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.DividedPowers.Padic | {
"line": 53,
"column": 28
} | {
"line": 53,
"column": 30
} | {
"line": 54,
"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)\nn : ℕ\nx y : A\nhx : x ∈ I\nhy : y ∈... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 58,
"column": 25
} | {
"line": 58,
"column": 27
} | {
"line": 59,
"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)\nn : ℕ\na x : A\nhx : x ∈ I\n⊢ (if hx... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 103,
"column": 34
} | {
"line": 103,
"column": 36
} | {
"line": 104,
"column": 6
} | [
{
"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)!\nhnI : I ^ n = 0\nm : ℕ\nx : A\nhx : x ∈ I\ny : A\nhy : y ∈ I\nhmn : n ≤ m\nh_sub : I ^ m ≤ I ^ n\n⊢ (x + y) ^ m = 0",
"ppTerm": "?m.121",
"assigned": true,
"usedConstan... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 62,
"column": 17
} | {
"line": 62,
"column": 19
} | {
"line": 62,
"column": 20
} | [
{
"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)\nm✝ n✝ : ℕ\nx✝ : A\nhx : x✝ ∈ I\n⊢ ((... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 417,
"column": 21
} | {
"line": 417,
"column": 23
} | {
"line": 418,
"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\nhn : n✝ ≠ 0\nhx : x✝ ∈ J\n⊢ e (hI.dpow m✝ (e.symm (e (hI.dpow n✝ (e.symm x✝))))) = ↑(m✝.uniformBell n✝) * e (hI.dpow (m✝ * ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 428,
"column": 57
} | {
"line": 428,
"column": 59
} | {
"line": 429,
"column": 2
} | [
{
"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 : A\n⊢ (ofRingEquiv h hI).dpow n (e a) = e (hI.dpow n a)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 435,
"column": 23
} | {
"line": 435,
"column": 25
} | {
"line": 435,
"column": 26
} | [
{
"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\n⊢ ofRingEquiv ⋯ (ofRingEquiv h hI) = hI",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"DividedPowers.ofRingEquiv... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 436,
"column": 24
} | {
"line": 436,
"column": 26
} | {
"line": 436,
"column": 27
} | [
{
"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\nhJ : DividedPowers J\n⊢ ofRingEquiv h (ofRingEquiv ⋯ hJ) = hJ",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"DividedPowers.ofRingEquiv... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 562,
"column": 2
} | {
"line": 562,
"column": 24
} | {
"line": 563,
"column": 4
} | [
{
"pp": "case cons\nR : Type u_1\nM₄ : Type u_6\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M₄\ninst✝⁸ : Module R M₄\nrs✝ : List R\nM✝ : Type u_6\ninst✝⁷ : AddCommGroup M✝\ninst✝⁶ : Module R M✝\nr : R\nrs : List R\nh₄ : IsSMulRegular M✝ r\nh2✝ : IsWeaklyRegular (QuotSMulTop r M✝) rs\nih :\n ∀ {M : Type u_3} {... | [] | | cons r rs h₄ _ ih => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 97,
"column": 90
} | {
"line": 97,
"column": 92
} | {
"line": 98,
"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)!\nhnI : I ^ n = 0\nm : ℕ\nx : A\nhx : x ∈ I\ny : A\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.57",
"assigned":... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 64,
"column": 29
} | {
"line": 64,
"column": 31
} | {
"line": 65,
"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)\nn m : ℕ\nx : A\nhm : m ≠ 0\nhx : x ∈... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 555,
"column": 76
} | {
"line": 555,
"column": 78
} | {
"line": 556,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nM₄ : Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M₂\ninst✝⁵ : AddCommGroup M₃\ninst✝⁴ : AddCommGroup M₄\ninst✝³ : Module R M\ninst✝² : Module R M₂\ninst✝¹ : Module R M₃\ninst✝ : Module R M₄\nrs : List R\nf₁ : M ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 116,
"column": 91
} | {
"line": 116,
"column": 93
} | {
"line": 117,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nm : ℕ\na x : A\nhx : x ∈ I\n⊢ dpow I m (a * x) = a ^ m * dpow I m x",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semi... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 103,
"column": 21
} | {
"line": 103,
"column": 23
} | {
"line": 103,
"column": 24
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : hI.IsSubDPIdeal J\ninst✝ : (x : A) → Decidable (x ∈ J)\nn✝ : ℕ\nx✝ : A\nhx : x✝ ∉ J\n⊢ (if x✝ ∈ J then hI.dpow n✝ x✝ else 0) = 0",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Semir... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 104,
"column": 21
} | {
"line": 104,
"column": 23
} | {
"line": 104,
"column": 24
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : hI.IsSubDPIdeal J\ninst✝ : (x : A) → Decidable (x ∈ J)\nx✝ : A\nhx : x✝ ∈ J\n⊢ (if x✝ ∈ J then hI.dpow 0 x✝ else 0) = 1",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"NonAssocSemiri... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 93,
"column": 54
} | {
"line": 93,
"column": 56
} | {
"line": 94,
"column": 6
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nhn : n ≠ 0\nx : ℤ_[p]\nhx : x ∈ Ideal.span {↑p}\nhx0 : ¬x = 0\n⊢ ↑(padicValNat p n !) < ↑n",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Nat.cast_lt._simp_1",
"PartialOrder.toPreorder",
"AddGroupWit... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 105,
"column": 21
} | {
"line": 105,
"column": 23
} | {
"line": 105,
"column": 24
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : hI.IsSubDPIdeal J\ninst✝ : (x : A) → Decidable (x ∈ J)\nx✝ : A\nhx : x✝ ∈ J\n⊢ (if x✝ ∈ J then hI.dpow 1 x✝ else 0) = x✝",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Semiring.toMo... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 106,
"column": 21
} | {
"line": 106,
"column": 23
} | {
"line": 106,
"column": 24
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : hI.IsSubDPIdeal J\ninst✝ : (x : A) → Decidable (x ∈ J)\nn✝ : ℕ\nx✝ : A\nhn : n✝ ≠ 0\nhx : x✝ ∈ J\n⊢ (if x✝ ∈ J then hI.dpow n✝ x✝ else 0) ∈ J",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 95,
"column": 40
} | {
"line": 95,
"column": 42
} | {
"line": 96,
"column": 6
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nhn : n ≠ 0\nx : ℤ_[p]\nhx : x ∈ Ideal.span {↑p}\nhx0 : ¬x = 0\nhlt : ↑(padicValNat p n !) < ↑n\n⊢ 0 < ‖↑n !‖",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 107,
"column": 21
} | {
"line": 107,
"column": 23
} | {
"line": 107,
"column": 24
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : hI.IsSubDPIdeal J\ninst✝ : (x : A) → Decidable (x ∈ J)\nn✝ : ℕ\nx✝ y✝ : A\nhx : x✝ ∈ J\nhy : y✝ ∈ J\n⊢ (if x✝ + y✝ ∈ J then hI.dpow n✝ (x✝ + y✝) else 0) =\n ∑ k ∈ Finset.antidiagonal n✝, (if x✝ ∈ J then hI.dp... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 109,
"column": 21
} | {
"line": 109,
"column": 23
} | {
"line": 110,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : hI.IsSubDPIdeal J\ninst✝ : (x : A) → Decidable (x ∈ J)\nn✝ : ℕ\na✝ x✝ : A\nhx : x✝ ∈ J\n⊢ (if a✝ * x✝ ∈ J then hI.dpow n✝ (a✝ * x✝) else 0) = a✝ ^ n✝ * if x✝ ∈ J then hI.dpow n✝ x✝ else 0",
"ppTerm": "?m.73"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 111,
"column": 21
} | {
"line": 111,
"column": 23
} | {
"line": 111,
"column": 24
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : hI.IsSubDPIdeal J\ninst✝ : (x : A) → Decidable (x ∈ J)\nm✝ n✝ : ℕ\nx✝ : A\nhx : x✝ ∈ J\n⊢ ((if x✝ ∈ J then hI.dpow m✝ x✝ else 0) * if x✝ ∈ J then hI.dpow n✝ x✝ else 0) =\n ↑((m✝ + n✝).choose m✝) * if x✝ ∈ J t... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 112,
"column": 21
} | {
"line": 112,
"column": 23
} | {
"line": 113,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : hI.IsSubDPIdeal J\ninst✝ : (x : A) → Decidable (x ∈ J)\nm✝ n✝ : ℕ\nx✝ : A\nhn : n✝ ≠ 0\nhx : x✝ ∈ J\n⊢ (if (if x✝ ∈ J then hI.dpow n✝ x✝ else 0) ∈ J then hI.dpow m✝ (if x✝ ∈ J then hI.dpow n✝ x✝ else 0) else 0) ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 121,
"column": 65
} | {
"line": 121,
"column": 67
} | {
"line": 121,
"column": 68
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhJ : hI.IsSubDPIdeal J\ninst✝ : (x : A) → Decidable (x ∈ J)\nn : ℕ\na : A\nha : a ∈ J\n⊢ (dividedPowers hI hJ).dpow n a = hI.dpow n a",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 122,
"column": 72
} | {
"line": 122,
"column": 74
} | {
"line": 123,
"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 k : ℕ\nhkm : m + k < n\nx : A\nhx : x ∈ I\n⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 578,
"column": 32
} | {
"line": 578,
"column": 34
} | {
"line": 579,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na b : R\nh1 : IsWeaklyRegular M [a, b]\nh2 : torsionBy R M b = a • torsionBy R M b → torsionBy R M b = ⊥\n⊢ IsWeaklyRegular M [b, a]",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 122,
"column": 72
} | {
"line": 134,
"column": 100
} | {
"line": 136,
"column": 0
} | [
{
"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 k : ℕ\nhkm : m + k < n\nx : A\nhx : x ∈ I\n⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": ... | [] | by
have hm : m < n := lt_of_le_of_lt le_self_add hkm
have hk : k < n := lt_of_le_of_lt le_add_self hkm
rw [dpow_eq_of_mem hx, dpow_eq_of_mem hx, dpow_eq_of_mem hx,
mul_assoc, ← mul_assoc (x ^ m), mul_comm (x ^ m), mul_assoc _ (x ^ m),
← pow_add, ← mul_assoc, ← mul_assoc]
apply congr_arg₂ _ _ rfl
rw [e... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 126,
"column": 32
} | {
"line": 126,
"column": 34
} | {
"line": 126,
"column": 35
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\ninst✝ : (x : A) → Decidable (x ∈ J)\nhJ : hI.IsSubDPIdeal J\nx✝² : ℕ\nx✝¹ : x✝² ≠ 0\nx✝ : A\nha : x✝ ∈ J\n⊢ hI.dpow x✝² x✝ = (dividedPowers hI hJ).dpow x✝² x✝",
"ppTerm": "?m.47",
"assigned": true,
"usedC... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 124,
"column": 73
} | {
"line": 124,
"column": 75
} | {
"line": 125,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\ninst✝ : (x : A) → Decidable (x ∈ J)\nhJ : hI.IsSubDPIdeal J\n⊢ (dividedPowers hI hJ).IsDPMorphism hI (RingHom.id A)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semirin... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 138,
"column": 72
} | {
"line": 138,
"column": 74
} | {
"line": 139,
"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)!\nhnI : I ^ n = 0\nm k : ℕ\nx : A\nhx : x ∈ I\n⊢ dpow I m x * dpow I k x = ↑((m + k).choose m) * dpow I (m + k) x",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 87,
"column": 71
} | {
"line": 87,
"column": 73
} | {
"line": 88,
"column": 2
} | [
{
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Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
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Mathlib.RingTheory.DividedPowers.Padic | {
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Mathlib.RingTheory.DividedPowers.RatAlgebra | {
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Mathlib.RingTheory.DividedPowers.Padic | {
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Mathlib.RingTheory.DividedPowers.RatAlgebra | {
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Mathlib.RingTheory.DividedPowers.RatAlgebra | {
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Mathlib.RingTheory.DividedPowers.Padic | {
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Mathlib.RingTheory.DividedPowers.RatAlgebra | {
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Mathlib.RingTheory.DividedPowers.Padic | {
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{
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Mathlib.RingTheory.DividedPowers.RatAlgebra | {
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Mathlib.RingTheory.DividedPowers.RatAlgebra | {
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Mathlib.RingTheory.DividedPowers.Padic | {
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Mathlib.RingTheory.DividedPowers.RatAlgebra | {
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
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Mathlib.RingTheory.DividedPowers.Padic | {
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{
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Mathlib.RingTheory.DividedPowers.RatAlgebra | {
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Mathlib.RingTheory.DividedPowers.RatAlgebra | {
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Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Padic | {
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{
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Mathlib.RingTheory.DividedPowers.RatAlgebra | {
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Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
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Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DualNumber | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DualNumber | {
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{
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Mathlib.RingTheory.DualNumber | {
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{
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Mathlib.RingTheory.DualNumber | {
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Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
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"Su... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
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{
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Mathlib.RingTheory.DualNumber | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DualNumber | {
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{
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Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
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{
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Mathlib.RingTheory.Regular.RegularSequence | {
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{
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Mathlib.RingTheory.DualNumber | {
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{
"pp": "k : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝⁵ : Field k\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra k R\ninst✝¹ : Algebra R S\ninst✝ : IsFiniteSplit R S\n⊢ Etale R S",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Algebra.Etale.inst",
"AlgEquiv.symm... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TotallySplit | {
"line": 87,
"column": 49
} | {
"line": 87,
"column": 51
} | {
"line": 88,
"column": 2
} | [
{
"pp": "k : Type u_1\nR : Type u_2\ninst✝⁴ : Field k\ninst✝³ : CommRing R\ninst✝² : Algebra k R\ninst✝¹ : IsFiniteSplit k R\np : Ideal R\ninst✝ : p.IsPrime\n⊢ Function.Bijective ⇑(algebraMap k (R ⧸ p))",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.mk... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TotallySplit | {
"line": 110,
"column": 16
} | {
"line": 110,
"column": 18
} | {
"line": 111,
"column": 4
} | [
{
"pp": "k : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝⁵ : Field k\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra k R\ninst✝¹ : Algebra R S\ninst✝ : IsFiniteSplit k R\nf : R →ₐ[k] k\n⊢ (fun p ↦ (↑(AlgEquiv.ofBijective (ofId k (R ⧸ p.asIdeal)) ⋯).symm).comp (Ideal.Quotient.mkₐ k p.asIdeal))\n ... | [] | by | [anonymous] | by |
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