≤ degree p + degree (p ^ n)", "tactic": "exact degree_mul_le _ _" }, { "state_after": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\nn : ℕ\n⊢ degree p + degree (p ^ n) ≤ degree p + n • degree p", "state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\nn : ℕ\n⊢ degree p + degree (p ^ n) ≤ (n + 1) • degree p", "tactic": "rw [succ_nsmul]" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\nn : ℕ\n⊢ degree p + degree (p ^ n) ≤ degree p + n • degree p", "tactic": "exact add_le_add le_rfl (degree_pow_le _ _)" } ]

[ 791, 79 ]

[ 785, 1 ]

[]

[ 223, 33 ]

[ 222, 1 ]

[ { "state_after": "o : Ordinal\n⊢ alephIdx (aleph' (succ o)) ≤ alephIdx (succ (aleph' o))", "state_before": "o : Ordinal\n⊢ aleph' (succ o) = succ (aleph' o)", "tactic": "apply (succ_le_of_lt <| aleph'_lt.2 <| lt_succ o).antisymm' (Cardinal.alephIdx_le.1 <| _)" }, { "state_after": "o : Ordinal\n⊢ aleph' o < succ (aleph' o)", "state_before": "o : Ordinal\n⊢ alephIdx (aleph' (succ o)) ≤ alephIdx (succ (aleph' o))", "tactic": "rw [alephIdx_aleph', succ_le_iff, ← aleph'_lt, aleph'_alephIdx]" }, { "state_after": "no goals", "state_before": "o : Ordinal\n⊢ aleph' o < succ (aleph' o)", "tactic": "apply lt_succ" } ]

[ 206, 16 ]

[ 203, 1 ]

[]

[ 1533, 83 ]

[ 1533, 1 ]

[]

[ 713, 41 ]

[ 712, 1 ]

[]

[ 195, 6 ]

[ 194, 1 ]

[]

[ 301, 13 ]

[ 298, 1 ]

[ { "state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m < j ∧ n ≤ m\n⊢ n ≤ m", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m ∈ filter (fun k => n ≤ k) (range j)\n⊢ n ≤ m", "tactic": "simp at hm" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m < j ∧ n ≤ m\n⊢ n ≤ m", "tactic": "tauto" }, { "state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m < j ∧ n ≤ m\n⊢ m < j", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m ∈ filter (fun k => n ≤ k) (range j)\n⊢ m < j", "tactic": "simp at hm" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m < j ∧ n ≤ m\n⊢ m < j", "tactic": "tauto" }, { "state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m ∈ filter (fun k => n ≤ k) (range j)\n⊢ n ≤ m", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m ∈ filter (fun k => n ≤ k) (range j)\n⊢ 1 / ↑(Nat.factorial m) = 1 / ↑(Nat.factorial ((fun m x => m - n) m hm + n))", "tactic": "rw [tsub_add_cancel_of_le]" }, { "state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m < j ∧ n ≤ m\n⊢ n ≤ m", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m ∈ filter (fun k => n ≤ k) (range j)\n⊢ n ≤ m", "tactic": "simp at *" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m < j ∧ n ≤ m\n⊢ n ≤ m", "tactic": "tauto" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\na₁ a₂ : ℕ\nha₁ : a₁ ∈ filter (fun k => n ≤ k) (range j)\nha₂ : a₂ ∈ filter (fun k => n ≤ k) (range j)\nh : (fun m x => m - n) a₁ ha₁ = (fun m x => m - n) a₂ ha₂\n⊢ a₁ = a₂", "tactic": "rwa [tsub_eq_iff_eq_add_of_le, tsub_add_eq_add_tsub, eq_comm, tsub_eq_iff_eq_add_of_le,\n add_left_inj, eq_comm] at h <;>\nsimp at * <;> aesop" }, { "state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nb : ℕ\nhb : b ∈ range (j - n)\n⊢ b = b + n - n", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nb : ℕ\nhb : b ∈ range (j - n)\n⊢ b = (fun m x => m - n) (b + n) (_ : b + n ∈ filter (fun k => n ≤ k) (range j))", "tactic": "dsimp" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nb : ℕ\nhb : b ∈ range (j - n)\n⊢ b = b + n - n", "tactic": "rw [add_tsub_cancel_right]" }, { "state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\n⊢ ∑ x in range (j - n), (↑(Nat.factorial (x + n)))⁻¹ ≤ ∑ m in range (j - n), (↑(Nat.factorial n) * ↑(Nat.succ n) ^ m)⁻¹", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\n⊢ ∑ m in range (j - n), 1 / ↑(Nat.factorial (m + n)) ≤ ∑ m in range (j - n), (↑(Nat.factorial n) * ↑(Nat.succ n) ^ m)⁻¹", "tactic": "simp_rw [one_div]" }, { "state_after": "case h.h\nα : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\ni✝ : ℕ\na✝ : i✝ ∈ range (j - n)\n⊢ ↑(Nat.factorial n) * ↑(Nat.succ n) ^ i✝ ≤ ↑(Nat.factorial (i✝ + n))", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\n⊢ ∑ x in range (j - n), (↑(Nat.factorial (x + n)))⁻¹ ≤ ∑ m in range (j - n), (↑(Nat.factorial n) * ↑(Nat.succ n) ^ m)⁻¹", "tactic": "gcongr" }, { "state_after": "case h.h\nα : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\ni✝ : ℕ\na✝ : i✝ ∈ range (j - n)\n⊢ Nat.factorial n * Nat.succ n ^ i✝ ≤ Nat.factorial (n + i✝)", "state_before": "case h.h\nα : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\ni✝ : ℕ\na✝ : i✝ ∈ range (j - n)\n⊢ ↑(Nat.factorial n) * ↑(Nat.succ n) ^ i✝ ≤ ↑(Nat.factorial (i✝ + n))", "tactic": "rw [← Nat.cast_pow, ← Nat.cast_mul, Nat.cast_le, add_comm]" }, { "state_after": "no goals", "state_before": "case h.h\nα : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\ni✝ : ℕ\na✝ : i✝ ∈ range (j - n)\n⊢ Nat.factorial n * Nat.succ n ^ i✝ ≤ Nat.factorial (n + i✝)", "tactic": "exact Nat.factorial_mul_pow_le_factorial" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\n⊢ ∑ m in range (j - n), (↑(Nat.factorial n) * ↑(Nat.succ n) ^ m)⁻¹ =\n (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m", "tactic": "simp [mul_inv, mul_sum.symm, sum_mul.symm, -Nat.factorial_succ, mul_comm, inv_pow]" }, { "state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)", "tactic": "have h₁ : (n.succ : α) ≠ 1 :=\n @Nat.cast_one α _ ▸ mt Nat.cast_inj.1 (mt Nat.succ.inj (pos_iff_ne_zero.1 hn))" }, { "state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\nh₂ : ↑(Nat.succ n) ≠ 0\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)", "tactic": "have h₂ : (n.succ : α) ≠ 0 := by positivity" }, { "state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\nh₂ : ↑(Nat.succ n) ≠ 0\nh₃ : ↑(Nat.factorial n) * ↑n ≠ 0\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\nh₂ : ↑(Nat.succ n) ≠ 0\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)", "tactic": "have h₃ : (n.factorial * n : α) ≠ 0 := by positivity" }, { "state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\nh₂ : ↑(Nat.succ n) ≠ 0\nh₃ : ↑(Nat.factorial n) * ↑n ≠ 0\nh₄ : ↑(Nat.succ n) - 1 = ↑n\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\nh₂ : ↑(Nat.succ n) ≠ 0\nh₃ : ↑(Nat.factorial n) * ↑n ≠ 0\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)", "tactic": "have h₄ : (n.succ - 1 : α) = n := by simp" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\nh₂ : ↑(Nat.succ n) ≠ 0\nh₃ : ↑(Nat.factorial n) * ↑n ≠ 0\nh₄ : ↑(Nat.succ n) - 1 = ↑n\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)", "tactic": "rw [geom_sum_inv h₁ h₂, eq_div_iff_mul_eq h₃, mul_comm _ (n.factorial * n : α),\n ← mul_assoc (n.factorial⁻¹ : α), ← mul_inv_rev, h₄, ← mul_assoc (n.factorial * n : α),\n mul_comm (n : α) n.factorial, mul_inv_cancel h₃, one_mul, mul_comm]" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\n⊢ ↑(Nat.succ n) ≠ 0", "tactic": "positivity" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\nh₂ : ↑(Nat.succ n) ≠ 0\n⊢ ↑(Nat.factorial n) * ↑n ≠ 0", "tactic": "positivity" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\nh₂ : ↑(Nat.succ n) ≠ 0\nh₃ : ↑(Nat.factorial n) * ↑n ≠ 0\n⊢ ↑(Nat.succ n) - 1 = ↑n", "tactic": "simp" }, { "state_after": "case h\nα : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\n⊢ ↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n) ≤ ↑(Nat.succ n)", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\n⊢ (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n) ≤\n ↑(Nat.succ n) / (↑(Nat.factorial n) * ↑n)", "tactic": "gcongr" }, { "state_after": "case h.a\nα : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\n⊢ 0 ≤ ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)", "state_before": "case h\nα : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\n⊢ ↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n) ≤ ↑(Nat.succ n)", "tactic": "apply sub_le_self" }, { "state_after": "no goals", "state_before": "case h.a\nα : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\n⊢ 0 ≤ ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)", "tactic": "positivity" } ]

[ 1608, 83 ]

[ 1573, 1 ]

[ { "state_after": "no goals", "state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.6513\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t : Set E\nx y : E\n⊢ x ∈ ↑(convexHull 𝕜).toOrderHom s ↔ ∀ (t : Set E), s ⊆ t → Convex 𝕜 t → x ∈ t", "tactic": "simp_rw [convexHull_eq_iInter, mem_iInter]" } ]

[ 72, 45 ]

[ 71, 1 ]

[ { "state_after": "R : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nM : Type u_3\nM' : Type u_2\nM'' : Type ?u.844419\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nm : M\ns₁ s₂ : { x // x ∈ S }\n⊢ ↑(fromLocalizedModule S f) (LocalizedModule.mk (s₂ • m) (s₁ * s₂)) =\n ↑(fromLocalizedModule S f) (LocalizedModule.mk m s₁)", "state_before": "R : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nM : Type u_3\nM' : Type u_2\nM'' : Type ?u.844419\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nm : M\ns₁ s₂ : { x // x ∈ S }\n⊢ mk' f (s₂ • m) (s₁ * s₂) = mk' f m s₁", "tactic": "delta mk'" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nM : Type u_3\nM' : Type u_2\nM'' : Type ?u.844419\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nm : M\ns₁ s₂ : { x // x ∈ S }\n⊢ ↑(fromLocalizedModule S f) (LocalizedModule.mk (s₂ • m) (s₁ * s₂)) =\n ↑(fromLocalizedModule S f) (LocalizedModule.mk m s₁)", "tactic": "rw [LocalizedModule.mk_cancel_common_right]" } ]

[ 969, 46 ]

[ 967, 1 ]

[]

[ 2283, 80 ]

[ 2282, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nhg : AEMeasurable g\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\n⊢ (∫⁻ (a : α), g a ∂Measure.withDensity μ f) = ∫⁻ (a : α), (f * g) a ∂μ", "state_before": "α : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nhg : AEMeasurable g\n⊢ (∫⁻ (a : α), g a ∂Measure.withDensity μ f) = ∫⁻ (a : α), (f * g) a ∂μ", "tactic": "let f' := hf.mk f" }, { "state_after": "α : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nhg : AEMeasurable g\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\n⊢ (∫⁻ (a : α), g a ∂Measure.withDensity μ f) = ∫⁻ (a : α), (f * g) a ∂μ", "state_before": "α : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nhg : AEMeasurable g\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\n⊢ (∫⁻ (a : α), g a ∂Measure.withDensity μ f) = ∫⁻ (a : α), (f * g) a ∂μ", "tactic": "have : μ.withDensity f = μ.withDensity f' := withDensity_congr_ae hf.ae_eq_mk" }, { "state_after": "α : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\n⊢ (∫⁻ (a : α), g a ∂Measure.withDensity μ f') = ∫⁻ (a : α), (f * g) a ∂μ", "state_before": "α : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nhg : AEMeasurable g\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\n⊢ (∫⁻ (a : α), g a ∂Measure.withDensity μ f) = ∫⁻ (a : α), (f * g) a ∂μ", "tactic": "rw [this] at hg⊢" }, { "state_after": "α : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ (∫⁻ (a : α), g a ∂Measure.withDensity μ f') = ∫⁻ (a : α), (f * g) a ∂μ", "state_before": "α : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\n⊢ (∫⁻ (a : α), g a ∂Measure.withDensity μ f') = ∫⁻ (a : α), (f * g) a ∂μ", "tactic": "let g' := hg.mk g" }, { "state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ (fun a => (f' * g') a) =ᵐ[μ] fun a => (f' * g) a", "state_before": "α : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ (∫⁻ (a : α), (f' * g') a ∂μ) = ∫⁻ (a : α), (f' * g) a ∂μ", "tactic": "apply lintegral_congr_ae" }, { "state_after": "case h.ht\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ {x | f' x ≠ 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x\n\ncase h.htc\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ ({x | f' x ≠ 0}ᶜ), (fun a => (f' * g') a) x = (fun a => (f' * g) a) x", "state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ (fun a => (f' * g') a) =ᵐ[μ] fun a => (f' * g) a", "tactic": "apply ae_of_ae_restrict_of_ae_restrict_compl { x | f' x ≠ 0 }" }, { "state_after": "case h.ht\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nZ : g =ᵐ[Measure.withDensity μ f'] AEMeasurable.mk g hg\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ {x | f' x ≠ 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x", "state_before": "case h.ht\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ {x | f' x ≠ 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x", "tactic": "have Z := hg.ae_eq_mk" }, { "state_after": "case h.ht\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nZ : ∀ᵐ (x : α) ∂Measure.restrict μ {x | AEMeasurable.mk f hf x ≠ 0}, g x = AEMeasurable.mk g hg x\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ {x | f' x ≠ 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x", "state_before": "case h.ht\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nZ : g =ᵐ[Measure.withDensity μ f'] AEMeasurable.mk g hg\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ {x | f' x ≠ 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x", "tactic": "rw [EventuallyEq, ae_withDensity_iff_ae_restrict hf.measurable_mk] at Z" }, { "state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nZ : ∀ᵐ (x : α) ∂Measure.restrict μ {x | AEMeasurable.mk f hf x ≠ 0}, g x = AEMeasurable.mk g hg x\n⊢ ∀ (a : α),\n g a = AEMeasurable.mk g hg a → (AEMeasurable.mk f hf * AEMeasurable.mk g hg) a = (AEMeasurable.mk f hf * g) a", "state_before": "case h.ht\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nZ : ∀ᵐ (x : α) ∂Measure.restrict μ {x | AEMeasurable.mk f hf x ≠ 0}, g x = AEMeasurable.mk g hg x\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ {x | f' x ≠ 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x", "tactic": "filter_upwards [Z]" }, { "state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nZ : ∀ᵐ (x : α) ∂Measure.restrict μ {x | AEMeasurable.mk f hf x ≠ 0}, g x = AEMeasurable.mk g hg x\nx : α\nhx : g x = AEMeasurable.mk g hg x\n⊢ (AEMeasurable.mk f hf * AEMeasurable.mk g hg) x = (AEMeasurable.mk f hf * g) x", "state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nZ : ∀ᵐ (x : α) ∂Measure.restrict μ {x | AEMeasurable.mk f hf x ≠ 0}, g x = AEMeasurable.mk g hg x\n⊢ ∀ (a : α),\n g a = AEMeasurable.mk g hg a → (AEMeasurable.mk f hf * AEMeasurable.mk g hg) a = (AEMeasurable.mk f hf * g) a", "tactic": "intro x hx" }, { "state_after": "no goals", "state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nZ : ∀ᵐ (x : α) ∂Measure.restrict μ {x | AEMeasurable.mk f hf x ≠ 0}, g x = AEMeasurable.mk g hg x\nx : α\nhx : g x = AEMeasurable.mk g hg x\n⊢ (AEMeasurable.mk f hf * AEMeasurable.mk g hg) x = (AEMeasurable.mk f hf * g) x", "tactic": "simp only [hx, Pi.mul_apply]" }, { "state_after": "case h.htc\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ ({x | f' x ≠ 0}ᶜ), (fun a => (f' * g') a) x = (fun a => (f' * g) a) x", "state_before": "case h.htc\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ ({x | f' x ≠ 0}ᶜ), (fun a => (f' * g') a) x = (fun a => (f' * g) a) x", "tactic": "have M : MeasurableSet ({ x : α | f' x ≠ 0 }ᶜ) :=\n (hf.measurable_mk (measurableSet_singleton 0).compl).compl" }, { "state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\n⊢ ∀ (a : α),\n a ∈ {x | AEMeasurable.mk f hf x ≠ 0}ᶜ →\n (AEMeasurable.mk f hf * AEMeasurable.mk g hg) a = (AEMeasurable.mk f hf * g) a", "state_before": "case h.htc\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ ({x | f' x ≠ 0}ᶜ), (fun a => (f' * g') a) x = (fun a => (f' * g) a) x", "tactic": "filter_upwards [ae_restrict_mem M]" }, { "state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\nx : α\nhx : x ∈ {x | AEMeasurable.mk f hf x ≠ 0}ᶜ\n⊢ (AEMeasurable.mk f hf * AEMeasurable.mk g hg) x = (AEMeasurable.mk f hf * g) x", "state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\n⊢ ∀ (a : α),\n a ∈ {x | AEMeasurable.mk f hf x ≠ 0}ᶜ →\n (AEMeasurable.mk f hf * AEMeasurable.mk g hg) a = (AEMeasurable.mk f hf * g) a", "tactic": "intro x hx" }, { "state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\nx : α\nhx : AEMeasurable.mk f hf x = 0\n⊢ (AEMeasurable.mk f hf * AEMeasurable.mk g hg) x = (AEMeasurable.mk f hf * g) x", "state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\nx : α\nhx : x ∈ {x | AEMeasurable.mk f hf x ≠ 0}ᶜ\n⊢ (AEMeasurable.mk f hf * AEMeasurable.mk g hg) x = (AEMeasurable.mk f hf * g) x", "tactic": "simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx" }, { "state_after": "no goals", "state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\nx : α\nhx : AEMeasurable.mk f hf x = 0\n⊢ (AEMeasurable.mk f hf * AEMeasurable.mk g hg) x = (AEMeasurable.mk f hf * g) x", "tactic": "simp only [hx, MulZeroClass.zero_mul, Pi.mul_apply]" }, { "state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ (fun a => (f' * g) a) =ᵐ[μ] fun a => (f * g) a", "state_before": "α : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ (∫⁻ (a : α), (f' * g) a ∂μ) = ∫⁻ (a : α), (f * g) a ∂μ", "tactic": "apply lintegral_congr_ae" }, { "state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ ∀ (a : α), f a = AEMeasurable.mk f hf a → (AEMeasurable.mk f hf * g) a = (f * g) a", "state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ (fun a => (f' * g) a) =ᵐ[μ] fun a => (f * g) a", "tactic": "filter_upwards [hf.ae_eq_mk]" }, { "state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nx : α\nhx : f x = AEMeasurable.mk f hf x\n⊢ (AEMeasurable.mk f hf * g) x = (f * g) x", "state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ ∀ (a : α), f a = AEMeasurable.mk f hf a → (AEMeasurable.mk f hf * g) a = (f * g) a", "tactic": "intro x hx" }, { "state_after": "no goals", "state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nx : α\nhx : f x = AEMeasurable.mk f hf x\n⊢ (AEMeasurable.mk f hf * g) x = (f * g) x", "tactic": "simp only [hx, Pi.mul_apply]" } ]

[ 1817, 35 ]

[ 1788, 1 ]

[ { "state_after": "α : Type u_2\nβ : α → Type u_1\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\ni : ℕ\n⊢ ↑(approxChain f) i ≤ ↑f (↑(approxChain f) i)", "state_before": "α : Type u_2\nβ : α → Type u_1\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\nx : (a : α) → Part (β a)\n⊢ ∀ (x_1 : ℕ), x = ↑(approxChain f) x_1 → x ≤ ↑f x", "tactic": "rintro i rfl" }, { "state_after": "no goals", "state_before": "α : Type u_2\nβ : α → Type u_1\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\ni : ℕ\n⊢ ↑(approxChain f) i ≤ ↑f (↑(approxChain f) i)", "tactic": "apply approx_mono'" } ]

[ 126, 21 ]

[ 123, 1 ]

[]

[ 124, 21 ]

[ 121, 1 ]

[ { "state_after": "no goals", "state_before": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np1 : P\np : ι → P\nh : p1 ∈ affineSpan k (Set.range p)\n⊢ ∃ w x, p1 = ↑(Finset.affineCombination k Finset.univ p) w", "tactic": "classical\n obtain ⟨s, w, hw, rfl⟩ := eq_affineCombination_of_mem_affineSpan h\n refine'\n ⟨(s : Set ι).indicator w, _, Finset.affineCombination_indicator_subset w p s.subset_univ⟩\n simp only [Finset.mem_coe, Set.indicator_apply, ← hw]\n rw [Fintype.sum_extend_by_zero s w]" }, { "state_after": "case intro.intro.intro\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 1\nh : ↑(Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)\n⊢ ∃ w_1 x, ↑(Finset.affineCombination k s p) w = ↑(Finset.affineCombination k Finset.univ p) w_1", "state_before": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np1 : P\np : ι → P\nh : p1 ∈ affineSpan k (Set.range p)\n⊢ ∃ w x, p1 = ↑(Finset.affineCombination k Finset.univ p) w", "tactic": "obtain ⟨s, w, hw, rfl⟩ := eq_affineCombination_of_mem_affineSpan h" }, { "state_after": "case intro.intro.intro\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 1\nh : ↑(Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)\n⊢ ∑ i : ι, Set.indicator (↑s) w i = 1", "state_before": "case intro.intro.intro\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 1\nh : ↑(Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)\n⊢ ∃ w_1 x, ↑(Finset.affineCombination k s p) w = ↑(Finset.affineCombination k Finset.univ p) w_1", "tactic": "refine'\n ⟨(s : Set ι).indicator w, _, Finset.affineCombination_indicator_subset w p s.subset_univ⟩" }, { "state_after": "case intro.intro.intro\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 1\nh : ↑(Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)\n⊢ (∑ x : ι, if x ∈ s then w x else 0) = ∑ i in s, w i", "state_before": "case intro.intro.intro\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 1\nh : ↑(Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)\n⊢ ∑ i : ι, Set.indicator (↑s) w i = 1", "tactic": "simp only [Finset.mem_coe, Set.indicator_apply, ← hw]" }, { "state_after": "no goals", "state_before": "case intro.intro.intro\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 1\nh : ↑(Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)\n⊢ (∑ x : ι, if x ∈ s then w x else 0) = ∑ i in s, w i", "tactic": "rw [Fintype.sum_extend_by_zero s w]" } ]

[ 1136, 40 ]

[ 1128, 1 ]

[ { "state_after": "K : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\n⊢ ↑(↑(Basis.constr (Basis.singleton Unit K) K) fun x =>\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x)\n 1 =\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V), ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x", "state_before": "K : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\n⊢ ↑(coevaluation K V) 1 =\n let bV := Basis.ofVectorSpace K V;\n ∑ i : ↑(Basis.ofVectorSpaceIndex K V), ↑bV i ⊗ₜ[K] Basis.coord bV i", "tactic": "simp only [coevaluation, id]" }, { "state_after": "K : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\n⊢ ∑ i : Unit,\n ↑(Basis.equivFun (Basis.singleton Unit K)) 1 i •\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x =\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V), ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x", "state_before": "K : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\n⊢ ↑(↑(Basis.constr (Basis.singleton Unit K) K) fun x =>\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x)\n 1 =\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V), ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x", "tactic": "rw [(Basis.singleton Unit K).constr_apply_fintype K]" }, { "state_after": "no goals", "state_before": "K : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\n⊢ ∑ i : Unit,\n ↑(Basis.equivFun (Basis.singleton Unit K)) 1 i •\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x =\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V), ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x", "tactic": "simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,\n Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]" } ]

[ 58, 85 ]

[ 51, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ✝ : Type ?u.20071\ninst✝¹ : LinearOrder α\nx y : α\nβ : Type u_2\ninst✝ : LinearOrder β\nx' y' : β\nh : cmp x y = cmp x' y'\n⊢ x < y ↔ x' < y'", "tactic": "rw [← cmp_eq_lt_iff, ← cmp_eq_lt_iff, h]" } ]

[ 255, 43 ]

[ 254, 1 ]

[]

[ 110, 72 ]

[ 109, 1 ]

[]

[ 1128, 23 ]

[ 1127, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.227043\nι : Type ?u.227046\nM : Type u_3\nM' : Type ?u.227052\nN : Type ?u.227055\nP : Type ?u.227058\nG : Type ?u.227061\nH : Type ?u.227064\nR : Type ?u.227067\nS : Type ?u.227070\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\nl₁ l₂ : α →₀ M\nh : embDomain f l₁ = embDomain f l₂\na : α\n⊢ ↑l₁ a = ↑l₂ a", "tactic": "simpa only [embDomain_apply] using FunLike.ext_iff.1 h (f a)" } ]

[ 872, 94 ]

[ 871, 1 ]

[]

[ 273, 71 ]

[ 272, 1 ]

[]

[ 819, 6 ]

[ 818, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nm : ℤ\n⊢ toIcoMod hp (m • p + a) b = m • p + toIcoMod hp a b", "tactic": "rw [add_comm, toIcoMod_add_zsmul', add_comm]" } ]

[ 436, 47 ]

[ 434, 1 ]

[ { "state_after": "case a.h\nn : ℕ\nα : TypeVec (n + 1)\np : α ⟹ repeat (n + 1) Prop\ni : Fin2 n\nx✝ : drop (fun i => { x // ofRepeat (p i x) }) i\n⊢ dropFun (toSubtype p) i x✝ = toSubtype (fun i x => p (Fin2.fs i) x) i x✝", "state_before": "n : ℕ\nα : TypeVec (n + 1)\np : α ⟹ repeat (n + 1) Prop\n⊢ dropFun (toSubtype p) = toSubtype fun i x => p (Fin2.fs i) x", "tactic": "ext i" }, { "state_after": "no goals", "state_before": "case a.h\nn : ℕ\nα : TypeVec (n + 1)\np : α ⟹ repeat (n + 1) Prop\ni : Fin2 n\nx✝ : drop (fun i => { x // ofRepeat (p i x) }) i\n⊢ dropFun (toSubtype p) i x✝ = toSubtype (fun i x => p (Fin2.fs i) x) i x✝", "tactic": "induction i <;> simp [dropFun, *] <;> rfl" } ]

[ 709, 44 ]

[ 706, 1 ]