module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 178,
"column": 77
} | {
"line": 178,
"column": 79
} | {
"line": 179,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nx y : M\n⊢ dp R n (x + y) = ∑ k ∈ antidiagonal n, dp R k.1 x * dp R k.2 y",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingCon.toCon",
"Nat.ins... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 192,
"column": 60
} | {
"line": 192,
"column": 62
} | {
"line": 193,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Type u_4\ninst✝ : DecidableEq ι\ns : Finset ι\nq : ℕ\na : ι → R\nx : ι → M\n⊢ dp R q (∑ i ∈ s, a i • x i) =\n ∑ k ∈ s.sym q, (∏ i ∈ s, a i ^ Multiset.count i ↑k) • ∏ i ∈ s, dp R (Multiset.count i ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 66,
"column": 52
} | {
"line": 66,
"column": 54
} | {
"line": 67,
"column": 2
} | [
{
"pp": "R : Type u_7\ninst✝² : CommSemiring R\nr : R\nrs : List R\nM : Type u_8\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\n⊢ ofList (r :: rs) • N = r • N ⊔ ofList rs • N",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule.pointwiseDistri... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 82,
"column": 25
} | {
"line": 82,
"column": 27
} | {
"line": 82,
"column": 28
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nM₄ : Type u_6\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\nr : R\nrs : List R\n⊢ map (r • ⊤).mkQ (Ideal.ofList rs • ⊤) = Ideal.ofList rs • ⊤",
"ppTerm": "?m.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 90,
"column": 25
} | {
"line": 90,
"column": 27
} | {
"line": 90,
"column": 28
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nM₄ : Type u_6\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\nr : R\nrs : List R\n⊢ map (Ideal.ofList rs • ⊤).mkQ (r • ⊤) = r • ⊤",
"ppTerm": "?m.209",
"ass... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 197,
"column": 74
} | {
"line": 197,
"column": 76
} | {
"line": 198,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Type u_4\ns : Finset ι\nn : ι → ℕ\nm : M\n⊢ ∏ i ∈ s, dp R (n i) m = ↑(multinomial s n) * dp R (s.sum n) m",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZe... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Depth.Rees | {
"line": 135,
"column": 79
} | {
"line": 135,
"column": 81
} | {
"line": 135,
"column": 82
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\nNfin : Module.Finite R ↑N\nNsupp : Module.support R ↑N ⊆ PrimeSpectrum.zeroLocus ↑I\nn : ℕ\nih :\n ∀ (M : ModuleCat R) [Module.Finite R ↑M],\n I • ⊤ < ⊤ →\n ∀ (rs : List R),\n ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 103,
"column": 67
} | {
"line": 103,
"column": 69
} | {
"line": 104,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nrs : List R\n⊢ ⊤ = Ideal.ofList (r :: rs) • ⊤ ↔ ⊤ = Ideal.ofList rs • ⊤",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule.pointwiseDistribMulAction",... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Depth.Rees | {
"line": 132,
"column": 32
} | {
"line": 132,
"column": 34
} | {
"line": 133,
"column": 10
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\nNfin : Module.Finite R ↑N\nNsupp : Module.support R ↑N ⊆ PrimeSpectrum.zeroLocus ↑I\nn : ℕ\nih :\n ∀ (M : ModuleCat R) [Module.Finite R ↑M],\n I • ⊤ < ⊤ →\n ∀ (rs : List R),\n ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 211,
"column": 39
} | {
"line": 211,
"column": 41
} | {
"line": 212,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nx : M\n⊢ ↑n ! * dp R n x = dp R 1 x ^ n",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"NonAssocSemiring.toAddCommMono... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 222,
"column": 18
} | {
"line": 222,
"column": 20
} | {
"line": 222,
"column": 21
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx✝¹ x✝ : M\n⊢ dp R 1 (x✝¹ + x✝) = dp R 1 x✝¹ + dp R 1 x✝",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Nat.instCanonicallyOrderedAdd",
"False",
"Na... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 223,
"column": 19
} | {
"line": 223,
"column": 21
} | {
"line": 223,
"column": 22
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx✝¹ : R\nx✝ : M\n⊢ dp R 1 (x✝¹ • x✝) = (RingHom.id R) x✝¹ • dp R 1 x✝",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlge... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 230,
"column": 22
} | {
"line": 230,
"column": 24
} | {
"line": 230,
"column": 25
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf g : DividedPowerAlgebra R M →ₐ[R] A\nh : f = g\nx✝¹ : ℕ\nx✝ : M\n⊢ f (dp R x✝¹ x✝) = g (dp R x✝¹ x✝)",
"ppTerm": "?m.41",
"assigned":... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Depth.Rees | {
"line": 139,
"column": 34
} | {
"line": 139,
"column": 36
} | {
"line": 140,
"column": 10
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\nNfin : Module.Finite R ↑N\nNsupp : Module.support R ↑N ⊆ PrimeSpectrum.zeroLocus ↑I\nn : ℕ\nih :\n ∀ (M : ModuleCat R) [Module.Finite R ↑M],\n I • ⊤ < ⊤ →\n ∀ (rs : List R),\n ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 229,
"column": 50
} | {
"line": 229,
"column": 52
} | {
"line": 230,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf g : DividedPowerAlgebra R M →ₐ[R] A\n⊢ f = g ↔ ∀ (n : ℕ) (m : M), f (dp R n m) = g (dp R n m)",
"ppTerm": "?m.35",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 260,
"column": 40
} | {
"line": 260,
"column": 42
} | {
"line": 260,
"column": 43
} | [
{
"pp": "R : Type u_4\nM : Type u_5\nι : Type u_6\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nr : R\n⊢ (fun n ↦ n.prod fun i k ↦ dp R k (v i)) 0 = 1",
"ppTerm": "?m.91",
"assigned": true,
"usedConstants": [
"MulOne.toOn... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 263,
"column": 42
} | {
"line": 263,
"column": 44
} | {
"line": 263,
"column": 45
} | [
{
"pp": "R : Type u_4\nM : Type u_5\nι : Type u_6\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nx : DividedPowerAlgebra R M\nk : ℕ\nm : M\nhx : x ∈ Submodule.span R (Set.range fun n ↦ n.prod fun i k ↦ dp R k (v i))\n⊢ m ∈ Submodule.span R ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 304,
"column": 18
} | {
"line": 304,
"column": 20
} | {
"line": 305,
"column": 8
} | [
{
"pp": "A : Type u_1\ninst✝² : CommSemiring A\nM : Type u_2\ninst✝¹ : AddCommMonoid M\nI : AddSubmonoid M\ndpow : ℕ → M → A\ndpow_zero : ∀ {x : M}, x ∈ I → dpow 0 x = 1\ndpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0\nι : Type u_3\ninst✝ : DecidableEq ι\nx : ι → M\ndpow_add : ∀ {n : ℕ} {x y : M}, x ∈ I → y ∈... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Depth.Rees | {
"line": 106,
"column": 49
} | {
"line": 106,
"column": 51
} | {
"line": 107,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\nNfin : Module.Finite R ↑N\nNsupp : Module.support R ↑N ⊆ PrimeSpectrum.zeroLocus ↑I\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nsmul_lt : I • ⊤ < ⊤\nrs : List R\nmem : ∀ r ∈ rs, r ∈ I\... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Depth.Rees | {
"line": 168,
"column": 36
} | {
"line": 168,
"column": 38
} | {
"line": 169,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\nn : ℕ\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nsmul_lt : I • ⊤ < ⊤\n⊢ Nontrivial (R ⧸ I)",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Iff.mpr",
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Depth.Rees | {
"line": 172,
"column": 84
} | {
"line": 172,
"column": 86
} | {
"line": 173,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\nn : ℕ\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nsmul_lt : I • ⊤ < ⊤\nntrQ : Nontrivial (R ⧸ I)\n⊢ Module.support R (Shrink.{v, u} (R ⧸ I)) = PrimeSpectrum.zeroLocus ↑I",
"ppTerm": "?m.119",
"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 273,
"column": 8
} | {
"line": 273,
"column": 17
} | {
"line": 274,
"column": 8
} | [
{
"pp": "case dp.mem.mem\nR : Type u_4\nM : Type u_5\nι : Type u_6\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nm : M\nx : DividedPowerAlgebra R M\nk : ℕ\ni : ι\nn : ι →₀ ℕ\n⊢ (fun n ↦ n.prod fun i k ↦ dp R k (v i)) n * dp R k (v i) ∈\n ... | [
"case dp.mem.mem\nR : Type u_4\nM : Type u_5\nι : Type u_6\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nm : M\nx : DividedPowerAlgebra R M\nk : ℕ\ni : ι\nn : ι →₀ ℕ\n⊢ (n.prod fun i k ↦ dp R k (v i)) * dp R k (v i) ∈ Submodule.span R (Set.ran... | simp only | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 277,
"column": 8
} | {
"line": 277,
"column": 17
} | {
"line": 278,
"column": 8
} | [
{
"pp": "case dp.mem.mem\nR : Type u_4\nM : Type u_5\nι : Type u_6\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nm : M\nx : DividedPowerAlgebra R M\nk : ℕ\ni : ι\nn : ι →₀ ℕ\n⊢ (fun n ↦ n.prod fun i k ↦ dp R k (v i)) (Finsupp.single i k + ... | [
"case dp.mem.mem\nR : Type u_4\nM : Type u_5\nι : Type u_6\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nm : M\nx : DividedPowerAlgebra R M\nk : ℕ\ni : ι\nn : ι →₀ ℕ\n⊢ ((Finsupp.single i k + n).prod fun i k ↦ dp R k (v i)) =\n dp R (k + n ... | simp only | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Depth.Rees | {
"line": 166,
"column": 16
} | {
"line": 166,
"column": 18
} | {
"line": 168,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\nn : ℕ\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nsmul_lt : I • ⊤ < ⊤\n⊢ [∀ (N : ModuleCat R),\n Nontrivial ↑N →\n Module.Finite R ↑N → Module.support R ↑N ⊆ PrimeSpectrum.zeroLocus ↑I ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 75,
"column": 96
} | {
"line": 75,
"column": 98
} | {
"line": 76,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers J\nf : A →+* B\n⊢ hI.IsDPMorphism hJ f ↔ map f I ≤ J ∧ ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)",
"ppTerm": "?m.42",
"assigned": t... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 91,
"column": 37
} | {
"line": 91,
"column": 39
} | {
"line": 92,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝² : CommSemiring A\ninst✝¹ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers J\nC : Type u_3\ninst✝ : CommSemiring C\nK : Ideal C\nhK : DividedPowers K\nf : A →+* B\ng : B →+* C\nhg : hJ.IsDPMorphism hK g\nhf : hI.IsDPMorphism hJ f\n⊢ ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 113,
"column": 28
} | {
"line": 113,
"column": 30
} | {
"line": 114,
"column": 4
} | [
{
"pp": "A✝ : Type u_1\nB✝ : Type u_2\ninst✝³ : CommSemiring A✝\ninst✝² : CommSemiring B✝\nI✝ : Ideal A✝\nJ✝ : Ideal B✝\nhI✝ : DividedPowers I✝\nhJ✝ : DividedPowers J✝\nA : Type u_3\nB : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 169,
"column": 4
} | {
"line": 175,
"column": 33
} | {
"line": 177,
"column": 0
} | [
{
"pp": "case cons\nR : Type u_1\nS : Type u_2\nM : Type u_3\nM₂ : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nf : M →+ M₂\nas : List R\nbs : List S\nhf : Surjective ⇑f\nr : R\ns : S\nl₁✝ : List R\nl₂✝ : List S\... | [] | conv => congr <;> rw [Ideal.ofList_cons, sup_smul, sup_toAddSubgroup,
ideal_span_singleton_smul, pointwise_smul_toAddSubgroup,
top_toAddSubgroup, AddSubgroup.pointwise_smul_def]
apply DFunLike.ext (f.comp (toAddMonoidEnd R M r))
((toAddMonoidEnd S M₂ s).comp f) at h
rw [AddSubgroup.map_sup, ih... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 169,
"column": 4
} | {
"line": 175,
"column": 33
} | {
"line": 177,
"column": 0
} | [
{
"pp": "case cons\nR : Type u_1\nS : Type u_2\nM : Type u_3\nM₂ : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nf : M →+ M₂\nas : List R\nbs : List S\nhf : Surjective ⇑f\nr : R\ns : S\nl₁✝ : List R\nl₂✝ : List S\... | [] | conv => congr <;> rw [Ideal.ofList_cons, sup_smul, sup_toAddSubgroup,
ideal_span_singleton_smul, pointwise_smul_toAddSubgroup,
top_toAddSubgroup, AddSubgroup.pointwise_smul_def]
apply DFunLike.ext (f.comp (toAddMonoidEnd R M r))
((toAddMonoidEnd S M₂ s).comp f) at h
rw [AddSubgroup.map_sup, ih... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 163,
"column": 62
} | {
"line": 163,
"column": 64
} | {
"line": 164,
"column": 2
} | [
{
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.DividedPowers.DPMorphism | {
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.DividedPowers.Basic | {
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.DividedPowers.Basic | {
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Mathlib.RingTheory.DividedPowers.DPMorphism | {
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Mathlib.RingTheory.Regular.RegularSequence | {
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.DividedPowers.DPMorphism | {
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.DividedPowers.Basic | {
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.DividedPowers.DPMorphism | {
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Mathlib.RingTheory.Regular.RegularSequence | {
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Mathlib.RingTheory.DividedPowers.DPMorphism | {
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Mathlib.RingTheory.DividedPowers.DPMorphism | {
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.Regular.RegularSequence | {
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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Mathlib.RingTheory.DividedPowers.DPMorphism | {
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{
"pp": "A : Type u_3\nB : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers J\nf : A →+* B\nS : Set A\nhS : I = span S\nhS' : ∀ s ∈ S, f s ∈ J\nhdp : ∀ {n : ℕ}, ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)\nh : map f I ≤ J\nn✝ : ℕ\na : A... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 432,
"column": 76
} | {
"line": 432,
"column": 78
} | {
"line": 433,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nS : Type u_4\ninst✝⁵ : CommSemiring S\nN : Type u_5\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R N\ninst✝² : Module S N\nf : M →ₗ[R] N\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R S N\nn : ℕ\na : M\n⊢ ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 199,
"column": 27
} | {
"line": 199,
"column": 29
} | {
"line": 200,
"column": 4
} | [
{
"pp": "A : Type u_3\nB : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers J\nf : A →+* B\nS : Set A\nhS : I = span S\nhS' : ∀ s ∈ S, f s ∈ J\nhdp : ∀ {n : ℕ}, ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)\nh : map f I ≤ J\n⊢ hI.IsDPMorp... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 436,
"column": 77
} | {
"line": 436,
"column": 79
} | {
"line": 437,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nS : Type u_4\ninst✝⁵ : CommSemiring S\nN : Type u_5\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R N\ninst✝² : Module S N\nf : M →ₗ[R] N\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R S N\nm : M\n⊢ (map S ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 370,
"column": 34
} | {
"line": 370,
"column": 36
} | {
"line": 370,
"column": 37
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nι : Type u_3\nr : ι → A\nn : ℕ\na : ι\ns : Finset ι\nhas : a ∉ s\nhrec : s.Nonempty → (∀ i ∈ s, r i ∈ I) → hI.dpow n (∏ i ∈ s, r i) = n ! ^ (#s - 1) • ∏ i ∈ s, hI.dpow n (r i)\nhs : (insert a s).Nonempty\nhs' : ∀ i ∈ insert a s, r... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 440,
"column": 87
} | {
"line": 440,
"column": 89
} | {
"line": 441,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nS : Type u_4\ninst✝⁵ : CommSemiring S\nN : Type u_5\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R N\ninst✝² : Module S N\nf : M →ₗ[R] N\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R S N\n⊢ (map S f).toLi... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 444,
"column": 37
} | {
"line": 444,
"column": 39
} | {
"line": 445,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_5\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\n⊢ Function.Surjective ⇑(map R f)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 198,
"column": 92
} | {
"line": 198,
"column": 94
} | {
"line": 199,
"column": 2
} | [
{
"pp": "A : Type u_3\nB : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers J\nf : A →+* B\nS : Set A\nhS : I = span S\nhS' : ∀ s ∈ S, f s ∈ J\nhdp : ∀ {n : ℕ}, ∀ a ∈ S, f (hI.dpow n a) = hJ.dpow n (f a)\n⊢ hI.IsDPMorphism hJ f",
"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 470,
"column": 51
} | {
"line": 470,
"column": 53
} | {
"line": 471,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nN : Type u_5\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\nP : Type u_6\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\n⊢ map R (g ∘ₗ f) = (map R g).comp (map R f)",
"p... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 475,
"column": 73
} | {
"line": 475,
"column": 75
} | {
"line": 476,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ map R LinearMap.id = AlgHom.id R (DividedPowerAlgebra R M)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"RingCon.instCommSemiringQuotient",
"Eq... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 488,
"column": 5
} | {
"line": 488,
"column": 7
} | {
"line": 488,
"column": 8
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nS : Type u_4\ninst✝⁷ : CommSemiring S\nN : Type u_5\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : Module R N\ninst✝⁴ : Module S N\nf : M →ₗ[R] N\ninst✝³ : Algebra R S\ninst✝² : IsScalarTower R S N\n... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 256,
"column": 59
} | {
"line": 256,
"column": 61
} | {
"line": 257,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nrs : List R\n⊢ IsRegular M (r :: rs) ↔\n IsSMulRegular M r ∧ IsRegular (QuotSMulTop r M) (List.map (⇑(Ideal.Quotient.mk (Ideal.span {r}))) rs)",
"ppTerm": "?m.35",
"assigned": true,
"used... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 488,
"column": 36
} | {
"line": 488,
"column": 38
} | {
"line": 488,
"column": 39
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nS : Type u_4\ninst✝⁷ : CommSemiring S\nN : Type u_5\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : Module R N\ninst✝⁴ : Module S N\nf : M →ₗ[R] N\ninst✝³ : Algebra R S\ninst✝² : IsScalarTower R S N\n... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 358,
"column": 90
} | {
"line": 358,
"column": 92
} | {
"line": 359,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nι : Type u_3\nr : ι → A\ns : Finset ι\nhs : s.Nonempty\nhs' : ∀ i ∈ s, r i ∈ I\nn : ℕ\n⊢ hI.dpow n (∏ i ∈ s, r i) = n ! ^ (#s - 1) • ∏ i ∈ s, hI.dpow n (r i)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 66,
"column": 96
} | {
"line": 66,
"column": 98
} | {
"line": 67,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nm : ℕ\nx : A\nhx : x ∈ I\n⊢ dpow I m x = (↑m !)⁻¹ʳ * x ^ m",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"HM... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 69,
"column": 76
} | {
"line": 69,
"column": 78
} | {
"line": 69,
"column": 79
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nm : ℕ\nx : A\nhx : x ∉ I\n⊢ dpow I m x = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"HMul.hMul",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 71,
"column": 67
} | {
"line": 71,
"column": 69
} | {
"line": 71,
"column": 70
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nm : ℕ\nx : A\nhx : x ∉ I\n⊢ dpow I m x = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"HMul.hMul",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 73,
"column": 59
} | {
"line": 73,
"column": 61
} | {
"line": 73,
"column": 62
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nx : A\nhx : x ∈ I\n⊢ dpow I 0 x = 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Semiring.toModule",
"HMu... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 75,
"column": 58
} | {
"line": 75,
"column": 60
} | {
"line": 75,
"column": 61
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nx : A\nhx : x ∈ I\n⊢ dpow I 1 x = x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"HMul.hMul",
"congrArg",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 77,
"column": 79
} | {
"line": 77,
"column": 81
} | {
"line": 78,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nm : ℕ\nhm : m ≠ 0\nx : A\nhx : x ∈ I\n⊢ dpow I m x ∈ I",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 390,
"column": 18
} | {
"line": 390,
"column": 20
} | {
"line": 391,
"column": 4
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : CommSemiring B\nJ : Ideal B\ne : A ≃+* B\nh : Ideal.map e I = J\nhI : DividedPowers I\nn✝ : ℕ\nx✝ : B\nhx : x✝ ∉ J\n⊢ e (hI.dpow n✝ (e.symm x✝)) = 0",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Eq.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 393,
"column": 18
} | {
"line": 393,
"column": 20
} | {
"line": 394,
"column": 4
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : CommSemiring B\nJ : Ideal B\ne : A ≃+* B\nh : Ideal.map e I = J\nhI : DividedPowers I\nx✝ : B\nhx : x✝ ∈ J\n⊢ e (hI.dpow 0 (e.symm x✝)) = 1",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 207,
"column": 67
} | {
"line": 207,
"column": 69
} | {
"line": 208,
"column": 2
} | [
{
"pp": "A : Type u_3\nB : Type u_4\nC : Type u_5\ninst✝² : CommSemiring A\ninst✝¹ : CommSemiring B\ninst✝ : CommSemiring C\nI : Ideal A\nJ : Ideal B\nK : Ideal C\nhI : DividedPowers I\nhJ : DividedPowers J\nhK : DividedPowers K\nf : A →+* B\ng : B →+* C\nheq : J = map f I\nhf : hI.IsDPMorphism hJ f\nhh : hI.Is... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 370,
"column": 76
} | {
"line": 370,
"column": 78
} | {
"line": 371,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nrs₁ rs₂ : List R\n⊢ IsWeaklyRegular M (rs₁ ++ rs₂) ↔ IsWeaklyRegular M rs₁ ∧ IsWeaklyRegular (M ⧸ Ideal.ofList rs₁ • ⊤) rs₂",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 396,
"column": 19
} | {
"line": 396,
"column": 21
} | {
"line": 397,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Nontrivial M\nh : ⊤ = Ideal.ofList [] • ⊤\n⊢ False",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Submodule",
"instHSMul",
"Semiring.toModule",
"Submodu... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 396,
"column": 17
} | {
"line": 396,
"column": 19
} | {
"line": 397,
"column": 4
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : CommSemiring B\nJ : Ideal B\ne : A ≃+* B\nh : Ideal.map e I = J\nhI : DividedPowers I\nx✝ : B\nhx : x✝ ∈ J\n⊢ e (hI.dpow 1 (e.symm x✝)) = x✝",
"ppTerm": "?m.124",
"assigned": true,
"usedConstants": [
"RingEquiv.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 399,
"column": 20
} | {
"line": 399,
"column": 22
} | {
"line": 400,
"column": 4
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : CommSemiring B\nJ : Ideal B\ne : A ≃+* B\nh : Ideal.map e I = J\nhI : DividedPowers I\nn✝ : ℕ\nx✝ : B\nhn : n✝ ≠ 0\nhx : x✝ ∈ J\n⊢ e (hI.dpow n✝ (e.symm x✝)) ∈ J",
"ppTerm": "?m.149",
"assigned": true,
"usedConstants"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 524,
"column": 18
} | {
"line": 524,
"column": 20
} | {
"line": 525,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nr : R\nrs : List R\nh : IsRegular M (r :: rs)\n⊢ r :: rs = []",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Submodule.pointwiseDistribMu... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 533,
"column": 38
} | {
"line": 533,
"column": 40
} | {
"line": 534,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nM₂ : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module.Flat R M₂\nrs : List R\nh : IsWeaklyRegular M rs\n⊢ IsWeaklyRegular (M₂ ⊗[R] M) rs",
"ppTerm": "?m.25",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 403,
"column": 20
} | {
"line": 403,
"column": 22
} | {
"line": 404,
"column": 4
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : CommSemiring B\nJ : Ideal B\ne : A ≃+* B\nh : Ideal.map e I = J\nhI : DividedPowers I\nn✝ : ℕ\nx✝ y✝ : B\nhx : x✝ ∈ J\nhy : y✝ ∈ J\n⊢ e (hI.dpow n✝ (e.symm (x✝ + y✝))) = ∑ k ∈ antidiagonal n✝, e (hI.dpow k.1 (e.symm x✝)) * e (hI.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 542,
"column": 38
} | {
"line": 542,
"column": 40
} | {
"line": 543,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nM₂ : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module.Flat R M₂\nrs : List R\nh : IsWeaklyRegular M rs\n⊢ IsWeaklyRegular (M ⊗[R] M₂) rs",
"ppTerm": "?m.25",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 61
} | {
"line": 95,
"column": 0
} | [
{
"pp": "case e_a.e_a\nA : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nm : ℕ\nhmn : m < n\nx y : A\nhx : x ∈ I\nhy : y ∈ I\nk : ℕ × ℕ\nhk : k ∈ Finset.antidiagonal m\n⊢ ↑(m.choose k.1) = ↑m ! * (↑k.1!)⁻¹ʳ * (↑k.2!)⁻¹ʳ",
"ppTerm": "?e... | [] | exact castChoose_eq (hn_fac.natCast_factorial_of_lt hmn) hk | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 408,
"column": 17
} | {
"line": 408,
"column": 19
} | {
"line": 409,
"column": 4
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : CommSemiring B\nJ : Ideal B\ne : A ≃+* B\nh : Ideal.map e I = J\nhI : DividedPowers I\nn✝ : ℕ\na✝ x✝ : B\nhx : x✝ ∈ J\n⊢ e (hI.dpow n✝ (e.symm (a✝ * x✝))) = a✝ ^ n✝ * e (hI.dpow n✝ (e.symm x✝))",
"ppTerm": "?m.201",
"assi... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 83,
"column": 92
} | {
"line": 83,
"column": 94
} | {
"line": 84,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nm : ℕ\nhmn : m < n\nx y : A\nhx : x ∈ I\nhy : y ∈ I\n⊢ dpow I m (x + y) = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y",
"ppTerm": "?m.52",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 246,
"column": 73
} | {
"line": 246,
"column": 75
} | {
"line": 247,
"column": 2
} | [
{
"pp": "A : Type u_3\ninst✝ : CommSemiring A\nI : Ideal A\nhI hI' : DividedPowers I\nS : Set A\nhS : I = span S\nhdp : ∀ {n : ℕ}, ∀ a ∈ S, hI.dpow n a = hI'.dpow n a\n⊢ hI' = hI",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ideal.subset_span",
"Semiring.toMo... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 413,
"column": 17
} | {
"line": 413,
"column": 19
} | {
"line": 414,
"column": 4
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : CommSemiring B\nJ : Ideal B\ne : A ≃+* B\nh : Ideal.map e I = J\nhI : DividedPowers I\nm✝ n✝ : ℕ\nx✝ : B\nhx : x✝ ∈ J\n⊢ e (hI.dpow m✝ (e.symm x✝)) * e (hI.dpow n✝ (e.symm x✝)) = ↑((m✝ + n✝).choose m✝) * e (hI.dpow (m✝ + n✝) (e.s... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 45,
"column": 18
} | {
"line": 45,
"column": 20
} | {
"line": 45,
"column": 21
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nn✝ : ℕ\nx✝ : A\nhx : x✝ ∉ I\n⊢ (if h... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 46,
"column": 22
} | {
"line": 46,
"column": 24
} | {
"line": 47,
"column": 4
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nx : A\nhx : x ∈ I\n⊢ (if hx : x ∈ I ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 49,
"column": 17
} | {
"line": 49,
"column": 19
} | {
"line": 50,
"column": 4
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nx✝ : A\nhx : x✝ ∈ I\n⊢ (if hx : x✝ ∈... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 52,
"column": 26
} | {
"line": 52,
"column": 28
} | {
"line": 52,
"column": 29
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nf : A →+* B\nhf : Injective ⇑f\nhJ : DividedPowers J\nhIJ : Ideal.map f I = J\nhmem : ∀ (n : ℕ) {x : A}, x ∈ I → ∃ y, ∃ (_ : n ≠ 0 → y ∈ I), f y = hJ.dpow n (f x)\nn : ℕ\nx : A\nhn : n ≠ 0\nhx : x ∈ I... | [] | by | [anonymous] | by |
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