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Mathlib/Topology/Constructions.lean | nhds_inl | [] | [
952,
41
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951,
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Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | ProjectiveSpectrum.zeroLocus_anti_mono_homogeneousIdeal | [] | [
191,
39
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189,
1
] |
Mathlib/Algebra/RingQuot.lean | RingQuot.liftAlgHom_mkAlgHom_apply | [
{
"state_after": "R : Type u₁\ninst✝⁶ : Semiring R\nS : Type u₂\ninst✝⁵ : CommSemiring S\nA : Type u₃\ninst✝⁴ : Semiring A\ninst✝³ : Algebra S A\nr : R → R → Prop\nT : Type u₄\ninst✝² : Semiring T\nB : Type u₄\ninst✝¹ : Semiring B\ninst✝ : Algebra S B\nf : A →ₐ[S] B\ns : A → A → Prop\nw : ∀ ⦃x y : A⦄, s x y → ↑f x = ↑f y\nx : A\n⊢ ↑(↑{\n toFun := fun f' =>\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) },\n map_zero' :=\n (_ :\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) 0.toQuot = 0),\n map_add' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n y) },\n commutes' :=\n (_ :\n ∀ (x : S),\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n (↑(algebraMap S (RingQuot fun x y => s x y)) x).toQuot =\n ↑(algebraMap S B) x) },\n invFun := fun F =>\n {\n val :=\n AlgHom.comp F\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) },\n commutes' :=\n (_ :\n ∀ (r : S),\n OneHom.toFun\n (↑↑{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) })\n (↑(algebraMap S A) r) =\n ↑(algebraMap S (RingQuot fun x y => s x y)) r) },\n property :=\n (_ :\n (fun f => ∀ ⦃x y : A⦄, s x y → ↑f x = ↑f y)\n (AlgHom.comp F\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) },\n commutes' :=\n (_ :\n ∀ (r : S),\n OneHom.toFun\n (↑↑{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) })\n (↑(algebraMap S A) r) =\n ↑(algebraMap S (RingQuot fun x y => s x y)) r) })) },\n left_inv :=\n (_ :\n Function.LeftInverse\n (fun F =>\n {\n val :=\n AlgHom.comp F\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) },\n commutes' :=\n (_ :\n ∀ (r : S),\n OneHom.toFun\n (↑↑{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) })\n (↑(algebraMap S A) r) =\n ↑(algebraMap S (RingQuot fun x y => s x y)) r) },\n property :=\n (_ :\n (fun f => ∀ ⦃x y : A⦄, s x y → ↑f x = ↑f y)\n (AlgHom.comp F\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) },\n commutes' :=\n (_ :\n ∀ (r : S),\n OneHom.toFun\n (↑↑{\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y)\n (a * a_1) } =\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y)\n (a * a_1) } =\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) })\n (↑(algebraMap S A) r) =\n ↑(algebraMap S (RingQuot fun x y => s x y)) r) })) })\n fun f' =>\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) },\n map_zero' :=\n (_ :\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) 0.toQuot =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n y) },\n commutes' :=\n (_ :\n ∀ (x : S),\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n (↑(algebraMap S (RingQuot fun x y => s x y)) x).toQuot =\n ↑(algebraMap S B) x) }),\n right_inv :=\n (_ :\n Function.RightInverse\n (fun F =>\n {\n val :=\n AlgHom.comp F\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) },\n commutes' :=\n (_ :\n ∀ (r : S),\n OneHom.toFun\n (↑↑{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) })\n (↑(algebraMap S A) r) =\n ↑(algebraMap S (RingQuot fun x y => s x y)) r) },\n property :=\n (_ :\n (fun f => ∀ ⦃x y : A⦄, s x y → ↑f x = ↑f y)\n (AlgHom.comp F\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) },\n commutes' :=\n (_ :\n ∀ (r : S),\n OneHom.toFun\n (↑↑{\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y)\n (a * a_1) } =\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y)\n (a * a_1) } =\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) })\n (↑(algebraMap S A) r) =\n ↑(algebraMap S (RingQuot fun x y => s x y)) r) })) })\n fun f' =>\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) },\n map_zero' :=\n (_ :\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) 0.toQuot =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n y) },\n commutes' :=\n (_ :\n ∀ (x : S),\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n (↑(algebraMap S (RingQuot fun x y => s x y)) x).toQuot =\n ↑(algebraMap S B) x) }) }\n { val := f, property := w })\n (↑{\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } * { toQuot := Quot.mk (Rel s) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } * { toQuot := Quot.mk (Rel s) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } * { toQuot := Quot.mk (Rel s) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } * { toQuot := Quot.mk (Rel s) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } * { toQuot := Quot.mk (Rel s) a_1 }) })\n y) },\n commutes' :=\n (_ :\n ∀ (r : S),\n OneHom.toFun\n (↑↑{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } * { toQuot := Quot.mk (Rel s) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } * { toQuot := Quot.mk (Rel s) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } *\n { toQuot := Quot.mk (Rel s) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } *\n { toQuot := Quot.mk (Rel s) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } *\n { toQuot := Quot.mk (Rel s) a_1 }) })\n y) })\n (↑(algebraMap S A) r) =\n ↑(algebraMap S (RingQuot s)) r) }\n x) =\n ↑f x",
"state_before": "R : Type u₁\ninst✝⁶ : Semiring R\nS : Type u₂\ninst✝⁵ : CommSemiring S\nA : Type u₃\ninst✝⁴ : Semiring A\ninst✝³ : Algebra S A\nr : R → R → Prop\nT : Type u₄\ninst✝² : Semiring T\nB : Type u₄\ninst✝¹ : Semiring B\ninst✝ : Algebra S B\nf : A →ₐ[S] B\ns : A → A → Prop\nw : ∀ ⦃x y : A⦄, s x y → ↑f x = ↑f y\nx : A\n⊢ ↑(↑(liftAlgHom S) { val := f, property := w }) (↑(mkAlgHom S s) x) = ↑f x",
"tactic": "simp_rw [liftAlgHom_def, preLiftAlgHom_def, mkAlgHom_def, mkRingHom_def]"
},
{
"state_after": "no goals",
"state_before": "R : Type u₁\ninst✝⁶ : Semiring R\nS : Type u₂\ninst✝⁵ : CommSemiring S\nA : Type u₃\ninst✝⁴ : Semiring A\ninst✝³ : Algebra S A\nr : R → R → Prop\nT : Type u₄\ninst✝² : Semiring T\nB : Type u₄\ninst✝¹ : Semiring B\ninst✝ : Algebra S B\nf : A →ₐ[S] B\ns : A → A → Prop\nw : ∀ ⦃x y : A⦄, s x y → ↑f x = ↑f y\nx : A\n⊢ ↑(↑{\n toFun := fun f' =>\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) },\n map_zero' :=\n (_ :\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) 0.toQuot = 0),\n map_add' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n y) },\n commutes' :=\n (_ :\n ∀ (x : S),\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n (↑(algebraMap S (RingQuot fun x y => s x y)) x).toQuot =\n ↑(algebraMap S B) x) },\n invFun := fun F =>\n {\n val :=\n AlgHom.comp F\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) },\n commutes' :=\n (_ :\n ∀ (r : S),\n OneHom.toFun\n (↑↑{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) })\n (↑(algebraMap S A) r) =\n ↑(algebraMap S (RingQuot fun x y => s x y)) r) },\n property :=\n (_ :\n (fun f => ∀ ⦃x y : A⦄, s x y → ↑f x = ↑f y)\n (AlgHom.comp F\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) },\n commutes' :=\n (_ :\n ∀ (r : S),\n OneHom.toFun\n (↑↑{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) })\n (↑(algebraMap S A) r) =\n ↑(algebraMap S (RingQuot fun x y => s x y)) r) })) },\n left_inv :=\n (_ :\n Function.LeftInverse\n (fun F =>\n {\n val :=\n AlgHom.comp F\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) },\n commutes' :=\n (_ :\n ∀ (r : S),\n OneHom.toFun\n (↑↑{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) })\n (↑(algebraMap S A) r) =\n ↑(algebraMap S (RingQuot fun x y => s x y)) r) },\n property :=\n (_ :\n (fun f => ∀ ⦃x y : A⦄, s x y → ↑f x = ↑f y)\n (AlgHom.comp F\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) },\n commutes' :=\n (_ :\n ∀ (r : S),\n OneHom.toFun\n (↑↑{\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y)\n (a * a_1) } =\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y)\n (a * a_1) } =\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) })\n (↑(algebraMap S A) r) =\n ↑(algebraMap S (RingQuot fun x y => s x y)) r) })) })\n fun f' =>\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) },\n map_zero' :=\n (_ :\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) 0.toQuot =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n y) },\n commutes' :=\n (_ :\n ∀ (x : S),\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n (↑(algebraMap S (RingQuot fun x y => s x y)) x).toQuot =\n ↑(algebraMap S B) x) }),\n right_inv :=\n (_ :\n Function.RightInverse\n (fun F =>\n {\n val :=\n AlgHom.comp F\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) },\n commutes' :=\n (_ :\n ∀ (r : S),\n OneHom.toFun\n (↑↑{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) })\n (↑(algebraMap S A) r) =\n ↑(algebraMap S (RingQuot fun x y => s x y)) r) },\n property :=\n (_ :\n (fun f => ∀ ⦃x y : A⦄, s x y → ↑f x = ↑f y)\n (AlgHom.comp F\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) },\n commutes' :=\n (_ :\n ∀ (r : S),\n OneHom.toFun\n (↑↑{\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ : { toQuot := Quot.mk (Rel fun x y => s x y) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n { toQuot := Quot.mk (Rel fun x y => s x y) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n { toQuot := Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) (a * a_1) } =\n { toQuot := Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y)\n (a * a_1) } =\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n { toQuot := Quot.mk (Rel fun x y => s x y) x },\n map_one' :=\n (_ :\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) 1 } =\n 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y)\n (a * a_1) } =\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a } *\n {\n toQuot :=\n Quot.mk (Rel fun x y => s x y) a_1 }) })\n y) })\n (↑(algebraMap S A) r) =\n ↑(algebraMap S (RingQuot fun x y => s x y)) r) })) })\n fun f' =>\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) },\n map_zero' :=\n (_ :\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b) 0.toQuot =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : RingQuot fun x y => s x y),\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n x.toQuot,\n map_one' :=\n (_ :\n Quot.lift ↑↑f'\n (_ :\n ∀ (a b : A),\n Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n 1.toQuot =\n 1) }\n y) })\n y) },\n commutes' :=\n (_ :\n ∀ (x : S),\n Quot.lift ↑↑f' (_ : ∀ (a b : A), Rel (fun x y => s x y) a b → ↑↑f' a = ↑↑f' b)\n (↑(algebraMap S (RingQuot fun x y => s x y)) x).toQuot =\n ↑(algebraMap S B) x) }) }\n { val := f, property := w })\n (↑{\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } * { toQuot := Quot.mk (Rel s) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } * { toQuot := Quot.mk (Rel s) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } * { toQuot := Quot.mk (Rel s) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } * { toQuot := Quot.mk (Rel s) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } * { toQuot := Quot.mk (Rel s) a_1 }) })\n y) },\n commutes' :=\n (_ :\n ∀ (r : S),\n OneHom.toFun\n (↑↑{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } * { toQuot := Quot.mk (Rel s) a_1 }) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } * { toQuot := Quot.mk (Rel s) a_1 }) })\n 0 =\n 0),\n map_add' :=\n (_ :\n ∀ (x y : A),\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } *\n { toQuot := Quot.mk (Rel s) a_1 }) })\n (x + y) =\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } *\n { toQuot := Quot.mk (Rel s) a_1 }) })\n x +\n OneHom.toFun\n (↑{\n toOneHom :=\n { toFun := fun x => { toQuot := Quot.mk (Rel s) x },\n map_one' := (_ : { toQuot := Quot.mk (Rel s) 1 } = 1) },\n map_mul' :=\n (_ :\n ∀ (a a_1 : A),\n { toQuot := Quot.mk (Rel s) (a * a_1) } =\n { toQuot := Quot.mk (Rel s) a } *\n { toQuot := Quot.mk (Rel s) a_1 }) })\n y) })\n (↑(algebraMap S A) r) =\n ↑(algebraMap S (RingQuot s)) r) }\n x) =\n ↑f x",
"tactic": "rfl"
}
] | [
670,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
667,
1
] |
Mathlib/LinearAlgebra/Matrix/Adjugate.lean | Matrix.cramer_row_self | [
{
"state_after": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\ni : n\nh : ∀ (j : n), b j = A j i\n⊢ ↑(cramer Aᵀᵀ) b = Pi.single i (det Aᵀ)",
"state_before": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\ni : n\nh : ∀ (j : n), b j = A j i\n⊢ ↑(cramer A) b = Pi.single i (det A)",
"tactic": "rw [← transpose_transpose A, det_transpose]"
},
{
"state_after": "case h.e'_2.h.e'_6\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\ni : n\nh : ∀ (j : n), b j = A j i\n⊢ b = Aᵀ i",
"state_before": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\ni : n\nh : ∀ (j : n), b j = A j i\n⊢ ↑(cramer Aᵀᵀ) b = Pi.single i (det Aᵀ)",
"tactic": "convert cramer_transpose_row_self Aᵀ i"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_6\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\ni : n\nh : ∀ (j : n), b j = A j i\n⊢ b = Aᵀ i",
"tactic": "exact funext h"
}
] | [
126,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
1
] |
Mathlib/Topology/Instances/ENNReal.lean | NNReal.tsum_eq_toNNReal_tsum | [
{
"state_after": "case pos\nα : Type ?u.309286\nβ : Type u_1\nγ : Type ?u.309292\nf : β → ℝ≥0\nh : Summable f\n⊢ (∑' (b : β), f b) = ENNReal.toNNReal (∑' (b : β), ↑(f b))\n\ncase neg\nα : Type ?u.309286\nβ : Type u_1\nγ : Type ?u.309292\nf : β → ℝ≥0\nh : ¬Summable f\n⊢ (∑' (b : β), f b) = ENNReal.toNNReal (∑' (b : β), ↑(f b))",
"state_before": "α : Type ?u.309286\nβ : Type u_1\nγ : Type ?u.309292\nf : β → ℝ≥0\n⊢ (∑' (b : β), f b) = ENNReal.toNNReal (∑' (b : β), ↑(f b))",
"tactic": "by_cases h : Summable f"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type ?u.309286\nβ : Type u_1\nγ : Type ?u.309292\nf : β → ℝ≥0\nh : Summable f\n⊢ (∑' (b : β), f b) = ENNReal.toNNReal (∑' (b : β), ↑(f b))",
"tactic": "rw [← ENNReal.coe_tsum h, ENNReal.toNNReal_coe]"
},
{
"state_after": "case neg\nα : Type ?u.309286\nβ : Type u_1\nγ : Type ?u.309292\nf : β → ℝ≥0\nh : ¬Summable f\nA : (∑' (b : β), f b) = 0\n⊢ (∑' (b : β), f b) = ENNReal.toNNReal (∑' (b : β), ↑(f b))",
"state_before": "case neg\nα : Type ?u.309286\nβ : Type u_1\nγ : Type ?u.309292\nf : β → ℝ≥0\nh : ¬Summable f\n⊢ (∑' (b : β), f b) = ENNReal.toNNReal (∑' (b : β), ↑(f b))",
"tactic": "have A := tsum_eq_zero_of_not_summable h"
},
{
"state_after": "case neg\nα : Type ?u.309286\nβ : Type u_1\nγ : Type ?u.309292\nf : β → ℝ≥0\nA : (∑' (b : β), f b) = 0\nh : (∑' (b : β), ↑(f b)) = ⊤\n⊢ (∑' (b : β), f b) = ENNReal.toNNReal (∑' (b : β), ↑(f b))",
"state_before": "case neg\nα : Type ?u.309286\nβ : Type u_1\nγ : Type ?u.309292\nf : β → ℝ≥0\nh : ¬Summable f\nA : (∑' (b : β), f b) = 0\n⊢ (∑' (b : β), f b) = ENNReal.toNNReal (∑' (b : β), ↑(f b))",
"tactic": "simp only [← ENNReal.tsum_coe_ne_top_iff_summable, Classical.not_not] at h"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type ?u.309286\nβ : Type u_1\nγ : Type ?u.309292\nf : β → ℝ≥0\nA : (∑' (b : β), f b) = 0\nh : (∑' (b : β), ↑(f b)) = ⊤\n⊢ (∑' (b : β), f b) = ENNReal.toNNReal (∑' (b : β), ↑(f b))",
"tactic": "simp only [h, ENNReal.top_toNNReal, A]"
}
] | [
1103,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1098,
1
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.inv_lt_inv | [] | [
1479,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1478,
11
] |
Mathlib/LinearAlgebra/Projection.lean | Submodule.existsUnique_add_of_isCompl | [] | [
211,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
208,
1
] |
Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | TopCat.Sheaf.existsUnique_gluing | [] | [
250,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
248,
1
] |
Mathlib/MeasureTheory/Measure/AEMeasurable.lean | MeasurableEmbedding.aemeasurable_map_iff | [
{
"state_after": "ι : Type ?u.3357463\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.3357475\nR : Type ?u.3357478\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nf g✝ : α → β\nμ ν : MeasureTheory.Measure α\ng : β → γ\nhf : MeasurableEmbedding f\n⊢ AEMeasurable (g ∘ f) → AEMeasurable g",
"state_before": "ι : Type ?u.3357463\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.3357475\nR : Type ?u.3357478\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nf g✝ : α → β\nμ ν : MeasureTheory.Measure α\ng : β → γ\nhf : MeasurableEmbedding f\n⊢ AEMeasurable g ↔ AEMeasurable (g ∘ f)",
"tactic": "refine' ⟨fun H => H.comp_measurable hf.measurable, _⟩"
},
{
"state_after": "case intro.intro\nι : Type ?u.3357463\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.3357475\nR : Type ?u.3357478\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nf g✝ : α → β\nμ ν : MeasureTheory.Measure α\ng : β → γ\nhf : MeasurableEmbedding f\ng₁ : α → γ\nhgm₁ : Measurable g₁\nheq : g ∘ f =ᶠ[ae μ] g₁\n⊢ AEMeasurable g",
"state_before": "ι : Type ?u.3357463\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.3357475\nR : Type ?u.3357478\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nf g✝ : α → β\nμ ν : MeasureTheory.Measure α\ng : β → γ\nhf : MeasurableEmbedding f\n⊢ AEMeasurable (g ∘ f) → AEMeasurable g",
"tactic": "rintro ⟨g₁, hgm₁, heq⟩"
},
{
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{
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"tactic": "exact ⟨g₂, hgm₂, hf.ae_map_iff.2 heq⟩"
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] | [
267,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
262,
1
] |
Mathlib/Order/Lattice.lean | Monotone.le_map_sup | [] | [
1099,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1097,
1
] |
Mathlib/Data/Fin/VecNotation.lean | Matrix.vecAlt1_vecAppend | [
{
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{
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},
{
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},
{
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"state_before": "case h.zero.mk\nα : Type u\nm o : ℕ\nm' : Type ?u.42128\nn' : Type ?u.42131\no' : Type ?u.42134\nv : Fin (Nat.zero + 1) → α\ni : ℕ\nhi : i < Nat.zero + 1\n⊢ (if h : ↑{ val := i, isLt := hi } + ↑{ val := i, isLt := hi } + 1 < Nat.zero + 1 then\n v\n { val := ↑{ val := i, isLt := hi } + ↑{ val := i, isLt := hi } + 1,\n isLt :=\n (_ :\n ↑{ val := ↑{ val := i, isLt := hi } + ↑{ val := i, isLt := hi } + 1,\n isLt :=\n (_ :\n ↑{ val := i, isLt := hi } + ↑{ val := i, isLt := hi } + 1 < Nat.zero + 1 + (Nat.zero + 1)) } <\n Nat.zero + 1) }\n else\n v\n { val := ↑{ val := i, isLt := hi } + ↑{ val := i, isLt := hi } + 1 - (Nat.zero + 1),\n isLt :=\n (_ :\n ↑{ val := ↑{ val := i, isLt := hi } + ↑{ val := i, isLt := hi } + 1,\n isLt :=\n (_ :\n ↑{ val := i, isLt := hi } + ↑{ val := i, isLt := hi } + 1 < Nat.zero + 1 + (Nat.zero + 1)) } -\n (Nat.zero + 1) <\n Nat.zero + 1) }) =\n v (bit1 { val := i, isLt := hi })",
"tactic": "simp only [Nat.zero_eq, zero_add, Nat.lt_one_iff] at hi"
},
{
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},
{
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"tactic": "rfl"
},
{
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"tactic": "split_ifs with h <;> simp_rw [bit1, bit0] <;> congr"
},
{
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"tactic": "simp only [Fin.ext_iff, Fin.val_add, Fin.val_mk]"
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{
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},
{
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"tactic": "rw [Fin.val_mk, not_lt] at h"
},
{
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"tactic": "simp only [Fin.ext_iff, Fin.val_add, Fin.val_mk, Nat.mod_add_mod, Fin.val_one,\n Nat.mod_eq_sub_mod h, show 1 % (n + 2) = 1 from Nat.mod_eq_of_lt (by simp)]"
},
{
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"tactic": "refine (Nat.mod_eq_of_lt ?_).symm"
},
{
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356,
40
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338,
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Mathlib/Topology/Algebra/Monoid.lean | continuous_pow | [
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47
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560,
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Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right | [
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456,
75
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448,
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Mathlib/Analysis/SpecialFunctions/Sqrt.lean | DifferentiableAt.sqrt | [] | [
140,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
138,
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Mathlib/Computability/Reduce.lean | OneOneEquiv.trans | [] | [
197,
53
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195,
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Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_floor_right_pure_floor | [] | [
73,
56
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71,
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Mathlib/Topology/Inseparable.lean | SeparationQuotient.inducing_mk | [] | [
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63
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467,
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Mathlib/NumberTheory/ArithmeticFunction.lean | Nat.ArithmeticFunction.toFun_eq | [] | [
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65
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Mathlib/Data/Multiset/Basic.lean | Multiset.sub_cons | [] | [
1650,
74
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1649,
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Mathlib/Algebra/DirectLimit.lean | Module.DirectLimit.lift_unique | [
{
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},
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{
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},
{
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}
] | [
171,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
167,
1
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Mathlib/LinearAlgebra/Eigenspace/Basic.lean | Module.End.hasEigenvalue_of_hasGeneralizedEigenvalue | [
{
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"tactic": "intro contra"
},
{
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"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nk : ℕ\nhμ : HasGeneralizedEigenvalue f μ k\ncontra : eigenspace f μ = ⊥\n⊢ False",
"tactic": "apply hμ"
},
{
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"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nk : ℕ\nhμ : HasGeneralizedEigenvalue f μ k\ncontra : eigenspace f μ = ⊥\n⊢ ↑(generalizedEigenspace f μ) k = ⊥",
"tactic": "erw [LinearMap.ker_eq_bot] at contra ⊢"
},
{
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"tactic": "rw [LinearMap.coe_pow]"
},
{
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"tactic": "exact Function.Injective.iterate contra k"
}
] | [
361,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
357,
1
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Mathlib/LinearAlgebra/Prod.lean | LinearMap.coprod_comp_prod | [] | [
259,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
257,
1
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Mathlib/Topology/Separation.lean | Continuous.ext_on | [] | [
1197,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1195,
1
] |
Mathlib/AlgebraicTopology/TopologicalSimplex.lean | SimplexCategory.toTopObj.ext | [] | [
50,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
49,
1
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Mathlib/Data/Sign.lean | SignType.neg_one_le | [] | [
206,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
205,
1
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Mathlib/Order/Heyting/Boundary.lean | Coheyting.boundary_sup_sup_boundary_inf | [] | [
131,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
128,
1
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Mathlib/GroupTheory/MonoidLocalization.lean | Submonoid.LocalizationMap.ofMulEquivOfDom_id | [
{
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"tactic": "ext"
},
{
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"tactic": "rfl"
}
] | [
1544,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/SetTheory/Ordinal/FixedPoint.lean | Ordinal.deriv_mul_eq_opow_omega_mul | [
{
"state_after": "a : Ordinal\nha : 0 < a\n⊢ ∀ (b : Ordinal), deriv (fun x => a * x) b = a ^ ω * b",
"state_before": "a : Ordinal\nha : 0 < a\nb : Ordinal\n⊢ deriv (fun x => a * x) b = a ^ ω * b",
"tactic": "revert b"
},
{
"state_after": "a : Ordinal\nha : 0 < a\n⊢ deriv (fun x => a * x) 0 = (fun x x_1 => x * x_1) (a ^ ω) 0 ∧\n ∀ (a_1 : Ordinal),\n deriv (fun x => a * x) a_1 = (fun x x_1 => x * x_1) (a ^ ω) a_1 →\n deriv (fun x => a * x) (succ a_1) = (fun x x_1 => x * x_1) (a ^ ω) (succ a_1)",
"state_before": "a : Ordinal\nha : 0 < a\n⊢ ∀ (b : Ordinal), deriv (fun x => a * x) b = a ^ ω * b",
"tactic": "rw [← funext_iff,\n IsNormal.eq_iff_zero_and_succ (deriv_isNormal _) (mul_isNormal (opow_pos omega ha))]"
},
{
"state_after": "case refine'_1\na : Ordinal\nha : 0 < a\n⊢ deriv (fun x => a * x) 0 = (fun x x_1 => x * x_1) (a ^ ω) 0\n\ncase refine'_2\na : Ordinal\nha : 0 < a\nc : Ordinal\nh : deriv (fun x => a * x) c = (fun x x_1 => x * x_1) (a ^ ω) c\n⊢ deriv (fun x => a * x) (succ c) = (fun x x_1 => x * x_1) (a ^ ω) (succ c)",
"state_before": "a : Ordinal\nha : 0 < a\n⊢ deriv (fun x => a * x) 0 = (fun x x_1 => x * x_1) (a ^ ω) 0 ∧\n ∀ (a_1 : Ordinal),\n deriv (fun x => a * x) a_1 = (fun x x_1 => x * x_1) (a ^ ω) a_1 →\n deriv (fun x => a * x) (succ a_1) = (fun x x_1 => x * x_1) (a ^ ω) (succ a_1)",
"tactic": "refine' ⟨_, fun c h => _⟩"
},
{
"state_after": "case refine'_1\na : Ordinal\nha : 0 < a\n⊢ deriv (fun x => a * x) 0 = a ^ ω * 0",
"state_before": "case refine'_1\na : Ordinal\nha : 0 < a\n⊢ deriv (fun x => a * x) 0 = (fun x x_1 => x * x_1) (a ^ ω) 0",
"tactic": "dsimp only"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\na : Ordinal\nha : 0 < a\n⊢ deriv (fun x => a * x) 0 = a ^ ω * 0",
"tactic": "rw [deriv_zero, nfp_mul_zero, mul_zero]"
},
{
"state_after": "case refine'_2\na : Ordinal\nha : 0 < a\nc : Ordinal\nh : deriv (fun x => a * x) c = (fun x x_1 => x * x_1) (a ^ ω) c\n⊢ nfp (fun x => a * x) (succ ((fun x x_1 => x * x_1) (a ^ ω) c)) = (fun x x_1 => x * x_1) (a ^ ω) (succ c)",
"state_before": "case refine'_2\na : Ordinal\nha : 0 < a\nc : Ordinal\nh : deriv (fun x => a * x) c = (fun x x_1 => x * x_1) (a ^ ω) c\n⊢ deriv (fun x => a * x) (succ c) = (fun x x_1 => x * x_1) (a ^ ω) (succ c)",
"tactic": "rw [deriv_succ, h]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\na : Ordinal\nha : 0 < a\nc : Ordinal\nh : deriv (fun x => a * x) c = (fun x x_1 => x * x_1) (a ^ ω) c\n⊢ nfp (fun x => a * x) (succ ((fun x x_1 => x * x_1) (a ^ ω) c)) = (fun x x_1 => x * x_1) (a ^ ω) (succ c)",
"tactic": "exact nfp_mul_opow_omega_add c ha zero_lt_one (one_le_iff_pos.2 (opow_pos _ ha))"
}
] | [
726,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
718,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean | EMetric.tendstoLocallyUniformlyOn_iff | [
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.29841\ninst✝¹ : PseudoEMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nf : β → α\np : Filter ι\ns : Set β\nH :\n ∀ (ε : ℝ≥0∞), ε > 0 → ∀ (x : β), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → edist (f y) (F n y) < ε\nu : Set (α × α)\nhu : u ∈ 𝓤 α\nx : β\nhx : x ∈ s\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → (f y, F n y) ∈ u",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.29841\ninst✝¹ : PseudoEMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nf : β → α\np : Filter ι\ns : Set β\n⊢ TendstoLocallyUniformlyOn F f p s ↔\n ∀ (ε : ℝ≥0∞),\n ε > 0 → ∀ (x : β), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → edist (f y) (F n y) < ε",
"tactic": "refine' ⟨fun H ε hε => H _ (edist_mem_uniformity hε), fun H u hu x hx => _⟩"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nX : Type ?u.29841\ninst✝¹ : PseudoEMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nf : β → α\np : Filter ι\ns : Set β\nH :\n ∀ (ε : ℝ≥0∞), ε > 0 → ∀ (x : β), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → edist (f y) (F n y) < ε\nu : Set (α × α)\nhu : u ∈ 𝓤 α\nx : β\nhx : x ∈ s\nε : ℝ≥0∞\nεpos : ε > 0\nhε : ∀ {a b : α}, edist a b < ε → (a, b) ∈ u\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → (f y, F n y) ∈ u",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.29841\ninst✝¹ : PseudoEMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nf : β → α\np : Filter ι\ns : Set β\nH :\n ∀ (ε : ℝ≥0∞), ε > 0 → ∀ (x : β), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → edist (f y) (F n y) < ε\nu : Set (α × α)\nhu : u ∈ 𝓤 α\nx : β\nhx : x ∈ s\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → (f y, F n y) ∈ u",
"tactic": "rcases mem_uniformity_edist.1 hu with ⟨ε, εpos, hε⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nX : Type ?u.29841\ninst✝¹ : PseudoEMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nf : β → α\np : Filter ι\ns : Set β\nH :\n ∀ (ε : ℝ≥0∞), ε > 0 → ∀ (x : β), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → edist (f y) (F n y) < ε\nu : Set (α × α)\nhu : u ∈ 𝓤 α\nx : β\nhx : x ∈ s\nε : ℝ≥0∞\nεpos : ε > 0\nhε : ∀ {a b : α}, edist a b < ε → (a, b) ∈ u\nt : Set β\nht : t ∈ 𝓝[s] x\nHt : ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → edist (f y) (F n y) < ε\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → (f y, F n y) ∈ u",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nX : Type ?u.29841\ninst✝¹ : PseudoEMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nf : β → α\np : Filter ι\ns : Set β\nH :\n ∀ (ε : ℝ≥0∞), ε > 0 → ∀ (x : β), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → edist (f y) (F n y) < ε\nu : Set (α × α)\nhu : u ∈ 𝓤 α\nx : β\nhx : x ∈ s\nε : ℝ≥0∞\nεpos : ε > 0\nhε : ∀ {a b : α}, edist a b < ε → (a, b) ∈ u\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → (f y, F n y) ∈ u",
"tactic": "rcases H ε εpos x hx with ⟨t, ht, Ht⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nX : Type ?u.29841\ninst✝¹ : PseudoEMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nf : β → α\np : Filter ι\ns : Set β\nH :\n ∀ (ε : ℝ≥0∞), ε > 0 → ∀ (x : β), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → edist (f y) (F n y) < ε\nu : Set (α × α)\nhu : u ∈ 𝓤 α\nx : β\nhx : x ∈ s\nε : ℝ≥0∞\nεpos : ε > 0\nhε : ∀ {a b : α}, edist a b < ε → (a, b) ∈ u\nt : Set β\nht : t ∈ 𝓝[s] x\nHt : ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → edist (f y) (F n y) < ε\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : β), y ∈ t → (f y, F n y) ∈ u",
"tactic": "exact ⟨t, ht, Ht.mono fun n hs x hx => hε (hs x hx)⟩"
}
] | [
349,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
342,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | Subgroup.le_centralizer_iff | [] | [
2300,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2299,
1
] |
Mathlib/RingTheory/Int/Basic.lean | Int.normalize_coe_nat | [] | [
116,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
115,
1
] |
Mathlib/Algebra/Algebra/Spectrum.lean | spectrum.mem_resolventSet_iff | [] | [
135,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
134,
1
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.coe_mono | [] | [
336,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
336,
1
] |
Mathlib/GroupTheory/Submonoid/Basic.lean | Submonoid.mem_sInf | [] | [
324,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
323,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Walk.copy_copy | [
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu'' v'' : V\np : Walk G u'' v''\n⊢ Walk.copy (Walk.copy p (_ : u'' = u'') (_ : v'' = v'')) (_ : u'' = u'') (_ : v'' = v'') =\n Walk.copy p (_ : u'' = u'') (_ : v'' = v'')",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v u' v' u'' v'' : V\np : Walk G u v\nhu : u = u'\nhv : v = v'\nhu' : u' = u''\nhv' : v' = v''\n⊢ Walk.copy (Walk.copy p hu hv) hu' hv' = Walk.copy p (_ : u = u'') (_ : v = v'')",
"tactic": "subst_vars"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu'' v'' : V\np : Walk G u'' v''\n⊢ Walk.copy (Walk.copy p (_ : u'' = u'') (_ : v'' = v'')) (_ : u'' = u'') (_ : v'' = v'') =\n Walk.copy p (_ : u'' = u'') (_ : v'' = v'')",
"tactic": "rfl"
}
] | [
146,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
142,
1
] |
Mathlib/Algebra/Module/Submodule/Pointwise.lean | Submodule.zero_eq_bot | [] | [
176,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
175,
1
] |
Mathlib/Topology/SubsetProperties.lean | IsCompact.elim_finite_subcover | [] | [
197,
99
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
194,
1
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean | FreeAbelianGroup.liftMonoid_symm_coe | [] | [
548,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
546,
1
] |
Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | CategoryTheory.Limits.coprodZeroIso_inv | [] | [
156,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
] |
Mathlib/Analysis/SpecialFunctions/Exp.lean | Real.isBoundedUnder_le_exp_comp | [] | [
213,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
211,
1
] |
Mathlib/Algebra/Associated.lean | dvdNotUnit_of_dvdNotUnit_associated | [
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.348972\nγ : Type ?u.348975\nδ : Type ?u.348978\ninst✝¹ : CommMonoidWithZero α\ninst✝ : Nontrivial α\np r : α\nu : αˣ\nh : DvdNotUnit p (r * ↑u)\nh' : r * ↑u ~ᵤ r\n⊢ DvdNotUnit p r",
"state_before": "α : Type u_1\nβ : Type ?u.348972\nγ : Type ?u.348975\nδ : Type ?u.348978\ninst✝¹ : CommMonoidWithZero α\ninst✝ : Nontrivial α\np q r : α\nh : DvdNotUnit p q\nh' : q ~ᵤ r\n⊢ DvdNotUnit p r",
"tactic": "obtain ⟨u, rfl⟩ := Associated.symm h'"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.348972\nγ : Type ?u.348975\nδ : Type ?u.348978\ninst✝¹ : CommMonoidWithZero α\ninst✝ : Nontrivial α\np r : α\nu : αˣ\nh' : r * ↑u ~ᵤ r\nhp : p ≠ 0\nx : α\nhx : ¬IsUnit x ∧ r * ↑u = p * x\n⊢ DvdNotUnit p r",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.348972\nγ : Type ?u.348975\nδ : Type ?u.348978\ninst✝¹ : CommMonoidWithZero α\ninst✝ : Nontrivial α\np r : α\nu : αˣ\nh : DvdNotUnit p (r * ↑u)\nh' : r * ↑u ~ᵤ r\n⊢ DvdNotUnit p r",
"tactic": "obtain ⟨hp, x, hx⟩ := h"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.348972\nγ : Type ?u.348975\nδ : Type ?u.348978\ninst✝¹ : CommMonoidWithZero α\ninst✝ : Nontrivial α\np r : α\nu : αˣ\nh' : r * ↑u ~ᵤ r\nhp : p ≠ 0\nx : α\nhx : ¬IsUnit x ∧ r * ↑u = p * x\n⊢ r = p * (x * ↑u⁻¹)",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.348972\nγ : Type ?u.348975\nδ : Type ?u.348978\ninst✝¹ : CommMonoidWithZero α\ninst✝ : Nontrivial α\np r : α\nu : αˣ\nh' : r * ↑u ~ᵤ r\nhp : p ≠ 0\nx : α\nhx : ¬IsUnit x ∧ r * ↑u = p * x\n⊢ DvdNotUnit p r",
"tactic": "refine' ⟨hp, x * ↑u⁻¹, DvdNotUnit.not_unit ⟨u⁻¹.ne_zero, x, hx.left, mul_comm _ _⟩, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.348972\nγ : Type ?u.348975\nδ : Type ?u.348978\ninst✝¹ : CommMonoidWithZero α\ninst✝ : Nontrivial α\np r : α\nu : αˣ\nh' : r * ↑u ~ᵤ r\nhp : p ≠ 0\nx : α\nhx : ¬IsUnit x ∧ r * ↑u = p * x\n⊢ r = p * (x * ↑u⁻¹)",
"tactic": "rw [← mul_assoc, ← hx.right, mul_assoc, Units.mul_inv, mul_one]"
}
] | [
1178,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1173,
1
] |
Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean | AlgebraicTopology.DoldKan.Γ₀_obj_termwise_mapMono_comp_PInfty | [
{
"state_after": "case h\nC : Type u_2\ninst✝³ : Category C\ninst✝² : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝ : Δ ⟶ Δ'\ninst✝¹ : Mono i✝\nn : ℕ\ni : [n] ⟶ Δ'\ninst✝ : Mono i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty (len [n]) =\n HomologicalComplex.Hom.f PInfty (len Δ') ≫ X.map i.op",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni : Δ ⟶ Δ'\ninst✝ : Mono i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty (len Δ) =\n HomologicalComplex.Hom.f PInfty (len Δ') ≫ X.map i.op",
"tactic": "induction' Δ using SimplexCategory.rec with n"
},
{
"state_after": "case h.h\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty (len [n]) =\n HomologicalComplex.Hom.f PInfty (len [n']) ≫ X.map i.op",
"state_before": "case h\nC : Type u_2\ninst✝³ : Category C\ninst✝² : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝ : Δ ⟶ Δ'\ninst✝¹ : Mono i✝\nn : ℕ\ni : [n] ⟶ Δ'\ninst✝ : Mono i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty (len [n]) =\n HomologicalComplex.Hom.f PInfty (len Δ') ≫ X.map i.op",
"tactic": "induction' Δ' using SimplexCategory.rec with n'"
},
{
"state_after": "case h.h\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op",
"state_before": "case h.h\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty (len [n]) =\n HomologicalComplex.Hom.f PInfty (len [n']) ≫ X.map i.op",
"tactic": "dsimp"
},
{
"state_after": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : n = n'\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op\n\ncase neg\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op",
"state_before": "case h.h\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op",
"tactic": "by_cases n = n'"
},
{
"state_after": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : Isδ₀ i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op\n\ncase neg\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op",
"state_before": "case neg\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op",
"tactic": "by_cases hi : Isδ₀ i"
},
{
"state_after": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n]\ninst✝ : Mono i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n ≫ X.map i.op",
"state_before": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : n = n'\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op",
"tactic": "subst h"
},
{
"state_after": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n]\ninst✝ : Mono i\n⊢ 𝟙 (HomologicalComplex.X K[X] (len [n])) ≫ HomologicalComplex.Hom.f PInfty n =\n HomologicalComplex.Hom.f PInfty n ≫ 𝟙 (X.obj [n].op)",
"state_before": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n]\ninst✝ : Mono i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n ≫ X.map i.op",
"tactic": "simp only [SimplexCategory.eq_id_of_mono i, Γ₀.Obj.Termwise.mapMono_id, op_id, X.map_id]"
},
{
"state_after": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n]\ninst✝ : Mono i\n⊢ 𝟙 (X.obj [n].op) ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n ≫ 𝟙 (X.obj [n].op)",
"state_before": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n]\ninst✝ : Mono i\n⊢ 𝟙 (HomologicalComplex.X K[X] (len [n])) ≫ HomologicalComplex.Hom.f PInfty n =\n HomologicalComplex.Hom.f PInfty n ≫ 𝟙 (X.obj [n].op)",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n]\ninst✝ : Mono i\n⊢ 𝟙 (X.obj [n].op) ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n ≫ 𝟙 (X.obj [n].op)",
"tactic": "simp only [id_comp, comp_id]"
},
{
"state_after": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : Isδ₀ i\nh' : n' = n + 1\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op",
"state_before": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : Isδ₀ i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op",
"tactic": "have h' : n' = n + 1 := hi.left"
},
{
"state_after": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n =\n HomologicalComplex.Hom.f PInfty (n + 1) ≫ X.map i.op",
"state_before": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : Isδ₀ i\nh' : n' = n + 1\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op",
"tactic": "subst h'"
},
{
"state_after": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ HomologicalComplex.d K[X] (len [n + 1]) (len [n]) ≫ HomologicalComplex.Hom.f PInfty n =\n HomologicalComplex.Hom.f PInfty (n + 1) ≫ X.map i.op",
"state_before": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n =\n HomologicalComplex.Hom.f PInfty (n + 1) ≫ X.map i.op",
"tactic": "simp only [Γ₀.Obj.Termwise.mapMono_δ₀' _ i hi]"
},
{
"state_after": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ HomologicalComplex.d K[X] (n + 1) n ≫ HomologicalComplex.Hom.f PInfty n =\n HomologicalComplex.Hom.f PInfty (n + 1) ≫ X.map i.op",
"state_before": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ HomologicalComplex.d K[X] (len [n + 1]) (len [n]) ≫ HomologicalComplex.Hom.f PInfty n =\n HomologicalComplex.Hom.f PInfty (n + 1) ≫ X.map i.op",
"tactic": "dsimp"
},
{
"state_after": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ (HomologicalComplex.Hom.f PInfty (n + 1) ≫ Finset.sum Finset.univ fun i => (-1) ^ ↑i • SimplicialObject.δ X i) =\n HomologicalComplex.Hom.f PInfty (n + 1) ≫ X.map i.op",
"state_before": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ HomologicalComplex.d K[X] (n + 1) n ≫ HomologicalComplex.Hom.f PInfty n =\n HomologicalComplex.Hom.f PInfty (n + 1) ≫ X.map i.op",
"tactic": "rw [← PInfty.comm _ n, AlternatingFaceMapComplex.obj_d_eq]"
},
{
"state_after": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ (Finset.sum Finset.univ fun j => HomologicalComplex.Hom.f PInfty (n + 1) ≫ ((-1) ^ ↑j • SimplicialObject.δ X j)) =\n HomologicalComplex.Hom.f PInfty (n + 1) ≫ X.map i.op",
"state_before": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ (HomologicalComplex.Hom.f PInfty (n + 1) ≫ Finset.sum Finset.univ fun i => (-1) ^ ↑i • SimplicialObject.δ X i) =\n HomologicalComplex.Hom.f PInfty (n + 1) ≫ X.map i.op",
"tactic": "simp only [eq_self_iff_true, id_comp, if_true, Preadditive.comp_sum]"
},
{
"state_after": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ HomologicalComplex.Hom.f PInfty (n + 1) ≫ ((-1) ^ ↑0 • SimplicialObject.δ X 0) =\n HomologicalComplex.Hom.f PInfty (n + 1) ≫ X.map i.op\n\ncase pos.h₀\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ ∀ (b : Fin (n + 2)),\n b ∈ Finset.univ → b ≠ 0 → HomologicalComplex.Hom.f PInfty (n + 1) ≫ ((-1) ^ ↑b • SimplicialObject.δ X b) = 0\n\ncase pos.h₁\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ ¬0 ∈ Finset.univ → HomologicalComplex.Hom.f PInfty (n + 1) ≫ ((-1) ^ ↑0 • SimplicialObject.δ X 0) = 0",
"state_before": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ (Finset.sum Finset.univ fun j => HomologicalComplex.Hom.f PInfty (n + 1) ≫ ((-1) ^ ↑j • SimplicialObject.δ X j)) =\n HomologicalComplex.Hom.f PInfty (n + 1) ≫ X.map i.op",
"tactic": "rw [Finset.sum_eq_single (0 : Fin (n + 2))]"
},
{
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"state_before": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ HomologicalComplex.Hom.f PInfty (n + 1) ≫ ((-1) ^ ↑0 • SimplicialObject.δ X 0) =\n HomologicalComplex.Hom.f PInfty (n + 1) ≫ X.map i.op\n\ncase pos.h₀\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ ∀ (b : Fin (n + 2)),\n b ∈ Finset.univ → b ≠ 0 → HomologicalComplex.Hom.f PInfty (n + 1) ≫ ((-1) ^ ↑b • SimplicialObject.δ X b) = 0\n\ncase pos.h₁\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ ¬0 ∈ Finset.univ → HomologicalComplex.Hom.f PInfty (n + 1) ≫ ((-1) ^ ↑0 • SimplicialObject.δ X 0) = 0",
"tactic": "rotate_left"
},
{
"state_after": "case pos.h₀\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\nb : Fin (n + 2)\na✝ : b ∈ Finset.univ\nhb : b ≠ 0\n⊢ HomologicalComplex.Hom.f PInfty (n + 1) ≫ ((-1) ^ ↑b • SimplicialObject.δ X b) = 0",
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"tactic": "intro b _ hb"
},
{
"state_after": "case pos.h₀\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\nb : Fin (n + 2)\na✝ : b ∈ Finset.univ\nhb : b ≠ 0\n⊢ (-1) ^ ↑b • HomologicalComplex.Hom.f PInfty (n + 1) ≫ SimplicialObject.δ X b = 0",
"state_before": "case pos.h₀\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\nb : Fin (n + 2)\na✝ : b ∈ Finset.univ\nhb : b ≠ 0\n⊢ HomologicalComplex.Hom.f PInfty (n + 1) ≫ ((-1) ^ ↑b • SimplicialObject.δ X b) = 0",
"tactic": "rw [Preadditive.comp_zsmul]"
},
{
"state_after": "no goals",
"state_before": "case pos.h₀\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\nb : Fin (n + 2)\na✝ : b ∈ Finset.univ\nhb : b ≠ 0\n⊢ (-1) ^ ↑b • HomologicalComplex.Hom.f PInfty (n + 1) ≫ SimplicialObject.δ X b = 0",
"tactic": "erw [PInfty_comp_map_mono_eq_zero X (SimplexCategory.δ b) h\n (by\n rw [Isδ₀.iff]\n exact hb),\n zsmul_zero]"
},
{
"state_after": "C : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\nb : Fin (n + 2)\na✝ : b ∈ Finset.univ\nhb : b ≠ 0\n⊢ ¬b = 0",
"state_before": "C : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\nb : Fin (n + 2)\na✝ : b ∈ Finset.univ\nhb : b ≠ 0\n⊢ ¬Isδ₀ (SimplexCategory.δ b)",
"tactic": "rw [Isδ₀.iff]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\nb : Fin (n + 2)\na✝ : b ∈ Finset.univ\nhb : b ≠ 0\n⊢ ¬b = 0",
"tactic": "exact hb"
},
{
"state_after": "no goals",
"state_before": "case pos.h₁\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ ¬0 ∈ Finset.univ → HomologicalComplex.Hom.f PInfty (n + 1) ≫ ((-1) ^ ↑0 • SimplicialObject.δ X 0) = 0",
"tactic": "simp only [Finset.mem_univ, not_true, IsEmpty.forall_iff]"
},
{
"state_after": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ HomologicalComplex.Hom.f PInfty (n + 1) ≫ SimplicialObject.δ X 0 =\n HomologicalComplex.Hom.f PInfty (n + 1) ≫ X.map (SimplexCategory.δ 0).op",
"state_before": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ HomologicalComplex.Hom.f PInfty (n + 1) ≫ ((-1) ^ ↑0 • SimplicialObject.δ X 0) =\n HomologicalComplex.Hom.f PInfty (n + 1) ≫ X.map i.op",
"tactic": "simp only [hi.eq_δ₀, Fin.val_zero, pow_zero, one_zsmul]"
},
{
"state_after": "no goals",
"state_before": "case pos\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\ni✝ : Δ ⟶ [n + 1]\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n + 1]\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ HomologicalComplex.Hom.f PInfty (n + 1) ≫ SimplicialObject.δ X 0 =\n HomologicalComplex.Hom.f PInfty (n + 1) ≫ X.map (SimplexCategory.δ 0).op",
"tactic": "rfl"
},
{
"state_after": "case neg\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\n⊢ 0 = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op\n\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\n⊢ [n'] ≠ [n]",
"state_before": "case neg\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op",
"tactic": "rw [Γ₀.Obj.Termwise.mapMono_eq_zero _ i _ hi, zero_comp]"
},
{
"state_after": "C : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\n⊢ [n'] ≠ [n]\n\ncase neg\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\n⊢ 0 = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op",
"state_before": "case neg\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\n⊢ 0 = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op\n\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\n⊢ [n'] ≠ [n]",
"tactic": "swap"
},
{
"state_after": "case neg.h₁\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\n⊢ len [n] ≠ n'\n\ncase neg.h₂\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\n⊢ ¬Isδ₀ i",
"state_before": "case neg\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\n⊢ 0 = HomologicalComplex.Hom.f PInfty n' ≫ X.map i.op",
"tactic": "rw [PInfty_comp_map_mono_eq_zero]"
},
{
"state_after": "C : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\nh' : [n'] = [n]\n⊢ False",
"state_before": "C : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\n⊢ [n'] ≠ [n]",
"tactic": "by_contra h'"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\nh' : [n'] = [n]\n⊢ False",
"tactic": "exact h (congr_arg SimplexCategory.len h'.symm)"
},
{
"state_after": "no goals",
"state_before": "case neg.h₁\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\n⊢ len [n] ≠ n'",
"tactic": "exact h"
},
{
"state_after": "case neg.h₂\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\nh' : Isδ₀ i\n⊢ False",
"state_before": "case neg.h₂\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\n⊢ ¬Isδ₀ i",
"tactic": "by_contra h'"
},
{
"state_after": "no goals",
"state_before": "case neg.h₂\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nX : SimplicialObject C\nΔ Δ' : SimplexCategory\ni✝² : Δ ⟶ Δ'\ninst✝³ : Mono i✝²\nn : ℕ\ni✝¹ : [n] ⟶ Δ'\ninst✝² : Mono i✝¹\nn' : ℕ\ni✝ : Δ ⟶ [n']\ninst✝¹ : Mono i✝\ni : [n] ⟶ [n']\ninst✝ : Mono i\nh : ¬n = n'\nhi : ¬Isδ₀ i\nh' : Isδ₀ i\n⊢ False",
"tactic": "exact hi h'"
}
] | [
125,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
] |
Mathlib/Order/Bounds/Basic.lean | AntitoneOn.map_bddBelow | [] | [
1245,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1244,
1
] |
Mathlib/Topology/MetricSpace/Gluing.lean | Metric.Sigma.isometry_mk | [
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nE : ι → Type u_1\ninst✝ : (i : ι) → MetricSpace (E i)\ni : ι\nx y : E i\n⊢ dist { fst := i, snd := x } { fst := i, snd := y } = dist x y",
"tactic": "simp"
}
] | [
439,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
438,
1
] |
Mathlib/Data/List/MinMax.lean | List.argmin_cons | [] | [
202,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
199,
1
] |
Mathlib/RingTheory/HahnSeries.lean | HahnSeries.addVal_le_of_coeff_ne_zero | [
{
"state_after": "Γ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : Ring R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\ng : Γ\nh : coeff x g ≠ 0\n⊢ order x ≤ g",
"state_before": "Γ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : Ring R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\ng : Γ\nh : coeff x g ≠ 0\n⊢ ↑(addVal Γ R) x ≤ ↑g",
"tactic": "rw [addVal_apply_of_ne (ne_zero_of_coeff_ne_zero h), WithTop.coe_le_coe]"
},
{
"state_after": "no goals",
"state_before": "Γ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : Ring R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\ng : Γ\nh : coeff x g ≠ 0\n⊢ order x ≤ g",
"tactic": "exact order_le_of_coeff_ne_zero h"
}
] | [
1379,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1376,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.bind_def | [] | [
2045,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2044,
1
] |
Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | MeasureTheory.tendstoInMeasure_of_tendsto_snorm | [
{
"state_after": "case pos\nα : Type u_2\nι : Type u_1\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nl : Filter ι\nhp_ne_zero : p ≠ 0\nhf : ∀ (n : ι), AEStronglyMeasurable (f n) μ\nhg : AEStronglyMeasurable g μ\nhfg : Tendsto (fun n => snorm (f n - g) p μ) l (𝓝 0)\nhp_ne_top : p = ⊤\n⊢ TendstoInMeasure μ f l g\n\ncase neg\nα : Type u_2\nι : Type u_1\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nl : Filter ι\nhp_ne_zero : p ≠ 0\nhf : ∀ (n : ι), AEStronglyMeasurable (f n) μ\nhg : AEStronglyMeasurable g μ\nhfg : Tendsto (fun n => snorm (f n - g) p μ) l (𝓝 0)\nhp_ne_top : ¬p = ⊤\n⊢ TendstoInMeasure μ f l g",
"state_before": "α : Type u_2\nι : Type u_1\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nl : Filter ι\nhp_ne_zero : p ≠ 0\nhf : ∀ (n : ι), AEStronglyMeasurable (f n) μ\nhg : AEStronglyMeasurable g μ\nhfg : Tendsto (fun n => snorm (f n - g) p μ) l (𝓝 0)\n⊢ TendstoInMeasure μ f l g",
"tactic": "by_cases hp_ne_top : p = ∞"
},
{
"state_after": "case pos\nα : Type u_2\nι : Type u_1\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : ι → α → E\ng : α → E\nl : Filter ι\nhf : ∀ (n : ι), AEStronglyMeasurable (f n) μ\nhg : AEStronglyMeasurable g μ\nhp_ne_zero : ⊤ ≠ 0\nhfg : Tendsto (fun n => snorm (f n - g) ⊤ μ) l (𝓝 0)\n⊢ TendstoInMeasure μ f l g",
"state_before": "case pos\nα : Type u_2\nι : Type u_1\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nl : Filter ι\nhp_ne_zero : p ≠ 0\nhf : ∀ (n : ι), AEStronglyMeasurable (f n) μ\nhg : AEStronglyMeasurable g μ\nhfg : Tendsto (fun n => snorm (f n - g) p μ) l (𝓝 0)\nhp_ne_top : p = ⊤\n⊢ TendstoInMeasure μ f l g",
"tactic": "subst hp_ne_top"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_2\nι : Type u_1\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : ι → α → E\ng : α → E\nl : Filter ι\nhf : ∀ (n : ι), AEStronglyMeasurable (f n) μ\nhg : AEStronglyMeasurable g μ\nhp_ne_zero : ⊤ ≠ 0\nhfg : Tendsto (fun n => snorm (f n - g) ⊤ μ) l (𝓝 0)\n⊢ TendstoInMeasure μ f l g",
"tactic": "exact tendstoInMeasure_of_tendsto_snorm_top hfg"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_2\nι : Type u_1\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nl : Filter ι\nhp_ne_zero : p ≠ 0\nhf : ∀ (n : ι), AEStronglyMeasurable (f n) μ\nhg : AEStronglyMeasurable g μ\nhfg : Tendsto (fun n => snorm (f n - g) p μ) l (𝓝 0)\nhp_ne_top : ¬p = ⊤\n⊢ TendstoInMeasure μ f l g",
"tactic": "exact tendstoInMeasure_of_tendsto_snorm_of_ne_top hp_ne_zero hp_ne_top hf hg hfg"
}
] | [
355,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
349,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean | HomologicalComplex.hom_f_injective | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC C₁ C₂ : HomologicalComplex V c\n⊢ Function.Injective fun f => f.f",
"tactic": "aesop_cat"
}
] | [
246,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
245,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean | closure_eq_inter_uniformity | [
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.88389\ninst✝ : UniformSpace α\nt : Set (α × α)\n⊢ (⋂ (V : Set (α × α)) (_ : V ∈ 𝓤 α), V ○ t ○ V) = ⋂ (V : Set (α × α)) (_ : V ∈ 𝓤 α), V ○ (t ○ V)",
"tactic": "simp only [compRel_assoc]"
}
] | [
979,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
972,
1
] |
Mathlib/Data/Int/Parity.lean | Int.even_iff | [
{
"state_after": "no goals",
"state_before": "m✝ n : ℤ\nx✝ : Even n\nm : ℤ\nhm : n = m + m\n⊢ n % 2 = 0",
"tactic": "simp [← two_mul, hm]"
},
{
"state_after": "no goals",
"state_before": "m n : ℤ\nh : n % 2 = 0\n⊢ n % 2 + 2 * (n / 2) = n / 2 + n / 2",
"tactic": "simp [← two_mul, h]"
}
] | [
42,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
40,
1
] |
Mathlib/LinearAlgebra/Dfinsupp.lean | Dfinsupp.coprodMap_apply_single | [
{
"state_after": "no goals",
"state_before": "ι : Type u_4\nR : Type u_1\nS : Type ?u.308054\nM : ι → Type u_2\nN : Type u_3\ndec_ι : DecidableEq ι\ninst✝⁶ : Semiring R\ninst✝⁵ : (i : ι) → AddCommMonoid (M i)\ninst✝⁴ : (i : ι) → Module R (M i)\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : DecidableEq ι\ninst✝ : (x : N) → Decidable (x ≠ 0)\nf : (i : ι) → M i →ₗ[R] N\ni : ι\nx : M i\n⊢ ↑(coprodMap f) (single i x) = ↑(f i) x",
"tactic": "simp [coprodMap_apply]"
}
] | [
282,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
280,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean | DifferentiableOn.arctan | [] | [
205,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
204,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | IsBoundedBilinearMap.map_sub_right | [] | [
396,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
394,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean | Valuation.ne_zero_iff | [] | [
238,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
237,
1
] |
Mathlib/GroupTheory/Perm/Option.lean | Equiv.optionCongr_sign | [
{
"state_after": "case refine_1\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ne : Perm α\n⊢ ↑Perm.sign (optionCongr 1) = ↑Perm.sign 1\n\ncase refine_2\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ne : Perm α\n⊢ ∀ (f : Perm α) (x y : α),\n x ≠ y →\n ↑Perm.sign (optionCongr f) = ↑Perm.sign f → ↑Perm.sign (optionCongr (swap x y * f)) = ↑Perm.sign (swap x y * f)",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ne : Perm α\n⊢ ↑Perm.sign (optionCongr e) = ↑Perm.sign e",
"tactic": "refine Perm.swap_induction_on e ?_ ?_"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ne : Perm α\n⊢ ↑Perm.sign (optionCongr 1) = ↑Perm.sign 1",
"tactic": "simp [Perm.one_def]"
},
{
"state_after": "case refine_2\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ne f : Perm α\nx y : α\nhne : x ≠ y\nh : ↑Perm.sign (optionCongr f) = ↑Perm.sign f\n⊢ ↑Perm.sign (optionCongr (swap x y * f)) = ↑Perm.sign (swap x y * f)",
"state_before": "case refine_2\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ne : Perm α\n⊢ ∀ (f : Perm α) (x y : α),\n x ≠ y →\n ↑Perm.sign (optionCongr f) = ↑Perm.sign f → ↑Perm.sign (optionCongr (swap x y * f)) = ↑Perm.sign (swap x y * f)",
"tactic": "intro f x y hne h"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ne f : Perm α\nx y : α\nhne : x ≠ y\nh : ↑Perm.sign (optionCongr f) = ↑Perm.sign f\n⊢ ↑Perm.sign (optionCongr (swap x y * f)) = ↑Perm.sign (swap x y * f)",
"tactic": "simp [h, hne, Perm.mul_def, ← Equiv.optionCongr_trans]"
}
] | [
43,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
38,
1
] |
Mathlib/Algebra/BigOperators/Multiset/Basic.lean | Multiset.prod_eq_zero | [
{
"state_after": "case intro\nι : Type ?u.89646\nα : Type u_1\nβ : Type ?u.89652\nγ : Type ?u.89655\ninst✝ : CommMonoidWithZero α\ns : Multiset α\nh : 0 ∈ s\ns' : Multiset α\nhs' : s = 0 ::ₘ s'\n⊢ prod s = 0",
"state_before": "ι : Type ?u.89646\nα : Type u_1\nβ : Type ?u.89652\nγ : Type ?u.89655\ninst✝ : CommMonoidWithZero α\ns : Multiset α\nh : 0 ∈ s\n⊢ prod s = 0",
"tactic": "rcases Multiset.exists_cons_of_mem h with ⟨s', hs'⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type ?u.89646\nα : Type u_1\nβ : Type ?u.89652\nγ : Type ?u.89655\ninst✝ : CommMonoidWithZero α\ns : Multiset α\nh : 0 ∈ s\ns' : Multiset α\nhs' : s = 0 ::ₘ s'\n⊢ prod s = 0",
"tactic": "simp [hs', Multiset.prod_cons]"
}
] | [
285,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
283,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.measure_ne_top | [] | [
3060,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3059,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | Pmf.mem_support_filter_iff | [] | [
284,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
283,
1
] |
Mathlib/Topology/Constructions.lean | interior_pi_set | [
{
"state_after": "case h\nα : Type u\nβ : Type v\nγ : Type ?u.313054\nδ : Type ?u.313057\nε : Type ?u.313060\nζ : Type ?u.313063\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.313074\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\nI : Set ι\nhI : Set.Finite I\ns : (i : ι) → Set (π i)\na : (i : ι) → π i\n⊢ a ∈ interior (Set.pi I s) ↔ a ∈ Set.pi I fun i => interior (s i)",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.313054\nδ : Type ?u.313057\nε : Type ?u.313060\nζ : Type ?u.313063\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.313074\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\nI : Set ι\nhI : Set.Finite I\ns : (i : ι) → Set (π i)\n⊢ interior (Set.pi I s) = Set.pi I fun i => interior (s i)",
"tactic": "ext a"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\nβ : Type v\nγ : Type ?u.313054\nδ : Type ?u.313057\nε : Type ?u.313060\nζ : Type ?u.313063\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.313074\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\nI : Set ι\nhI : Set.Finite I\ns : (i : ι) → Set (π i)\na : (i : ι) → π i\n⊢ a ∈ interior (Set.pi I s) ↔ a ∈ Set.pi I fun i => interior (s i)",
"tactic": "simp only [Set.mem_pi, mem_interior_iff_mem_nhds, set_pi_mem_nhds_iff hI]"
}
] | [
1357,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1354,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.comap_mono | [] | [
2162,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2161,
1
] |
Mathlib/Data/Complex/Exponential.lean | Complex.isCauSeq_exp | [] | [
369,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
368,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/IsOpenComapC.lean | AlgebraicGeometry.Polynomial.isOpenMap_comap_C | [
{
"state_after": "case intro\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nU : Set (PrimeSpectrum R[X])\ns : Set R[X]\nz : zeroLocus s = Uᶜ\n⊢ IsOpen (↑(PrimeSpectrum.comap C) '' U)",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nf : R[X]\n⊢ IsOpenMap ↑(PrimeSpectrum.comap C)",
"tactic": "rintro U ⟨s, z⟩"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nU : Set (PrimeSpectrum R[X])\ns : Set R[X]\nz : zeroLocus s = Uᶜ\n⊢ IsOpen (⋃ (i : ↑s), ↑(PrimeSpectrum.comap C) '' zeroLocus {↑i}ᶜ)",
"state_before": "case intro\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nU : Set (PrimeSpectrum R[X])\ns : Set R[X]\nz : zeroLocus s = Uᶜ\n⊢ IsOpen (↑(PrimeSpectrum.comap C) '' U)",
"tactic": "rw [← compl_compl U, ← z, ← iUnion_of_singleton_coe s, zeroLocus_iUnion, compl_iInter,\n image_iUnion]"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nU : Set (PrimeSpectrum R[X])\ns : Set R[X]\nz : zeroLocus s = Uᶜ\n⊢ IsOpen (⋃ (i : ↑s), imageOfDf ↑i)",
"state_before": "case intro\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nU : Set (PrimeSpectrum R[X])\ns : Set R[X]\nz : zeroLocus s = Uᶜ\n⊢ IsOpen (⋃ (i : ↑s), ↑(PrimeSpectrum.comap C) '' zeroLocus {↑i}ᶜ)",
"tactic": "simp_rw [← imageOfDf_eq_comap_C_compl_zeroLocus]"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nU : Set (PrimeSpectrum R[X])\ns : Set R[X]\nz : zeroLocus s = Uᶜ\n⊢ IsOpen (⋃ (i : ↑s), imageOfDf ↑i)",
"tactic": "exact isOpen_iUnion fun f => isOpen_imageOfDf"
}
] | [
82,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
77,
1
] |
Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | EMetric.exists_forall_closedBall_subset_aux₁ | [
{
"state_after": "ι : Type u_2\nX : Type u_1\ninst✝ : EMetricSpace X\nK U : ι → Set X\nhK : ∀ (i : ι), IsClosed (K i)\nhU : ∀ (i : ι), IsOpen (U i)\nhKU : ∀ (i : ι), K i ⊆ U i\nhfin : LocallyFinite K\nx : X\nthis :\n ∀ᶠ (x_1 : ℝ) in 𝓝 0,\n ∀ᶠ (y : X) in 𝓝 x,\n ∀ (i : ι),\n (ENNReal.ofReal x_1, y).snd ∈ K i → closedBall (ENNReal.ofReal x_1, y).snd (ENNReal.ofReal x_1, y).fst ⊆ U i\n⊢ ∃ r, ∀ᶠ (y : X) in 𝓝 x, r ∈ Ioi 0 ∩ ENNReal.ofReal ⁻¹' ⋂ (i : ι) (_ : y ∈ K i), {r | closedBall y r ⊆ U i}",
"state_before": "ι : Type u_2\nX : Type u_1\ninst✝ : EMetricSpace X\nK U : ι → Set X\nhK : ∀ (i : ι), IsClosed (K i)\nhU : ∀ (i : ι), IsOpen (U i)\nhKU : ∀ (i : ι), K i ⊆ U i\nhfin : LocallyFinite K\nx : X\n⊢ ∃ r, ∀ᶠ (y : X) in 𝓝 x, r ∈ Ioi 0 ∩ ENNReal.ofReal ⁻¹' ⋂ (i : ι) (_ : y ∈ K i), {r | closedBall y r ⊆ U i}",
"tactic": "have := (ENNReal.continuous_ofReal.tendsto' 0 0 ENNReal.ofReal_zero).eventually\n (eventually_nhds_zero_forall_closedBall_subset hK hU hKU hfin x).curry"
},
{
"state_after": "case intro.intro\nι : Type u_2\nX : Type u_1\ninst✝ : EMetricSpace X\nK U : ι → Set X\nhK : ∀ (i : ι), IsClosed (K i)\nhU : ∀ (i : ι), IsOpen (U i)\nhKU : ∀ (i : ι), K i ⊆ U i\nhfin : LocallyFinite K\nx : X\nthis :\n ∀ᶠ (x_1 : ℝ) in 𝓝 0,\n ∀ᶠ (y : X) in 𝓝 x,\n ∀ (i : ι),\n (ENNReal.ofReal x_1, y).snd ∈ K i → closedBall (ENNReal.ofReal x_1, y).snd (ENNReal.ofReal x_1, y).fst ⊆ U i\nr : ℝ\nhr0 : r > 0\nhr :\n ∀ᶠ (y : X) in 𝓝 x,\n ∀ (i : ι), (ENNReal.ofReal r, y).snd ∈ K i → closedBall (ENNReal.ofReal r, y).snd (ENNReal.ofReal r, y).fst ⊆ U i\n⊢ ∃ r, ∀ᶠ (y : X) in 𝓝 x, r ∈ Ioi 0 ∩ ENNReal.ofReal ⁻¹' ⋂ (i : ι) (_ : y ∈ K i), {r | closedBall y r ⊆ U i}",
"state_before": "ι : Type u_2\nX : Type u_1\ninst✝ : EMetricSpace X\nK U : ι → Set X\nhK : ∀ (i : ι), IsClosed (K i)\nhU : ∀ (i : ι), IsOpen (U i)\nhKU : ∀ (i : ι), K i ⊆ U i\nhfin : LocallyFinite K\nx : X\nthis :\n ∀ᶠ (x_1 : ℝ) in 𝓝 0,\n ∀ᶠ (y : X) in 𝓝 x,\n ∀ (i : ι),\n (ENNReal.ofReal x_1, y).snd ∈ K i → closedBall (ENNReal.ofReal x_1, y).snd (ENNReal.ofReal x_1, y).fst ⊆ U i\n⊢ ∃ r, ∀ᶠ (y : X) in 𝓝 x, r ∈ Ioi 0 ∩ ENNReal.ofReal ⁻¹' ⋂ (i : ι) (_ : y ∈ K i), {r | closedBall y r ⊆ U i}",
"tactic": "rcases this.exists_gt with ⟨r, hr0, hr⟩"
},
{
"state_after": "case intro.intro\nι : Type u_2\nX : Type u_1\ninst✝ : EMetricSpace X\nK U : ι → Set X\nhK : ∀ (i : ι), IsClosed (K i)\nhU : ∀ (i : ι), IsOpen (U i)\nhKU : ∀ (i : ι), K i ⊆ U i\nhfin : LocallyFinite K\nx : X\nthis :\n ∀ᶠ (x_1 : ℝ) in 𝓝 0,\n ∀ᶠ (y : X) in 𝓝 x,\n ∀ (i : ι),\n (ENNReal.ofReal x_1, y).snd ∈ K i → closedBall (ENNReal.ofReal x_1, y).snd (ENNReal.ofReal x_1, y).fst ⊆ U i\nr : ℝ\nhr0 : r > 0\nhr :\n ∀ᶠ (y : X) in 𝓝 x,\n ∀ (i : ι), (ENNReal.ofReal r, y).snd ∈ K i → closedBall (ENNReal.ofReal r, y).snd (ENNReal.ofReal r, y).fst ⊆ U i\ny : X\nhy : ∀ (i : ι), (ENNReal.ofReal r, y).snd ∈ K i → closedBall (ENNReal.ofReal r, y).snd (ENNReal.ofReal r, y).fst ⊆ U i\n⊢ r ∈ ENNReal.ofReal ⁻¹' ⋂ (i : ι) (_ : y ∈ K i), {r | closedBall y r ⊆ U i}",
"state_before": "case intro.intro\nι : Type u_2\nX : Type u_1\ninst✝ : EMetricSpace X\nK U : ι → Set X\nhK : ∀ (i : ι), IsClosed (K i)\nhU : ∀ (i : ι), IsOpen (U i)\nhKU : ∀ (i : ι), K i ⊆ U i\nhfin : LocallyFinite K\nx : X\nthis :\n ∀ᶠ (x_1 : ℝ) in 𝓝 0,\n ∀ᶠ (y : X) in 𝓝 x,\n ∀ (i : ι),\n (ENNReal.ofReal x_1, y).snd ∈ K i → closedBall (ENNReal.ofReal x_1, y).snd (ENNReal.ofReal x_1, y).fst ⊆ U i\nr : ℝ\nhr0 : r > 0\nhr :\n ∀ᶠ (y : X) in 𝓝 x,\n ∀ (i : ι), (ENNReal.ofReal r, y).snd ∈ K i → closedBall (ENNReal.ofReal r, y).snd (ENNReal.ofReal r, y).fst ⊆ U i\n⊢ ∃ r, ∀ᶠ (y : X) in 𝓝 x, r ∈ Ioi 0 ∩ ENNReal.ofReal ⁻¹' ⋂ (i : ι) (_ : y ∈ K i), {r | closedBall y r ⊆ U i}",
"tactic": "refine' ⟨r, hr.mono fun y hy => ⟨hr0, _⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nι : Type u_2\nX : Type u_1\ninst✝ : EMetricSpace X\nK U : ι → Set X\nhK : ∀ (i : ι), IsClosed (K i)\nhU : ∀ (i : ι), IsOpen (U i)\nhKU : ∀ (i : ι), K i ⊆ U i\nhfin : LocallyFinite K\nx : X\nthis :\n ∀ᶠ (x_1 : ℝ) in 𝓝 0,\n ∀ᶠ (y : X) in 𝓝 x,\n ∀ (i : ι),\n (ENNReal.ofReal x_1, y).snd ∈ K i → closedBall (ENNReal.ofReal x_1, y).snd (ENNReal.ofReal x_1, y).fst ⊆ U i\nr : ℝ\nhr0 : r > 0\nhr :\n ∀ᶠ (y : X) in 𝓝 x,\n ∀ (i : ι), (ENNReal.ofReal r, y).snd ∈ K i → closedBall (ENNReal.ofReal r, y).snd (ENNReal.ofReal r, y).fst ⊆ U i\ny : X\nhy : ∀ (i : ι), (ENNReal.ofReal r, y).snd ∈ K i → closedBall (ENNReal.ofReal r, y).snd (ENNReal.ofReal r, y).fst ⊆ U i\n⊢ r ∈ ENNReal.ofReal ⁻¹' ⋂ (i : ι) (_ : y ∈ K i), {r | closedBall y r ⊆ U i}",
"tactic": "rwa [mem_preimage, mem_iInter₂]"
}
] | [
74,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
66,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean | LinearOrderedRing.orderOf_le_two | [
{
"state_after": "case inl\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝ : LinearOrderedRing G\nh : abs x ≠ 1\n⊢ orderOf x ≤ 2\n\ncase inr\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝ : LinearOrderedRing G\nh : abs x = 1\n⊢ orderOf x ≤ 2",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝ : LinearOrderedRing G\n⊢ orderOf x ≤ 2",
"tactic": "cases' ne_or_eq (|x|) 1 with h h"
},
{
"state_after": "case inr.inl\nG : Type u\nA : Type v\ny : G\na b : A\nn m : ℕ\ninst✝ : LinearOrderedRing G\nh : abs 1 = 1\n⊢ orderOf 1 ≤ 2\n\ncase inr.inr\nG : Type u\nA : Type v\ny : G\na b : A\nn m : ℕ\ninst✝ : LinearOrderedRing G\nh : abs (-1) = 1\n⊢ orderOf (-1) ≤ 2",
"state_before": "case inr\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝ : LinearOrderedRing G\nh : abs x = 1\n⊢ orderOf x ≤ 2",
"tactic": "rcases eq_or_eq_neg_of_abs_eq h with (rfl | rfl)"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nG : Type u\nA : Type v\ny : G\na b : A\nn m : ℕ\ninst✝ : LinearOrderedRing G\nh : abs (-1) = 1\n⊢ orderOf (-1) ≤ 2",
"tactic": "apply orderOf_le_of_pow_eq_one <;> norm_num"
},
{
"state_after": "no goals",
"state_before": "case inl\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝ : LinearOrderedRing G\nh : abs x ≠ 1\n⊢ orderOf x ≤ 2",
"tactic": "simp [orderOf_abs_ne_one h]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nG : Type u\nA : Type v\ny : G\na b : A\nn m : ℕ\ninst✝ : LinearOrderedRing G\nh : abs 1 = 1\n⊢ orderOf 1 ≤ 2",
"tactic": "simp"
}
] | [
1096,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1091,
1
] |
Mathlib/GroupTheory/DoubleCoset.lean | Doset.right_bot_eq_right_quot | [
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nα : Type ?u.70749\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\n⊢ _root_.Quotient (setoid ↑H.toSubmonoid ↑⊥) = _root_.Quotient (QuotientGroup.rightRel H)",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nα : Type ?u.70749\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\n⊢ Quotient ↑H.toSubmonoid ↑⊥ = _root_.Quotient (QuotientGroup.rightRel H)",
"tactic": "unfold Quotient"
},
{
"state_after": "case e_s\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.70749\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\n⊢ setoid ↑H.toSubmonoid ↑⊥ = QuotientGroup.rightRel H",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nα : Type ?u.70749\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\n⊢ _root_.Quotient (setoid ↑H.toSubmonoid ↑⊥) = _root_.Quotient (QuotientGroup.rightRel H)",
"tactic": "congr"
},
{
"state_after": "case e_s.H\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.70749\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na✝ b✝ : G\n⊢ Setoid.Rel (setoid ↑H.toSubmonoid ↑⊥) a✝ b✝ ↔ Setoid.Rel (QuotientGroup.rightRel H) a✝ b✝",
"state_before": "case e_s\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.70749\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\n⊢ setoid ↑H.toSubmonoid ↑⊥ = QuotientGroup.rightRel H",
"tactic": "ext"
},
{
"state_after": "case e_s.H\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.70749\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na✝ b✝ : G\n⊢ Setoid.Rel (setoid ↑H.toSubmonoid ↑⊥) a✝ b✝ ↔ Setoid.Rel (setoid ↑H ↑⊥) a✝ b✝",
"state_before": "case e_s.H\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.70749\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na✝ b✝ : G\n⊢ Setoid.Rel (setoid ↑H.toSubmonoid ↑⊥) a✝ b✝ ↔ Setoid.Rel (QuotientGroup.rightRel H) a✝ b✝",
"tactic": "simp_rw [← rel_bot_eq_right_group_rel H]"
},
{
"state_after": "no goals",
"state_before": "case e_s.H\nG : Type u_1\ninst✝¹ : Group G\nα : Type ?u.70749\ninst✝ : Mul α\nJ : Subgroup G\ng : G\nH : Subgroup G\na✝ b✝ : G\n⊢ Setoid.Rel (setoid ↑H.toSubmonoid ↑⊥) a✝ b✝ ↔ Setoid.Rel (setoid ↑H ↑⊥) a✝ b✝",
"tactic": "rfl"
}
] | [
215,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
209,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Linear.lean | ContinuousLinearMap.differentiableAt | [] | [
83,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
82,
11
] |
Mathlib/Data/Multiset/FinsetOps.lean | Multiset.length_ndinsert_of_not_mem | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nh : ¬a ∈ s\n⊢ ↑card (ndinsert a s) = ↑card s + 1",
"tactic": "simp [h]"
}
] | [
83,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
82,
1
] |
Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean | HasDerivAt.cexp | [] | [
97,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
95,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean | AlgEquiv.toRingEquiv_symm | [] | [
379,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
378,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.power_lt_aleph0 | [
{
"state_after": "α β : Type u\na b : Cardinal\nm n : ℕ\nha : ↑m < ℵ₀\nhb : ↑n < ℵ₀\n⊢ ↑(m ^ n) < ℵ₀",
"state_before": "α β : Type u\na b : Cardinal\nm n : ℕ\nha : ↑m < ℵ₀\nhb : ↑n < ℵ₀\n⊢ ↑m ^ ↑n < ℵ₀",
"tactic": "rw [← natCast_pow]"
},
{
"state_after": "no goals",
"state_before": "α β : Type u\na b : Cardinal\nm n : ℕ\nha : ↑m < ℵ₀\nhb : ↑n < ℵ₀\n⊢ ↑(m ^ n) < ℵ₀",
"tactic": "apply nat_lt_aleph0"
}
] | [
1595,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1593,
1
] |
Mathlib/Order/Filter/Bases.lean | Filter.asBasis_filter | [] | [
411,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
410,
1
] |
Mathlib/Analysis/Calculus/FDerivMeasurable.lean | RightDerivMeasurableAux.measurableSet_b | [] | [
510,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
509,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean | MulPosReflectLT.toMulPosMonoRev | [] | [
1015,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1013,
1
] |
Mathlib/Data/Set/Image.lean | Set.subset_preimage_univ | [] | [
85,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
] |
Mathlib/FieldTheory/Subfield.lean | Subfield.sum_mem | [] | [
317,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
315,
11
] |
Mathlib/Algebra/Order/Ring/Defs.lean | lt_mul_right | [
{
"state_after": "case h.e'_3\nα : Type u\nβ : Type ?u.63411\ninst✝ : StrictOrderedSemiring α\na b c d : α\nhn : 0 < a\nhm : 1 < b\n⊢ a = a * 1",
"state_before": "α : Type u\nβ : Type ?u.63411\ninst✝ : StrictOrderedSemiring α\na b c d : α\nhn : 0 < a\nhm : 1 < b\n⊢ a < a * b",
"tactic": "convert mul_lt_mul_of_pos_left hm hn"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nα : Type u\nβ : Type ?u.63411\ninst✝ : StrictOrderedSemiring α\na b c d : α\nhn : 0 < a\nhm : 1 < b\n⊢ a = a * 1",
"tactic": "rw [mul_one]"
}
] | [
570,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
568,
1
] |
Mathlib/Order/SymmDiff.lean | bihimp_left_comm | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.80811\nα : Type u_1\nβ : Type ?u.80817\nπ : ι → Type ?u.80822\ninst✝ : BooleanAlgebra α\na b c d : α\n⊢ a ⇔ (b ⇔ c) = b ⇔ (a ⇔ c)",
"tactic": "simp_rw [← bihimp_assoc, bihimp_comm]"
}
] | [
634,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
634,
1
] |
Mathlib/Order/Filter/AtTopBot.lean | Filter.Tendsto.atBot_div_const | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.249256\nι' : Type ?u.249259\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.249268\ninst✝ : LinearOrderedField α\nl : Filter β\nf : β → α\nr : α\nhr : 0 < r\nhf : Tendsto f l atBot\n⊢ Tendsto (fun x => f x / r) l atBot",
"tactic": "simpa only [div_eq_mul_inv] using hf.atBot_mul_const (inv_pos.2 hr)"
}
] | [
1222,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1220,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | Continuous.dist | [] | [
1809,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1807,
11
] |
Mathlib/Data/Nat/Digits.lean | Nat.digits_one_succ | [] | [
115,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
114,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean | contDiff_smul | [] | [
1493,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1492,
1
] |
Mathlib/Analysis/InnerProductSpace/Adjoint.lean | ContinuousLinearMap.adjoint_adjoint | [] | [
136,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
135,
1
] |
Mathlib/Algebra/Quaternion.lean | QuaternionAlgebra.star_mk | [] | [
641,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
640,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean | MeasureTheory.SimpleFunc.setToSimpleFunc_congr_left | [
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.205114\nG : Type ?u.205117\n𝕜 : Type ?u.205120\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT T' : Set α → E →L[ℝ] F\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T s = T' s\nf : α →ₛ E\nhf : Integrable ↑f\n⊢ ∑ x in SimpleFunc.range f, ↑(T (↑f ⁻¹' {x})) x = ∑ x in SimpleFunc.range f, ↑(T' (↑f ⁻¹' {x})) x",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.205114\nG : Type ?u.205117\n𝕜 : Type ?u.205120\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT T' : Set α → E →L[ℝ] F\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T s = T' s\nf : α →ₛ E\nhf : Integrable ↑f\n⊢ setToSimpleFunc T f = setToSimpleFunc T' f",
"tactic": "simp_rw [setToSimpleFunc]"
},
{
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{
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{
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{
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"tactic": "rw [h (f ⁻¹' {x}) (SimpleFunc.measurableSet_fiber _ _)\n (SimpleFunc.measure_preimage_lt_top_of_integrable _ hf hx0)]"
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403,
69
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1
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Mathlib/Data/Finset/Card.lean | Finset.card_union_add_card_inter | [
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{
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"tactic": "by_cases a ∈ s <;>\nsimp [*, ← add_assoc, add_right_comm _ 1]"
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411,
46
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408,
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Mathlib/Data/List/Sigma.lean | List.dlookup_kunion_right | [
{
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"tactic": "induction l₁ generalizing l₂"
},
{
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"tactic": "case nil => simp"
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{
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"tactic": "case cons _ _ ih => simp [not_or] at h; simp [h.1, ih h.2]"
},
{
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{
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"tactic": "simp [not_or] at h"
},
{
"state_after": "no goals",
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"tactic": "simp [h.1, ih h.2]"
}
] | [
763,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
759,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean | linearIndependent_inl_union_inr' | [
{
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"tactic": "refine' isCompl_range_inl_inr.disjoint.mono _ _ <;>\n simp only [span_le, range_coe, range_comp_subset_range]"
}
] | [
995,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
990,
1
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Mathlib/LinearAlgebra/AffineSpace/Independent.lean | Affine.Simplex.reindex_symm_reindex | [
{
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"tactic": "rw [← reindex_trans, Equiv.symm_trans_self, reindex_refl]"
}
] | [
897,
101
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
896,
1
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Mathlib/MeasureTheory/Integral/Lebesgue.lean | MeasureTheory.SimpleFunc.lintegral_eq_lintegral | [
{
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"tactic": "rw [MeasureTheory.lintegral]"
},
{
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"tactic": "exact\n le_antisymm (iSup₂_le fun g hg => lintegral_mono hg <| le_rfl) (le_iSup₂_of_le f le_rfl le_rfl)"
}
] | [
124,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
120,
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Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | StructureGroupoid.LocalInvariantProp.liftPropOn_congr_iff | [] | [
444,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
443,
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Mathlib/CategoryTheory/Extensive.lean | CategoryTheory.finitaryExtensive_iff_of_isTerminal | [
{
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"tactic": "refine' ⟨fun H => H.2 c₀ hc₀, fun H => _⟩"
},
{
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"tactic": "constructor"
},
{
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"tactic": "simp_rw [BinaryCofan.isVanKampen_iff] at H⊢"
},
{
"state_after": "case van_kampen'\nJ : Type v'\ninst✝³ : Category J\nC✝ : Type u\ninst✝² : Category C✝\nX✝ Y✝ : C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nT : C\nHT : IsTerminal T\nc₀ : BinaryCofan T T\nhc₀ : IsColimit c₀\nH :\n ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),\n αX ≫ BinaryCofan.inl c₀ = BinaryCofan.inl c' ≫ f →\n αY ≫ BinaryCofan.inr c₀ = BinaryCofan.inr c' ≫ f →\n (Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c₀) ∧\n IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c₀))\nX Y : C\nc : BinaryCofan X Y\nhc : IsColimit c\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\n⊢ Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"state_before": "case van_kampen'\nJ : Type v'\ninst✝³ : Category J\nC✝ : Type u\ninst✝² : Category C✝\nX Y : C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nT : C\nHT : IsTerminal T\nc₀ : BinaryCofan T T\nhc₀ : IsColimit c₀\nH :\n ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),\n αX ≫ BinaryCofan.inl c₀ = BinaryCofan.inl c' ≫ f →\n αY ≫ BinaryCofan.inr c₀ = BinaryCofan.inr c' ≫ f →\n (Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c₀) ∧\n IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c₀))\n⊢ ∀ {X Y : C} (c : BinaryCofan X Y),\n IsColimit c →\n ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : c'.pt ⟶ c.pt),\n αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f →\n αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f →\n (Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧\n IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c))",
"tactic": "intro X Y c hc X' Y' c' αX αY f hX hY"
},
{
"state_after": "case van_kampen'.mk.intro\nJ : Type v'\ninst✝³ : Category J\nC✝ : Type u\ninst✝² : Category C✝\nX✝ Y✝ : C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nT : C\nHT : IsTerminal T\nc₀ : BinaryCofan T T\nhc₀ : IsColimit c₀\nH :\n ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),\n αX ≫ BinaryCofan.inl c₀ = BinaryCofan.inl c' ≫ f →\n αY ≫ BinaryCofan.inr c₀ = BinaryCofan.inr c' ≫ f →\n (Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c₀) ∧\n IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c₀))\nX Y : C\nc : BinaryCofan X Y\nhc : IsColimit c\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nd : c.pt ⟶ ((Functor.const (Discrete WalkingPair)).obj c₀.pt).obj { as := WalkingPair.left }\nhd : BinaryCofan.inl c ≫ d = IsTerminal.from HT X ≫ BinaryCofan.inl c₀\nhd' : BinaryCofan.inr c ≫ d = IsTerminal.from HT Y ≫ BinaryCofan.inr c₀\n⊢ Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"state_before": "case van_kampen'\nJ : Type v'\ninst✝³ : Category J\nC✝ : Type u\ninst✝² : Category C✝\nX✝ Y✝ : C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nT : C\nHT : IsTerminal T\nc₀ : BinaryCofan T T\nhc₀ : IsColimit c₀\nH :\n ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),\n αX ≫ BinaryCofan.inl c₀ = BinaryCofan.inl c' ≫ f →\n αY ≫ BinaryCofan.inr c₀ = BinaryCofan.inr c' ≫ f →\n (Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c₀) ∧\n IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c₀))\nX Y : C\nc : BinaryCofan X Y\nhc : IsColimit c\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\n⊢ Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"tactic": "obtain ⟨d, hd, hd'⟩ :=\n Limits.BinaryCofan.IsColimit.desc' hc (HT.from _ ≫ c₀.inl) (HT.from _ ≫ c₀.inr)"
},
{
"state_after": "case van_kampen'.mk.intro\nJ : Type v'\ninst✝³ : Category J\nC✝ : Type u\ninst✝² : Category C✝\nX✝ Y✝ : C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nT : C\nHT : IsTerminal T\nc₀ : BinaryCofan T T\nhc₀ : IsColimit c₀\nH :\n ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),\n αX ≫ BinaryCofan.inl c₀ = BinaryCofan.inl c' ≫ f →\n αY ≫ BinaryCofan.inr c₀ = BinaryCofan.inr c' ≫ f →\n (Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c₀) ∧\n IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c₀))\nX Y : C\nc : BinaryCofan X Y\nhc : IsColimit c\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nd : c.pt ⟶ ((Functor.const (Discrete WalkingPair)).obj c₀.pt).obj { as := WalkingPair.left }\nhd : BinaryCofan.inl c ≫ d = IsTerminal.from HT X ≫ BinaryCofan.inl c₀\nhd' : BinaryCofan.inr c ≫ d = IsTerminal.from HT Y ≫ BinaryCofan.inr c₀\n⊢ IsPullback (BinaryCofan.inl c') (αX ≫ IsTerminal.from HT X) (f ≫ d) (BinaryCofan.inl c₀) ∧\n IsPullback (BinaryCofan.inr c') (αY ≫ IsTerminal.from HT Y) (f ≫ d) (BinaryCofan.inr c₀) ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"state_before": "case van_kampen'.mk.intro\nJ : Type v'\ninst✝³ : Category J\nC✝ : Type u\ninst✝² : Category C✝\nX✝ Y✝ : C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nT : C\nHT : IsTerminal T\nc₀ : BinaryCofan T T\nhc₀ : IsColimit c₀\nH :\n ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),\n αX ≫ BinaryCofan.inl c₀ = BinaryCofan.inl c' ≫ f →\n αY ≫ BinaryCofan.inr c₀ = BinaryCofan.inr c' ≫ f →\n (Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c₀) ∧\n IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c₀))\nX Y : C\nc : BinaryCofan X Y\nhc : IsColimit c\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nd : c.pt ⟶ ((Functor.const (Discrete WalkingPair)).obj c₀.pt).obj { as := WalkingPair.left }\nhd : BinaryCofan.inl c ≫ d = IsTerminal.from HT X ≫ BinaryCofan.inl c₀\nhd' : BinaryCofan.inr c ≫ d = IsTerminal.from HT Y ≫ BinaryCofan.inr c₀\n⊢ Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"tactic": "rw [H c' (αX ≫ HT.from _) (αY ≫ HT.from _) (f ≫ d) (by rw [← reassoc_of% hX, hd, Category.assoc])\n (by rw [← reassoc_of% hY, hd', Category.assoc])]"
},
{
"state_after": "case van_kampen'.mk.intro.intro\nJ : Type v'\ninst✝³ : Category J\nC✝ : Type u\ninst✝² : Category C✝\nX✝ Y✝ : C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nT : C\nHT : IsTerminal T\nc₀ : BinaryCofan T T\nhc₀ : IsColimit c₀\nH :\n ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),\n αX ≫ BinaryCofan.inl c₀ = BinaryCofan.inl c' ≫ f →\n αY ≫ BinaryCofan.inr c₀ = BinaryCofan.inr c' ≫ f →\n (Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c₀) ∧\n IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c₀))\nX Y : C\nc : BinaryCofan X Y\nhc : IsColimit c\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nd : c.pt ⟶ ((Functor.const (Discrete WalkingPair)).obj c₀.pt).obj { as := WalkingPair.left }\nhd : BinaryCofan.inl c ≫ d = IsTerminal.from HT X ≫ BinaryCofan.inl c₀\nhd' : BinaryCofan.inr c ≫ d = IsTerminal.from HT Y ≫ BinaryCofan.inr c₀\nhl : IsPullback (BinaryCofan.inl c) (IsTerminal.from HT X) d (BinaryCofan.inl c₀)\nhr : IsPullback (BinaryCofan.inr c) (IsTerminal.from HT Y) d (BinaryCofan.inr c₀)\n⊢ IsPullback (BinaryCofan.inl c') (αX ≫ IsTerminal.from HT X) (f ≫ d) (BinaryCofan.inl c₀) ∧\n IsPullback (BinaryCofan.inr c') (αY ≫ IsTerminal.from HT Y) (f ≫ d) (BinaryCofan.inr c₀) ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"state_before": "case van_kampen'.mk.intro\nJ : Type v'\ninst✝³ : Category J\nC✝ : Type u\ninst✝² : Category C✝\nX✝ Y✝ : C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nT : C\nHT : IsTerminal T\nc₀ : BinaryCofan T T\nhc₀ : IsColimit c₀\nH :\n ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),\n αX ≫ BinaryCofan.inl c₀ = BinaryCofan.inl c' ≫ f →\n αY ≫ BinaryCofan.inr c₀ = BinaryCofan.inr c' ≫ f →\n (Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c₀) ∧\n IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c₀))\nX Y : C\nc : BinaryCofan X Y\nhc : IsColimit c\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nd : c.pt ⟶ ((Functor.const (Discrete WalkingPair)).obj c₀.pt).obj { as := WalkingPair.left }\nhd : BinaryCofan.inl c ≫ d = IsTerminal.from HT X ≫ BinaryCofan.inl c₀\nhd' : BinaryCofan.inr c ≫ d = IsTerminal.from HT Y ≫ BinaryCofan.inr c₀\n⊢ IsPullback (BinaryCofan.inl c') (αX ≫ IsTerminal.from HT X) (f ≫ d) (BinaryCofan.inl c₀) ∧\n IsPullback (BinaryCofan.inr c') (αY ≫ IsTerminal.from HT Y) (f ≫ d) (BinaryCofan.inr c₀) ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"tactic": "obtain ⟨hl, hr⟩ := (H c (HT.from _) (HT.from _) d hd.symm hd'.symm).mp ⟨hc⟩"
},
{
"state_after": "no goals",
"state_before": "case van_kampen'.mk.intro.intro\nJ : Type v'\ninst✝³ : Category J\nC✝ : Type u\ninst✝² : Category C✝\nX✝ Y✝ : C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nT : C\nHT : IsTerminal T\nc₀ : BinaryCofan T T\nhc₀ : IsColimit c₀\nH :\n ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),\n αX ≫ BinaryCofan.inl c₀ = BinaryCofan.inl c' ≫ f →\n αY ≫ BinaryCofan.inr c₀ = BinaryCofan.inr c' ≫ f →\n (Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c₀) ∧\n IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c₀))\nX Y : C\nc : BinaryCofan X Y\nhc : IsColimit c\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nd : c.pt ⟶ ((Functor.const (Discrete WalkingPair)).obj c₀.pt).obj { as := WalkingPair.left }\nhd : BinaryCofan.inl c ≫ d = IsTerminal.from HT X ≫ BinaryCofan.inl c₀\nhd' : BinaryCofan.inr c ≫ d = IsTerminal.from HT Y ≫ BinaryCofan.inr c₀\nhl : IsPullback (BinaryCofan.inl c) (IsTerminal.from HT X) d (BinaryCofan.inl c₀)\nhr : IsPullback (BinaryCofan.inr c) (IsTerminal.from HT Y) d (BinaryCofan.inr c₀)\n⊢ IsPullback (BinaryCofan.inl c') (αX ≫ IsTerminal.from HT X) (f ≫ d) (BinaryCofan.inl c₀) ∧\n IsPullback (BinaryCofan.inr c') (αY ≫ IsTerminal.from HT Y) (f ≫ d) (BinaryCofan.inr c₀) ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c) ∧ IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c)",
"tactic": "rw [hl.paste_vert_iff hX.symm, hr.paste_vert_iff hY.symm]"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝³ : Category J\nC✝ : Type u\ninst✝² : Category C✝\nX✝ Y✝ : C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nT : C\nHT : IsTerminal T\nc₀ : BinaryCofan T T\nhc₀ : IsColimit c₀\nH :\n ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),\n αX ≫ BinaryCofan.inl c₀ = BinaryCofan.inl c' ≫ f →\n αY ≫ BinaryCofan.inr c₀ = BinaryCofan.inr c' ≫ f →\n (Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c₀) ∧\n IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c₀))\nX Y : C\nc : BinaryCofan X Y\nhc : IsColimit c\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nd : c.pt ⟶ ((Functor.const (Discrete WalkingPair)).obj c₀.pt).obj { as := WalkingPair.left }\nhd : BinaryCofan.inl c ≫ d = IsTerminal.from HT X ≫ BinaryCofan.inl c₀\nhd' : BinaryCofan.inr c ≫ d = IsTerminal.from HT Y ≫ BinaryCofan.inr c₀\n⊢ (αX ≫ IsTerminal.from HT X) ≫ BinaryCofan.inl c₀ = BinaryCofan.inl c' ≫ f ≫ d",
"tactic": "rw [← reassoc_of% hX, hd, Category.assoc]"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝³ : Category J\nC✝ : Type u\ninst✝² : Category C✝\nX✝ Y✝ : C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nT : C\nHT : IsTerminal T\nc₀ : BinaryCofan T T\nhc₀ : IsColimit c₀\nH :\n ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),\n αX ≫ BinaryCofan.inl c₀ = BinaryCofan.inl c' ≫ f →\n αY ≫ BinaryCofan.inr c₀ = BinaryCofan.inr c' ≫ f →\n (Nonempty (IsColimit c') ↔\n IsPullback (BinaryCofan.inl c') αX f (BinaryCofan.inl c₀) ∧\n IsPullback (BinaryCofan.inr c') αY f (BinaryCofan.inr c₀))\nX Y : C\nc : BinaryCofan X Y\nhc : IsColimit c\nX' Y' : C\nc' : BinaryCofan X' Y'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : c'.pt ⟶ c.pt\nhX : αX ≫ BinaryCofan.inl c = BinaryCofan.inl c' ≫ f\nhY : αY ≫ BinaryCofan.inr c = BinaryCofan.inr c' ≫ f\nd : c.pt ⟶ ((Functor.const (Discrete WalkingPair)).obj c₀.pt).obj { as := WalkingPair.left }\nhd : BinaryCofan.inl c ≫ d = IsTerminal.from HT X ≫ BinaryCofan.inl c₀\nhd' : BinaryCofan.inr c ≫ d = IsTerminal.from HT Y ≫ BinaryCofan.inr c₀\n⊢ (αY ≫ IsTerminal.from HT Y) ≫ BinaryCofan.inr c₀ = BinaryCofan.inr c' ≫ f ≫ d",
"tactic": "rw [← reassoc_of% hY, hd', Category.assoc]"
}
] | [
311,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
299,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.inter_biInter | [
{
"state_after": "α✝ : Type ?u.105347\nβ : Type ?u.105350\nγ : Type ?u.105353\nι✝ : Sort ?u.105356\nι' : Sort ?u.105359\nι₂ : Sort ?u.105362\nκ : ι✝ → Sort ?u.105367\nκ₁ : ι✝ → Sort ?u.105372\nκ₂ : ι✝ → Sort ?u.105377\nκ' : ι' → Sort ?u.105382\nι : Type u_1\nα : Type u_2\ns : Set ι\nhs : Set.Nonempty s\nf : ι → Set α\nt : Set α\n⊢ (⋂ (i : ι) (_ : i ∈ s), t ∩ f i) = ⋂ (i : ι) (_ : i ∈ s), f i ∩ t",
"state_before": "α✝ : Type ?u.105347\nβ : Type ?u.105350\nγ : Type ?u.105353\nι✝ : Sort ?u.105356\nι' : Sort ?u.105359\nι₂ : Sort ?u.105362\nκ : ι✝ → Sort ?u.105367\nκ₁ : ι✝ → Sort ?u.105372\nκ₂ : ι✝ → Sort ?u.105377\nκ' : ι' → Sort ?u.105382\nι : Type u_1\nα : Type u_2\ns : Set ι\nhs : Set.Nonempty s\nf : ι → Set α\nt : Set α\n⊢ (⋂ (i : ι) (_ : i ∈ s), t ∩ f i) = t ∩ ⋂ (i : ι) (_ : i ∈ s), f i",
"tactic": "rw [inter_comm, ← biInter_inter hs]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.105347\nβ : Type ?u.105350\nγ : Type ?u.105353\nι✝ : Sort ?u.105356\nι' : Sort ?u.105359\nι₂ : Sort ?u.105362\nκ : ι✝ → Sort ?u.105367\nκ₁ : ι✝ → Sort ?u.105372\nκ₂ : ι✝ → Sort ?u.105377\nκ' : ι' → Sort ?u.105382\nι : Type u_1\nα : Type u_2\ns : Set ι\nhs : Set.Nonempty s\nf : ι → Set α\nt : Set α\n⊢ (⋂ (i : ι) (_ : i ∈ s), t ∩ f i) = ⋂ (i : ι) (_ : i ∈ s), f i ∩ t",
"tactic": "simp [inter_comm]"
}
] | [
960,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
957,
1
] |
Mathlib/Probability/Independence/Basic.lean | ProbabilityTheory.IndepSets.symm | [
{
"state_after": "Ω : Type u_1\nι : Type ?u.432097\ns₁ s₂ : Set (Set Ω)\ninst✝ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nh : IndepSets s₁ s₂\nt1 t2 : Set Ω\nht1 : t1 ∈ s₂\nht2 : t2 ∈ s₁\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"state_before": "Ω : Type u_1\nι : Type ?u.432097\ns₁ s₂ : Set (Set Ω)\ninst✝ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nh : IndepSets s₁ s₂\n⊢ IndepSets s₂ s₁",
"tactic": "intro t1 t2 ht1 ht2"
},
{
"state_after": "Ω : Type u_1\nι : Type ?u.432097\ns₁ s₂ : Set (Set Ω)\ninst✝ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nh : IndepSets s₁ s₂\nt1 t2 : Set Ω\nht1 : t1 ∈ s₂\nht2 : t2 ∈ s₁\n⊢ ↑↑μ (t2 ∩ t1) = ↑↑μ t2 * ↑↑μ t1",
"state_before": "Ω : Type u_1\nι : Type ?u.432097\ns₁ s₂ : Set (Set Ω)\ninst✝ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nh : IndepSets s₁ s₂\nt1 t2 : Set Ω\nht1 : t1 ∈ s₂\nht2 : t2 ∈ s₁\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"tactic": "rw [Set.inter_comm, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\nι : Type ?u.432097\ns₁ s₂ : Set (Set Ω)\ninst✝ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nh : IndepSets s₁ s₂\nt1 t2 : Set Ω\nht1 : t1 ∈ s₂\nht2 : t2 ∈ s₁\n⊢ ↑↑μ (t2 ∩ t1) = ↑↑μ t2 * ↑↑μ t1",
"tactic": "exact h t2 t1 ht2 ht1"
}
] | [
151,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
148,
1
] |
Mathlib/Data/Polynomial/Expand.lean | Polynomial.isLocalRingHom_expand | [
{
"state_after": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : ℕ\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (↑↑(expand R p) f)\n⊢ IsUnit f",
"state_before": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : ℕ\nhp : 0 < p\n⊢ IsLocalRingHom ↑(expand R p)",
"tactic": "refine' ⟨fun f hf1 => _⟩"
},
{
"state_after": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : ℕ\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (↑(expand R p) f)\n⊢ IsUnit f",
"state_before": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : ℕ\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (↑↑(expand R p) f)\n⊢ IsUnit f",
"tactic": "norm_cast at hf1"
},
{
"state_after": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : ℕ\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (↑(expand R p) f)\nhf2 : ↑(expand R p) f = ↑C (coeff (↑(expand R p) f) 0)\n⊢ IsUnit f",
"state_before": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : ℕ\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (↑(expand R p) f)\n⊢ IsUnit f",
"tactic": "have hf2 := eq_C_of_degree_eq_zero (degree_eq_zero_of_isUnit hf1)"
},
{
"state_after": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : ℕ\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (↑(expand R p) f)\nhf2 : ↑(expand R p) f = ↑C (coeff f 0)\n⊢ IsUnit f",
"state_before": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : ℕ\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (↑(expand R p) f)\nhf2 : ↑(expand R p) f = ↑C (coeff (↑(expand R p) f) 0)\n⊢ IsUnit f",
"tactic": "rw [coeff_expand hp, if_pos (dvd_zero _), p.zero_div] at hf2"
},
{
"state_after": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : ℕ\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (coeff f 0)\nhf2 : ↑(expand R p) f = ↑C (coeff f 0)\n⊢ IsUnit f",
"state_before": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : ℕ\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (↑(expand R p) f)\nhf2 : ↑(expand R p) f = ↑C (coeff f 0)\n⊢ IsUnit f",
"tactic": "rw [hf2, isUnit_C] at hf1"
},
{
"state_after": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : ℕ\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (coeff f 0)\nhf2 : f = ↑C (coeff f 0)\n⊢ IsUnit f",
"state_before": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : ℕ\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (coeff f 0)\nhf2 : ↑(expand R p) f = ↑C (coeff f 0)\n⊢ IsUnit f",
"tactic": "rw [expand_eq_C hp] at hf2"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : ℕ\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (coeff f 0)\nhf2 : f = ↑C (coeff f 0)\n⊢ IsUnit f",
"tactic": "rwa [hf2, isUnit_C]"
}
] | [
278,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
273,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | AffineSubspace.subtype_apply | [] | [
431,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
430,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean | Cubic.disc_ne_zero_iff_roots_nodup | [
{
"state_after": "R : Type ?u.1296918\nS : Type ?u.1296921\nF : Type u_1\nK : Type u_2\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nx y z : K\nha : P.a ≠ 0\nh3 : roots (map φ P) = {x, y, z}\n⊢ x ≠ y ∧ x ≠ z ∧ y ≠ z ↔ Nodup {x, y, z}",
"state_before": "R : Type ?u.1296918\nS : Type ?u.1296921\nF : Type u_1\nK : Type u_2\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nx y z : K\nha : P.a ≠ 0\nh3 : roots (map φ P) = {x, y, z}\n⊢ disc P ≠ 0 ↔ Nodup (roots (map φ P))",
"tactic": "rw [disc_ne_zero_iff_roots_ne ha h3, h3]"
},
{
"state_after": "R : Type ?u.1296918\nS : Type ?u.1296921\nF : Type u_1\nK : Type u_2\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nx y z : K\nha : P.a ≠ 0\nh3 : roots (map φ P) = {x, y, z}\n⊢ x ≠ y ∧ x ≠ z ∧ y ≠ z ↔ Nodup (x ::ₘ y ::ₘ {z})",
"state_before": "R : Type ?u.1296918\nS : Type ?u.1296921\nF : Type u_1\nK : Type u_2\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nx y z : K\nha : P.a ≠ 0\nh3 : roots (map φ P) = {x, y, z}\n⊢ x ≠ y ∧ x ≠ z ∧ y ≠ z ↔ Nodup {x, y, z}",
"tactic": "change _ ↔ (x ::ₘ y ::ₘ {z}).Nodup"
},
{
"state_after": "R : Type ?u.1296918\nS : Type ?u.1296921\nF : Type u_1\nK : Type u_2\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nx y z : K\nha : P.a ≠ 0\nh3 : roots (map φ P) = {x, y, z}\n⊢ x ≠ y ∧ x ≠ z ∧ y ≠ z ↔ ¬(x = y ∨ x = z) ∧ ¬y = z ∧ Nodup {z}",
"state_before": "R : Type ?u.1296918\nS : Type ?u.1296921\nF : Type u_1\nK : Type u_2\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nx y z : K\nha : P.a ≠ 0\nh3 : roots (map φ P) = {x, y, z}\n⊢ x ≠ y ∧ x ≠ z ∧ y ≠ z ↔ Nodup (x ::ₘ y ::ₘ {z})",
"tactic": "rw [nodup_cons, nodup_cons, mem_cons, mem_singleton, mem_singleton]"
},
{
"state_after": "R : Type ?u.1296918\nS : Type ?u.1296921\nF : Type u_1\nK : Type u_2\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nx y z : K\nha : P.a ≠ 0\nh3 : roots (map φ P) = {x, y, z}\n⊢ x ≠ y ∧ x ≠ z ∧ y ≠ z ↔ ¬(x = y ∨ x = z) ∧ ¬y = z ∧ True",
"state_before": "R : Type ?u.1296918\nS : Type ?u.1296921\nF : Type u_1\nK : Type u_2\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nx y z : K\nha : P.a ≠ 0\nh3 : roots (map φ P) = {x, y, z}\n⊢ x ≠ y ∧ x ≠ z ∧ y ≠ z ↔ ¬(x = y ∨ x = z) ∧ ¬y = z ∧ Nodup {z}",
"tactic": "simp only [nodup_singleton]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1296918\nS : Type ?u.1296921\nF : Type u_1\nK : Type u_2\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nx y z : K\nha : P.a ≠ 0\nh3 : roots (map φ P) = {x, y, z}\n⊢ x ≠ y ∧ x ≠ z ∧ y ≠ z ↔ ¬(x = y ∨ x = z) ∧ ¬y = z ∧ True",
"tactic": "tauto"
}
] | [
594,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
588,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean | MultilinearMap.mkPiRing_apply_one_eq_self | [
{
"state_after": "case H\nR : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : Module R M₂\nf✝ f' : MultilinearMap R M₁ M₂\ninst✝ : Fintype ι\nf : MultilinearMap R (fun x => R) M₂\nm : ι → R\n⊢ ↑(MultilinearMap.mkPiRing R ι (↑f fun x => 1)) m = ↑f m",
"state_before": "R : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : Module R M₂\nf✝ f' : MultilinearMap R M₁ M₂\ninst✝ : Fintype ι\nf : MultilinearMap R (fun x => R) M₂\n⊢ MultilinearMap.mkPiRing R ι (↑f fun x => 1) = f",
"tactic": "ext m"
},
{
"state_after": "case H\nR : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : Module R M₂\nf✝ f' : MultilinearMap R M₁ M₂\ninst✝ : Fintype ι\nf : MultilinearMap R (fun x => R) M₂\nm : ι → R\nthis : m = fun i => m i • 1\n⊢ ↑(MultilinearMap.mkPiRing R ι (↑f fun x => 1)) m = ↑f m",
"state_before": "case H\nR : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : Module R M₂\nf✝ f' : MultilinearMap R M₁ M₂\ninst✝ : Fintype ι\nf : MultilinearMap R (fun x => R) M₂\nm : ι → R\n⊢ ↑(MultilinearMap.mkPiRing R ι (↑f fun x => 1)) m = ↑f m",
"tactic": "have : m = fun i => m i • (1 : R) := by\n ext j\n simp"
},
{
"state_after": "no goals",
"state_before": "case H\nR : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : Module R M₂\nf✝ f' : MultilinearMap R M₁ M₂\ninst✝ : Fintype ι\nf : MultilinearMap R (fun x => R) M₂\nm : ι → R\nthis : m = fun i => m i • 1\n⊢ ↑(MultilinearMap.mkPiRing R ι (↑f fun x => 1)) m = ↑f m",
"tactic": "conv_rhs => rw [this, f.map_smul_univ]"
},
{
"state_after": "case h\nR : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : Module R M₂\nf✝ f' : MultilinearMap R M₁ M₂\ninst✝ : Fintype ι\nf : MultilinearMap R (fun x => R) M₂\nm : ι → R\nj : ι\n⊢ m j = m j • 1",
"state_before": "R : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : Module R M₂\nf✝ f' : MultilinearMap R M₁ M₂\ninst✝ : Fintype ι\nf : MultilinearMap R (fun x => R) M₂\nm : ι → R\n⊢ m = fun i => m i • 1",
"tactic": "ext j"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : Module R M₂\nf✝ f' : MultilinearMap R M₁ M₂\ninst✝ : Fintype ι\nf : MultilinearMap R (fun x => R) M₂\nm : ι → R\nj : ι\n⊢ m j = m j • 1",
"tactic": "simp"
}
] | [
1072,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1066,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean | IsometryEquiv.preimage_sphere | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1073874\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nh✝ h : α ≃ᵢ β\nx : β\nr : ℝ\n⊢ ↑h ⁻¹' Metric.sphere x r = Metric.sphere (↑(IsometryEquiv.symm h) x) r",
"tactic": "rw [← h.isometry.preimage_sphere (h.symm x) r, h.apply_symm_apply]"
}
] | [
618,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
616,
1
] |
Mathlib/GroupTheory/Coset.lean | Subgroup.quotientEquivOfEq_mk | [] | [
632,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
630,
1
] |
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