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.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 17
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\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\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\ninst... | contrapose hk | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.LinearAlgebra.Eigenspace.Minpoly | {
"line": 83,
"column": 4
} | {
"line": 85,
"column": 30
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nf : End R M\nμ : R\ninst✝¹ : IsDomain R\ninst✝ : Module.Finite R M\nh : (minpoly R f).IsRoot μ\nq : R[X]\nhq : minpoly R f = (X - C μ) * q\nh_contra : ∀ (v : M), ((aeval f) q) v = 0\n⊢ False",
"usedConstants"... | have := minpoly.min R f
((monic_X_sub_C μ).of_mul_monic_left (hq ▸ minpoly.monic (Algebra.IsIntegral.isIntegral f)))
(LinearMap.ext h_contra) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 383,
"column": 2
} | {
"line": 387,
"column": 79
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nn m : ℕ\n⊢ descPochhammer R n * (descPochhammer R m).comp (X - ↑n) = descPochhammer R (n + m)",
"usedConstants": [
"Eq.mpr",
"Polynomial.instOne",
"Semigroup.toMul",
"Nat.recAux",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"congrAr... | induction m with
| zero => simp
| succ m ih =>
rw [descPochhammer_succ_right, Polynomial.mul_X_sub_intCast_comp, ← mul_assoc, ih,
← add_assoc, descPochhammer_succ_right, Nat.cast_add, sub_add_eq_sub_sub] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 383,
"column": 2
} | {
"line": 387,
"column": 79
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nn m : ℕ\n⊢ descPochhammer R n * (descPochhammer R m).comp (X - ↑n) = descPochhammer R (n + m)",
"usedConstants": [
"Eq.mpr",
"Polynomial.instOne",
"Semigroup.toMul",
"Nat.recAux",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"congrAr... | induction m with
| zero => simp
| succ m ih =>
rw [descPochhammer_succ_right, Polynomial.mul_X_sub_intCast_comp, ← mul_assoc, ih,
← add_assoc, descPochhammer_succ_right, Nat.cast_add, sub_add_eq_sub_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 383,
"column": 2
} | {
"line": 387,
"column": 79
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nn m : ℕ\n⊢ descPochhammer R n * (descPochhammer R m).comp (X - ↑n) = descPochhammer R (n + m)",
"usedConstants": [
"Eq.mpr",
"Polynomial.instOne",
"Semigroup.toMul",
"Nat.recAux",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"congrAr... | induction m with
| zero => simp
| succ m ih =>
rw [descPochhammer_succ_right, Polynomial.mul_X_sub_intCast_comp, ← mul_assoc, ih,
← add_assoc, descPochhammer_succ_right, Nat.cast_add, sub_add_eq_sub_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 40
} | [
{
"pp": "case e_g.h\nσ : Type u_1\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring T\nf : R →+* MvPolynomial σ S\ng : S →+* T\nφ : MvPolynomial σ R\nx✝ : σ\n⊢ (⇑(map g) ∘ X) x✝ = X x✝",
"usedConstants": [
"Nat.instMulZeroClass",
"c... | simp only [Function.comp_apply, map_X] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 248,
"column": 2
} | {
"line": 249,
"column": 21
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\nS : Type u_4\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\ng : σ → S\n⊢ (eval₂Hom f g).comp C = f",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Nat.instMulZeroClass",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
... | ext1 r
exact eval₂_C f g r | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 248,
"column": 2
} | {
"line": 249,
"column": 21
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\nS : Type u_4\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\ng : σ → S\n⊢ (eval₂Hom f g).comp C = f",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Nat.instMulZeroClass",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
... | ext1 r
exact eval₂_C f g r | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Charpoly.Univ | {
"line": 59,
"column": 6
} | {
"line": 59,
"column": 11
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nn : Type u_3\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nf : R →+* S\nM : n × n → S\n⊢ Polynomial.map (eval₂Hom f M) (univ R n) = (of (Function.curry M)).charpoly",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",... | univ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 378,
"column": 2
} | {
"line": 378,
"column": 60
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\nn : ℕ\nhσ : Finite σ\np : Polynomial (MvPolynomial σ R)\nhp : ((optionEquivLeft R σ).symm p).IsHomogeneous n\ni j : ℕ\nh : i + j = n\nk : ℕ\ne : σ ≃ Fin k\ne' : Option σ ≃ Fin (k + 1) := e.optionCongr.trans (_root_.finSuccEquiv k).symm\nF : MvPolynomi... | convert! hφ.finSuccEquiv_coeff_isHomogeneous i j h using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Algebra.Lie.Basis | {
"line": 327,
"column": 39
} | {
"line": 327,
"column": 60
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁴ : Finite ι\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nb : Basis ι R L\ninst✝ : Fintype ι\ni : ι\nx : ↥b.cartan\n⊢ -(b.baseSupp i) x • b.f i = ⁅x, b.f i⁆",
"usedConstants": [
"LieAlgebra.toModule",
"LieAlgebra.Basis... | ← LinearMap.neg_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 301,
"column": 96
} | {
"line": 303,
"column": 71
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\nι : Type u_5\nιM : Type u_7\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup L\ninst✝⁸ : Module R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁵ : Fintype ι\ninst✝⁴ : Fintype ιM\ninst✝³ : DecidableEq ι\ninst✝² : DecidableEq ιM\nb : Bas... | by
nontriviality R
rw [← polyCharpolyAux_map_eq_charpoly φ b bₘ x, Polynomial.coeff_map] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.SymplecticGroup | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 22
} | [
{
"pp": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ∈ symplecticGroup l R\n⊢ Aᵀ ∈ symplecticGroup l R",
"usedConstants": [
"instFintypeSum",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
... | rw [mem_iff] at hA ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.SymplecticGroup | {
"line": 168,
"column": 10
} | {
"line": 168,
"column": 18
} | [
{
"pp": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ∈ symplecticGroup l R\n⊢ -J l R * (Aᵀ * J l R * A) = -J l R * J l R",
"usedConstants": [
"instFintypeSum",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | mem_iff' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Basis | {
"line": 522,
"column": 4
} | {
"line": 522,
"column": 62
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁷ : Fintype ι\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra K L\ninst✝² : FiniteDimensional K L\nb : Basis ι K L\ninst✝¹ : IsTriangularizable K (↥b.cartan) L\ninst✝ : IsKilling K L\nχ : ↥root\nthis✝ : ∀ (n : ι → ℕ), ∑ i, n... | simpa [iSup_and, iSup_comm (ι := b.cartan → K)] using this | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Lie.Free | {
"line": 209,
"column": 25
} | {
"line": 209,
"column": 56
} | [
{
"pp": "case add_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c' : lib R X\na✝ : Rel R X a✝¹ b✝\nh₂ : (liftAux R f) a✝¹ = (liftAux R f) b✝\n⊢ (liftAux R f) (a✝¹ + c') = (liftAux R f) (b✝ + c')",
"usedConstants": [
"L... | simp only [liftAux_map_add, h₂] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Free | {
"line": 209,
"column": 25
} | {
"line": 209,
"column": 56
} | [
{
"pp": "case add_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c' : lib R X\na✝ : Rel R X a✝¹ b✝\nh₂ : (liftAux R f) a✝¹ = (liftAux R f) b✝\n⊢ (liftAux R f) (a✝¹ + c') = (liftAux R f) (b✝ + c')",
"usedConstants": [
"L... | simp only [liftAux_map_add, h₂] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Free | {
"line": 209,
"column": 25
} | {
"line": 209,
"column": 56
} | [
{
"pp": "case add_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c' : lib R X\na✝ : Rel R X a✝¹ b✝\nh₂ : (liftAux R f) a✝¹ = (liftAux R f) b✝\n⊢ (liftAux R f) (a✝¹ + c') = (liftAux R f) (b✝ + c')",
"usedConstants": [
"L... | simp only [liftAux_map_add, h₂] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Free | {
"line": 258,
"column": 26
} | {
"line": 258,
"column": 59
} | [
{
"pp": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nF : FreeLieAlgebra R X →ₗ⁅R⁆ L\n⊢ (lift R) ((lift R).symm F) = F",
"usedConstants": [
"LieHom",
"Equiv.apply_symm_apply",
"FreeLieAlgebra.instLieRing",
"FreeLieAlgebra.lift",... | exact (lift R).apply_symm_apply F | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.SemiDirect | {
"line": 150,
"column": 21
} | {
"line": 150,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : LieRing K\ninst✝² : LieAlgebra R K\nL : Type u_3\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nψ : L →ₗ⁅R⁆ LieDerivation R K K\n⊢ (projr ψ).range = ⊤",
"usedConstants": [
"LieHom",
"LieAlgebra.SemiDirectSum.projr_surjective._simp_... | simp [LieHom.range_eq_top] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.SemiDirect | {
"line": 150,
"column": 21
} | {
"line": 150,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : LieRing K\ninst✝² : LieAlgebra R K\nL : Type u_3\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nψ : L →ₗ⁅R⁆ LieDerivation R K K\n⊢ (projr ψ).range = ⊤",
"usedConstants": [
"LieHom",
"LieAlgebra.SemiDirectSum.projr_surjective._simp_... | simp [LieHom.range_eq_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.SemiDirect | {
"line": 150,
"column": 21
} | {
"line": 150,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : LieRing K\ninst✝² : LieAlgebra R K\nL : Type u_3\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nψ : L →ₗ⁅R⁆ LieDerivation R K K\n⊢ (projr ψ).range = ⊤",
"usedConstants": [
"LieHom",
"LieAlgebra.SemiDirectSum.projr_surjective._simp_... | simp [LieHom.range_eq_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 303,
"column": 6
} | {
"line": 303,
"column": 24
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\nn : ℕ\nx✝ : x ∈ p.support\n⊢ p.SameCycle ((p ^ n) x) x",
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"Equiv.Perm.instPowNat",
"id",
"Equiv.Perm.SameCycle",
"HPow... | sameCycle_pow_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 447,
"column": 6
} | {
"line": 449,
"column": 20
} | [
{
"pp": "case intro.refine_2.mk.refine_2\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : DecidableEq α\nval✝ : Fintype α\nx : α\ny✝ : Cycle α\nl : List α\nhn : Cycle.Nodup (Quot.mk (⇑(IsRotated.setoid α)) l)\nhf : (Cycle.formPerm (Quot.mk (⇑(IsRotated.setoid α)) l) hn).IsCycle\nhx : (Cycle.formPerm (Quot.mk (⇑(IsRota... | rw [← mem_toFinset]
refine support_formPerm_le l ?_
simpa using hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 447,
"column": 6
} | {
"line": 449,
"column": 20
} | [
{
"pp": "case intro.refine_2.mk.refine_2\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : DecidableEq α\nval✝ : Fintype α\nx : α\ny✝ : Cycle α\nl : List α\nhn : Cycle.Nodup (Quot.mk (⇑(IsRotated.setoid α)) l)\nhf : (Cycle.formPerm (Quot.mk (⇑(IsRotated.setoid α)) l) hn).IsCycle\nhx : (Cycle.formPerm (Quot.mk (⇑(IsRota... | rw [← mem_toFinset]
refine support_formPerm_le l ?_
simpa using hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Extension | {
"line": 261,
"column": 22
} | {
"line": 265,
"column": 12
} | [
{
"pp": "R : Type u_1\nN : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : LieRing M\ninst✝³ : LieAlgebra R M\ninst✝² : IsLieAbelian M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nc : ↥(twoCocycle R L M)\n⊢ { toFun := fun x ↦ ((ofProd c).... | by
rw [LieHom.range_eq_top]
intro x
use (ofProd c (x, 0))
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 195,
"column": 56
} | {
"line": 195,
"column": 77
} | [
{
"pp": "k : Type u_1\ninst✝¹⁰ : Field k\nL : Type u_2\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra k L\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : LieRingModule L V\ninst✝⁴ : LieModule k L V\ninst✝³ : CharZero k\ninst✝² : Module.Finite k V\ninst✝¹ : IsTriangularizable k L V\nA : LieIdeal ... | rw [hv' (π₁ x), this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 195,
"column": 56
} | {
"line": 195,
"column": 77
} | [
{
"pp": "k : Type u_1\ninst✝¹⁰ : Field k\nL : Type u_2\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra k L\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : LieRingModule L V\ninst✝⁴ : LieModule k L V\ninst✝³ : CharZero k\ninst✝² : Module.Finite k V\ninst✝¹ : IsTriangularizable k L V\nA : LieIdeal ... | rw [hv' (π₁ x), this] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 195,
"column": 56
} | {
"line": 195,
"column": 77
} | [
{
"pp": "k : Type u_1\ninst✝¹⁰ : Field k\nL : Type u_2\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra k L\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : LieRingModule L V\ninst✝⁴ : LieModule k L V\ninst✝³ : CharZero k\ninst✝² : Module.Finite k V\ninst✝¹ : IsTriangularizable k L V\nA : LieIdeal ... | rw [hv' (π₁ x), this] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Extension | {
"line": 348,
"column": 47
} | {
"line": 348,
"column": 63
} | [
{
"pp": "R : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : LieRing M\ninst✝¹ : LieAlgebra R M\ninst✝ : IsLieAbelian M\nE : Extension R M L\ny : M\nz : E.L\nx : M\nhx : ↑E.incl x = Exists.choose ⋯ (E.proj z) - z\n⊢ ⁅Exists.choose ⋯ (E.proj z) - z... | lie_toKer_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.LinearRecurrence | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : StrongRankCondition R\nE : LinearRecurrence R\n⊢ Module.rank R ↥E.solSpace = ↑E.order",
"usedConstants": [
"Pi.Function.module",
"Submodule",
"Fintype.card_fin",
"Semiring.toModule",
"Pi.addCommMonoid",
"rank_eq_card_bas... | simp [rank_eq_card_basis E.basis] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.LinearRecurrence | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : StrongRankCondition R\nE : LinearRecurrence R\n⊢ Module.rank R ↥E.solSpace = ↑E.order",
"usedConstants": [
"Pi.Function.module",
"Submodule",
"Fintype.card_fin",
"Semiring.toModule",
"Pi.addCommMonoid",
"rank_eq_card_bas... | simp [rank_eq_card_basis E.basis] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.LinearRecurrence | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : StrongRankCondition R\nE : LinearRecurrence R\n⊢ Module.rank R ↥E.solSpace = ↑E.order",
"usedConstants": [
"Pi.Function.module",
"Submodule",
"Fintype.card_fin",
"Semiring.toModule",
"Pi.addCommMonoid",
"rank_eq_card_bas... | simp [rank_eq_card_basis E.basis] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.LinearRecurrence | {
"line": 233,
"column": 2
} | {
"line": 234,
"column": 33
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝ : CommRing R\nE : LinearRecurrence R\nq : R\n⊢ (E.IsSolution fun n ↦ q ^ n) → q ^ E.order - ∑ x, E.coeffs x * q ^ ↑x = 0",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Finset.univ",
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemirin... | · intro h
simpa [sub_eq_zero] using h 0 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 73,
"column": 2
} | {
"line": 78,
"column": 51
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : CharZero K\ninst✝¹ : IsKilling K L\ninst✝ : IsTriangularizable K (↥H) L\nI : LieIdeal K L\nα : Weight K (↥H) L\nhα : ... | intro y hy
have : γ (coroot α) • y ∈ I.toSubmodule := by
rw [← lie_eq_smul_of_mem_rootSpace hy (coroot α)]
exact lie_mem_left K L I _ y
(I.corootSubmodule_le hα (coe_coroot_mem_corootSubmodule α))
exact I.toSubmodule.smul_mem_iff hγ_ne |>.mp this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 73,
"column": 2
} | {
"line": 78,
"column": 51
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : CharZero K\ninst✝¹ : IsKilling K L\ninst✝ : IsTriangularizable K (↥H) L\nI : LieIdeal K L\nα : Weight K (↥H) L\nhα : ... | intro y hy
have : γ (coroot α) • y ∈ I.toSubmodule := by
rw [← lie_eq_smul_of_mem_rootSpace hy (coroot α)]
exact lie_mem_left K L I _ y
(I.corootSubmodule_le hα (coe_coroot_mem_corootSubmodule α))
exact I.toSubmodule.smul_mem_iff hγ_ne |>.mp this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 260,
"column": 6
} | {
"line": 260,
"column": 14
} | [
{
"pp": "case neg\nM : Type u_1\ninst✝⁴ : CommMonoidWithZero M\ninst✝³ : IsCancelMulZero M\nN : Type u_2\ninst✝² : CommMonoidWithZero N\ninst✝¹ : UniqueFactorizationMonoid N\ninst✝ : UniqueFactorizationMonoid M\nm : Associates M\nn : Associates N\nhn : n ≠ 0\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\ns : ℕ\nhs : ¬s = 0... | ← c₂_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 401,
"column": 8
} | {
"line": 401,
"column": 50
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : CommMonoidWithZero M\ninst✝⁵ : IsCancelMulZero M\nN : Type u_2\ninst✝⁴ : CommMonoidWithZero N\ninst✝³ : Subsingleton Mˣ\ninst✝² : Subsingleton Nˣ\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : UniqueFactorizationMonoid N\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalize... | mkFactorOrderIsoOfFactorDvdEquiv_apply_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 427,
"column": 8
} | {
"line": 427,
"column": 50
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : CommMonoidWithZero M\ninst✝⁵ : IsCancelMulZero M\nN : Type u_2\ninst✝⁴ : CommMonoidWithZero N\ninst✝³ : Subsingleton Mˣ\ninst✝² : Subsingleton Nˣ\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : UniqueFactorizationMonoid N\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalize... | mkFactorOrderIsoOfFactorDvdEquiv_apply_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 259,
"column": 2
} | {
"line": 269,
"column": 21
} | [
{
"pp": "case refine_2\nA : Type u_2\nK : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI✝ : Ideal A\nhNF : ¬IsField A\nI : Ideal A\nhI0✝ : I ≠ ⊥\nhI1 : I ≠ ⊤\nhM : I.IsMaximal\nhI0 : ⊥ < I\na : A\nhaI : a ∈ I\nha0 : a ≠ 0\nJ : Id... | · rintro y₀ hy₀
obtain ⟨y, h_Iy, rfl⟩ := (mem_coeIdeal _).mp hy₀
rw [mul_comm, ← mul_assoc, ← map_mul]
have h_yb : y * b ∈ J := by
apply hle
rw [Multiset.prod_cons]
exact Submodule.smul_mem_smul h_Iy hbZ
rw [Ideal.mem_span_singleton'] at h_yb
rcases h_yb with ⟨c, hc⟩
rw [← hc, ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 323,
"column": 2
} | {
"line": 325,
"column": 53
} | [
{
"pp": "case neg\nA : Type u_2\nK : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhJ0 : ¬↑I * (↑I)⁻¹ = 0\nx : K\nhx : x ∈ (↑I * (↑I)⁻¹)⁻¹\nx_mul_mem : ∀ b ∈ (↑I)⁻¹, x * b ∈ (↑I)⁻¹\np : A[X]\nhy : (Polyno... | induction i with
| zero => rw [pow_zero]; exact one_mem_inv_coe_ideal hI0
| succ i ih => rw [pow_succ']; exact x_mul_mem _ ih | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.Module.DedekindDomain | {
"line": 47,
"column": 36
} | {
"line": 47,
"column": 55
} | [
{
"pp": "case h.e'_6.h.e'_4\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ ⊥\nhM : Module.IsTorsionBySet R M ↑I\nP : Multiset (Ideal R) := factors I\nprime_of_mem : ∀ p ∈ P.toFinset, Prime p\n⊢ (f... | ← associated_iff_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 244,
"column": 2
} | {
"line": 244,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nP' : Type u_3\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\ninst✝¹ : IsLocalization S P\ninst✝ : IsLocalization S P'\nI : FractionalIdeal S P'\n⊢ (canonicalEquiv S P P') ((canonicalEquiv S P' P)... | rw [← canonicalEquiv_symm, RingEquiv.symm_apply_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 244,
"column": 2
} | {
"line": 244,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nP' : Type u_3\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\ninst✝¹ : IsLocalization S P\ninst✝ : IsLocalization S P'\nI : FractionalIdeal S P'\n⊢ (canonicalEquiv S P P') ((canonicalEquiv S P' P)... | rw [← canonicalEquiv_symm, RingEquiv.symm_apply_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 244,
"column": 2
} | {
"line": 244,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nP' : Type u_3\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\ninst✝¹ : IsLocalization S P\ninst✝ : IsLocalization S P'\nI : FractionalIdeal S P'\n⊢ (canonicalEquiv S P P') ((canonicalEquiv S P' P)... | rw [← canonicalEquiv_symm, RingEquiv.symm_apply_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 679,
"column": 4
} | {
"line": 683,
"column": 34
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\ninst✝³ : IsLocalization S P\nP' : Type u_5\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\ny : P'\nh : y ∈ spanSingleton S ((IsLocalization.map P' (Ri... | rw [mem_canonicalEquiv_apply]
obtain ⟨z, rfl⟩ := (mem_spanSingleton _).mp h
use z • x
use (mem_spanSingleton _).mpr ⟨z, rfl⟩
simp [IsLocalization.map_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 679,
"column": 4
} | {
"line": 683,
"column": 34
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\ninst✝³ : IsLocalization S P\nP' : Type u_5\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\ny : P'\nh : y ∈ spanSingleton S ((IsLocalization.map P' (Ri... | rw [mem_canonicalEquiv_apply]
obtain ⟨z, rfl⟩ := (mem_spanSingleton _).mp h
use z • x
use (mem_spanSingleton _).mpr ⟨z, rfl⟩
simp [IsLocalization.map_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 357,
"column": 10
} | {
"line": 357,
"column": 24
} | [
{
"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 : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\nhq : ∀ (i : ↥Li... | h_pairing_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 460,
"column": 48
} | {
"line": 460,
"column": 70
} | [
{
"pp": "case a.a\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\ninst✝⁹ : AddCommGroup N\ninst✝⁸ : Module R N\nS : Submonoid R\nM' : Type u_1\ninst✝⁷ : AddCommGroup M'\ninst✝⁶ : Module R M'\nf : M →ₗ[R] M'\ninst✝⁵ : IsLocalizedModule S f\nN' : Ty... | Algebra.smul_def s₀.1, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 471,
"column": 48
} | {
"line": 471,
"column": 70
} | [
{
"pp": "case a.a\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\ninst✝⁹ : AddCommGroup N\ninst✝⁸ : Module R N\nS : Submonoid R\nM' : Type u_1\ninst✝⁷ : AddCommGroup M'\ninst✝⁶ : Module R M'\nf : M →ₗ[R] M'\ninst✝⁵ : IsLocalizedModule S f\nN' : Ty... | Algebra.smul_def s₀.1, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 555,
"column": 2
} | {
"line": 556,
"column": 75
} | [
{
"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 : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\na✝ : Nontrivial L\n⊢ (rootSystem H).IsIrredu... | have hL : ¬ IsLieAbelian L :=
(isLieAbelian_iff_subsingleton K (L := L)).not.mpr (not_subsingleton L) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Module.PID | {
"line": 213,
"column": 10
} | {
"line": 213,
"column": 59
} | [
{
"pp": "case h.succ.refine_3.refine_2.refine_1\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsPrincipalIdealRing R\ninst✝² : IsDomain R\np : R\nhp : Irreducible p\nd : ℕ\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nhM : IsTorsion' M ↥(Submonoid.powers p)\ns : Fin (d + 1) → M\nhs : Submodule.span R (... | rw [range_subtype, LinearEquiv.ker_comp, ker_mkQ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.PID | {
"line": 213,
"column": 10
} | {
"line": 213,
"column": 59
} | [
{
"pp": "case h.succ.refine_3.refine_2.refine_1\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsPrincipalIdealRing R\ninst✝² : IsDomain R\np : R\nhp : Irreducible p\nd : ℕ\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nhM : IsTorsion' M ↥(Submonoid.powers p)\ns : Fin (d + 1) → M\nhs : Submodule.span R (... | rw [range_subtype, LinearEquiv.ker_comp, ker_mkQ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.PID | {
"line": 213,
"column": 10
} | {
"line": 213,
"column": 59
} | [
{
"pp": "case h.succ.refine_3.refine_2.refine_1\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsPrincipalIdealRing R\ninst✝² : IsDomain R\np : R\nhp : Irreducible p\nd : ℕ\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nhM : IsTorsion' M ↥(Submonoid.powers p)\ns : Fin (d + 1) → M\nhs : Submodule.span R (... | rw [range_subtype, LinearEquiv.ker_comp, ker_mkQ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.PDeriv | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 33
} | [
{
"pp": "case refine_1\nR : Type u\nσ : Type v\na : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni j : σ\nx✝ : j ∈ s.support\nhne : j ≠ i\n⊢ (fun a_1 b ↦ (monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) j (s j) = 0",
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instCanonicallyOrderedAdd",... | · simp [Pi.single_eq_of_ne hne] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MvPolynomial.Localization | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 27
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na : R\nm : ↥M\n⊢ C (IsLocalization.mk' S a m * (algebraMap R S) ↑m) = C ((algebraMap R S) a)",
"usedConstants": [
"Finsupp.instAddZeroClass",
... | IsLocalization.mk'_spec | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 82
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Presentation R S ι σ\ni : σ\n⊢ (aeval P.val) (P.relation i) = 0",
"usedConstants": [
"Algebra.Generators.ker",
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgeb... | rw [← RingHom.mem_ker, ← P.ker_eq_ker_aeval_val, ← P.span_range_relation_eq_ker] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Extension.Generators | {
"line": 276,
"column": 29
} | {
"line": 276,
"column": 40
} | [
{
"pp": "case h\nR : Type u\nS : Type v\nι : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nP✝ : Generators R S ι\nι' : Type u_1\nT✝ : Type ?u.62457\ninst✝³ : CommRing T✝\ninst✝² : Algebra R T✝\nT : Type ?u.62538\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nP : Generators R S ι\na : T\nb ... | smul_tmul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Vanishing | {
"line": 157,
"column": 2
} | {
"line": 170,
"column": 9
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u_3\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_4\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nhm : span R (Set.range m) = ⊤\nhmn : ∑ i, m i ⊗ₜ[R] n i = 0\nG : (ι →₀ R) →ₗ[R... | · intro i
classical
apply_fun finsuppScalarLeft R N ι at hkn
apply_fun (· i) at hkn
symm at hkn
simp only [map_sum, finsuppScalarLeft_apply_tmul, zero_smul, Finsupp.single_zero,
Finsupp.sum_single_index, one_smul, Finsupp.finsetSum_apply, Finsupp.single_apply,
Finset.sum_ite_eq', Finset.... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 17
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝⁶ : CommRing R'\ninst✝⁵ : CommRing S'\ninst✝⁴ : Algebra R' S'\nP' : Extension R' S'\ninst✝³ : Algebra R R'\ninst✝² : Algebra S S'\ninst✝¹ : Algebra R S'\ninst✝ : I... | smul_tmul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 167,
"column": 47
} | {
"line": 167,
"column": 58
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝⁶ : CommRing R'\ninst✝⁵ : CommRing S'\ninst✝⁴ : Algebra R' S'\nP' : Extension R' S'\ninst✝³ : Algebra R R'\ninst✝² : Algebra S S'\ninst✝¹ : Algebra R S'\ninst✝ : I... | smul_tmul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 200,
"column": 6
} | {
"line": 200,
"column": 26
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝⁶ : CommRing R'\ninst✝⁵ : CommRing S'\ninst✝⁴ : Algebra R' S'\nP' : Extension R' S'\ninst✝³ : Algebra R R'\ninst✝² : Algebra S S'\ninst✝¹ : Algebra R S'\ninst✝ : I... | cotangentComplex_mk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Support | {
"line": 66,
"column": 26
} | {
"line": 66,
"column": 45
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : PrimeSpectrum R\n⊢ ¬p ∉ support R M ↔ ∃ m, ∀ r ∉ p.asIdeal, r • m ≠ 0",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring... | notMem_support_iff' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 209,
"column": 2
} | {
"line": 210,
"column": 87
} | [
{
"pp": "case h\nR : Type u\ninst✝¹⁰ : CommSemiring R\nM : Type w\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nt : Finset R\nht : Ideal.span ↑t = ⊤\nMₚ : ↥t → Type u_1\ninst✝⁷ : (g : ↥t) → AddCommMonoid (Mₚ g)\ninst✝⁶ : (g : ↥t) → Module R (Mₚ g)\nRₚ : ↥t → Type u_2\ninst✝⁵ : (g : ↥t) → CommSemiring (Rₚ g)\n... | obtain ⟨⟨_, n₁, rfl⟩, hn₁⟩ := multiple_mem_span_of_mem_localization_span S (Rₚ r)
(s₁ r : Set (Mₚ r)) (IsLocalizedModule.mk' (f r) x (1 : S)) (by rw [s₂ r]; trivial) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Support | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 93
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : PrimeSpectrum R\nhp : p ∈ support R M\nm : M\nhm : (R ∙ m).annihilator ≤ p.asIdeal\n⊢ annihilator R M ≤ p.asIdeal",
"usedConstants": [
"Submodule",
"Semiring.toModule",
"Module.annihi... | exact le_trans ((Submodule.subtype _).annihilator_le_of_injective Subtype.val_injective) hm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.ExteriorPower.Basis | {
"line": 181,
"column": 50
} | {
"line": 181,
"column": 66
} | [
{
"pp": "case h.h\nK : Type u_2\nE : Type u_4\nn : ℕ\ninst✝³ : Field K\ninst✝² : AddCommGroup E\ninst✝¹ : Module K E\nI : Type u_5\ninst✝ : LinearOrder I\nv : I → E\nhv : LinearIndependent K v\nW : Submodule K E := Submodule.span K (range v)\ni : I\n⊢ v i = ↑((Basis.span hv) i)",
"usedConstants": [
"S... | Basis.span_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorPower.Basic | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 20
} | [
{
"pp": "case h.e_6.h.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb : ℕ\na : Fin na → M\nb : Fin nb → M\n⊢ ∀ (x : Fin (na + nb)), Sum.elim a b (Fin.addCases Sum.inl Sum.inr x) = Fin.addCases a b x",
"usedConstants": [
"Sum",
"Fin.addC... | apply Fin.addCases | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.LinearDisjoint | {
"line": 426,
"column": 2
} | {
"line": 427,
"column": 31
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nM N : Submodule R S\nκ : Type u_1\nι : Type u_2\nm : Basis κ R ↥M\nn : Basis ι R ↥N\nH : LinearIndependent R fun i ↦ ↑(m i.1) * ↑(n i.2)\n⊢ M.LinearDisjoint N",
"usedConstants": [
"Submodule",
"Semiring.t... | rw [LinearIndependent] at H
exact of_basis_mul' M N m n H | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearDisjoint | {
"line": 426,
"column": 2
} | {
"line": 427,
"column": 31
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nM N : Submodule R S\nκ : Type u_1\nι : Type u_2\nm : Basis κ R ↥M\nn : Basis ι R ↥N\nH : LinearIndependent R fun i ↦ ↑(m i.1) * ↑(n i.2)\n⊢ M.LinearDisjoint N",
"usedConstants": [
"Submodule",
"Semiring.t... | rw [LinearIndependent] at H
exact of_basis_mul' M N m n H | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.LocalRing.Module | {
"line": 280,
"column": 32
} | {
"line": 280,
"column": 34
} | [
{
"pp": "case insert.specialize_1\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : Flat R M\nι : Type u\nf : ι → R\nn✝ : ι\ns : Finset ι\nhn : n✝ ∉ s\nih : ∀ (v : ι → M), LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v) → ∑ i ... | c, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.LocalRing.Module | {
"line": 282,
"column": 56
} | {
"line": 282,
"column": 58
} | [
{
"pp": "case insert.specialize_2\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : Flat R M\nι : Type u\nf : ι → R\nn✝ : ι\ns : Finset ι\nhn : n✝ ∉ s\nih : ∀ (v : ι → M), LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v) → ∑ i ... | c, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Ideal.MinimalPrime.Colon | {
"line": 33,
"column": 2
} | {
"line": 98,
"column": 59
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nN : Submodule R M\nI : Ideal R\nx : M\ninst✝ : IsNoetherianRing R\nhI : I ∈ (N.colon {x}).minimalPrimes\n⊢ ∃ x', I = N.colon {x'}",
"usedConstants": [
"Ideal.fg_of_isNoetherianRing",
"Mat... | by_cases hx : x ∈ N
· simp [show (colon N {x}) = ⊤ by simpa, Ideal.minimalPrimes_top] at hI
classical
-- `I` is a minimal prime over `ann = colon N {x}`
set ann := colon N {x}
-- there exists an integer `n ≠ 0` and an ideal `J` satisfying `I ^ n * J ≤ ann` and `¬ J ≠ I`
have key : ∃ n ≠ 0, ∃ J : Ideal R, I ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.MinimalPrime.Colon | {
"line": 33,
"column": 2
} | {
"line": 98,
"column": 59
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nN : Submodule R M\nI : Ideal R\nx : M\ninst✝ : IsNoetherianRing R\nhI : I ∈ (N.colon {x}).minimalPrimes\n⊢ ∃ x', I = N.colon {x'}",
"usedConstants": [
"Ideal.fg_of_isNoetherianRing",
"Mat... | by_cases hx : x ∈ N
· simp [show (colon N {x}) = ⊤ by simpa, Ideal.minimalPrimes_top] at hI
classical
-- `I` is a minimal prime over `ann = colon N {x}`
set ann := colon N {x}
-- there exists an integer `n ≠ 0` and an ideal `J` satisfying `I ^ n * J ≤ ann` and `¬ J ≠ I`
have key : ∃ n ≠ 0, ∃ J : Ideal R, I ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 243,
"column": 4
} | {
"line": 243,
"column": 66
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsNoetherianRing R\n⊢ {r | ∃ x, x ≠ 0 ∧ r • x = 0} = (↑(nonZeroDivisors R))ᶜ",
"usedConstants": [
"Set.ext",
"SetLike.mem_coe._simp_1",
"instHSMul",
"Semiring.toModule",
"HMul.hMul",
"MulZeroClass.toMul",
"cong... | ext; simp [← nonZeroDivisorsLeft_eq_nonZeroDivisors, and_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 243,
"column": 4
} | {
"line": 243,
"column": 66
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsNoetherianRing R\n⊢ {r | ∃ x, x ≠ 0 ∧ r • x = 0} = (↑(nonZeroDivisors R))ᶜ",
"usedConstants": [
"Set.ext",
"SetLike.mem_coe._simp_1",
"instHSMul",
"Semiring.toModule",
"HMul.hMul",
"MulZeroClass.toMul",
"cong... | ext; simp [← nonZeroDivisorsLeft_eq_nonZeroDivisors, and_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.UniqueFactorizationDomain.ClassGroup | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Nonempty (NormalizedGCDMonoid R)\nJ : Ideal R\nthis : NormalizedGCDMonoid R := Classical.choice ⋯\nK : Ideal R\nhJK0 : J * K ≠ 0\nhK : Submodule.IsPrincipal (J * K)\nx : R\nhJK : J * K = R ∙ x\nhxmemJK : x ∈ J * K\nT : Finset R\nhTK : ↑T ⊆... | apply dvd_mul_of_dvd_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.LocallyConstant.Basic | {
"line": 147,
"column": 2
} | {
"line": 149,
"column": 52
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : PreconnectedSpace X\ninst✝ : Nonempty Y\nf : X → Y\nhf : IsLocallyConstant f\n⊢ ∃ y, f = Function.const X y",
"usedConstants": [
"IsEmpty.elim",
"Exists",
"isEmpty_or_nonempty",
"IsEmpty",
"Or.casesOn",
... | rcases isEmpty_or_nonempty X with h | h
· exact ⟨Classical.arbitrary Y, funext <| h.elim⟩
· exact ⟨f (Classical.arbitrary X), hf.eq_const _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocallyConstant.Basic | {
"line": 147,
"column": 2
} | {
"line": 149,
"column": 52
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : PreconnectedSpace X\ninst✝ : Nonempty Y\nf : X → Y\nhf : IsLocallyConstant f\n⊢ ∃ y, f = Function.const X y",
"usedConstants": [
"IsEmpty.elim",
"Exists",
"isEmpty_or_nonempty",
"IsEmpty",
"Or.casesOn",
... | rcases isEmpty_or_nonempty X with h | h
· exact ⟨Classical.arbitrary Y, funext <| h.elim⟩
· exact ⟨f (Classical.arbitrary X), hf.eq_const _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 136,
"column": 57
} | {
"line": 136,
"column": 92
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nS : Submonoid R\np : PrimeSpectrum (Localization S)\np' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal\nhp' : S ≤ p'.primeCompl\nRₚ : Type uR := Localization.AtPrime p'\nMₚ : Type (max uR uM)... | rw [← mul_smul, ← Algebra.smul_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 136,
"column": 57
} | {
"line": 136,
"column": 92
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nS : Submonoid R\np : PrimeSpectrum (Localization S)\np' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal\nhp' : S ≤ p'.primeCompl\nRₚ : Type uR := Localization.AtPrime p'\nMₚ : Type (max uR uM)... | rw [← mul_smul, ← Algebra.smul_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 136,
"column": 57
} | {
"line": 136,
"column": 92
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nS : Submonoid R\np : PrimeSpectrum (Localization S)\np' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal\nhp' : S ≤ p'.primeCompl\nRₚ : Type uR := Localization.AtPrime p'\nMₚ : Type (max uR uM)... | rw [← mul_smul, ← Algebra.smul_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 194,
"column": 2
} | {
"line": 196,
"column": 36
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : FinitePresentation R M\nx✝ : ↑(freeLocus R M)\nx : PrimeSpectrum R\nhx : x ∈ freeLocus R M\nthis : Free (Localization.AtPrime x.asIdeal) (LocalizedModule x.asIdeal.primeCompl M)\n⊢ ∃ U, IsOpen U ∧ ⟨x, h... | obtain ⟨f, hf, hf', hf''⟩ := Module.FinitePresentation.exists_free_localizedModule_powers
x.asIdeal.primeCompl (LocalizedModule.mkLinearMap x.asIdeal.primeCompl M)
(Localization.AtPrime x.asIdeal) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 213,
"column": 57
} | {
"line": 213,
"column": 92
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : FinitePresentation R M\nx✝ : ↑(freeLocus R M)\nx : PrimeSpectrum R\nhx : x ∈ freeLocus R M\nthis✝³ : Free (Localization.AtPrime x.asIdeal) (LocalizedModule x.asIdeal.primeCompl M)\nf : R\nhf : f ∈ x.asI... | rw [← mul_smul, ← Algebra.smul_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 213,
"column": 57
} | {
"line": 213,
"column": 92
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : FinitePresentation R M\nx✝ : ↑(freeLocus R M)\nx : PrimeSpectrum R\nhx : x ∈ freeLocus R M\nthis✝³ : Free (Localization.AtPrime x.asIdeal) (LocalizedModule x.asIdeal.primeCompl M)\nf : R\nhf : f ∈ x.asI... | rw [← mul_smul, ← Algebra.smul_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 213,
"column": 57
} | {
"line": 213,
"column": 92
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : FinitePresentation R M\nx✝ : ↑(freeLocus R M)\nx : PrimeSpectrum R\nhx : x ∈ freeLocus R M\nthis✝³ : Free (Localization.AtPrime x.asIdeal) (LocalizedModule x.asIdeal.primeCompl M)\nf : R\nhf : f ∈ x.asI... | rw [← mul_smul, ← Algebra.smul_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Int.Basic | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 46
} | [
{
"pp": "m : ℤ\np : ℕ\nhp : Nat.Prime p\nh : ↑p ∣ 2 * m ^ 2\n⊢ p = 2 ∨ p ∣ m.natAbs",
"usedConstants": [
"Int.Prime.dvd_mul",
"instOfNatNat",
"Int",
"Int.instMonoid",
"Monoid.toPow",
"HPow.hPow",
"instOfNat",
"Nat",
"instHPow",
"OfNat.ofNat"
]
... | rcases Int.Prime.dvd_mul hp h with hp2 | hpp | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.RingTheory.PicardGroup | {
"line": 120,
"column": 70
} | {
"line": 120,
"column": 97
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\nP : Type u_2\nQ : Type u_3\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : AddCommMonoid Q\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\ne : M ⊗[R] N ≃ₗ[R] R\nx✝ ... | LinearEquiv.coe_toLinearMap | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 309,
"column": 4
} | {
"line": 319,
"column": 59
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\ni : ℕ\nh : i < q - 1\nhi : ¬i = 0\n⊢ ∑ x, x ^ i = 0",
"usedConstants": [
"Units.val",
"Eq.mpr",
"NegZeroClass.toNeg",
"Trans.trans",
"Preorder.toLT",
"Dvd.dvd",
"NonUnitalCommRing.toNonUnitalNonAs... | have hiq : ¬q - 1 ∣ i := by contrapose! h; exact Nat.le_of_dvd (Nat.pos_of_ne_zero hi) h
let φ : Kˣ ↪ K := ⟨fun x ↦ x, Units.val_injective⟩
have : univ.map φ = univ \ {0} := by
ext x
simpa only [mem_map, mem_univ, Function.Embedding.coeFn_mk, true_and, mem_sdiff,
mem_singleton, φ] using isUn... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Finite.Basic | {
"line": 309,
"column": 4
} | {
"line": 319,
"column": 59
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\ni : ℕ\nh : i < q - 1\nhi : ¬i = 0\n⊢ ∑ x, x ^ i = 0",
"usedConstants": [
"Units.val",
"Eq.mpr",
"NegZeroClass.toNeg",
"Trans.trans",
"Preorder.toLT",
"Dvd.dvd",
"NonUnitalCommRing.toNonUnitalNonAs... | have hiq : ¬q - 1 ∣ i := by contrapose! h; exact Nat.le_of_dvd (Nat.pos_of_ne_zero hi) h
let φ : Kˣ ↪ K := ⟨fun x ↦ x, Units.val_injective⟩
have : univ.map φ = univ \ {0} := by
ext x
simpa only [mem_map, mem_univ, Function.Embedding.coeFn_mk, true_and, mem_sdiff,
mem_singleton, φ] using isUn... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Finite.Basic | {
"line": 479,
"column": 2
} | {
"line": 479,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\nf : K[X]\np : ℕ\nhp✝ : CharP K p\nn : ℕ\nnpos : 0 < n\nhp : Nat.Prime p\nhn : q = p ^ ↑⟨n, npos⟩\n⊢ (expand K q) f = f ^ q",
"usedConstants": [
"Nat.Prime",
"Fact.mk"
]
}
] | haveI : Fact p.Prime := ⟨hp⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.FieldTheory.Finite.Basic | {
"line": 635,
"column": 2
} | {
"line": 635,
"column": 30
} | [
{
"pp": "p : ℕ\nhp : Nat.Prime p\nn : ℤ\nhpn : IsCoprime n ↑p\n⊢ n ^ (p - 1) ≡ 1 [ZMOD ↑p]",
"usedConstants": [
"Nat.Prime",
"Fact.mk"
]
}
] | haveI : Fact p.Prime := ⟨hp⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.FieldTheory.Finite.Basic | {
"line": 646,
"column": 2
} | {
"line": 646,
"column": 30
} | [
{
"pp": "p : ℕ\nhp : Nat.Prime p\nn : ℤ\n⊢ n ^ p ≡ n [ZMOD ↑p]",
"usedConstants": [
"Nat.Prime",
"Fact.mk"
]
}
] | haveI : Fact p.Prime := ⟨hp⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative | {
"line": 78,
"column": 69
} | {
"line": 91,
"column": 37
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\nP : α → Prop\na : α\nh0 : P 0\nh1 : ∀ {x : α}, IsUnit x → P x\nhpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)\nhcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)\n⊢ P a",
"usedConstants": [
"UniqueFactoriza... | by
letI := Classical.decEq α
have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
rintro x y ⟨u, rfl⟩ hx
exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)
by_cases ha0 : a = 0
· rwa [ha0]
haveI : Nontrivial α := ⟨⟨_, _, ha0⟩⟩
letI : NormalizationMonoid α := Uniq... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PicardGroup | {
"line": 785,
"column": 44
} | {
"line": 785,
"column": 55
} | [
{
"pp": "case h.e'_3.h\nR : Type u\nM : Type v\nN : Type u_1\nP : Type u_2\nQ : Type u_3\nA : Type u_4\ninst✝¹² : CommSemiring R\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid N\ninst✝⁹ : AddCommMonoid P\ninst✝⁸ : AddCommMonoid Q\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module R P\ninst✝⁴ : Modu... | smul_tmul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative | {
"line": 102,
"column": 2
} | {
"line": 112,
"column": 79
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_3\ninst✝ : CommMonoidWithZero β\nf : α → β\ns : Finset α\ni j : α → ℕ\nis_prime : ∀ p ∈ s, Prime p\nis_coprime : ∀ p ∈ s, ∀ q ∈ s, p ∣ q → p = q\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p ... | induction s using Finset.induction_on with
| empty => simpa using h1 isUnit_one
| insert p s hps ih =>
have hpr_p := is_prime _ (Finset.mem_insert_self _ _)
have hpr_s : ∀ p ∈ s, Prime p := fun p hp => is_prime _ (Finset.mem_insert_of_mem hp)
have hcp_p := fun i => prime_pow_coprime_prod_of_coprime_inse... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.Norm.Basic | {
"line": 232,
"column": 8
} | {
"line": 232,
"column": 45
} | [
{
"pp": "case refine_2\nA₁ : Type u_8\nB₁ : Type u_9\nA₂ : Type u_10\nB₂ : Type u_11\ninst✝⁵ : CommRing A₁\ninst✝⁴ : Ring B₁\ninst✝³ : CommRing A₂\ninst✝² : Ring B₂\ninst✝¹ : Algebra A₁ B₁\ninst✝ : Algebra A₂ B₂\ne₁ : A₁ ≃+* A₂\ne₂ : B₁ ≃+* B₂\nhe : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)\nx... | ← Algebra.norm_eq_of_ringEquiv e₁ he, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 156,
"column": 6
} | {
"line": 156,
"column": 59
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsSeparable F E\nhα : IsIntegral F α\nhβ : IsIntegral F β\nf : F[X] := minpoly F α\ng : F[X] := minpoly F β\nιFE : F →+* E := algebraMap F E\nιEE' : E →+* (Polynomial.map ... | exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 179,
"column": 2
} | {
"line": 199,
"column": 62
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Finite (IntermediateField F E)\n⊢ ∃ γ, F⟮α, β⟯ = F⟮γ⟯",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"IntermediateField.inst... | let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_s... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 179,
"column": 2
} | {
"line": 199,
"column": 62
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Finite (IntermediateField F E)\n⊢ ∃ γ, F⟮α, β⟯ = F⟮γ⟯",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"IntermediateField.inst... | let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯
obtain ⟨x, y, hneq, heq⟩ := Finite.exists_ne_map_eq_of_infinite f
use α + x • β
apply le_antisymm
· rw [adjoin_le_iff]
have αxβ_in_K : α + x • β ∈ F⟮α + x • β⟯ := mem_adjoin_simple_self F _
have αyβ_in_K : α + y • β ∈ F⟮α + y • β⟯ := mem_adjoin_s... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 170,
"column": 53
} | {
"line": 170,
"column": 75
} | [
{
"pp": "case succ.refine_1\nS : Type u_1\ninst✝¹ : CommRing S\nP : Ideal S\nP_prime : P.IsPrime\ninst✝ : IsDedekindDomain S\nhP : P ≠ ⊥\ni : ℕ\nih : cardQuot (P ^ i) = cardQuot P ^ i\nthis : P ^ (i + 1) < P ^ i\na : S\na_mem : a ∈ P ^ i\na_notMem : a ∉ P ^ (i + 1)\nf g : (c : S) → c ∈ P ^ i → S\nhg : ∀ (c : S)... | ← hf _ (hk_mem _ hc₂') | Lean.Elab.Tactic.evalRewriteSeq | null |
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