module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 203,
"column": 6
} | {
"line": 205,
"column": 15
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsCancelMulZero R\nhR : HasUnitMulPowIrreducibleFactorization R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x ≠ 0... | · simp only [hm, one_mul, pow_zero] at h ⊢
right
exact h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 49
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nn : ℕ\nu : Rˣ\nhxI : p ^ n * ↑u ∈ I\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 49
} | [
{
"pp": "case neg.h.h\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 283,
"column": 4
} | {
"line": 283,
"column": 19
} | [
{
"pp": "case refine_3\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\nI : Ideal R\n⊢ (fun P ↦ P ≠ ⊥ ∧ P.IsPrime) (R ∙ Submodule.IsPrincipal.gene... | rintro ⟨I0, hI⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 60
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhR : HasUnitMulPowIrreducibleFactorization R\nthis : UniqueFactorizationMonoid R := HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid hR\n⊢ IsDiscreteValuationRing R",
"usedConstants": [
"IsDiscreteValuationRing.HasUnitMulPo... | apply of_ufd_of_unique_irreducible _ hR.unique_irreducible | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 355,
"column": 56
} | {
"line": 355,
"column": 67
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nϖ : R\nhϖ : Irreducible ϖ\ny : R\nhy : y ∈ nonZeroDivisors R\nn : ℕ\nu : Rˣ\nhx : (algebraMap R K) (↑u * ϖ ^ n) / (algebraMap R K... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 356,
"column": 40
} | {
"line": 356,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nϖ : R\nhϖ : Irreducible ϖ\nn : ℕ\nu : Rˣ\nm : ℕ\nv : Rˣ\nhy : ↑v * ϖ ^ m ∈ nonZeroDivisors R\nhx : (algebraMap R K) (↑u * ϖ ^ n) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 306,
"column": 42
} | {
"line": 306,
"column": 67
} | [
{
"pp": "case refine_1.inl.refine_1\nR : Type u_1\ninst✝⁶ : CommRing R\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nP : Ideal R[X]\npX : R[X]\nhpX : pX ∈ P\ninst✝³ : Algebra (R ⧸ comap C P) Rₘ\ninst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ\ni... | Quotient.eq_zero_iff_mem, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 431,
"column": 2
} | {
"line": 432,
"column": 9
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nϖ : R\nhϖ : Irreducible ϖ\n⊢ (addVal R) ϖ = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 495,
"column": 4
} | {
"line": 496,
"column": 38
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nx y : R\nh : (addVal R) x = (addVal R) y\nhx : x = 0\n⊢ Associated x y",
"usedConstants": [
"Eq.mpr",
"instTopENat",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrde... | · simp_all only [AddValuation.map_zero]
rw [addVal_eq_top_iff.mp h.symm] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 553,
"column": 2
} | {
"line": 553,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nn : ℕ\n⊢ Order.coheight (maximalIdeal R ^ n) = ↑n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝ : Nontrivial Γ₀\nv : Valuation K Γ₀\nx : K\nhx : IsUnit x\n⊢ v x ≠ 0",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"ValuationClass.toMonoidWithZeroHomClass",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 329,
"column": 2
} | {
"line": 329,
"column": 13
} | [
{
"pp": "R : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nx y : R\nh : v (y - x) < v x\nthis : v (y - x + x) = v x\n⊢ v y = v x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 333,
"column": 4
} | {
"line": 333,
"column": 19
} | [
{
"pp": "case pos\nR : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nx y : R\nhx : v x = 0\nhy : v y = 0\n⊢ v (x - y) = v y",
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"HSub.hSub",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 30
} | [
{
"pp": "R : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nh : v x < v 1\n⊢ v (1 + x) = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 352,
"column": 2
} | {
"line": 352,
"column": 41
} | [
{
"pp": "R : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nh : v (-x) < v 1\n⊢ v (1 + -x) = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 469,
"column": 8
} | {
"line": 469,
"column": 19
} | [
{
"pp": "case pos\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx✝ y✝ x y : R\nh : v x ≠ 0\nH : ¬v (x + y) = 0\nhy : v y = 0\n⊢ v (x + y) ≤ v x",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 440,
"column": 4
} | {
"line": 440,
"column": 39
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nP : Ideal R[X]\nhP : P.IsMaximal\ninst✝¹ : IsJacobsonRing R\ninst✝ : Nontrivial R\nhP' : ∀ (x : R), C x ∈ P → x = 0\nP' : Ideal R := comap C P\nhP'_prime : P'.IsPrime\nm : R[X]\nhmem_P : m ∈ P\nhm : ⟨m, hmem_P⟩ ≠ 0\n⊢ m ≠ 0",
"usedConstants": [
"CommSemiring... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 449,
"column": 15
} | {
"line": 449,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nP : Ideal R[X]\nhP : P.IsMaximal\ninst✝¹ : IsJacobsonRing R\ninst✝ : Nontrivial R\nhP' : ∀ (x : R), C x ∈ P → x = 0\nP' : Ideal R := comap C P\nhP'_prime : P'.IsPrime\nm : R[X]\nhmem_P : m ∈ P\nhm : ⟨m, hmem_P⟩ ≠ 0\nhm' : m ≠ 0\nφ : R ⧸ P' →+* R[X] ⧸ P := quotientMap ... | Quotient.eq_zero_iff_mem, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 469,
"column": 8
} | {
"line": 469,
"column": 19
} | [
{
"pp": "case pos\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx✝ y✝ x y : R\nh : v y ≠ 0\nH : ¬v (x + y) = 0\nhy : v x = 0\n⊢ v (x + y) ≤ v y",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 450,
"column": 12
} | {
"line": 450,
"column": 75
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nP : Ideal R[X]\nhP : P.IsMaximal\ninst✝¹ : IsJacobsonRing R\ninst✝ : Nontrivial R\nhP' : ∀ (x : R), C x ∈ P → x = 0\nP' : Ideal R := comap C P\nhP'_prime : P'.IsPrime\nm : R[X]\nhmem_P : m ∈ P\nhm : ⟨m, hmem_P⟩ ≠ 0\nhm' : m ≠ 0\nφ : R ⧸ P' →+* R[X] ⧸ P := quotientMap ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 473,
"column": 6
} | {
"line": 473,
"column": 17
} | [
{
"pp": "case neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx✝ y✝ x y : R\nH : v x = 0 ∧ v y = 0\n⊢ v (x + y) = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DiscreteValuationRing.TFAE | {
"line": 92,
"column": 4
} | {
"line": 93,
"column": 28
} | [
{
"pp": "case h.a\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\ninst✝¹ : IsLocalRing R\ninst✝ : IsDomain R\nI : Ideal R\nhI : I ≠ ⊥\nh : ¬IsField R\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : ¬I = ⊤\nH : ∀ (r : R), ¬IsUnit r ↔ x ∣ r\nthis✝ : x ≠ 0\nhx' : Irreducible x\nH' : ∀ (r : R),... | rw [hx, Ideal.span_singleton_pow, Ideal.span_le, Set.singleton_subset_iff]
exact Nat.find_spec this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DiscreteValuationRing.TFAE | {
"line": 92,
"column": 4
} | {
"line": 93,
"column": 28
} | [
{
"pp": "case h.a\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\ninst✝¹ : IsLocalRing R\ninst✝ : IsDomain R\nI : Ideal R\nhI : I ≠ ⊥\nh : ¬IsField R\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : ¬I = ⊤\nH : ∀ (r : R), ¬IsUnit r ↔ x ∣ r\nthis✝ : x ≠ 0\nhx' : Irreducible x\nH' : ∀ (r : R),... | rw [hx, Ideal.span_singleton_pow, Ideal.span_le, Set.singleton_subset_iff]
exact Nat.find_spec this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 527,
"column": 4
} | {
"line": 527,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R[X]\nhP : P.IsMaximal\ninst✝ : IsJacobsonRing R\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Ideal.Quotient.mk P')\nhf : Function.Surjective ⇑f\np : R[X]\nhp : p ∈ comap f ⊥\nn : ℕ\n⊢ (Ideal.Quotient.mk (comap C P))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 538,
"column": 7
} | {
"line": 538,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R[X]\nhP : P.IsMaximal\ninst✝ : IsJacobsonRing R\nP' : Ideal R := comap C P\nthis✝ : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Ideal.Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (Ideal.map f P)\nthis : (Ideal.map (mapRingHom (Ide... | Quotient.eq_zero_iff_mem, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 622,
"column": 2
} | {
"line": 622,
"column": 29
} | [
{
"pp": "case right\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝³ : DivisionRing K\nΓ₀✝ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀✝\nv✝ : Valuation R Γ₀✝\nΓ₀ : Type u_7\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nhv : v.IsNon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 629,
"column": 2
} | {
"line": 629,
"column": 29
} | [
{
"pp": "case right\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝³ : DivisionRing K\nΓ₀✝ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀✝\nv✝ : Valuation R Γ₀✝\nΓ₀ : Type u_7\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nhv : v.IsNon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 659,
"column": 14
} | {
"line": 659,
"column": 25
} | [
{
"pp": "K : Type u_7\ninst✝¹ : DivisionRing K\nΓ₀ : Type u_8\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation K Γ₀\nh : v.IsNontrivial\n⊢ ∃ x, x ≠ 0 ∧ v x < 1",
"usedConstants": [
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"Preorder.toLT",
"InvOneClass.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 670,
"column": 14
} | {
"line": 670,
"column": 25
} | [
{
"pp": "K : Type u_7\ninst✝¹ : DivisionRing K\nΓ₀ : Type u_8\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation K Γ₀\nh : v.IsNontrivial\n⊢ ∃ x, 1 < v x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 732,
"column": 2
} | {
"line": 732,
"column": 36
} | [
{
"pp": "R : Type u_3\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝ : LinearOrderedCommMonoidWithZero Γ'₀\nv₁ : Valuation R Γ₀\nv₂ : Valuation R Γ'₀\nh : v₁.IsEquiv v₂\nr s : R\n⊢ v₁ r = v₁ s ↔ v₂ r = v₂ s",
"usedConstants": [
"Eq.mpr",
"cong... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Tensor | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 48
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(lTensor M (Submodule.subtype I))",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"TensorProduct.comm",
"congrAr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.IsPrincipal | {
"line": 74,
"column": 38
} | {
"line": 74,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx✝ : R\n⊢ (fun I ↦ Associates.mk (IsPrincipal.generator ↑I)) (Quotient.lift (fun x ↦ ⟨span {x}, ⋯⟩) ⋯ ⟦x✝⟧) = ⟦x✝⟧",
"usedConstants": [
"Eq.mpr",
"Associates.mk",
"Submodule",
"Semiring.toModule",
"CommSemiring.toS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Tensor | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 48
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ Flat R M ↔ ∀ ⦃I : Ideal R⦄, I.FG → Function.Injective ⇑(lTensor M (Submodule.subtype I))",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"TensorProduct.comm",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.TorsionFree | {
"line": 89,
"column": 48
} | {
"line": 89,
"column": 59
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Flat R M\nm : M\nr : R\nhr : r ∈ R⁰\nh : ⟨r, hr⟩ • m = 0\n⊢ (fun x ↦ r • x) m = (fun x ↦ r • x) 0",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"CommSemiring.toSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.TorsionFree | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 12
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsDomain R\nh✝ : ∀ (P : Ideal R) [inst : P.IsMaximal], ValuationRing (Localization P.primeCompl)\nh : torsion R M = ⊥\n⊢ ∀ (P : Ideal R) [inst : P.IsMaximal], Flat R (LocalizedModule P.primeCo... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddZeroClass R\ninst✝ : Mul R\na : R\n⊢ IsSumSq (a * a)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝¹ : AddCommMonoid R\ninst✝ : Mul R\nι : Type u_2\nI : Finset ι\ns : ι → R\nhs : ∀ i ∈ I, IsSumSq (s i)\n⊢ IsSumSq (∑ i ∈ I, s i)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 978,
"column": 19
} | {
"line": 978,
"column": 54
} | [
{
"pp": "case neg.mp.inl\nK : Type u_1\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, v x < 1 ↔ v' x < 1\nx : K\nhx : ¬x = 0\nhh : v x = 1\nh_1 : ¬v' x = 1\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalCommSemiring R\ns₁ s₂ : R\nh₁ : IsSumSq s₁\nh₂ : IsSumSq s₂\n⊢ IsSumSq (s₁ * s₂)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nι : Type u_2\nI : Finset ι\nx : ι → R\nhx : ∀ i ∈ I, IsSumSq (x i)\n⊢ IsSumSq (∏ i ∈ I, x i)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 985,
"column": 19
} | {
"line": 985,
"column": 54
} | [
{
"pp": "case neg.mpr.inl\nK : Type u_1\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\nh : ∀ {x : K}, v x < 1 ↔ v' x < 1\nx : K\nhx : ¬x = 0\nhh : v' x = 1\nh_1 : ¬v x = 1\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 65
} | [
{
"pp": "case a\nι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\nS : Type u_6\ninst✝³ : CommRing S\ninst✝² : Algebra S R\ninst✝¹ : FaithfulSMul S R\ninst✝ : P.IsValue... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 65
} | [
{
"pp": "case a\nι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\nS : Type u_6\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra S R\ninst✝³ : FaithfulSMul S R\ninst✝² : P.IsVal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 15
} | [
{
"pp": "case mem\nι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\nS : Type u_6\ninst✝⁴ : CommRing S\ninst✝³ : Algebra S R\ninst✝² : Module S M\ninst✝¹ : IsScalarTowe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.RootPositive | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 21
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nS : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁹ : CommRing S\ninst✝⁸ : LinearOrder S\ninst✝⁷ : CommRing R\ninst✝⁶ : Algebra S R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsValuedIn S\nB :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 157,
"column": 25
} | {
"line": 157,
"column": 51
} | [
{
"pp": "case add\nι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\nS : Type u_6\ninst✝⁴ : CommRing S\ninst✝³ : Algebra S R\ninst✝² : Module S M\ninst✝¹ : IsScalarTowe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 158,
"column": 21
} | {
"line": 158,
"column": 67
} | [
{
"pp": "case smul\nι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\nS : Type u_6\ninst✝⁴ : CommRing S\ninst✝³ : Algebra S R\ninst✝² : Module S M\ninst✝¹ : IsScalarTow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.RootPositive | {
"line": 138,
"column": 6
} | {
"line": 138,
"column": 17
} | [
{
"pp": "case hB\nι : Type u_1\nR : Type u_2\nS : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : LinearOrder S\ninst✝⁹ : CommRing R\ninst✝⁸ : Algebra S R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsValu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.BaseChange | {
"line": 77,
"column": 52
} | {
"line": 77,
"column": 63
} | [
{
"pp": "ι : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field L\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : Module L M\ninst✝⁸ : Module L N\nP : RootPairing ι L M N\nK : Type u_5\ninst✝⁷ : Field K\ninst✝⁶ : Algebra K L\ninst✝⁵ : Module K M\ninst✝⁴ : Module K N\ninst✝³ : IsSc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.BaseChange | {
"line": 78,
"column": 71
} | {
"line": 78,
"column": 82
} | [
{
"pp": "ι : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field L\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : Module L M\ninst✝⁸ : Module L N\nP : RootPairing ι L M N\nK : Type u_5\ninst✝⁷ : Field K\ninst✝⁶ : Algebra K L\ninst✝⁵ : Module K M\ninst✝⁴ : Module K N\ninst✝³ : IsSc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.BaseChange | {
"line": 79,
"column": 73
} | {
"line": 79,
"column": 84
} | [
{
"pp": "ι : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field L\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : Module L M\ninst✝⁸ : Module L N\nP : RootPairing ι L M N\nK : Type u_5\ninst✝⁷ : Field K\ninst✝⁶ : Algebra K L\ninst✝⁵ : Module K M\ninst✝⁴ : Module K N\ninst✝³ : IsSc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 24
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\nS : Type u_6\ninst✝³ : CommRing S\ninst✝² : Module S M\ninst✝¹ : Nonempty ι\ninst✝ : NeZero 2\n⊢ P.rootSpan S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.BaseChange | {
"line": 86,
"column": 9
} | {
"line": 86,
"column": 48
} | [
{
"pp": "case a\nι : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field L\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : Module L M\ninst✝⁸ : Module L N\nP : RootPairing ι L M N\nK : Type u_5\ninst✝⁷ : Field K\ninst✝⁶ : Algebra K L\ninst✝⁵ : Module K M\ninst✝⁴ : Module K N\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.BaseChange | {
"line": 88,
"column": 9
} | {
"line": 88,
"column": 48
} | [
{
"pp": "case a\nι : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field L\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : Module L M\ninst✝⁸ : Module L N\nP : RootPairing ι L M N\nK : Type u_5\ninst✝⁷ : Field K\ninst✝⁶ : Algebra K L\ninst✝⁵ : Module K M\ninst✝⁴ : Module K N\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 254,
"column": 29
} | {
"line": 254,
"column": 40
} | [
{
"pp": "case add\nι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni : ι\nx y z : M\nhy : y ∈ span R (range ⇑P.root)\nhz : z ∈ span R (range ⇑P.root)\nhy' : y ∈ Submod... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 255,
"column": 23
} | {
"line": 255,
"column": 34
} | [
{
"pp": "case smul\nι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni : ι\nx : M\ny : R\nt : M\nhy : t ∈ span R (range ⇑P.root)\nhy' : t ∈ Submodule.comap (↑(P.reflect... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.RootPositive | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 40
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nS : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : LinearOrder S\ninst✝⁹ : CommRing R\ninst✝⁸ : Algebra S R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsValuedIn S\nB... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.RootPositive | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 15
} | [
{
"pp": "case a\nι : Type u_1\nR : Type u_2\nS : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : LinearOrder S\ninst✝⁹ : CommRing R\ninst✝⁸ : Algebra S R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsValue... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.RootPositive | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 13
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nS : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : LinearOrder S\ninst✝⁹ : CommRing R\ninst✝⁸ : Algebra S R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsValuedIn S\nB... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.RootPositive | {
"line": 186,
"column": 2
} | {
"line": 187,
"column": 34
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nS : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : LinearOrder S\ninst✝⁹ : CommRing R\ninst✝⁸ : Algebra S R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsValuedIn S\nB... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 84
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R N\nP : RootPairing ι R M N\nS : Type u_7\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra S R\ninst✝⁴ : FaithfulSMul S R\ninst✝³ : P.IsValuedIn S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.RootPositive | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 27
} | [
{
"pp": "case a\nι : Type u_1\nR : Type u_2\nS : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹² : CommRing S\ninst✝¹¹ : LinearOrder S\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Algebra S R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.IsValu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 339,
"column": 8
} | {
"line": 339,
"column": 19
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R N\nP : RootPairing ι R M N\nS : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra S R\ninst✝⁴ : P.IsValuedIn S\ninst✝³ : Module S M\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 13
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : AddCommGroup N\ninst✝¹³ : CommRing R\ninst✝¹² : Module R M\ninst✝¹¹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹⁰ : CommRing S\ninst✝⁹ : LinearOrder S\ninst✝⁸ : IsStrictOrdere... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 262,
"column": 2
} | {
"line": 263,
"column": 50
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Fintype ι\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : Module R M\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsAnisotropic\ninst✝ : P.IsRootSystem\n⊢ P.RootForm.Nonde... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 313,
"column": 6
} | {
"line": 313,
"column": 56
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : Fintype ι\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Field R\ninst✝² : Module R M\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsAnisotropic\n⊢ IsCompl (P.rootSpan R) (Submodule.map (↑P.toPerfPair.symm) (P.cor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 315,
"column": 6
} | {
"line": 315,
"column": 58
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : Fintype ι\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Field R\ninst✝² : Module R M\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsAnisotropic\n⊢ IsCompl (P.corootSpan R) (Submodule.map (↑P.flip.toPerfPair.symm)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 50
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : Fintype ι\nm : M\nh : P.flip.toPerfPair (P.Polarization m) = 0\n⊢ P.Polarization m = 0",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 61
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : CharZero R\ninst✝² : IsAddTorsionFree M\ninst✝¹ : P.IsReduced\nn : ℕ\ninst✝ : n.AtLeastTwo\ni : ι\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 81
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : CharZero R\ninst✝² : IsAddTorsionFree M\ninst✝¹ : P.IsReduced\nn : ℕ\ninst✝ : n.AtLeastTwo\ni : ι\nt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 55
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : Fintype ι\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : Module R M\ninst✝ : Module R N\nP : RootPairing ι R M N\nx : M\nhx : x ∈ P.rootSpan R\nhx' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 111,
"column": 77
} | {
"line": 111,
"column": 88
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : CharZero R\ninst✝¹ : IsAddTorsionFree M\ninst✝ : P.IsReduced\ni j : ι\nh : P.root i - P.root j ∈ ran... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 30
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : CharZero R\ninst✝¹ : IsAddTorsionFree M\ninst✝ : P.IsReduced\ni j : ι\nh : P.root i - P.root j ∈ ran... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 156,
"column": 84
} | {
"line": 156,
"column": 95
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ni j : ι\ninst✝² : NeZero 2\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nt : R\nh₁ : 0 • P.root i + t • P.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 160,
"column": 80
} | {
"line": 160,
"column": 91
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ni j : ι\ninst✝² : NeZero 2\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\ns : R\nh₂ : s ≠ 0\nh₁ : s • P.roo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 163,
"column": 47
} | {
"line": 163,
"column": 58
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ni j : ι\ninst✝² : NeZero 2\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\ns t : R\nh₂ : s ≠ 0\nh₃ : t ≠ 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 163,
"column": 47
} | {
"line": 163,
"column": 85
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ni j : ι\ninst✝² : NeZero 2\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\ns t : R\nh₂ : s ≠ 0\nh₃ : t ≠ 0\n... | simpa using congr_arg (P.coroot' i) h₁ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 58
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ni j : ι\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\n⊢ -P.pairing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 45
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ni j : ι\ninst✝⁴ : Finite ι\ninst✝³ : CharZero R\ninst✝² : IsDomain R\ninst✝¹ : IsTorsionFree R M\ninst✝ : IsT... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 392,
"column": 2
} | {
"line": 392,
"column": 13
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹³ : CommRing R\ninst✝¹² : AddCommGroup M\ninst✝¹¹ : Module R M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝⁸ : CommRing S\ninst✝⁷ : LinearOrder S\ninst✝⁶ : IsStrictOrderedRing S\ninst✝⁵ : Algeb... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 43,
"column": 6
} | {
"line": 43,
"column": 17
} | [
{
"pp": "case insert.inl\nι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup M\ninst✝ : Module K M\np : ι → Submodule K M\nh₁ : ∀ (i : ι), p i ≠ ⊤\nj : ι\nhj : j ∉ ∅\nh₂ : ↑(insert j ∅).card < ENat.card K\nhj' : ↑∅.card < ENat.card K → ⋃ i ∈ ∅, ↑(p i) ≠ univ\n⊢ ⋃ i ∈ insert j ∅, ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 44,
"column": 48
} | {
"line": 44,
"column": 93
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup M\ninst✝ : Module K M\np : ι → Submodule K M\nh₁ : ∀ (i : ι), p i ≠ ⊤\nj : ι\ns : Finset ι\nhj : j ∉ s\nh₂ : ↑(insert j s).card < ENat.card K\nhj' : ↑s.card < ENat.card K → ⋃ i ∈ s, ↑(p i) ≠ univ\nhs : s.Nonempty\n⊢ ↑s.ca... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 48,
"column": 6
} | {
"line": 48,
"column": 68
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup M\ninst✝ : Module K M\np : ι → Submodule K M\nh₁ : ∀ (i : ι), p i ≠ ⊤\nj : ι\ns : Finset ι\nhj : j ∉ s\nhs : s.Nonempty\nh₂ : ↑s.card + 1 < ENat.card K\nhj' : ⋃ i ∈ insert j s, ↑(p i) = univ\n⊢ ↑(p j) ∪ ⋃ i ∈ s, ↑(p i) = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 52,
"column": 6
} | {
"line": 52,
"column": 17
} | [
{
"pp": "case insert.inr.inl\nι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup M\ninst✝ : Module K M\np : ι → Submodule K M\nh₁ : ∀ (i : ι), p i ≠ ⊤\nj : ι\ns : Finset ι\nhj : j ∉ s\nhs : s.Nonempty\nh₂ : ↑s.card + 1 < ENat.card K\nhj' : ↑(p j) ∪ ⋃ i ∈ s, ↑(p i) = univ\nhx : 0 ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.QuadraticForm.Prod | {
"line": 145,
"column": 4
} | {
"line": 147,
"column": 31
} | [
{
"pp": "case right\nR : Type u_2\nM₁ : Type u_3\nM₂ : Type u_4\nP : Type u_7\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R P\nQ₁ : QuadraticMap R M₁ P\nQ₂ : QuadraticMap R M₂ P\nh : ∀ (a : M... | intro x hx
refine (h 0 x ?_).2
rw [hx, add_zero, map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.QuadraticForm.Prod | {
"line": 145,
"column": 4
} | {
"line": 147,
"column": 31
} | [
{
"pp": "case right\nR : Type u_2\nM₁ : Type u_3\nM₂ : Type u_4\nP : Type u_7\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R P\nQ₁ : QuadraticMap R M₁ P\nQ₂ : QuadraticMap R M₂ P\nh : ∀ (a : M... | intro x hx
refine (h 0 x ?_).2
rw [hx, add_zero, map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Prod | {
"line": 156,
"column": 15
} | {
"line": 156,
"column": 52
} | [
{
"pp": "case mp.left\nR : Type u_2\nM₁ : Type u_3\nM₂ : Type u_4\nP : Type u_7\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Module R P\ninst✝¹ : Preorder P\ninst✝ : AddLeftMono P\nQ₁ : QuadraticMap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.QuadraticForm.Prod | {
"line": 157,
"column": 15
} | {
"line": 157,
"column": 52
} | [
{
"pp": "case mp.right\nR : Type u_2\nM₁ : Type u_3\nM₂ : Type u_4\nP : Type u_7\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Module R P\ninst✝¹ : Preorder P\ninst✝ : AddLeftMono P\nQ₁ : QuadraticMa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 76,
"column": 57
} | {
"line": 76,
"column": 68
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup M\ninst✝ : Module K M\np : ι → Submodule K M\nh₁ : ∀ (i : ι), p i ≠ ⊤\nj : ι\ns : Finset ι\nhj : j ∉ s\nhs : s.Nonempty\nh₂ : ↑s.card + 1 < ENat.card K\nhj' : ↑(p j) ∪ ⋃ i ∈ s, ↑(p i) = univ\nx : M\nhx : x ∈ p j\nhx₀ : x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.QuadraticForm.Prod | {
"line": 328,
"column": 4
} | {
"line": 328,
"column": 38
} | [
{
"pp": "case mp\nι : Type u_1\nR : Type u_2\nMᵢ : ι → Type u_8\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → Module R (Mᵢ i)\nP : Type u_10\ninst✝⁴ : Fintype ι\ninst✝³ : AddCommMonoid P\ninst✝² : PartialOrder P\ninst✝¹ : IsOrderedAddMonoid P\ninst✝ : Module R P\nQ : (i :... | convert! h (Pi.single i x) using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 15
} | [
{
"pp": "case insert.inr.inr\nι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup M\ninst✝ : Module K M\np : ι → Submodule K M\nh₁ : ∀ (i : ι), p i ≠ ⊤\nj : ι\ns : Finset ι\nhj : j ∉ s\nhs : s.Nonempty\nh₂ : ↑s.card + 1 < ENat.card K\nhj' : ↑(p j) ∪ ⋃ i ∈ s, ↑(p i) = univ\nx : M\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.QuadraticForm.Prod | {
"line": 327,
"column": 4
} | {
"line": 330,
"column": 41
} | [
{
"pp": "case mp\nι : Type u_1\nR : Type u_2\nMᵢ : ι → Type u_8\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → Module R (Mᵢ i)\nP : Type u_10\ninst✝⁴ : Fintype ι\ninst✝³ : AddCommMonoid P\ninst✝² : PartialOrder P\ninst✝¹ : IsOrderedAddMonoid P\ninst✝ : Module R P\nQ : (i :... | classical
convert! h (Pi.single i x) using 1
rw [Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hji => ?_, Pi.single_eq_same]
rw [Pi.single_eq_of_ne hji, map_zero] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 93,
"column": 40
} | {
"line": 93,
"column": 90
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup M\ninst✝² : Module K M\ninst✝¹ : Finite ι\ninst✝ : Infinite K\np : ι → Submodule K M\nh : ∀ (i : ι), p i ≠ ⊤\n_i : Fintype ι := Fintype.ofFinite ι\nthis : ⋃ i, ↑(p i) ⊂ univ\n⊢ ∃ x, ∀ (i : ι), x ∉ p i",
"usedConstants... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 13
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup M\ninst✝² : Module K M\ninst✝¹ : Finite ι\ninst✝ : Infinite K\np : ι → Submodule K M\nh : ∀ (i : ι), p i ≠ ⊤\n_i : Fintype ι := Fintype.ofFinite ι\n⊢ ⋃ i, ↑(p i) ⊂ univ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 15
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nx : L\nhx : x ∈ rootS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 85,
"column": 62
} | {
"line": 85,
"column": 73
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nx : L\nhx : x ∈ rootS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 102,
"column": 15
} | {
"line": 102,
"column": 30
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup M\ninst✝² : Module K M\ninst✝¹ : Finite ι\ninst✝ : Infinite K\nf : ι → Dual K M\np : ι → Submodule K M := fun i ↦ LinearMap.ker (f i)\nh : ∀ (i : ι), p i ≠ ⊤\nx : M\nhx : ∀ (i : ι), x ∉ p i\n⊢ ∀ (i : ι), (f i) x ≠ 0",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 102,
"column": 12
} | {
"line": 102,
"column": 33
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup M\ninst✝² : Module K M\ninst✝¹ : Finite ι\ninst✝ : Infinite K\nf : ι → Dual K M\np : ι → Submodule K M := fun i ↦ LinearMap.ker (f i)\nh : ∀ (i : ι), p i ≠ ⊤\nx : M\nhx : ∀ (i : ι), x ∉ p i\n⊢ ∀ (i : ι), (f i) x ≠ 0",
... | by simpa [p] using hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 81
} | [
{
"pp": "case pos\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nn : ℕ\nhn :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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