file_path
stringlengths 11
79
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stringlengths 2
100
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list | end
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stringclasses 4
values | url
stringclasses 4
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Mathlib/Topology/SubsetProperties.lean
|
isClopen_discrete
|
[] |
[
1639,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1638,
1
] |
Mathlib/Algebra/Hom/Units.lean
|
IsUnit.div_mul_left
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.59663\nG : Type ?u.59666\nα : Type u_1\nM : Type ?u.59672\nN : Type ?u.59675\ninst✝ : DivisionMonoid α\na b c : α\nh : IsUnit b\n⊢ b / (a * b) = 1 / a",
"tactic": "rw [div_eq_mul_inv, mul_inv_rev, h.mul_inv_cancel_left, one_div]"
}
] |
[
443,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
11
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.Term.realize_constantsVarsEquivLeft
|
[
{
"state_after": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.22479\nP : Type ?u.22482\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nn : ℕ\nt : Term (L[[α]]) (β ⊕ Fin n)\nv : β → M\nxs : Fin n → M\n⊢ realize (Sum.elim (Sum.elim (fun a => ↑(Language.con L a)) v) xs ∘ ↑(Equiv.sumAssoc α β (Fin n)).symm)\n (constantsToVars t) =\n realize (Sum.elim v xs) t",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.22479\nP : Type ?u.22482\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nn : ℕ\nt : Term (L[[α]]) (β ⊕ Fin n)\nv : β → M\nxs : Fin n → M\n⊢ realize (Sum.elim (Sum.elim (fun a => ↑(Language.con L a)) v) xs) (↑constantsVarsEquivLeft t) =\n realize (Sum.elim v xs) t",
"tactic": "simp only [constantsVarsEquivLeft, realize_relabel, Equiv.coe_trans, Function.comp_apply,\n constantsVarsEquiv_apply, relabelEquiv_symm_apply]"
},
{
"state_after": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.22479\nP : Type ?u.22482\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nn : ℕ\nt : Term (L[[α]]) (β ⊕ Fin n)\nv : β → M\nxs : Fin n → M\n⊢ realize (Sum.elim (Sum.elim (fun a => ↑(Language.con L a)) v) xs ∘ ↑(Equiv.sumAssoc α β (Fin n)).symm)\n (constantsToVars t) =\n realize (Sum.elim (fun a => ↑(Language.con L a)) (Sum.elim v xs)) (constantsToVars t)",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.22479\nP : Type ?u.22482\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nn : ℕ\nt : Term (L[[α]]) (β ⊕ Fin n)\nv : β → M\nxs : Fin n → M\n⊢ realize (Sum.elim (Sum.elim (fun a => ↑(Language.con L a)) v) xs ∘ ↑(Equiv.sumAssoc α β (Fin n)).symm)\n (constantsToVars t) =\n realize (Sum.elim v xs) t",
"tactic": "refine' _root_.trans _ realize_constantsToVars"
},
{
"state_after": "case e_v.h\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.22479\nP : Type ?u.22482\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nn : ℕ\nt : Term (L[[α]]) (β ⊕ Fin n)\nv : β → M\nxs : Fin n → M\nx : α ⊕ β ⊕ Fin n\n⊢ (Sum.elim (Sum.elim (fun a => ↑(Language.con L a)) v) xs ∘ ↑(Equiv.sumAssoc α β (Fin n)).symm) x =\n Sum.elim (fun a => ↑(Language.con L a)) (Sum.elim v xs) x",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.22479\nP : Type ?u.22482\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nn : ℕ\nt : Term (L[[α]]) (β ⊕ Fin n)\nv : β → M\nxs : Fin n → M\n⊢ realize (Sum.elim (Sum.elim (fun a => ↑(Language.con L a)) v) xs ∘ ↑(Equiv.sumAssoc α β (Fin n)).symm)\n (constantsToVars t) =\n realize (Sum.elim (fun a => ↑(Language.con L a)) (Sum.elim v xs)) (constantsToVars t)",
"tactic": "rcongr x"
},
{
"state_after": "no goals",
"state_before": "case e_v.h\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.22479\nP : Type ?u.22482\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nn : ℕ\nt : Term (L[[α]]) (β ⊕ Fin n)\nv : β → M\nxs : Fin n → M\nx : α ⊕ β ⊕ Fin n\n⊢ (Sum.elim (Sum.elim (fun a => ↑(Language.con L a)) v) xs ∘ ↑(Equiv.sumAssoc α β (Fin n)).symm) x =\n Sum.elim (fun a => ↑(Language.con L a)) (Sum.elim v xs) x",
"tactic": "rcases x with (a | (b | i)) <;> simp"
}
] |
[
204,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.comap_id
|
[] |
[
267,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
Convex.affine_preimage
|
[] |
[
498,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
ENNReal.ofReal_rpow_of_nonneg
|
[
{
"state_after": "case pos\nx p : ℝ\nhx_nonneg : 0 ≤ x\nhp_nonneg : 0 ≤ p\nhp0 : p = 0\n⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)\n\ncase neg\nx p : ℝ\nhx_nonneg : 0 ≤ x\nhp_nonneg : 0 ≤ p\nhp0 : ¬p = 0\n⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)",
"state_before": "x p : ℝ\nhx_nonneg : 0 ≤ x\nhp_nonneg : 0 ≤ p\n⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)",
"tactic": "by_cases hp0 : p = 0"
},
{
"state_after": "case pos\nx p : ℝ\nhx_nonneg : 0 ≤ x\nhp_nonneg : 0 ≤ p\nhp0 : ¬p = 0\nhx0 : x = 0\n⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)\n\ncase neg\nx p : ℝ\nhx_nonneg : 0 ≤ x\nhp_nonneg : 0 ≤ p\nhp0 : ¬p = 0\nhx0 : ¬x = 0\n⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)",
"state_before": "case neg\nx p : ℝ\nhx_nonneg : 0 ≤ x\nhp_nonneg : 0 ≤ p\nhp0 : ¬p = 0\n⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)",
"tactic": "by_cases hx0 : x = 0"
},
{
"state_after": "case neg\nx p : ℝ\nhx_nonneg : 0 ≤ x\nhp_nonneg : 0 ≤ p\nhp0 : ¬p = 0\nhx0 : x ≠ 0\n⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)",
"state_before": "case neg\nx p : ℝ\nhx_nonneg : 0 ≤ x\nhp_nonneg : 0 ≤ p\nhp0 : ¬p = 0\nhx0 : ¬x = 0\n⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)",
"tactic": "rw [← Ne.def] at hx0"
},
{
"state_after": "no goals",
"state_before": "case neg\nx p : ℝ\nhx_nonneg : 0 ≤ x\nhp_nonneg : 0 ≤ p\nhp0 : ¬p = 0\nhx0 : x ≠ 0\n⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)",
"tactic": "exact ofReal_rpow_of_pos (hx_nonneg.lt_of_ne hx0.symm)"
},
{
"state_after": "no goals",
"state_before": "case pos\nx p : ℝ\nhx_nonneg : 0 ≤ x\nhp_nonneg : 0 ≤ p\nhp0 : p = 0\n⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)",
"tactic": "simp [hp0]"
},
{
"state_after": "case pos\nx p : ℝ\nhx_nonneg : 0 ≤ x\nhp_nonneg : 0 ≤ p\nhp0 : p ≠ 0\nhx0 : x = 0\n⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)",
"state_before": "case pos\nx p : ℝ\nhx_nonneg : 0 ≤ x\nhp_nonneg : 0 ≤ p\nhp0 : ¬p = 0\nhx0 : x = 0\n⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)",
"tactic": "rw [← Ne.def] at hp0"
},
{
"state_after": "case pos\nx p : ℝ\nhx_nonneg : 0 ≤ x\nhp_nonneg : 0 ≤ p\nhp0 : p ≠ 0\nhx0 : x = 0\nhp_pos : 0 < p\n⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)",
"state_before": "case pos\nx p : ℝ\nhx_nonneg : 0 ≤ x\nhp_nonneg : 0 ≤ p\nhp0 : p ≠ 0\nhx0 : x = 0\n⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)",
"tactic": "have hp_pos : 0 < p := lt_of_le_of_ne hp_nonneg hp0.symm"
},
{
"state_after": "no goals",
"state_before": "case pos\nx p : ℝ\nhx_nonneg : 0 ≤ x\nhp_nonneg : 0 ≤ p\nhp0 : p ≠ 0\nhx0 : x = 0\nhp_pos : 0 < p\n⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)",
"tactic": "simp [hx0, hp_pos, hp_pos.ne.symm]"
}
] |
[
755,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
746,
1
] |
Mathlib/Analysis/Fourier/FourierTransform.lean
|
Real.vector_fourierIntegral_eq_integral_exp_smul
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : CompleteSpace E\ninst✝⁵ : NormedSpace ℂ E\nV : Type u_1\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module ℝ V\ninst✝² : MeasurableSpace V\nW : Type u_2\ninst✝¹ : AddCommGroup W\ninst✝ : Module ℝ W\nL : V →ₗ[ℝ] W →ₗ[ℝ] ℝ\nμ : MeasureTheory.Measure V\nf : V → E\nw : W\n⊢ VectorFourier.fourierIntegral fourierChar μ L f w =\n ∫ (v : V), Complex.exp (↑(-2 * π * ↑(↑L v) w) * Complex.I) • f v ∂μ",
"tactic": "simp_rw [VectorFourier.fourierIntegral, Real.fourierChar_apply, mul_neg, neg_mul]"
}
] |
[
259,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
isIntegral_of_mem_of_FG
|
[
{
"state_after": "case intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\n⊢ IsIntegral R x",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nHS : FG (↑Subalgebra.toSubmodule S)\nx : A\nhx : x ∈ S\n⊢ IsIntegral R x",
"tactic": "cases' HS with y hy"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R (?m.368651 lx)\nhlx2 : ↑(Finsupp.total A A R id) lx = x\n⊢ IsIntegral R x",
"state_before": "case intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\n⊢ IsIntegral R x",
"tactic": "obtain ⟨lx, hlx1, hlx2⟩ :\n ∃ (l : A →₀ R), l ∈ Finsupp.supported R R ↑y ∧ (Finsupp.total A A R id) l = x := by\n rwa [← @Finsupp.mem_span_image_iff_total A A R _ _ _ id (↑y) x, Set.image_id (y : Set A), hy]"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\n⊢ IsIntegral R x",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R (?m.368651 lx)\nhlx2 : ↑(Finsupp.total A A R id) lx = x\n⊢ IsIntegral R x",
"tactic": "have hyS : ∀ {p}, p ∈ y → p ∈ S := fun {p} hp =>\n show p ∈ Subalgebra.toSubmodule S by\n rw [← hy]\n exact subset_span hp"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nthis : ∀ (jk : { x // x ∈ y ×ˢ y }), (↑jk).fst * (↑jk).snd ∈ ↑Subalgebra.toSubmodule S\n⊢ IsIntegral R x",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\n⊢ IsIntegral R x",
"tactic": "have : ∀ jk : (y ×ˢ y : Finset (A × A)),\n jk.1.1 * jk.1.2 ∈ (Subalgebra.toSubmodule S) := fun jk =>\n S.mul_mem (hyS (Finset.mem_product.1 jk.2).1) (hyS (Finset.mem_product.1 jk.2).2)"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nthis : ∀ (jk : { x // x ∈ y ×ˢ y }), (↑jk).fst * (↑jk).snd ∈ span R (id '' ↑y)\n⊢ IsIntegral R x",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nthis : ∀ (jk : { x // x ∈ y ×ˢ y }), (↑jk).fst * (↑jk).snd ∈ ↑Subalgebra.toSubmodule S\n⊢ IsIntegral R x",
"tactic": "rw [← hy, ← Set.image_id (y : Set A)] at this"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nthis :\n ∀ (jk : { x // x ∈ y ×ˢ y }), ∃ l, l ∈ Finsupp.supported R R ↑y ∧ ↑(Finsupp.total A A R id) l = (↑jk).fst * (↑jk).snd\n⊢ IsIntegral R x",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nthis : ∀ (jk : { x // x ∈ y ×ˢ y }), (↑jk).fst * (↑jk).snd ∈ span R (id '' ↑y)\n⊢ IsIntegral R x",
"tactic": "simp only [Finsupp.mem_span_image_iff_total] at this"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\n⊢ IsIntegral R x",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nthis :\n ∀ (jk : { x // x ∈ y ×ˢ y }), ∃ l, l ∈ Finsupp.supported R R ↑y ∧ ↑(Finsupp.total A A R id) l = (↑jk).fst * (↑jk).snd\n⊢ IsIntegral R x",
"tactic": "choose ly hly1 hly2 using this"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\n⊢ IsIntegral R x",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\n⊢ IsIntegral R x",
"tactic": "let S₀ : Subring R :=\n Subring.closure ↑(lx.frange ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\n⊢ IsIntegral { x // x ∈ S₀ } x",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\n⊢ IsIntegral R x",
"tactic": "refine' isIntegral_ofSubring S₀ _"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\n⊢ IsIntegral { x // x ∈ S₀ } x",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\n⊢ IsIntegral { x // x ∈ S₀ } x",
"tactic": "letI : CommRing S₀ := SubringClass.toCommRing S₀"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\n⊢ IsIntegral { x // x ∈ S₀ } x",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\n⊢ IsIntegral { x // x ∈ S₀ } x",
"tactic": "letI : Algebra S₀ A := Algebra.ofSubring S₀"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\n⊢ IsIntegral { x // x ∈ S₀ } x",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\n⊢ IsIntegral { x // x ∈ S₀ } x",
"tactic": "let S₁ : Subring A :=\n { carrier := span S₀ (insert 1 ↑y : Set A)\n one_mem' := subset_span <| Or.inl rfl\n mul_mem' := fun {p q} hp hq => this <| mul_mem_mul hp hq\n zero_mem' := (span S₀ (insert 1 ↑y : Set A)).zero_mem\n add_mem' := fun {_ _} => (span S₀ (insert 1 ↑y : Set A)).add_mem\n neg_mem' := fun {_} => (span S₀ (insert 1 ↑y : Set A)).neg_mem }"
},
{
"state_after": "case foo\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝² : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝¹ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\n⊢ ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\n\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝² : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝¹ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\n⊢ IsIntegral { x // x ∈ S₀ } x",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝² : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝¹ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\n⊢ IsIntegral { x // x ∈ S₀ } x",
"tactic": "have foo : ∀ z, z ∈ S₁ ↔ z ∈ Algebra.adjoin (↥S₀) (y : Set A)"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝² : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝¹ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\n⊢ IsIntegral { x // x ∈ S₀ } x",
"state_before": "case foo\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝² : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝¹ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\n⊢ ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\n\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝² : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝¹ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\n⊢ IsIntegral { x // x ∈ S₀ } x",
"tactic": "simp [this]"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝³ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝² : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝¹ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis : IsNoetherianRing { x // x ∈ S₀ }\n⊢ IsIntegral { x // x ∈ S₀ } x",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝² : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝¹ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\n⊢ IsIntegral { x // x ∈ S₀ } x",
"tactic": "haveI : IsNoetherianRing S₀ := is_noetherian_subring_closure _ (Finset.finite_toSet _)"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝³ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝² : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝¹ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis : IsNoetherianRing { x // x ∈ S₀ }\n⊢ x ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝³ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝² : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝¹ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis : IsNoetherianRing { x // x ∈ S₀ }\n⊢ IsIntegral { x // x ∈ S₀ } x",
"tactic": "refine'\n isIntegral_of_submodule_noetherian (Algebra.adjoin S₀ ↑y)\n (isNoetherian_of_fg_of_noetherian _\n ⟨insert 1 y, by\n rw [Finset.coe_insert]\n ext z\n simp only [Finset.coe_sort_coe, Finset.univ_eq_attach, Finset.mem_coe,\n Subalgebra.mem_toSubmodule]\n convert foo z⟩)\n _ _"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝³ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝² : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝¹ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis : IsNoetherianRing { x // x ∈ S₀ }\n⊢ ∑ a in lx.support, ↑lx a • id a ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝³ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝² : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝¹ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis : IsNoetherianRing { x // x ∈ S₀ }\n⊢ x ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y",
"tactic": "rw [← hlx2, Finsupp.total_apply, Finsupp.sum]"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝³ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝² : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝¹ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis : IsNoetherianRing { x // x ∈ S₀ }\nr : A\nhr : r ∈ lx.support\n⊢ ↑lx r • id r ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝³ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝² : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝¹ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis : IsNoetherianRing { x // x ∈ S₀ }\n⊢ ∑ a in lx.support, ↑lx a • id a ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y",
"tactic": "refine' Subalgebra.sum_mem _ fun r hr => _"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝⁴ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝³ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝² : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝¹ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis✝ : IsNoetherianRing { x // x ∈ S₀ }\nr : A\nhr : r ∈ lx.support\nthis : ↑lx r ∈ S₀\n⊢ ↑lx r • id r ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝³ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝² : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝¹ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis : IsNoetherianRing { x // x ∈ S₀ }\nr : A\nhr : r ∈ lx.support\n⊢ ↑lx r • id r ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y",
"tactic": "have : lx r ∈ S₀ :=\n Subring.subset_closure (Finset.mem_union_left _ (Finset.mem_image_of_mem _ hr))"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝⁴ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝³ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝² : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝¹ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis✝ : IsNoetherianRing { x // x ∈ S₀ }\nr : A\nhr : r ∈ lx.support\nthis : ↑lx r ∈ S₀\n⊢ { val := ↑lx r, property := this } • r ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝⁴ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝³ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝² : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝¹ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis✝ : IsNoetherianRing { x // x ∈ S₀ }\nr : A\nhr : r ∈ lx.support\nthis : ↑lx r ∈ S₀\n⊢ ↑lx r • id r ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y",
"tactic": "change (⟨_, this⟩ : S₀) • r ∈ _"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : ↑lx.support ⊆ ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝⁴ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝³ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝² : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝¹ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis✝ : IsNoetherianRing { x // x ∈ S₀ }\nr : A\nhr : r ∈ lx.support\nthis : ↑lx r ∈ S₀\n⊢ { val := ↑lx r, property := this } • r ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝⁴ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝³ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝² : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝¹ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis✝ : IsNoetherianRing { x // x ∈ S₀ }\nr : A\nhr : r ∈ lx.support\nthis : ↑lx r ∈ S₀\n⊢ { val := ↑lx r, property := this } • r ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y",
"tactic": "rw [Finsupp.mem_supported] at hlx1"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : ↑lx.support ⊆ ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝⁴ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝³ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝² : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝¹ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis✝ : IsNoetherianRing { x // x ∈ S₀ }\nr : A\nhr : r ∈ lx.support\nthis : ↑lx r ∈ S₀\n⊢ { val := ↑lx r, property := this } • r ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y",
"tactic": "exact Subalgebra.smul_mem _ (Algebra.subset_adjoin <| hlx1 hr) _"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\n⊢ ∃ l, l ∈ Finsupp.supported R R ↑y ∧ ↑(Finsupp.total A A R id) l = x",
"tactic": "rwa [← @Finsupp.mem_span_image_iff_total A A R _ _ _ id (↑y) x, Set.image_id (y : Set A), hy]"
},
{
"state_after": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\np : A\nhp : p ∈ y\n⊢ p ∈ span R ↑y",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\np : A\nhp : p ∈ y\n⊢ p ∈ ↑Subalgebra.toSubmodule S",
"tactic": "rw [← hy]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\np : A\nhp : p ∈ y\n⊢ p ∈ span R ↑y",
"tactic": "exact subset_span hp"
},
{
"state_after": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\n⊢ span { x // x ∈ S₀ } (insert 1 ↑y * insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\n⊢ span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)",
"tactic": "rw [span_mul_span]"
},
{
"state_after": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nz : A\nhz : z ∈ insert 1 ↑y * insert 1 ↑y\n⊢ z ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\n⊢ span { x // x ∈ S₀ } (insert 1 ↑y * insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)",
"tactic": "refine' span_le.2 fun z hz => _"
},
{
"state_after": "case intro.intro.intro.inl.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nq : A\nhq : q ∈ insert 1 ↑y\nhz : 1 * q ∈ insert 1 ↑y * insert 1 ↑y\n⊢ 1 * q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))\n\ncase intro.intro.intro.inr.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np q : A\nhp : p ∈ ↑y\nhq : q ∈ insert 1 ↑y\nhz : p * q ∈ insert 1 ↑y * insert 1 ↑y\n⊢ p * q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nz : A\nhz : z ∈ insert 1 ↑y * insert 1 ↑y\n⊢ z ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"tactic": "rcases Set.mem_mul.1 hz with ⟨p, q, rfl | hp, hq, rfl⟩"
},
{
"state_after": "case intro.intro.intro.inr.intro.inl\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np : A\nhp : p ∈ ↑y\nhz : p * 1 ∈ insert 1 ↑y * insert 1 ↑y\n⊢ p * 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))\n\ncase intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np q : A\nhp : p ∈ ↑y\nhz : p * q ∈ insert 1 ↑y * insert 1 ↑y\nhq : q ∈ ↑y\n⊢ p * q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"state_before": "case intro.intro.intro.inr.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np q : A\nhp : p ∈ ↑y\nhq : q ∈ insert 1 ↑y\nhz : p * q ∈ insert 1 ↑y * insert 1 ↑y\n⊢ p * q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"tactic": "rcases hq with (rfl | hq)"
},
{
"state_after": "case intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np q : A\nhp : p ∈ ↑y\nhz : p * q ∈ insert 1 ↑y * insert 1 ↑y\nhq : q ∈ ↑y\n⊢ ↑(Finsupp.total A A R id) (ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }) ∈\n ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"state_before": "case intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np q : A\nhp : p ∈ ↑y\nhz : p * q ∈ insert 1 ↑y * insert 1 ↑y\nhq : q ∈ ↑y\n⊢ p * q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"tactic": "erw [← hly2 ⟨(p, q), Finset.mem_product.2 ⟨hp, hq⟩⟩]"
},
{
"state_after": "case intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np q : A\nhp : p ∈ ↑y\nhz : p * q ∈ insert 1 ↑y * insert 1 ↑y\nhq : q ∈ ↑y\n⊢ ∑ a in (ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }).support,\n ↑(ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }) a • id a ∈\n ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"state_before": "case intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np q : A\nhp : p ∈ ↑y\nhz : p * q ∈ insert 1 ↑y * insert 1 ↑y\nhq : q ∈ ↑y\n⊢ ↑(Finsupp.total A A R id) (ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }) ∈\n ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"tactic": "rw [Finsupp.total_apply, Finsupp.sum]"
},
{
"state_after": "case intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np q : A\nhp : p ∈ ↑y\nhz : p * q ∈ insert 1 ↑y * insert 1 ↑y\nhq : q ∈ ↑y\nt : A\nht : t ∈ (ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }).support\n⊢ ↑(ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }) t • id t ∈ span { x // x ∈ S₀ } (insert 1 ↑y)",
"state_before": "case intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np q : A\nhp : p ∈ ↑y\nhz : p * q ∈ insert 1 ↑y * insert 1 ↑y\nhq : q ∈ ↑y\n⊢ ∑ a in (ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }).support,\n ↑(ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }) a • id a ∈\n ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"tactic": "refine' (span S₀ (insert 1 ↑y : Set A)).sum_mem fun t ht => _"
},
{
"state_after": "case intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np q : A\nhp : p ∈ ↑y\nhz : p * q ∈ insert 1 ↑y * insert 1 ↑y\nhq : q ∈ ↑y\nt : A\nht : t ∈ (ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }).support\nthis : ↑(ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }) t ∈ S₀\n⊢ ↑(ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }) t • id t ∈ span { x // x ∈ S₀ } (insert 1 ↑y)",
"state_before": "case intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np q : A\nhp : p ∈ ↑y\nhz : p * q ∈ insert 1 ↑y * insert 1 ↑y\nhq : q ∈ ↑y\nt : A\nht : t ∈ (ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }).support\n⊢ ↑(ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }) t • id t ∈ span { x // x ∈ S₀ } (insert 1 ↑y)",
"tactic": "have : ly ⟨(p, q), Finset.mem_product.2 ⟨hp, hq⟩⟩ t ∈ S₀ :=\n Subring.subset_closure\n (Finset.mem_union_right _ <|\n Finset.mem_biUnion.2\n ⟨⟨(p, q), Finset.mem_product.2 ⟨hp, hq⟩⟩, Finset.mem_univ _,\n Finsupp.mem_frange.2 ⟨Finsupp.mem_support_iff.1 ht, _, rfl⟩⟩)"
},
{
"state_after": "case intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np q : A\nhp : p ∈ ↑y\nhz : p * q ∈ insert 1 ↑y * insert 1 ↑y\nhq : q ∈ ↑y\nt : A\nht : t ∈ (ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }).support\nthis : ↑(ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }) t ∈ S₀\n⊢ { val := ↑(ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }) t, property := this } • t ∈\n span { x // x ∈ S₀ } (insert 1 ↑y)",
"state_before": "case intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np q : A\nhp : p ∈ ↑y\nhz : p * q ∈ insert 1 ↑y * insert 1 ↑y\nhq : q ∈ ↑y\nt : A\nht : t ∈ (ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }).support\nthis : ↑(ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }) t ∈ S₀\n⊢ ↑(ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }) t • id t ∈ span { x // x ∈ S₀ } (insert 1 ↑y)",
"tactic": "change (⟨_, this⟩ : S₀) • t ∈ _"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np q : A\nhp : p ∈ ↑y\nhz : p * q ∈ insert 1 ↑y * insert 1 ↑y\nhq : q ∈ ↑y\nt : A\nht : t ∈ (ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }).support\nthis : ↑(ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }) t ∈ S₀\n⊢ { val := ↑(ly { val := (p, q), property := (_ : (p, q) ∈ y ×ˢ y) }) t, property := this } • t ∈\n span { x // x ∈ S₀ } (insert 1 ↑y)",
"tactic": "exact smul_mem _ _ (subset_span <| Or.inr <| hly1 _ ht)"
},
{
"state_after": "case intro.intro.intro.inl.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nq : A\nhq : q ∈ insert 1 ↑y\nhz : 1 * q ∈ insert 1 ↑y * insert 1 ↑y\n⊢ q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"state_before": "case intro.intro.intro.inl.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nq : A\nhq : q ∈ insert 1 ↑y\nhz : 1 * q ∈ insert 1 ↑y * insert 1 ↑y\n⊢ 1 * q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"tactic": "rw [one_mul]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.inl.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nq : A\nhq : q ∈ insert 1 ↑y\nhz : 1 * q ∈ insert 1 ↑y * insert 1 ↑y\n⊢ q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"tactic": "exact subset_span hq"
},
{
"state_after": "case intro.intro.intro.inr.intro.inl\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np : A\nhp : p ∈ ↑y\nhz : p * 1 ∈ insert 1 ↑y * insert 1 ↑y\n⊢ p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"state_before": "case intro.intro.intro.inr.intro.inl\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np : A\nhp : p ∈ ↑y\nhz : p * 1 ∈ insert 1 ↑y * insert 1 ↑y\n⊢ p * 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"tactic": "rw [mul_one]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.inr.intro.inl\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\np : A\nhp : p ∈ ↑y\nhz : p * 1 ∈ insert 1 ↑y * insert 1 ↑y\n⊢ p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))",
"tactic": "exact subset_span (Or.inr hp)"
},
{
"state_after": "case h\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\n⊢ z ∈ S₁ ↔ z ∈ Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\n⊢ S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)",
"tactic": "ext z"
},
{
"state_after": "case h\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\n⊢ z ∈ span { x // x ∈ S₀ } (insert 1 ↑y) ↔ z ∈ ↑Subalgebra.toSubmodule (Algebra.adjoin { x // x ∈ S₀ } ↑y)",
"state_before": "case h\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\n⊢ z ∈ S₁ ↔ z ∈ Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)",
"tactic": "suffices\n z ∈ span (↥S₀) (insert 1 ↑y : Set A) ↔\n z ∈ Subalgebra.toSubmodule (Algebra.adjoin (↥S₀) (y : Set A)) by\n simpa"
},
{
"state_after": "case h.mp\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ span { x // x ∈ S₀ } (insert 1 ↑y)\n⊢ z ∈ ↑Subalgebra.toSubmodule (Algebra.adjoin { x // x ∈ S₀ } ↑y)\n\ncase h.mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ ↑Subalgebra.toSubmodule (Algebra.adjoin { x // x ∈ S₀ } ↑y)\n⊢ z ∈ span { x // x ∈ S₀ } (insert 1 ↑y)",
"state_before": "case h\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\n⊢ z ∈ span { x // x ∈ S₀ } (insert 1 ↑y) ↔ z ∈ ↑Subalgebra.toSubmodule (Algebra.adjoin { x // x ∈ S₀ } ↑y)",
"tactic": "constructor <;> intro hz"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝² : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝¹ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nthis : z ∈ span { x // x ∈ S₀ } (insert 1 ↑y) ↔ z ∈ ↑Subalgebra.toSubmodule (Algebra.adjoin { x // x ∈ S₀ } ↑y)\n⊢ z ∈ S₁ ↔ z ∈ Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "case h.mp\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ span { x // x ∈ S₀ } (insert 1 ↑y)\n⊢ z ∈ ↑Subalgebra.toSubmodule (Algebra.adjoin { x // x ∈ S₀ } ↑y)",
"tactic": "exact\n (span_le.2\n (Set.insert_subset.2 ⟨(Algebra.adjoin S₀ (y : Set A)).one_mem, Algebra.subset_adjoin⟩)) hz"
},
{
"state_after": "case h.mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y)\n⊢ z ∈ span { x // x ∈ S₀ } (insert 1 ↑y)",
"state_before": "case h.mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ ↑Subalgebra.toSubmodule (Algebra.adjoin { x // x ∈ S₀ } ↑y)\n⊢ z ∈ span { x // x ∈ S₀ } (insert 1 ↑y)",
"tactic": "rw [Subalgebra.mem_toSubmodule, Algebra.mem_adjoin_iff] at hz"
},
{
"state_after": "case h.mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y)\n⊢ Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y) ≤ S₁",
"state_before": "case h.mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y)\n⊢ z ∈ span { x // x ∈ S₀ } (insert 1 ↑y)",
"tactic": "suffices Subring.closure (Set.range (algebraMap (↥S₀) A) ∪ ↑y) ≤ S₁ by exact this hz"
},
{
"state_after": "case h.mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y)\n⊢ Set.range ↑(algebraMap { x // x ∈ S₀ } A) ⊆ ↑S₁",
"state_before": "case h.mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y)\n⊢ Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y) ≤ S₁",
"tactic": "refine' Subring.closure_le.2 (Set.union_subset _ fun t ht => subset_span <| Or.inr ht)"
},
{
"state_after": "case h.mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y)\n⊢ ∀ (y : { x // x ∈ S₀ }), ↑(algebraMap { x // x ∈ S₀ } A) y ∈ ↑S₁",
"state_before": "case h.mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y)\n⊢ Set.range ↑(algebraMap { x // x ∈ S₀ } A) ⊆ ↑S₁",
"tactic": "rw [Set.range_subset_iff]"
},
{
"state_after": "case h.mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y)\ny' : { x // x ∈ S₀ }\n⊢ ↑(algebraMap { x // x ∈ S₀ } A) y' ∈ ↑S₁",
"state_before": "case h.mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y)\n⊢ ∀ (y : { x // x ∈ S₀ }), ↑(algebraMap { x // x ∈ S₀ } A) y ∈ ↑S₁",
"tactic": "intro y'"
},
{
"state_after": "case h.mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y)\ny' : { x // x ∈ S₀ }\n⊢ y' • 1 ∈ ↑S₁",
"state_before": "case h.mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y)\ny' : { x // x ∈ S₀ }\n⊢ ↑(algebraMap { x // x ∈ S₀ } A) y' ∈ ↑S₁",
"tactic": "rw [Algebra.algebraMap_eq_smul_one]"
},
{
"state_after": "no goals",
"state_before": "case h.mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝¹ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y)\ny' : { x // x ∈ S₀ }\n⊢ y' • 1 ∈ ↑S₁",
"tactic": "exact smul_mem (span S₀ (insert (1 : A) (y : Set A))) y' (subset_span (Or.inl rfl))"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝² : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝¹ : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nz : A\nhz : z ∈ Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y)\nthis : Subring.closure (Set.range ↑(algebraMap { x // x ∈ S₀ } A) ∪ ↑y) ≤ S₁\n⊢ z ∈ span { x // x ∈ S₀ } (insert 1 ↑y)",
"tactic": "exact this hz"
},
{
"state_after": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝³ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝² : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝¹ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis : IsNoetherianRing { x // x ∈ S₀ }\n⊢ span { x // x ∈ S₀ } (insert 1 ↑y) = ↑Subalgebra.toSubmodule (Algebra.adjoin { x // x ∈ S₀ } ↑y)",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝³ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝² : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝¹ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis : IsNoetherianRing { x // x ∈ S₀ }\n⊢ span { x // x ∈ S₀ } ↑(insert 1 y) = ↑Subalgebra.toSubmodule (Algebra.adjoin { x // x ∈ S₀ } ↑y)",
"tactic": "rw [Finset.coe_insert]"
},
{
"state_after": "case h\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝³ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝² : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝¹ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis : IsNoetherianRing { x // x ∈ S₀ }\nz : A\n⊢ z ∈ span { x // x ∈ S₀ } (insert 1 ↑y) ↔ z ∈ ↑Subalgebra.toSubmodule (Algebra.adjoin { x // x ∈ S₀ } ↑y)",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝³ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝² : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝¹ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis : IsNoetherianRing { x // x ∈ S₀ }\n⊢ span { x // x ∈ S₀ } (insert 1 ↑y) = ↑Subalgebra.toSubmodule (Algebra.adjoin { x // x ∈ S₀ } ↑y)",
"tactic": "ext z"
},
{
"state_after": "case h\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝³ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝² : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝¹ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis : IsNoetherianRing { x // x ∈ S₀ }\nz : A\n⊢ z ∈\n span\n { x //\n x ∈ Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion (Finset.attach (y ×ˢ y)) (Finsupp.frange ∘ ly)) }\n (insert 1 ↑y) ↔\n z ∈\n Algebra.adjoin\n { x //\n x ∈ Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion (Finset.attach (y ×ˢ y)) (Finsupp.frange ∘ ly)) }\n ↑y",
"state_before": "case h\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝³ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝² : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝¹ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis : IsNoetherianRing { x // x ∈ S₀ }\nz : A\n⊢ z ∈ span { x // x ∈ S₀ } (insert 1 ↑y) ↔ z ∈ ↑Subalgebra.toSubmodule (Algebra.adjoin { x // x ∈ S₀ } ↑y)",
"tactic": "simp only [Finset.coe_sort_coe, Finset.univ_eq_attach, Finset.mem_coe,\n Subalgebra.mem_toSubmodule]"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\nA : Type u_2\nB : Type ?u.347819\nS✝ : Type ?u.347822\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S✝\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S✝\nS : Subalgebra R A\nx : A\nhx : x ∈ S\ny : Finset A\nhy : span R ↑y = ↑Subalgebra.toSubmodule S\nlx : A →₀ R\nhlx1 : lx ∈ Finsupp.supported R R ↑y\nhlx2 : ↑(Finsupp.total A A R id) lx = x\nhyS : ∀ {p : A}, p ∈ y → p ∈ S\nly : { x // x ∈ y ×ˢ y } → A →₀ R\nhly1 : ∀ (jk : { x // x ∈ y ×ˢ y }), ly jk ∈ Finsupp.supported R R ↑y\nhly2 : ∀ (jk : { x // x ∈ y ×ˢ y }), ↑(Finsupp.total A A R id) (ly jk) = (↑jk).fst * (↑jk).snd\nS₀ : Subring R := Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))\nthis✝³ : CommRing { x // x ∈ S₀ } := SubringClass.toCommRing S₀\nthis✝² : Algebra { x // x ∈ S₀ } A := Algebra.ofSubring S₀\nthis✝¹ : span { x // x ∈ S₀ } (insert 1 ↑y) * span { x // x ∈ S₀ } (insert 1 ↑y) ≤ span { x // x ∈ S₀ } (insert 1 ↑y)\nS₁ : Subring A :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ↑(span { x // x ∈ S₀ } (insert 1 ↑y)),\n mul_mem' :=\n (_ :\n ∀ {p q : A},\n p ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) →\n q ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y)) → p * q ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n one_mem' := (_ : 1 ∈ ↑(span { x // x ∈ S₀ } (insert 1 ↑y))) },\n add_mem' :=\n (_ :\n ∀ {x x_1 : A},\n x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) →\n x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → x + x_1 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)),\n zero_mem' := (_ : 0 ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) },\n neg_mem' := (_ : ∀ {x : A}, x ∈ span { x // x ∈ S₀ } (insert 1 ↑y) → -x ∈ span { x // x ∈ S₀ } (insert 1 ↑y)) }\nthis✝ : S₁ = Subalgebra.toSubring (Algebra.adjoin { x // x ∈ S₀ } ↑y)\nfoo : ∀ (z : A), z ∈ S₁ ↔ z ∈ Algebra.adjoin { x // x ∈ S₀ } ↑y\nthis : IsNoetherianRing { x // x ∈ S₀ }\nz : A\n⊢ z ∈\n span\n { x //\n x ∈ Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion (Finset.attach (y ×ˢ y)) (Finsupp.frange ∘ ly)) }\n (insert 1 ↑y) ↔\n z ∈\n Algebra.adjoin\n { x //\n x ∈ Subring.closure ↑(Finsupp.frange lx ∪ Finset.biUnion (Finset.attach (y ×ˢ y)) (Finsupp.frange ∘ ly)) }\n ↑y",
"tactic": "convert foo z"
}
] |
[
351,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Data/Sym/Sym2.lean
|
Sym2.fromRel_irreflexive
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.46398\nγ : Type ?u.46401\nr : α → α → Prop\nsym : Symmetric r\nh : Irreflexive r\n⊢ ∀ {z : Sym2 α}, z ∈ fromRel sym → ¬IsDiag z",
"state_before": "α : Type u_1\nβ : Type ?u.46398\nγ : Type ?u.46401\nr : α → α → Prop\nsym : Symmetric r\n⊢ Irreflexive r → ∀ {z : Sym2 α}, z ∈ fromRel sym → ¬IsDiag z",
"tactic": "intro h"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.46398\nγ : Type ?u.46401\nr : α → α → Prop\nsym : Symmetric r\nh : Irreflexive r\n⊢ ∀ (x y : α), Quotient.mk (Rel.setoid α) (x, y) ∈ fromRel sym → ¬IsDiag (Quotient.mk (Rel.setoid α) (x, y))",
"state_before": "α : Type u_1\nβ : Type ?u.46398\nγ : Type ?u.46401\nr : α → α → Prop\nsym : Symmetric r\nh : Irreflexive r\n⊢ ∀ {z : Sym2 α}, z ∈ fromRel sym → ¬IsDiag z",
"tactic": "apply Sym2.ind"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.46398\nγ : Type ?u.46401\nr : α → α → Prop\nsym : Symmetric r\nh : Irreflexive r\n⊢ ∀ (x y : α), Quotient.mk (Rel.setoid α) (x, y) ∈ fromRel sym → ¬IsDiag (Quotient.mk (Rel.setoid α) (x, y))",
"tactic": "aesop"
}
] |
[
518,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/Analysis/InnerProductSpace/Orientation.lean
|
OrthonormalBasis.det_adjustToOrientation
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\n⊢ Basis.det (OrthonormalBasis.toBasis (adjustToOrientation e x)) = Basis.det (OrthonormalBasis.toBasis e) ∨\n Basis.det (OrthonormalBasis.toBasis (adjustToOrientation e x)) = -Basis.det (OrthonormalBasis.toBasis e)",
"tactic": "simpa using e.toBasis.det_adjustToOrientation x"
}
] |
[
141,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Data/Finset/Sym.lean
|
Finset.sym_nonempty
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\na b : α\nn : ℕ\nm : Sym α n\n⊢ ¬Finset.sym s n = ∅ ↔ n = 0 ∨ ¬s = ∅",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\na b : α\nn : ℕ\nm : Sym α n\n⊢ Finset.Nonempty (Finset.sym s n) ↔ n = 0 ∨ Finset.Nonempty s",
"tactic": "simp_rw [nonempty_iff_ne_empty, Ne.def]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\na b : α\nn : ℕ\nm : Sym α n\n⊢ ¬Finset.sym s n = ∅ ↔ n = 0 ∨ ¬s = ∅",
"tactic": "rwa [sym_eq_empty, not_and_or, not_ne_iff]"
}
] |
[
204,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.sumFinsuppLEquivProdFinsupp_symm_inl
|
[] |
[
956,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
954,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.add_conj
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.3261129\ninst✝ : IsROrC K\nz : K\n⊢ z + ↑(starRingEnd K) z = ↑(↑re z) + ↑(↑im z) * I + (↑(↑re z) - ↑(↑im z) * I)",
"tactic": "rw [re_add_im, conj_eq_re_sub_im]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.3261129\ninst✝ : IsROrC K\nz : K\n⊢ ↑(↑re z) + ↑(↑im z) * I + (↑(↑re z) - ↑(↑im z) * I) = 2 * ↑(↑re z)",
"tactic": "rw [add_add_sub_cancel, two_mul]"
}
] |
[
397,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
394,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
IsFractional.nsmul
|
[
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nx✝ : IsFractional S I\n⊢ IsFractional S 0",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nx✝ : IsFractional S I\n⊢ IsFractional S (0 • I)",
"tactic": "rw [zero_smul]"
},
{
"state_after": "case h.e'_7\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nx✝ : IsFractional S I\n⊢ 0 = ↑↑0",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nx✝ : IsFractional S I\n⊢ IsFractional S 0",
"tactic": "convert((0 : Ideal R) : FractionalIdeal S P).isFractional"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nx✝ : IsFractional S I\n⊢ 0 = ↑↑0",
"tactic": "simp"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nn : ℕ\nh : IsFractional S I\n⊢ IsFractional S (I + n • I)",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nn : ℕ\nh : IsFractional S I\n⊢ IsFractional S ((n + 1) • I)",
"tactic": "rw [succ_nsmul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nn : ℕ\nh : IsFractional S I\n⊢ IsFractional S (I + n • I)",
"tactic": "exact h.sup (IsFractional.nsmul n h)"
}
] |
[
497,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/NumberTheory/LucasLehmer.lean
|
LucasLehmer.X.ω_mul_ωb
|
[
{
"state_after": "q✝ q : ℕ+\n⊢ (2, 1) * (2, -1) = 1",
"state_before": "q✝ q : ℕ+\n⊢ ω * ωb = 1",
"tactic": "dsimp [ω, ωb]"
},
{
"state_after": "case h₁\nq✝ q : ℕ+\n⊢ 2 * 2 + -3 = 1",
"state_before": "q✝ q : ℕ+\n⊢ (2, 1) * (2, -1) = 1",
"tactic": "ext <;> simp"
},
{
"state_after": "no goals",
"state_before": "case h₁\nq✝ q : ℕ+\n⊢ 2 * 2 + -3 = 1",
"tactic": "ring"
}
] |
[
370,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
368,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.coe_int_inj
|
[
{
"state_after": "no goals",
"state_before": "d m n : ℤ\nh : ↑m = ↑n\n⊢ m = n",
"tactic": "simpa using congr_arg re h"
}
] |
[
357,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
11
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.adj_of_mem_incidenceSet
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.101184\n𝕜 : Type ?u.101187\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\nh : a ≠ b\nha : e ∈ incidenceSet G a\nhb : e ∈ incidenceSet G b\n⊢ Adj G a b",
"tactic": "rwa [← mk'_mem_incidenceSet_left_iff, ←\n Set.mem_singleton_iff.1 <| G.incidenceSet_inter_incidenceSet_subset h ⟨ha, hb⟩]"
}
] |
[
865,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
862,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.neg_eq_zero_iff
|
[] |
[
788,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
787,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
zpow_induction_right
|
[
{
"state_after": "case hz\nα : Type ?u.153793\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\ng : G\nP : G → Prop\nh_one : P 1\nh_mul : ∀ (a : G), P a → P (a * g)\nh_inv : ∀ (a : G), P a → P (a * g⁻¹)\n⊢ P (g ^ 0)\n\ncase hp\nα : Type ?u.153793\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\ng : G\nP : G → Prop\nh_one : P 1\nh_mul : ∀ (a : G), P a → P (a * g)\nh_inv : ∀ (a : G), P a → P (a * g⁻¹)\nn : ℕ\nih : P (g ^ ↑n)\n⊢ P (g ^ (↑n + 1))\n\ncase hn\nα : Type ?u.153793\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\ng : G\nP : G → Prop\nh_one : P 1\nh_mul : ∀ (a : G), P a → P (a * g)\nh_inv : ∀ (a : G), P a → P (a * g⁻¹)\nn : ℕ\nih : P (g ^ (-↑n))\n⊢ P (g ^ (-↑n - 1))",
"state_before": "α : Type ?u.153793\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\ng : G\nP : G → Prop\nh_one : P 1\nh_mul : ∀ (a : G), P a → P (a * g)\nh_inv : ∀ (a : G), P a → P (a * g⁻¹)\nn : ℤ\n⊢ P (g ^ n)",
"tactic": "induction' n using Int.induction_on with n ih n ih"
},
{
"state_after": "no goals",
"state_before": "case hz\nα : Type ?u.153793\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\ng : G\nP : G → Prop\nh_one : P 1\nh_mul : ∀ (a : G), P a → P (a * g)\nh_inv : ∀ (a : G), P a → P (a * g⁻¹)\n⊢ P (g ^ 0)",
"tactic": "rwa [zpow_zero]"
},
{
"state_after": "case hp\nα : Type ?u.153793\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\ng : G\nP : G → Prop\nh_one : P 1\nh_mul : ∀ (a : G), P a → P (a * g)\nh_inv : ∀ (a : G), P a → P (a * g⁻¹)\nn : ℕ\nih : P (g ^ ↑n)\n⊢ P (g ^ ↑n * g)",
"state_before": "case hp\nα : Type ?u.153793\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\ng : G\nP : G → Prop\nh_one : P 1\nh_mul : ∀ (a : G), P a → P (a * g)\nh_inv : ∀ (a : G), P a → P (a * g⁻¹)\nn : ℕ\nih : P (g ^ ↑n)\n⊢ P (g ^ (↑n + 1))",
"tactic": "rw [zpow_add_one]"
},
{
"state_after": "no goals",
"state_before": "case hp\nα : Type ?u.153793\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\ng : G\nP : G → Prop\nh_one : P 1\nh_mul : ∀ (a : G), P a → P (a * g)\nh_inv : ∀ (a : G), P a → P (a * g⁻¹)\nn : ℕ\nih : P (g ^ ↑n)\n⊢ P (g ^ ↑n * g)",
"tactic": "exact h_mul _ ih"
},
{
"state_after": "case hn\nα : Type ?u.153793\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\ng : G\nP : G → Prop\nh_one : P 1\nh_mul : ∀ (a : G), P a → P (a * g)\nh_inv : ∀ (a : G), P a → P (a * g⁻¹)\nn : ℕ\nih : P (g ^ (-↑n))\n⊢ P (g ^ (-↑n) * g⁻¹)",
"state_before": "case hn\nα : Type ?u.153793\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\ng : G\nP : G → Prop\nh_one : P 1\nh_mul : ∀ (a : G), P a → P (a * g)\nh_inv : ∀ (a : G), P a → P (a * g⁻¹)\nn : ℕ\nih : P (g ^ (-↑n))\n⊢ P (g ^ (-↑n - 1))",
"tactic": "rw [zpow_sub_one]"
},
{
"state_after": "no goals",
"state_before": "case hn\nα : Type ?u.153793\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : Group G\ng : G\nP : G → Prop\nh_one : P 1\nh_mul : ∀ (a : G), P a → P (a * g)\nh_inv : ∀ (a : G), P a → P (a * g⁻¹)\nn : ℕ\nih : P (g ^ (-↑n))\n⊢ P (g ^ (-↑n) * g⁻¹)",
"tactic": "exact h_inv _ ih"
}
] |
[
302,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Topology/Inseparable.lean
|
SeparationQuotient.isClosedMap_mk
|
[
{
"state_after": "X : Type u_1\nY : Type ?u.58987\nZ : Type ?u.58990\nα : Type ?u.58993\nι : Type ?u.58996\nπ : ι → Type ?u.59001\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\nf : X → Y\nt : Set (SeparationQuotient X)\n⊢ IsClosed univ",
"state_before": "X : Type u_1\nY : Type ?u.58987\nZ : Type ?u.58990\nα : Type ?u.58993\nι : Type ?u.58996\nπ : ι → Type ?u.59001\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\nf : X → Y\nt : Set (SeparationQuotient X)\n⊢ IsClosed (Set.range mk)",
"tactic": "rw [range_mk]"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.58987\nZ : Type ?u.58990\nα : Type ?u.58993\nι : Type ?u.58996\nπ : ι → Type ?u.59001\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\nf : X → Y\nt : Set (SeparationQuotient X)\n⊢ IsClosed univ",
"tactic": "exact isClosed_univ"
}
] |
[
473,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/Algebra/Algebra/Operations.lean
|
Submodule.map_unop_one
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ map (↑(LinearEquiv.symm (opLinearEquiv R))) 1 = 1",
"tactic": "rw [← comap_equiv_eq_map_symm, comap_op_one]"
}
] |
[
150,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Data/List/FinRange.lean
|
List.ofFn_eq_map
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nα : Type u_1\nn : ℕ\nf : Fin n → α\n⊢ ofFn f = map f (finRange n)",
"tactic": "rw [← ofFn_id, map_ofFn, Function.right_id]"
}
] |
[
54,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.mem_prod
|
[] |
[
1717,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1716,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Ring.lean
|
Commute.tsum_left
|
[] |
[
72,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.AEStronglyMeasurable.isSeparable_ae_range
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.348372\nι : Type ?u.348375\ninst✝² : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf g : α → β\nhf : AEStronglyMeasurable f μ\n⊢ ∀ᵐ (x : α) ∂μ, f x ∈ range (AEStronglyMeasurable.mk f hf)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.348372\nι : Type ?u.348375\ninst✝² : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf g : α → β\nhf : AEStronglyMeasurable f μ\n⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂μ, f x ∈ t",
"tactic": "refine' ⟨range (hf.mk f), hf.stronglyMeasurable_mk.isSeparable_range, _⟩"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.348372\nι : Type ?u.348375\ninst✝² : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf g : α → β\nhf : AEStronglyMeasurable f μ\nx : α\nhx : f x = AEStronglyMeasurable.mk f hf x\n⊢ f x ∈ range (AEStronglyMeasurable.mk f hf)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.348372\nι : Type ?u.348375\ninst✝² : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf g : α → β\nhf : AEStronglyMeasurable f μ\n⊢ ∀ᵐ (x : α) ∂μ, f x ∈ range (AEStronglyMeasurable.mk f hf)",
"tactic": "filter_upwards [hf.ae_eq_mk] with x hx"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.348372\nι : Type ?u.348375\ninst✝² : Countable ι\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf g : α → β\nhf : AEStronglyMeasurable f μ\nx : α\nhx : f x = AEStronglyMeasurable.mk f hf x\n⊢ f x ∈ range (AEStronglyMeasurable.mk f hf)",
"tactic": "simp [hx]"
}
] |
[
1569,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1565,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.zpow_of_gcd_eq_one
|
[
{
"state_after": "case pos\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\nh0 : 0 ≤ i\n⊢ IsPrimitiveRoot (ζ ^ i) k\n\ncase neg\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\nh0 : ¬0 ≤ i\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"state_before": "M : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"tactic": "by_cases h0 : 0 ≤ i"
},
{
"state_after": "case neg\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\nh0 : ¬0 ≤ i\nthis : 0 ≤ -i\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"state_before": "case neg\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\nh0 : ¬0 ≤ i\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"tactic": "have : 0 ≤ -i := by simp only [not_le, neg_nonneg] at h0 ⊢; exact le_of_lt h0"
},
{
"state_after": "case neg.intro\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\nh0 : ¬0 ≤ i\ni' : ℕ\nhi' : ↑i' = -i\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"state_before": "case neg\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\nh0 : ¬0 ≤ i\nthis : 0 ≤ -i\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"tactic": "lift -i to ℕ using this with i' hi'"
},
{
"state_after": "case neg.intro\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\nh0 : ¬0 ≤ i\ni' : ℕ\nhi' : ↑i' = -i\n⊢ IsPrimitiveRoot (ζ ^ i') k",
"state_before": "case neg.intro\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\nh0 : ¬0 ≤ i\ni' : ℕ\nhi' : ↑i' = -i\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"tactic": "rw [← inv_iff, ← zpow_neg, ← hi', zpow_ofNat]"
},
{
"state_after": "case neg.intro.hi\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\nh0 : ¬0 ≤ i\ni' : ℕ\nhi' : ↑i' = -i\n⊢ Nat.coprime i' k",
"state_before": "case neg.intro\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\nh0 : ¬0 ≤ i\ni' : ℕ\nhi' : ↑i' = -i\n⊢ IsPrimitiveRoot (ζ ^ i') k",
"tactic": "apply h.pow_of_coprime"
},
{
"state_after": "case neg.intro.hi\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nh0 : ¬0 ≤ i\ni' : ℕ\nhi : Nat.gcd (Int.natAbs ↑i') (Int.natAbs ↑k) = 1\nhi' : ↑i' = -i\n⊢ Nat.coprime i' k",
"state_before": "case neg.intro.hi\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\nh0 : ¬0 ≤ i\ni' : ℕ\nhi' : ↑i' = -i\n⊢ Nat.coprime i' k",
"tactic": "rw [Int.gcd, ← Int.natAbs_neg, ← hi'] at hi"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.hi\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nh0 : ¬0 ≤ i\ni' : ℕ\nhi : Nat.gcd (Int.natAbs ↑i') (Int.natAbs ↑k) = 1\nhi' : ↑i' = -i\n⊢ Nat.coprime i' k",
"tactic": "exact hi"
},
{
"state_after": "case pos.intro\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℕ\nhi : Int.gcd ↑i ↑k = 1\n⊢ IsPrimitiveRoot (ζ ^ ↑i) k",
"state_before": "case pos\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\nh0 : 0 ≤ i\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"tactic": "lift i to ℕ using h0"
},
{
"state_after": "case pos.intro\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℕ\nhi : Int.gcd ↑i ↑k = 1\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"state_before": "case pos.intro\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℕ\nhi : Int.gcd ↑i ↑k = 1\n⊢ IsPrimitiveRoot (ζ ^ ↑i) k",
"tactic": "rw [zpow_ofNat]"
},
{
"state_after": "no goals",
"state_before": "case pos.intro\nM : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℕ\nhi : Int.gcd ↑i ↑k = 1\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"tactic": "exact h.pow_of_coprime i hi"
},
{
"state_after": "M : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\nh0 : i < 0\n⊢ i ≤ 0",
"state_before": "M : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\nh0 : ¬0 ≤ i\n⊢ 0 ≤ -i",
"tactic": "simp only [not_le, neg_nonneg] at h0 ⊢"
},
{
"state_after": "no goals",
"state_before": "M : Type ?u.2707630\nN : Type ?u.2707633\nG : Type u_1\nR : Type ?u.2707639\nS : Type ?u.2707642\nF : Type ?u.2707645\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : Int.gcd i ↑k = 1\nh0 : i < 0\n⊢ i ≤ 0",
"tactic": "exact le_of_lt h0"
}
] |
[
595,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
584,
1
] |
Mathlib/MeasureTheory/Function/LpOrder.lean
|
MeasureTheory.Lp.coeFn_inf
|
[] |
[
101,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.integral_eq_zero_of_ae
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.968060\n𝕜 : Type ?u.968063\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : SMulCommClass ℝ 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nf : α → E\nhf : f =ᵐ[μ] 0\n⊢ (∫ (a : α), f a ∂μ) = 0",
"tactic": "simp [integral_congr_ae hf, integral_zero]"
}
] |
[
953,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
952,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
HomologicalComplex.congr_hom
|
[] |
[
283,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
WithSeminorms.withSeminorms_eq
|
[] |
[
279,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/Analysis/Normed/Group/Seminorm.lean
|
GroupSeminorm.add_comp
|
[] |
[
373,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
1
] |
Mathlib/MeasureTheory/Function/UniformIntegrable.lean
|
MeasureTheory.uniformIntegrable_of'
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\n⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\n⊢ UniformIntegrable f p μ",
"tactic": "refine' ⟨fun i => (hf i).aestronglyMeasurable,\n unifIntegrable_of μ hp hp' (fun i => (hf i).aestronglyMeasurable) h, _⟩"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\n⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\n⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C",
"tactic": "obtain ⟨C, hC⟩ := h 1 one_pos"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\n⊢ snorm (f i) p μ ≤ ↑(ENNReal.toNNReal (↑C * ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹ + 1))",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\n⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C",
"tactic": "refine' ⟨((C : ℝ≥0∞) * μ Set.univ ^ p.toReal⁻¹ + 1).toNNReal, fun i => _⟩"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nx : α\n⊢ ‖f i x‖ ≤ ‖(Set.indicator {x | ‖f i x‖₊ < C} (f i) + Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) x‖",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\n⊢ snorm (f i) p μ ≤\n snorm (Set.indicator {x | ‖f i x‖₊ < C} (f i)) p μ + snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ",
"tactic": "refine' le_trans (snorm_mono fun x => _) (snorm_add_le\n (StronglyMeasurable.aestronglyMeasurable\n ((hf i).indicator ((hf i).nnnorm.measurableSet_lt stronglyMeasurable_const)))\n (StronglyMeasurable.aestronglyMeasurable\n ((hf i).indicator (stronglyMeasurable_const.measurableSet_le (hf i).nnnorm))) hp)"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nx : α\n⊢ ‖f i x‖ ≤ ‖(if x ∈ {x | ‖f i x‖₊ < C} then f i x else 0) + Set.indicator {x | C ≤ ‖f i x‖₊} (f i) x‖",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nx : α\n⊢ ‖f i x‖ ≤ ‖(Set.indicator {x | ‖f i x‖₊ < C} (f i) + Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) x‖",
"tactic": "rw [Pi.add_apply, Set.indicator_apply]"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nx : α\nhx : x ∈ {x | ‖f i x‖₊ < C}\n⊢ ‖f i x‖ ≤ ‖f i x + Set.indicator {x | C ≤ ‖f i x‖₊} (f i) x‖\n\ncase inr\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nx : α\nhx : ¬x ∈ {x | ‖f i x‖₊ < C}\n⊢ ‖f i x‖ ≤ ‖0 + Set.indicator {x | C ≤ ‖f i x‖₊} (f i) x‖",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nx : α\n⊢ ‖f i x‖ ≤ ‖(if x ∈ {x | ‖f i x‖₊ < C} then f i x else 0) + Set.indicator {x | C ≤ ‖f i x‖₊} (f i) x‖",
"tactic": "split_ifs with hx"
},
{
"state_after": "case inl.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nx : α\nhx : x ∈ {x | ‖f i x‖₊ < C}\n⊢ ¬x ∈ {x | C ≤ ‖f i x‖₊}",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nx : α\nhx : x ∈ {x | ‖f i x‖₊ < C}\n⊢ ‖f i x‖ ≤ ‖f i x + Set.indicator {x | C ≤ ‖f i x‖₊} (f i) x‖",
"tactic": "rw [Set.indicator_of_not_mem, add_zero]"
},
{
"state_after": "no goals",
"state_before": "case inl.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nx : α\nhx : x ∈ {x | ‖f i x‖₊ < C}\n⊢ ¬x ∈ {x | C ≤ ‖f i x‖₊}",
"tactic": "simpa using hx"
},
{
"state_after": "case inr.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nx : α\nhx : ¬x ∈ {x | ‖f i x‖₊ < C}\n⊢ x ∈ {x | C ≤ ‖f i x‖₊}",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nx : α\nhx : ¬x ∈ {x | ‖f i x‖₊ < C}\n⊢ ‖f i x‖ ≤ ‖0 + Set.indicator {x | C ≤ ‖f i x‖₊} (f i) x‖",
"tactic": "rw [Set.indicator_of_mem, zero_add]"
},
{
"state_after": "no goals",
"state_before": "case inr.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nx : α\nhx : ¬x ∈ {x | ‖f i x‖₊ < C}\n⊢ x ∈ {x | C ≤ ‖f i x‖₊}",
"tactic": "simpa using hx"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nthis : ∀ᵐ (x : α) ∂μ, ‖Set.indicator {x | ‖f i x‖₊ < C} (f i) x‖₊ ≤ C\n⊢ snorm (Set.indicator {x | ‖f i x‖₊ < C} (f i)) p μ + snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤\n ↑C * ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹ + 1",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\n⊢ snorm (Set.indicator {x | ‖f i x‖₊ < C} (f i)) p μ + snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤\n ↑C * ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹ + 1",
"tactic": "have : ∀ᵐ x ∂μ, ‖{ x : α | ‖f i x‖₊ < C }.indicator (f i) x‖₊ ≤ C := by\n refine' eventually_of_forall _\n simp_rw [nnnorm_indicator_eq_indicator_nnnorm]\n exact Set.indicator_le fun x (hx : _ < _) => hx.le"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nthis : ∀ᵐ (x : α) ∂μ, ‖Set.indicator {x | ‖f i x‖₊ < C} (f i) x‖₊ ≤ C\n⊢ ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹ * ENNReal.ofReal ((fun a => ↑a) C) ≤ ↑C * ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nthis : ∀ᵐ (x : α) ∂μ, ‖Set.indicator {x | ‖f i x‖₊ < C} (f i) x‖₊ ≤ C\n⊢ snorm (Set.indicator {x | ‖f i x‖₊ < C} (f i)) p μ + snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤\n ↑C * ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹ + 1",
"tactic": "refine' add_le_add (le_trans (snorm_le_of_ae_bound this) _) (ENNReal.ofReal_one ▸ hC i)"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nthis : ∀ᵐ (x : α) ∂μ, ‖Set.indicator {x | ‖f i x‖₊ < C} (f i) x‖₊ ≤ C\n⊢ ↑C * ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹ ≤ ↑C * ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nthis : ∀ᵐ (x : α) ∂μ, ‖Set.indicator {x | ‖f i x‖₊ < C} (f i) x‖₊ ≤ C\n⊢ ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹ * ENNReal.ofReal ((fun a => ↑a) C) ≤ ↑C * ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹",
"tactic": "simp_rw [NNReal.val_eq_coe, ENNReal.ofReal_coe_nnreal, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\nthis : ∀ᵐ (x : α) ∂μ, ‖Set.indicator {x | ‖f i x‖₊ < C} (f i) x‖₊ ≤ C\n⊢ ↑C * ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹ ≤ ↑C * ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹",
"tactic": "exact le_rfl"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\n⊢ ∀ (x : α), ‖Set.indicator {x | ‖f i x‖₊ < C} (f i) x‖₊ ≤ C",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\n⊢ ∀ᵐ (x : α) ∂μ, ‖Set.indicator {x | ‖f i x‖₊ < C} (f i) x‖₊ ≤ C",
"tactic": "refine' eventually_of_forall _"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\n⊢ ∀ (x : α), Set.indicator {x | ‖f i x‖₊ < C} (fun a => ‖f i a‖₊) x ≤ C",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\n⊢ ∀ (x : α), ‖Set.indicator {x | ‖f i x‖₊ < C} (f i) x‖₊ ≤ C",
"tactic": "simp_rw [nnnorm_indicator_eq_indicator_nnnorm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\n⊢ ∀ (x : α), Set.indicator {x | ‖f i x‖₊ < C} (fun a => ‖f i a‖₊) x ≤ C",
"tactic": "exact Set.indicator_le fun x (hx : _ < _) => hx.le"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\n⊢ ↑C * ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹ + 1 ≠ ⊤",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\n⊢ ↑C * ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹ + 1 = ↑(ENNReal.toNNReal (↑C * ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹ + 1))",
"tactic": "rw [ENNReal.coe_toNNReal]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε\nC : ℝ≥0\nhC : ∀ (i : ι), snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1\ni : ι\n⊢ ↑C * ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹ + 1 ≠ ⊤",
"tactic": "exact ENNReal.add_ne_top.2\n ⟨ENNReal.mul_ne_top ENNReal.coe_ne_top (ENNReal.rpow_ne_top_of_nonneg\n (inv_nonneg.2 ENNReal.toReal_nonneg) (measure_lt_top _ _).ne),\n ENNReal.one_ne_top⟩"
}
] |
[
834,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
797,
1
] |
src/lean/Init/Data/AC.lean
|
Lean.Data.AC.Context.mergeIdem_head
|
[
{
"state_after": "no goals",
"state_before": "x : Nat\nxs : List Nat\n⊢ mergeIdem (x :: x :: xs) = mergeIdem (x :: xs)",
"tactic": "simp [mergeIdem, mergeIdem.loop]"
}
] |
[
125,
35
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
124,
1
] |
Mathlib/GroupTheory/Congruence.lean
|
Con.conGen_of_con
|
[
{
"state_after": "M : Type u_1\nN : Type ?u.27586\nP : Type ?u.27589\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nc✝ c : Con M\n⊢ sInf {s | ∀ (x y : M), ↑c x y → ↑s x y} ≤ c",
"state_before": "M : Type u_1\nN : Type ?u.27586\nP : Type ?u.27589\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nc✝ c : Con M\n⊢ conGen ↑c ≤ c",
"tactic": "rw [conGen_eq]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type ?u.27586\nP : Type ?u.27589\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nc✝ c : Con M\n⊢ sInf {s | ∀ (x y : M), ↑c x y → ↑s x y} ≤ c",
"tactic": "exact sInf_le fun _ _ => id"
}
] |
[
547,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
546,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Types.lean
|
CategoryTheory.Limits.Types.binaryProductCone_fst
|
[] |
[
125,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Init/CcLemmas.lean
|
not_eq_of_eq_true
|
[] |
[
73,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Combinatorics/Quiver/Cast.lean
|
Quiver.Hom.cast_rfl_rfl
|
[] |
[
49,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.IsNormal.isLimit
|
[] |
[
483,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
480,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Set.toFinset_eq_empty
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.92229\nγ : Type ?u.92232\ns t : Set α\ninst✝ : Fintype ↑s\n⊢ toFinset s = ∅ ↔ s = ∅",
"tactic": "rw [← toFinset_empty, toFinset_inj]"
}
] |
[
748,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
747,
1
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean
|
CategoryTheory.Presieve.isSeparated_of_isSheaf
|
[] |
[
749,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
748,
1
] |
Mathlib/CategoryTheory/IsConnected.lean
|
CategoryTheory.zag_symmetric
|
[] |
[
257,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.isCoseparator_op_iff
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nG : C\n⊢ IsCoseparator G.op ↔ IsSeparator G",
"tactic": "rw [IsSeparator, IsCoseparator, ← isCoseparating_op_iff, Set.singleton_op]"
}
] |
[
411,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean
|
Lagrange.eq_interpolate_of_eval_eq
|
[
{
"state_after": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nf : F[X]\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card s)\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = r i\n⊢ (↑(interpolate s v) fun i => eval (v i) f) = ↑(interpolate s v) r",
"state_before": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nf : F[X]\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card s)\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = r i\n⊢ f = ↑(interpolate s v) r",
"tactic": "rw [eq_interpolate hvs degree_f_lt]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nf : F[X]\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card s)\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = r i\n⊢ (↑(interpolate s v) fun i => eval (v i) f) = ↑(interpolate s v) r",
"tactic": "exact interpolate_eq_of_values_eq_on _ _ eval_f"
}
] |
[
383,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
380,
1
] |
Mathlib/Data/Analysis/Filter.lean
|
Filter.Realizer.map_σ
|
[] |
[
227,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
Matrix.toLinearMap₂'_apply
|
[] |
[
206,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.injective
|
[] |
[
994,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
993,
11
] |
Mathlib/Order/CompactlyGenerated.lean
|
Directed.disjoint_iSup_right
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort u_2\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : Directed (fun x x_1 => x ≤ x_1) f\n⊢ Disjoint a (⨆ (i : ι), f i) ↔ ∀ (i : ι), Disjoint a (f i)",
"tactic": "simp_rw [disjoint_iff, h.inf_iSup_eq, iSup_eq_bot]"
}
] |
[
404,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
402,
11
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean
|
CategoryTheory.Presieve.isSheafFor_subsieve_aux
|
[
{
"state_after": "C : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\n⊢ IsSeparatedFor P R ∧\n ∀ (x : FamilyOfElements P R), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t",
"state_before": "C : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\n⊢ IsSheafFor P R",
"tactic": "rw [← isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]"
},
{
"state_after": "case left\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\n⊢ IsSeparatedFor P R\n\ncase right\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\n⊢ ∀ (x : FamilyOfElements P R), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t",
"state_before": "C : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\n⊢ IsSeparatedFor P R ∧\n ∀ (x : FamilyOfElements P R), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t",
"tactic": "constructor"
},
{
"state_after": "case left\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nt₁ t₂ : P.obj X.op\nht₁ : FamilyOfElements.IsAmalgamation x t₁\nht₂ : FamilyOfElements.IsAmalgamation x t₂\n⊢ t₁ = t₂",
"state_before": "case left\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\n⊢ IsSeparatedFor P R",
"tactic": "intro x t₁ t₂ ht₁ ht₂"
},
{
"state_after": "no goals",
"state_before": "case left\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nt₁ t₂ : P.obj X.op\nht₁ : FamilyOfElements.IsAmalgamation x t₁\nht₂ : FamilyOfElements.IsAmalgamation x t₂\n⊢ t₁ = t₂",
"tactic": "exact\n hS.isSeparatedFor _ _ _ (isAmalgamation_restrict h x t₁ ht₁)\n (isAmalgamation_restrict h x t₂ ht₂)"
},
{
"state_after": "case right\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\n⊢ ∃ t, FamilyOfElements.IsAmalgamation x t",
"state_before": "case right\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\n⊢ ∀ (x : FamilyOfElements P R), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t",
"tactic": "intro x hx"
},
{
"state_after": "case right\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\n⊢ FamilyOfElements.IsAmalgamation x\n (IsSheafFor.amalgamate hS (FamilyOfElements.restrict h x)\n (_ : FamilyOfElements.Compatible (FamilyOfElements.restrict h x)))",
"state_before": "case right\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\n⊢ ∃ t, FamilyOfElements.IsAmalgamation x t",
"tactic": "use hS.amalgamate _ (hx.restrict h)"
},
{
"state_after": "case right\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\nW : C\nj : W ⟶ X\nhj : R j\n⊢ P.map j.op\n (IsSheafFor.amalgamate hS (FamilyOfElements.restrict h x)\n (_ : FamilyOfElements.Compatible (FamilyOfElements.restrict h x))) =\n x j hj",
"state_before": "case right\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\n⊢ FamilyOfElements.IsAmalgamation x\n (IsSheafFor.amalgamate hS (FamilyOfElements.restrict h x)\n (_ : FamilyOfElements.Compatible (FamilyOfElements.restrict h x)))",
"tactic": "intro W j hj"
},
{
"state_after": "case right\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\nW : C\nj : W ⟶ X\nhj : R j\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ W⦄,\n (Sieve.pullback j S).arrows f →\n P.map f.op\n (P.map j.op\n (IsSheafFor.amalgamate hS (FamilyOfElements.restrict h x)\n (_ : FamilyOfElements.Compatible (FamilyOfElements.restrict h x)))) =\n P.map f.op (x j hj)",
"state_before": "case right\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\nW : C\nj : W ⟶ X\nhj : R j\n⊢ P.map j.op\n (IsSheafFor.amalgamate hS (FamilyOfElements.restrict h x)\n (_ : FamilyOfElements.Compatible (FamilyOfElements.restrict h x))) =\n x j hj",
"tactic": "apply (trans hj).ext"
},
{
"state_after": "case right\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\nW : C\nj : W ⟶ X\nhj : R j\nY : C\nf : Y ⟶ W\nhf : (Sieve.pullback j S).arrows f\n⊢ P.map f.op\n (P.map j.op\n (IsSheafFor.amalgamate hS (FamilyOfElements.restrict h x)\n (_ : FamilyOfElements.Compatible (FamilyOfElements.restrict h x)))) =\n P.map f.op (x j hj)",
"state_before": "case right\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\nW : C\nj : W ⟶ X\nhj : R j\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ W⦄,\n (Sieve.pullback j S).arrows f →\n P.map f.op\n (P.map j.op\n (IsSheafFor.amalgamate hS (FamilyOfElements.restrict h x)\n (_ : FamilyOfElements.Compatible (FamilyOfElements.restrict h x)))) =\n P.map f.op (x j hj)",
"tactic": "intro Y f hf"
},
{
"state_after": "case right\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\nW : C\nj : W ⟶ X\nhj : R j\nY : C\nf : Y ⟶ W\nhf : (Sieve.pullback j S).arrows f\n⊢ x (f ≫ j) (_ : f ≫ j ∈ R) = P.map (𝟙 Y).op (x (f ≫ j) (_ : f ≫ j ∈ R))",
"state_before": "case right\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\nW : C\nj : W ⟶ X\nhj : R j\nY : C\nf : Y ⟶ W\nhf : (Sieve.pullback j S).arrows f\n⊢ P.map f.op\n (P.map j.op\n (IsSheafFor.amalgamate hS (FamilyOfElements.restrict h x)\n (_ : FamilyOfElements.Compatible (FamilyOfElements.restrict h x)))) =\n P.map f.op (x j hj)",
"tactic": "rw [← FunctorToTypes.map_comp_apply, ← op_comp, hS.valid_glue (hx.restrict h) _ hf,\n FamilyOfElements.restrict, ← hx (𝟙 _) f (h _ hf) _ (id_comp _)]"
},
{
"state_after": "no goals",
"state_before": "case right\nC : Type u₁\ninst✝ : Category C\nP✝ Q U : Cᵒᵖ ⥤ Type w\nX Y✝ : C\nS✝ : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\nW : C\nj : W ⟶ X\nhj : R j\nY : C\nf : Y ⟶ W\nhf : (Sieve.pullback j S).arrows f\n⊢ x (f ≫ j) (_ : f ≫ j ∈ R) = P.map (𝟙 Y).op (x (f ≫ j) (_ : f ≫ j ∈ R))",
"tactic": "simp"
}
] |
[
710,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
693,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
StrictMono.mul_const_of_neg
|
[] |
[
741,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
739,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.SimpleFunc.integral_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.97756\nF : Type u_2\n𝕜 : Type ?u.97762\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.97864\nF' : Type ?u.97867\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nμ : Measure α\nf : α →ₛ F\n⊢ integral μ f = ∑ x in SimpleFunc.range f, ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • x",
"tactic": "simp [integral, setToSimpleFunc, weightedSMul_apply]"
}
] |
[
322,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/FieldTheory/IsAlgClosed/Basic.lean
|
IsAlgClosure.equivOfEquiv_algebraMap
|
[] |
[
517,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
integral_sin_pow_odd_mul_cos_pow
|
[
{
"state_after": "no goals",
"state_before": "a b : ℝ\nn✝ m n : ℕ\nhc : Continuous fun u => u ^ n * (1 - u ^ 2) ^ m\n⊢ (∫ (x : ℝ) in a..b, sin x ^ (2 * m + 1) * cos x ^ n) = -∫ (x : ℝ) in b..a, sin x ^ (2 * m + 1) * cos x ^ n",
"tactic": "rw [integral_symm]"
},
{
"state_after": "a b : ℝ\nn✝ m n : ℕ\nhc : Continuous fun u => u ^ n * (1 - u ^ 2) ^ m\n⊢ (∫ (x : ℝ) in b..a, (sin x * sin x) ^ m * sin x * cos x ^ n) =\n ∫ (x : ℝ) in b..a, (1 - cos x * cos x) ^ m * sin x * cos x ^ n",
"state_before": "a b : ℝ\nn✝ m n : ℕ\nhc : Continuous fun u => u ^ n * (1 - u ^ 2) ^ m\n⊢ (-∫ (x : ℝ) in b..a, sin x ^ (2 * m + 1) * cos x ^ n) = ∫ (x : ℝ) in b..a, (1 - cos x ^ 2) ^ m * -sin x * cos x ^ n",
"tactic": "simp only [_root_.pow_succ', pow_mul, _root_.pow_zero, one_mul, mul_neg, neg_mul,\n integral_neg, neg_inj]"
},
{
"state_after": "case h.e'_5.h.h.e'_5.h.e'_5.h.e'_5\na b : ℝ\nn✝ m n : ℕ\nhc : Continuous fun u => u ^ n * (1 - u ^ 2) ^ m\nx✝ : ℝ\n⊢ sin x✝ * sin x✝ = 1 - cos x✝ * cos x✝",
"state_before": "a b : ℝ\nn✝ m n : ℕ\nhc : Continuous fun u => u ^ n * (1 - u ^ 2) ^ m\n⊢ (∫ (x : ℝ) in b..a, (sin x * sin x) ^ m * sin x * cos x ^ n) =\n ∫ (x : ℝ) in b..a, (1 - cos x * cos x) ^ m * sin x * cos x ^ n",
"tactic": "congr! 5"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.h.e'_5.h.e'_5.h.e'_5\na b : ℝ\nn✝ m n : ℕ\nhc : Continuous fun u => u ^ n * (1 - u ^ 2) ^ m\nx✝ : ℝ\n⊢ sin x✝ * sin x✝ = 1 - cos x✝ * cos x✝",
"tactic": "rw [← sq, ← sq, sin_sq]"
},
{
"state_after": "case e_f\na b : ℝ\nn✝ m n : ℕ\nhc : Continuous fun u => u ^ n * (1 - u ^ 2) ^ m\n⊢ (fun x => (1 - cos x ^ 2) ^ m * -sin x * cos x ^ n) = fun x => cos x ^ n * (1 - cos x ^ 2) ^ m * -sin x",
"state_before": "a b : ℝ\nn✝ m n : ℕ\nhc : Continuous fun u => u ^ n * (1 - u ^ 2) ^ m\n⊢ (∫ (x : ℝ) in b..a, (1 - cos x ^ 2) ^ m * -sin x * cos x ^ n) =\n ∫ (x : ℝ) in b..a, cos x ^ n * (1 - cos x ^ 2) ^ m * -sin x",
"tactic": "congr"
},
{
"state_after": "case e_f.h\na b : ℝ\nn✝ m n : ℕ\nhc : Continuous fun u => u ^ n * (1 - u ^ 2) ^ m\nx✝ : ℝ\n⊢ (1 - cos x✝ ^ 2) ^ m * -sin x✝ * cos x✝ ^ n = cos x✝ ^ n * (1 - cos x✝ ^ 2) ^ m * -sin x✝",
"state_before": "case e_f\na b : ℝ\nn✝ m n : ℕ\nhc : Continuous fun u => u ^ n * (1 - u ^ 2) ^ m\n⊢ (fun x => (1 - cos x ^ 2) ^ m * -sin x * cos x ^ n) = fun x => cos x ^ n * (1 - cos x ^ 2) ^ m * -sin x",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case e_f.h\na b : ℝ\nn✝ m n : ℕ\nhc : Continuous fun u => u ^ n * (1 - u ^ 2) ^ m\nx✝ : ℝ\n⊢ (1 - cos x✝ ^ 2) ^ m * -sin x✝ * cos x✝ ^ n = cos x✝ ^ n * (1 - cos x✝ ^ 2) ^ m * -sin x✝",
"tactic": "ring"
}
] |
[
802,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
787,
1
] |
Mathlib/Topology/Instances/Matrix.lean
|
Matrix.blockDiagonal'_tsum
|
[
{
"state_after": "case pos\nX : Type u_5\nα : Type ?u.110701\nl : Type u_1\nm : Type ?u.110707\nn : Type ?u.110710\np : Type ?u.110713\nS : Type ?u.110716\nR : Type u_2\nm' : l → Type u_3\nn' : l → Type u_4\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq l\ninst✝ : T2Space R\nf : X → (i : l) → Matrix (m' i) (n' i) R\nhf : Summable f\n⊢ blockDiagonal' (∑' (x : X), f x) = ∑' (x : X), blockDiagonal' (f x)\n\ncase neg\nX : Type u_5\nα : Type ?u.110701\nl : Type u_1\nm : Type ?u.110707\nn : Type ?u.110710\np : Type ?u.110713\nS : Type ?u.110716\nR : Type u_2\nm' : l → Type u_3\nn' : l → Type u_4\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq l\ninst✝ : T2Space R\nf : X → (i : l) → Matrix (m' i) (n' i) R\nhf : ¬Summable f\n⊢ blockDiagonal' (∑' (x : X), f x) = ∑' (x : X), blockDiagonal' (f x)",
"state_before": "X : Type u_5\nα : Type ?u.110701\nl : Type u_1\nm : Type ?u.110707\nn : Type ?u.110710\np : Type ?u.110713\nS : Type ?u.110716\nR : Type u_2\nm' : l → Type u_3\nn' : l → Type u_4\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq l\ninst✝ : T2Space R\nf : X → (i : l) → Matrix (m' i) (n' i) R\n⊢ blockDiagonal' (∑' (x : X), f x) = ∑' (x : X), blockDiagonal' (f x)",
"tactic": "by_cases hf : Summable f"
},
{
"state_after": "no goals",
"state_before": "case pos\nX : Type u_5\nα : Type ?u.110701\nl : Type u_1\nm : Type ?u.110707\nn : Type ?u.110710\np : Type ?u.110713\nS : Type ?u.110716\nR : Type u_2\nm' : l → Type u_3\nn' : l → Type u_4\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq l\ninst✝ : T2Space R\nf : X → (i : l) → Matrix (m' i) (n' i) R\nhf : Summable f\n⊢ blockDiagonal' (∑' (x : X), f x) = ∑' (x : X), blockDiagonal' (f x)",
"tactic": "exact hf.hasSum.matrix_blockDiagonal'.tsum_eq.symm"
},
{
"state_after": "case neg\nX : Type u_5\nα : Type ?u.110701\nl : Type u_1\nm : Type ?u.110707\nn : Type ?u.110710\np : Type ?u.110713\nS : Type ?u.110716\nR : Type u_2\nm' : l → Type u_3\nn' : l → Type u_4\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq l\ninst✝ : T2Space R\nf : X → (i : l) → Matrix (m' i) (n' i) R\nhf : ¬Summable f\nhft : ¬Summable fun x => blockDiagonal' (f x)\n⊢ blockDiagonal' (∑' (x : X), f x) = ∑' (x : X), blockDiagonal' (f x)",
"state_before": "case neg\nX : Type u_5\nα : Type ?u.110701\nl : Type u_1\nm : Type ?u.110707\nn : Type ?u.110710\np : Type ?u.110713\nS : Type ?u.110716\nR : Type u_2\nm' : l → Type u_3\nn' : l → Type u_4\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq l\ninst✝ : T2Space R\nf : X → (i : l) → Matrix (m' i) (n' i) R\nhf : ¬Summable f\n⊢ blockDiagonal' (∑' (x : X), f x) = ∑' (x : X), blockDiagonal' (f x)",
"tactic": "have hft := summable_matrix_blockDiagonal'.not.mpr hf"
},
{
"state_after": "case neg\nX : Type u_5\nα : Type ?u.110701\nl : Type u_1\nm : Type ?u.110707\nn : Type ?u.110710\np : Type ?u.110713\nS : Type ?u.110716\nR : Type u_2\nm' : l → Type u_3\nn' : l → Type u_4\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq l\ninst✝ : T2Space R\nf : X → (i : l) → Matrix (m' i) (n' i) R\nhf : ¬Summable f\nhft : ¬Summable fun x => blockDiagonal' (f x)\n⊢ blockDiagonal' 0 = 0",
"state_before": "case neg\nX : Type u_5\nα : Type ?u.110701\nl : Type u_1\nm : Type ?u.110707\nn : Type ?u.110710\np : Type ?u.110713\nS : Type ?u.110716\nR : Type u_2\nm' : l → Type u_3\nn' : l → Type u_4\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq l\ninst✝ : T2Space R\nf : X → (i : l) → Matrix (m' i) (n' i) R\nhf : ¬Summable f\nhft : ¬Summable fun x => blockDiagonal' (f x)\n⊢ blockDiagonal' (∑' (x : X), f x) = ∑' (x : X), blockDiagonal' (f x)",
"tactic": "rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft]"
},
{
"state_after": "no goals",
"state_before": "case neg\nX : Type u_5\nα : Type ?u.110701\nl : Type u_1\nm : Type ?u.110707\nn : Type ?u.110710\np : Type ?u.110713\nS : Type ?u.110716\nR : Type u_2\nm' : l → Type u_3\nn' : l → Type u_4\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq l\ninst✝ : T2Space R\nf : X → (i : l) → Matrix (m' i) (n' i) R\nhf : ¬Summable f\nhft : ¬Summable fun x => blockDiagonal' (f x)\n⊢ blockDiagonal' 0 = 0",
"tactic": "exact blockDiagonal'_zero"
}
] |
[
443,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
436,
1
] |
Mathlib/Data/Nat/Fib.lean
|
Nat.fib_add_two_strictMono
|
[
{
"state_after": "n : ℕ\n⊢ fib (n + 2) < fib (n + 1 + 2)",
"state_before": "⊢ StrictMono fun n => fib (n + 2)",
"tactic": "refine' strictMono_nat_of_lt_succ fun n => _"
},
{
"state_after": "n : ℕ\n⊢ fib (n + 2) < fib (n + 2 + 1)",
"state_before": "n : ℕ\n⊢ fib (n + 2) < fib (n + 1 + 2)",
"tactic": "rw [add_right_comm]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ fib (n + 2) < fib (n + 2 + 1)",
"tactic": "exact fib_lt_fib_succ (self_le_add_left _ _)"
}
] |
[
125,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.foldr_eq_foldrM
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β → β\nb : β\nl : List α\n⊢ foldr f b l = foldrM f b l",
"tactic": "induction l <;> simp [*]"
}
] |
[
183,
27
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
181,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
Differentiable.cosh
|
[] |
[
1087,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1086,
1
] |
Mathlib/Data/List/Rotate.lean
|
List.length_cyclicPermutations_cons
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nl✝ l' : List α\nx : α\nl : List α\n⊢ length (cyclicPermutations (x :: l)) = length l + 1",
"tactic": "simp [cyclicPermutations_cons]"
}
] |
[
591,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
590,
1
] |
Mathlib/Analysis/Calculus/DiffContOnCl.lean
|
DiffContOnCl.const_smul
|
[] |
[
123,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.mem_iInf
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.692912\nR : Type u_2\nA : Type u_3\nB : Type ?u.692921\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\nι : Sort u_1\nS : ι → StarSubalgebra R A\nx : A\n⊢ (x ∈ ⨅ (i : ι), S i) ↔ ∀ (i : ι), x ∈ S i",
"tactic": "simp only [iInf, mem_sInf, Set.forall_range_iff]"
}
] |
[
690,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
689,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.div_inter_subset
|
[] |
[
680,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
679,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Order.lean
|
HasSum.nonneg
|
[] |
[
129,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
QuadraticForm.polar_neg_right
|
[
{
"state_after": "no goals",
"state_before": "S : Type ?u.169206\nR : Type u_1\nR₁ : Type ?u.169212\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : CommRing R₁\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx y : M\n⊢ polar (↑Q) x (-y) = -polar (↑Q) x y",
"tactic": "rw [← neg_one_smul R y, polar_smul_right, neg_one_mul]"
}
] |
[
303,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/LinearAlgebra/Matrix/IsDiag.lean
|
Matrix.isDiag_fromBlocks_iff
|
[
{
"state_after": "case mp\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\n⊢ IsDiag (fromBlocks A B C D) → IsDiag A ∧ B = 0 ∧ C = 0 ∧ IsDiag D\n\ncase mpr\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\n⊢ IsDiag A ∧ B = 0 ∧ C = 0 ∧ IsDiag D → IsDiag (fromBlocks A B C D)",
"state_before": "α : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\n⊢ IsDiag (fromBlocks A B C D) ↔ IsDiag A ∧ B = 0 ∧ C = 0 ∧ IsDiag D",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nh : IsDiag (fromBlocks A B C D)\n⊢ IsDiag A ∧ B = 0 ∧ C = 0 ∧ IsDiag D",
"state_before": "case mp\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\n⊢ IsDiag (fromBlocks A B C D) → IsDiag A ∧ B = 0 ∧ C = 0 ∧ IsDiag D",
"tactic": "intro h"
},
{
"state_after": "case mp.refine'_1\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nh : IsDiag (fromBlocks A B C D)\ni j : m\nhij : i ≠ j\n⊢ A i j = 0\n\ncase mp.refine'_2\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nh : IsDiag (fromBlocks A B C D)\ni : m\nj : n\n⊢ B i j = OfNat.ofNat 0 i j\n\ncase mp.refine'_3\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nh : IsDiag (fromBlocks A B C D)\ni : n\nj : m\n⊢ C i j = OfNat.ofNat 0 i j\n\ncase mp.refine'_4\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nh : IsDiag (fromBlocks A B C D)\ni j : n\nhij : i ≠ j\n⊢ D i j = 0",
"state_before": "case mp\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nh : IsDiag (fromBlocks A B C D)\n⊢ IsDiag A ∧ B = 0 ∧ C = 0 ∧ IsDiag D",
"tactic": "refine' ⟨fun i j hij => _, ext fun i j => _, ext fun i j => _, fun i j hij => _⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_1\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nh : IsDiag (fromBlocks A B C D)\ni j : m\nhij : i ≠ j\n⊢ A i j = 0",
"tactic": "exact h (Sum.inl_injective.ne hij)"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_2\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nh : IsDiag (fromBlocks A B C D)\ni : m\nj : n\n⊢ B i j = OfNat.ofNat 0 i j",
"tactic": "exact h Sum.inl_ne_inr"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_3\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nh : IsDiag (fromBlocks A B C D)\ni : n\nj : m\n⊢ C i j = OfNat.ofNat 0 i j",
"tactic": "exact h Sum.inr_ne_inl"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_4\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nh : IsDiag (fromBlocks A B C D)\ni j : n\nhij : i ≠ j\n⊢ D i j = 0",
"tactic": "exact h (Sum.inr_injective.ne hij)"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nha : IsDiag A\nhb : B = 0\nhc : C = 0\nhd : IsDiag D\n⊢ IsDiag (fromBlocks A B C D)",
"state_before": "case mpr\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\n⊢ IsDiag A ∧ B = 0 ∧ C = 0 ∧ IsDiag D → IsDiag (fromBlocks A B C D)",
"tactic": "rintro ⟨ha, hb, hc, hd⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.22462\nR : Type ?u.22465\nn : Type u_3\nm : Type u_2\ninst✝ : Zero α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\nha : IsDiag A\nhb : B = 0\nhc : C = 0\nhd : IsDiag D\n⊢ IsDiag (fromBlocks A B C D)",
"tactic": "convert IsDiag.fromBlocks ha hd"
}
] |
[
182,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.direction_inf
|
[
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns1 s2 : AffineSubspace k P\n⊢ Submodule.span k (↑(s1 ⊓ s2) -ᵥ ↑(s1 ⊓ s2)) ≤ Submodule.span k (↑s1 -ᵥ ↑s1) ⊓ Submodule.span k (↑s2 -ᵥ ↑s2)",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns1 s2 : AffineSubspace k P\n⊢ direction (s1 ⊓ s2) ≤ direction s1 ⊓ direction s2",
"tactic": "simp only [direction_eq_vectorSpan, vectorSpan_def]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns1 s2 : AffineSubspace k P\n⊢ Submodule.span k (↑(s1 ⊓ s2) -ᵥ ↑(s1 ⊓ s2)) ≤ Submodule.span k (↑s1 -ᵥ ↑s1) ⊓ Submodule.span k (↑s2 -ᵥ ↑s2)",
"tactic": "exact\n le_inf (sInf_le_sInf fun p hp => trans (vsub_self_mono (inter_subset_left _ _)) hp)\n (sInf_le_sInf fun p hp => trans (vsub_self_mono (inter_subset_right _ _)) hp)"
}
] |
[
904,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
899,
1
] |
Mathlib/RingTheory/LaurentSeries.lean
|
LaurentSeries.single_order_mul_powerSeriesPart
|
[
{
"state_after": "case coeff.h\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\n⊢ HahnSeries.coeff (↑(single (order x)) 1 * ↑(ofPowerSeries ℤ R) (powerSeriesPart x)) n = HahnSeries.coeff x n",
"state_before": "R : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\n⊢ ↑(single (order x)) 1 * ↑(ofPowerSeries ℤ R) (powerSeriesPart x) = x",
"tactic": "ext n"
},
{
"state_after": "case coeff.h\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\n⊢ HahnSeries.coeff (↑(ofPowerSeries ℤ R) (powerSeriesPart x)) (n - order x) = HahnSeries.coeff x n",
"state_before": "case coeff.h\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\n⊢ HahnSeries.coeff (↑(single (order x)) 1 * ↑(ofPowerSeries ℤ R) (powerSeriesPart x)) n = HahnSeries.coeff x n",
"tactic": "rw [← sub_add_cancel n x.order, single_mul_coeff_add, sub_add_cancel, one_mul]"
},
{
"state_after": "case pos\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nh : order x ≤ n\n⊢ HahnSeries.coeff (↑(ofPowerSeries ℤ R) (powerSeriesPart x)) (n - order x) = HahnSeries.coeff x n\n\ncase neg\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nh : ¬order x ≤ n\n⊢ HahnSeries.coeff (↑(ofPowerSeries ℤ R) (powerSeriesPart x)) (n - order x) = HahnSeries.coeff x n",
"state_before": "case coeff.h\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\n⊢ HahnSeries.coeff (↑(ofPowerSeries ℤ R) (powerSeriesPart x)) (n - order x) = HahnSeries.coeff x n",
"tactic": "by_cases h : x.order ≤ n"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nh : order x ≤ n\n⊢ HahnSeries.coeff (↑(ofPowerSeries ℤ R) (powerSeriesPart x)) (n - order x) = HahnSeries.coeff x n",
"tactic": "rw [Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h), coeff_coe_powerSeries,\n powerSeriesPart_coeff, ← Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h),\n add_sub_cancel'_right]"
},
{
"state_after": "case neg\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nh : ¬order x ≤ n\n⊢ 0 = HahnSeries.coeff x n\n\ncase neg\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nh : ¬order x ≤ n\n⊢ ¬n - order x ∈\n Set.range\n ↑{ toEmbedding := { toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n ∀ {a b : ℕ},\n ↑{ toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) } a ≤\n ↑{ toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) } b ↔\n a ≤ b) }",
"state_before": "case neg\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nh : ¬order x ≤ n\n⊢ HahnSeries.coeff (↑(ofPowerSeries ℤ R) (powerSeriesPart x)) (n - order x) = HahnSeries.coeff x n",
"tactic": "rw [ofPowerSeries_apply, embDomain_notin_range]"
},
{
"state_after": "case neg\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nh : 0 ≠ HahnSeries.coeff x n\n⊢ order x ≤ n",
"state_before": "case neg\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nh : ¬order x ≤ n\n⊢ 0 = HahnSeries.coeff x n",
"tactic": "contrapose! h"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nh : 0 ≠ HahnSeries.coeff x n\n⊢ order x ≤ n",
"tactic": "exact order_le_of_coeff_ne_zero h.symm"
},
{
"state_after": "case neg\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nh :\n n - order x ∈\n Set.range\n ↑{ toEmbedding := { toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n ∀ {a b : ℕ},\n ↑{ toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) } a ≤\n ↑{ toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) } b ↔\n a ≤ b) }\n⊢ order x ≤ n",
"state_before": "case neg\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nh : ¬order x ≤ n\n⊢ ¬n - order x ∈\n Set.range\n ↑{ toEmbedding := { toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n ∀ {a b : ℕ},\n ↑{ toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) } a ≤\n ↑{ toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) } b ↔\n a ≤ b) }",
"tactic": "contrapose! h"
},
{
"state_after": "case neg\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nh : ∃ y, ↑y = n - order x\n⊢ order x ≤ n",
"state_before": "case neg\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nh :\n n - order x ∈\n Set.range\n ↑{ toEmbedding := { toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n ∀ {a b : ℕ},\n ↑{ toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) } a ≤\n ↑{ toFun := Nat.cast, inj' := (_ : Function.Injective Nat.cast) } b ↔\n a ≤ b) }\n⊢ order x ≤ n",
"tactic": "simp only [Set.mem_range, RelEmbedding.coe_mk, Function.Embedding.coeFn_mk] at h"
},
{
"state_after": "case neg.intro\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nm : ℕ\nhm : ↑m = n - order x\n⊢ order x ≤ n",
"state_before": "case neg\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nh : ∃ y, ↑y = n - order x\n⊢ order x ≤ n",
"tactic": "obtain ⟨m, hm⟩ := h"
},
{
"state_after": "case neg.intro\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nm : ℕ\nhm : ↑m = n - order x\n⊢ 0 ≤ ↑m",
"state_before": "case neg.intro\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nm : ℕ\nhm : ↑m = n - order x\n⊢ order x ≤ n",
"tactic": "rw [← sub_nonneg, ← hm]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro\nR : Type u\ninst✝ : Semiring R\nx : LaurentSeries R\nn : ℤ\nm : ℕ\nhm : ↑m = n - order x\n⊢ 0 ≤ ↑m",
"tactic": "simp only [Nat.cast_nonneg]"
}
] |
[
110,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/MeasureTheory/Group/Arithmetic.lean
|
List.measurable_prod
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\nα : Type u_2\ninst✝² : Monoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nμ : MeasureTheory.Measure α\nl : List (α → M)\nhl : ∀ (f : α → M), f ∈ l → Measurable f\n⊢ Measurable fun x => prod (map (fun f => f x) l)",
"tactic": "simpa only [← Pi.list_prod_apply] using l.measurable_prod' hl"
}
] |
[
883,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
881,
1
] |
Mathlib/Data/SetLike/Basic.lean
|
SetLike.coe_strictMono
|
[] |
[
207,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Logic/Encodable/Basic.lean
|
ULower.down_eq_down
|
[] |
[
540,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
539,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.mem_prod
|
[] |
[
655,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
653,
1
] |
Mathlib/Combinatorics/SetFamily/Intersecting.lean
|
Set.Intersecting.card_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : BooleanAlgebra α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\n⊢ 2 * card s ≤ Fintype.card α",
"tactic": "classical\n refine' (s.disjUnion _ hs.disjoint_map_compl).card_le_univ.trans_eq' _\n rw [two_mul, card_disjUnion, card_map]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : BooleanAlgebra α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\n⊢ 2 * card s =\n card\n (disjUnion s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)\n (_ : Disjoint s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)))",
"state_before": "α : Type u_1\ninst✝¹ : BooleanAlgebra α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\n⊢ 2 * card s ≤ Fintype.card α",
"tactic": "refine' (s.disjUnion _ hs.disjoint_map_compl).card_le_univ.trans_eq' _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : BooleanAlgebra α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\n⊢ 2 * card s =\n card\n (disjUnion s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)\n (_ : Disjoint s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)))",
"tactic": "rw [two_mul, card_disjUnion, card_map]"
}
] |
[
170,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/MeasureTheory/Measure/NullMeasurable.lean
|
MeasureTheory.measure_union_add_inter₀'
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.18521\nα : Type u_1\nβ : Type ?u.18527\nγ : Type ?u.18530\nm0 : MeasurableSpace α\nμ : Measure α\ns t✝ : Set α\nhs : NullMeasurableSet s\nt : Set α\n⊢ ↑↑μ (s ∪ t) + ↑↑μ (s ∩ t) = ↑↑μ s + ↑↑μ t",
"tactic": "rw [union_comm, inter_comm, measure_union_add_inter₀ t hs, add_comm]"
}
] |
[
332,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/Control/Basic.lean
|
seq_map_assoc
|
[
{
"state_after": "α β γ : Type u\nF : Type u → Type v\ninst✝¹ : Applicative F\ninst✝ : LawfulApplicative F\nx : F (α → β)\nf : γ → α\ny : F γ\n⊢ (Seq.seq x fun x => Seq.seq (pure f) fun x => y) = Seq.seq (Seq.seq (pure fun x => x ∘ f) fun x_1 => x) fun x => y",
"state_before": "α β γ : Type u\nF : Type u → Type v\ninst✝¹ : Applicative F\ninst✝ : LawfulApplicative F\nx : F (α → β)\nf : γ → α\ny : F γ\n⊢ (Seq.seq x fun x => f <$> y) = Seq.seq ((fun x => x ∘ f) <$> x) fun x => y",
"tactic": "simp [← pure_seq]"
},
{
"state_after": "no goals",
"state_before": "α β γ : Type u\nF : Type u → Type v\ninst✝¹ : Applicative F\ninst✝ : LawfulApplicative F\nx : F (α → β)\nf : γ → α\ny : F γ\n⊢ (Seq.seq ((fun x x_1 => x (f x_1)) <$> x) fun x => y) =\n Seq.seq (Seq.seq (pure fun x x_1 => x (f x_1)) fun x_1 => x) fun x => y",
"tactic": "simp [pure_seq]"
}
] |
[
71,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Data/List/Dedup.lean
|
List.mem_dedup
|
[
{
"state_after": "case refine_1\nα : Type u\ninst✝ : DecidableEq α\na : α\nl : List α\n⊢ ∀ {x y z : α}, (fun x x_1 => x ≠ x_1) x z → (fun x x_1 => x ≠ x_1) x y ∨ (fun x x_1 => x ≠ x_1) y z",
"state_before": "case refine_2\nα : Type u\ninst✝ : DecidableEq α\na : α\nl : List α\nthis : (¬∀ (b : α), b ∈ pwFilter (fun x x_1 => x ≠ x_1) l → a ≠ b) ↔ ¬∀ (b : α), b ∈ l → a ≠ b\n⊢ a ∈ dedup l ↔ a ∈ l\n\ncase refine_1\nα : Type u\ninst✝ : DecidableEq α\na : α\nl : List α\n⊢ ∀ {x y z : α}, (fun x x_1 => x ≠ x_1) x z → (fun x x_1 => x ≠ x_1) x y ∨ (fun x x_1 => x ≠ x_1) y z",
"tactic": "simpa only [dedup, forall_mem_ne, not_not] using this"
},
{
"state_after": "case refine_1\nα : Type u\ninst✝ : DecidableEq α\na : α\nl : List α\nx y z : α\nxz : x ≠ z\n⊢ (fun x x_1 => x ≠ x_1) x y ∨ (fun x x_1 => x ≠ x_1) y z",
"state_before": "case refine_1\nα : Type u\ninst✝ : DecidableEq α\na : α\nl : List α\n⊢ ∀ {x y z : α}, (fun x x_1 => x ≠ x_1) x z → (fun x x_1 => x ≠ x_1) x y ∨ (fun x x_1 => x ≠ x_1) y z",
"tactic": "intros x y z xz"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nα : Type u\ninst✝ : DecidableEq α\na : α\nl : List α\nx y z : α\nxz : x ≠ z\n⊢ (fun x x_1 => x ≠ x_1) x y ∨ (fun x x_1 => x ≠ x_1) y z",
"tactic": "exact not_and_or.1 <| mt (fun h ↦ h.1.trans h.2) xz"
}
] |
[
51,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
FractionalIdeal.mul_right_strictMono
|
[] |
[
567,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
565,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.not_mem_support
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nx : α\n⊢ ¬x ∈ support f ↔ ↑f x = x",
"tactic": "simp"
}
] |
[
297,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/CategoryTheory/Limits/IsLimit.lean
|
CategoryTheory.Limits.IsLimit.map_π
|
[] |
[
92,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.Sigma.univ
|
[] |
[
1290,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1288,
1
] |
Mathlib/Data/FunLike/Equiv.lean
|
EquivLike.surjective
|
[] |
[
170,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
11
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.span_singleton_mul_eq_span_singleton_mul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type ?u.246088\ninst✝ : CommSemiring R\nI✝ J✝ K L : Ideal R\nx y : R\nI J : Ideal R\n⊢ span {x} * I = span {y} * J ↔\n (∀ (zI : R), zI ∈ I → ∃ zJ, zJ ∈ J ∧ x * zI = y * zJ) ∧ ∀ (zJ : R), zJ ∈ J → ∃ zI, zI ∈ I ∧ x * zI = y * zJ",
"tactic": "simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm]"
}
] |
[
607,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
604,
1
] |
Mathlib/Data/List/Count.lean
|
List.countp_congr
|
[] |
[
162,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/Data/Int/Basic.lean
|
Int.coe_nat_ediv
|
[] |
[
234,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Order/Circular.lean
|
btw_of_sbtw
|
[] |
[
191,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
MultilinearMap.uncurrySum_aux_apply
|
[] |
[
1485,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1482,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.integral_union_ae
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3445\nE : Type u_2\nF : Type ?u.3451\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nhst : AEDisjoint μ s t\nht : NullMeasurableSet t\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\n⊢ (∫ (x : α) in s ∪ t, f x ∂μ) = (∫ (x : α) in s, f x ∂μ) + ∫ (x : α) in t, f x ∂μ",
"tactic": "simp only [IntegrableOn, Measure.restrict_union₀ hst ht, integral_add_measure hfs hft]"
}
] |
[
102,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/Order/BoundedOrder.lean
|
BoundedOrder.ext
|
[
{
"state_after": "α✝ : Type u\nβ : Type v\nγ : Type ?u.28802\nδ : Type ?u.28805\nα : Type u_1\ninst✝ : PartialOrder α\nA B : BoundedOrder α\nht : toOrderTop = toOrderTop\n⊢ A = B",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type ?u.28802\nδ : Type ?u.28805\nα : Type u_1\ninst✝ : PartialOrder α\nA B : BoundedOrder α\n⊢ A = B",
"tactic": "have ht : @BoundedOrder.toOrderTop α _ A = @BoundedOrder.toOrderTop α _ B := OrderTop.ext"
},
{
"state_after": "α✝ : Type u\nβ : Type v\nγ : Type ?u.28802\nδ : Type ?u.28805\nα : Type u_1\ninst✝ : PartialOrder α\nA B : BoundedOrder α\nht : toOrderTop = toOrderTop\nhb : toOrderBot = toOrderBot\n⊢ A = B",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type ?u.28802\nδ : Type ?u.28805\nα : Type u_1\ninst✝ : PartialOrder α\nA B : BoundedOrder α\nht : toOrderTop = toOrderTop\n⊢ A = B",
"tactic": "have hb : @BoundedOrder.toOrderBot α _ A = @BoundedOrder.toOrderBot α _ B := OrderBot.ext"
},
{
"state_after": "case mk\nα✝ : Type u\nβ : Type v\nγ : Type ?u.28802\nδ : Type ?u.28805\nα : Type u_1\ninst✝ : PartialOrder α\nB : BoundedOrder α\ntoOrderTop✝ : OrderTop α\ntoOrderBot✝ : OrderBot α\nht : toOrderTop = toOrderTop\nhb : toOrderBot = toOrderBot\n⊢ mk = B",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type ?u.28802\nδ : Type ?u.28805\nα : Type u_1\ninst✝ : PartialOrder α\nA B : BoundedOrder α\nht : toOrderTop = toOrderTop\nhb : toOrderBot = toOrderBot\n⊢ A = B",
"tactic": "cases A"
},
{
"state_after": "case mk.mk\nα✝ : Type u\nβ : Type v\nγ : Type ?u.28802\nδ : Type ?u.28805\nα : Type u_1\ninst✝ : PartialOrder α\ntoOrderTop✝¹ : OrderTop α\ntoOrderBot✝¹ : OrderBot α\ntoOrderTop✝ : OrderTop α\ntoOrderBot✝ : OrderBot α\nht : toOrderTop = toOrderTop\nhb : toOrderBot = toOrderBot\n⊢ mk = mk",
"state_before": "case mk\nα✝ : Type u\nβ : Type v\nγ : Type ?u.28802\nδ : Type ?u.28805\nα : Type u_1\ninst✝ : PartialOrder α\nB : BoundedOrder α\ntoOrderTop✝ : OrderTop α\ntoOrderBot✝ : OrderBot α\nht : toOrderTop = toOrderTop\nhb : toOrderBot = toOrderBot\n⊢ mk = B",
"tactic": "cases B"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nα✝ : Type u\nβ : Type v\nγ : Type ?u.28802\nδ : Type ?u.28805\nα : Type u_1\ninst✝ : PartialOrder α\ntoOrderTop✝¹ : OrderTop α\ntoOrderBot✝¹ : OrderBot α\ntoOrderTop✝ : OrderTop α\ntoOrderBot✝ : OrderBot α\nht : toOrderTop = toOrderTop\nhb : toOrderBot = toOrderBot\n⊢ mk = mk",
"tactic": "congr"
}
] |
[
540,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/CategoryTheory/Closed/Monoidal.lean
|
CategoryTheory.MonoidalClosed.eq_curry_iff
|
[] |
[
216,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
RingHom.toMonoidHom_eq_coe
|
[] |
[
474,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
473,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.extend_function
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ns : Set α\nf : ↑s ↪ β\nh : Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))\n⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x",
"state_before": "α : Type u_1\nβ : Type u_2\ns : Set α\nf : ↑s ↪ β\nh : Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))\n⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x",
"tactic": "intros"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ns : Set α\nf : ↑s ↪ β\nh this : Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))\n⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x",
"state_before": "α : Type u_1\nβ : Type u_2\ns : Set α\nf : ↑s ↪ β\nh : Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))\n⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x",
"tactic": "have := h"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ns : Set α\nf : ↑s ↪ β\nh : Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))\ng : ↑(sᶜ) ≃ ↑(range ↑fᶜ)\n⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x",
"state_before": "α : Type u_1\nβ : Type u_2\ns : Set α\nf : ↑s ↪ β\nh this : Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))\n⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x",
"tactic": "cases' this with g"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ns : Set α\nf : ↑s ↪ β\nh✝ : Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))\ng : ↑(sᶜ) ≃ ↑(range ↑fᶜ)\nh : α ≃ β :=\n (Set.sumCompl s).symm.trans ((sumCongr (ofInjective ↑f (_ : Injective f.toFun)) g).trans (Set.sumCompl (range ↑f)))\n⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ns : Set α\nf : ↑s ↪ β\nh : Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))\ng : ↑(sᶜ) ≃ ↑(range ↑fᶜ)\n⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x",
"tactic": "let h : α ≃ β :=\n (Set.sumCompl (s : Set α)).symm.trans\n ((sumCongr (Equiv.ofInjective f f.2) g).trans (Set.sumCompl (range f)))"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ns : Set α\nf : ↑s ↪ β\nh✝ : Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))\ng : ↑(sᶜ) ≃ ↑(range ↑fᶜ)\nh : α ≃ β :=\n (Set.sumCompl s).symm.trans ((sumCongr (ofInjective ↑f (_ : Injective f.toFun)) g).trans (Set.sumCompl (range ↑f)))\n⊢ ∀ (x : ↑s), ↑h ↑x = ↑f x",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ns : Set α\nf : ↑s ↪ β\nh✝ : Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))\ng : ↑(sᶜ) ≃ ↑(range ↑fᶜ)\nh : α ≃ β :=\n (Set.sumCompl s).symm.trans ((sumCongr (ofInjective ↑f (_ : Injective f.toFun)) g).trans (Set.sumCompl (range ↑f)))\n⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x",
"tactic": "refine' ⟨h, _⟩"
},
{
"state_after": "case intro.mk\nα : Type u_1\nβ : Type u_2\ns : Set α\nf : ↑s ↪ β\nh✝ : Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))\ng : ↑(sᶜ) ≃ ↑(range ↑fᶜ)\nh : α ≃ β :=\n (Set.sumCompl s).symm.trans ((sumCongr (ofInjective ↑f (_ : Injective f.toFun)) g).trans (Set.sumCompl (range ↑f)))\nx : α\nhx : x ∈ s\n⊢ ↑h ↑{ val := x, property := hx } = ↑f { val := x, property := hx }",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ns : Set α\nf : ↑s ↪ β\nh✝ : Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))\ng : ↑(sᶜ) ≃ ↑(range ↑fᶜ)\nh : α ≃ β :=\n (Set.sumCompl s).symm.trans ((sumCongr (ofInjective ↑f (_ : Injective f.toFun)) g).trans (Set.sumCompl (range ↑f)))\n⊢ ∀ (x : ↑s), ↑h ↑x = ↑f x",
"tactic": "rintro ⟨x, hx⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.mk\nα : Type u_1\nβ : Type u_2\ns : Set α\nf : ↑s ↪ β\nh✝ : Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))\ng : ↑(sᶜ) ≃ ↑(range ↑fᶜ)\nh : α ≃ β :=\n (Set.sumCompl s).symm.trans ((sumCongr (ofInjective ↑f (_ : Injective f.toFun)) g).trans (Set.sumCompl (range ↑f)))\nx : α\nhx : x ∈ s\n⊢ ↑h ↑{ val := x, property := hx } = ↑f { val := x, property := hx }",
"tactic": "simp [Set.sumCompl_symm_apply_of_mem, hx]"
}
] |
[
1202,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1196,
1
] |
Mathlib/Analysis/Calculus/Deriv/Inverse.lean
|
LocalHomeomorph.hasDerivAt_symm
|
[] |
[
106,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
|
MeasureTheory.Measure.add_haar_sphere
|
[
{
"state_after": "case inl\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E\ninst✝⁵ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝⁴ : IsAddHaarMeasure μ\nF : Type ?u.2019568\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\ns : Set E\ninst✝ : Nontrivial E\nx : E\n⊢ ↑↑μ (sphere x 0) = 0\n\ncase inr\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E\ninst✝⁵ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝⁴ : IsAddHaarMeasure μ\nF : Type ?u.2019568\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\ns : Set E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\nh : r ≠ 0\n⊢ ↑↑μ (sphere x r) = 0",
"state_before": "E : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E\ninst✝⁵ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝⁴ : IsAddHaarMeasure μ\nF : Type ?u.2019568\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\ns : Set E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\n⊢ ↑↑μ (sphere x r) = 0",
"tactic": "rcases eq_or_ne r 0 with (rfl | h)"
},
{
"state_after": "no goals",
"state_before": "case inl\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E\ninst✝⁵ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝⁴ : IsAddHaarMeasure μ\nF : Type ?u.2019568\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\ns : Set E\ninst✝ : Nontrivial E\nx : E\n⊢ ↑↑μ (sphere x 0) = 0",
"tactic": "rw [sphere_zero, measure_singleton]"
},
{
"state_after": "no goals",
"state_before": "case inr\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E\ninst✝⁵ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝⁴ : IsAddHaarMeasure μ\nF : Type ?u.2019568\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\ns : Set E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\nh : r ≠ 0\n⊢ ↑↑μ (sphere x r) = 0",
"tactic": "exact add_haar_sphere_of_ne_zero μ x h"
}
] |
[
503,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
500,
1
] |
Mathlib/Analysis/Convex/Integral.lean
|
ConvexOn.map_set_average_le
|
[] |
[
192,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Algebra/FreeAlgebra.lean
|
FreeAlgebra.ι_injective
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nX : Type u_2\nA : Type ?u.688668\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx y : X\nhoxy : ι R x = ι R y\n⊢ ¬x = y → False",
"tactic": "classical exact fun hxy : x ≠ y ↦\n let f : FreeAlgebra R X →ₐ[R] R := lift R fun z ↦ if x = z then (1 : R) else 0\n have hfx1 : f (ι R x) = 1 := (lift_ι_apply _ _).trans <| if_pos rfl\n have hfy1 : f (ι R y) = 1 := hoxy ▸ hfx1\n have hfy0 : f (ι R y) = 0 := (lift_ι_apply _ _).trans <| if_neg hxy\n one_ne_zero <| hfy1.symm.trans hfy0"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nX : Type u_2\nA : Type ?u.688668\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx y : X\nhoxy : ι R x = ι R y\n⊢ ¬x = y → False",
"tactic": "exact fun hxy : x ≠ y ↦\nlet f : FreeAlgebra R X →ₐ[R] R := lift R fun z ↦ if x = z then (1 : R) else 0\nhave hfx1 : f (ι R x) = 1 := (lift_ι_apply _ _).trans <| if_pos rfl\nhave hfy1 : f (ι R y) = 1 := hoxy ▸ hfx1\nhave hfy0 : f (ι R y) = 0 := (lift_ι_apply _ _).trans <| if_neg hxy\none_ne_zero <| hfy1.symm.trans hfy0"
}
] |
[
445,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/Data/Set/Sigma.lean
|
Set.image_sigmaMk_preimage_sigmaMap_subset
|
[] |
[
42,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/GroupTheory/GroupAction/Quotient.lean
|
MulAction.card_orbit_mul_card_stabilizer_eq_card_group
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁵ inst✝⁴ : Group α\ninst✝³ : MulAction α β\nx b : β\ninst✝² : Fintype α\ninst✝¹ : Fintype ↑(orbit α b)\ninst✝ : Fintype { x // x ∈ stabilizer α b }\n⊢ Fintype.card ↑(orbit α b) * Fintype.card { x // x ∈ stabilizer α b } = Fintype.card α",
"tactic": "rw [← Fintype.card_prod, Fintype.card_congr (orbitProdStabilizerEquivGroup α b)]"
}
] |
[
217,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean
|
Ultrafilter.comap_comap
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.19128\nf✝ g : Ultrafilter α\ns t : Set α\np q : α → Prop\nf : Ultrafilter γ\nm : α → β\nn : β → γ\ninj₀ : Injective n\nlarge₀ : range n ∈ f\ninj₁ : Injective m\nlarge₁ : range m ∈ comap f inj₀ large₀\ninj₂ : optParam (Injective (n ∘ m)) (_ : Injective (n ∘ m))\n⊢ n '' range m ∈ f",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.19128\nf✝ g : Ultrafilter α\ns t : Set α\np q : α → Prop\nf : Ultrafilter γ\nm : α → β\nn : β → γ\ninj₀ : Injective n\nlarge₀ : range n ∈ f\ninj₁ : Injective m\nlarge₁ : range m ∈ comap f inj₀ large₀\ninj₂ : optParam (Injective (n ∘ m)) (_ : Injective (n ∘ m))\n⊢ range (n ∘ m) ∈ f",
"tactic": "rw [range_comp]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.19128\nf✝ g : Ultrafilter α\ns t : Set α\np q : α → Prop\nf : Ultrafilter γ\nm : α → β\nn : β → γ\ninj₀ : Injective n\nlarge₀ : range n ∈ f\ninj₁ : Injective m\nlarge₁ : range m ∈ comap f inj₀ large₀\ninj₂ : optParam (Injective (n ∘ m)) (_ : Injective (n ∘ m))\n⊢ n '' range m ∈ f",
"tactic": "exact image_mem_of_mem_comap large₀ large₁"
}
] |
[
279,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
8
] |
Mathlib/NumberTheory/RamificationInertia.lean
|
Ideal.Factors.ne_bot
|
[] |
[
700,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
699,
1
] |
Mathlib/Topology/MetricSpace/Infsep.lean
|
Set.Nontrivial.infsep_of_fintype
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.68548\nβ : Type ?u.68551\ninst✝² : PseudoMetricSpace α\nx y z : α\ns t : Set α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype ↑s\nhs : Set.Nontrivial s\n⊢ Finset.Nonempty (toFinset (offDiag s))",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.68551\ninst✝² : PseudoMetricSpace α\nx y z : α\ns t : Set α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype ↑s\nhs : Set.Nontrivial s\n⊢ infsep s = Finset.inf' (toFinset (offDiag s)) (_ : Finset.Nonempty (toFinset (offDiag s))) (uncurry dist)",
"tactic": "classical rw [Set.infsep_of_fintype, dif_pos hs]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.68551\ninst✝² : PseudoMetricSpace α\nx y z : α\ns t : Set α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype ↑s\nhs : Set.Nontrivial s\n⊢ infsep s = Finset.inf' (toFinset (offDiag s)) (_ : Finset.Nonempty (toFinset (offDiag s))) (uncurry dist)",
"tactic": "rw [Set.infsep_of_fintype, dif_pos hs]"
}
] |
[
481,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
479,
1
] |
Mathlib/Init/Data/Nat/Basic.lean
|
Nat.bit1_ne_zero
|
[] |
[
39,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
11
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Memℒp.indicator
|
[] |
[
633,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
632,
1
] |
Mathlib/Data/Fin/Tuple/Reflection.lean
|
FinVec.sum_eq
|
[
{
"state_after": "no goals",
"state_before": "m n✝ : ℕ\nα : Type u_1\nβ : Type ?u.11237\nγ : Type ?u.11240\ninst✝ : AddCommMonoid α\nn : ℕ\na : Fin (n + 2) → α\n⊢ sum a = ∑ i : Fin (n + 2), a i",
"tactic": "rw [Fin.sum_univ_castSucc, sum, sum_eq]"
}
] |
[
163,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
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