module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 488,
"column": 6
} | {
"line": 489,
"column": 79
} | {
"line": 490,
"column": 4
} | [
{
"pp": "case inl\nG : Type u\nH : Type v\ninst✝² : Mul G\ninst✝¹ : Mul H\nf : H →ₙ* G\nhf : ∀ ⦃a b c d : H⦄, a * b = c * d → f a = f c ∧ f b = f d → a = c ∧ b = d\ninst✝ : TwoUniqueProds G\nA B : Finset H\nhc : 1 < #A * #B\nhc' : 1 < #(image (⇑f) A) * #(image (⇑f) B)\na1 : H\nha1 : a1 ∈ A\nb1 : H\nhb1 : b1 ∈ B... | [] | exact ⟨(a1, b1), ⟨ha1, hb1⟩, (a2, b2), ⟨ha2, hb2⟩, mt (congr_arg (Prod.map f f)) hne,
UniqueMul.of_mulHom_image f hf hu1, UniqueMul.of_mulHom_image f hf hu2⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.Submodule.Invariant | {
"line": 207,
"column": 44
} | {
"line": 207,
"column": 63
} | {
"line": 207,
"column": 63
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nG : Type u_3\ninst✝² : Monoid G\ninst✝¹ : DistribMulAction G M\ninst✝ : SMulCommClass G R M\nx : M\ng : G\n⊢ MulAction.orbit G x ⊆ ↑(Submodule.comap (DistribSMul.toLinearMap R M g) (span R (MulAction.orbit G... | [
"R : Type u_1\nM : Type u_2\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nG : Type u_3\ninst✝² : Monoid G\ninst✝¹ : DistribMulAction G M\ninst✝ : SMulCommClass G R M\nx : M\ng : G\n⊢ MulAction.orbit G x ⊆ ⇑(DistribSMul.toLinearMap R M g) ⁻¹' ↑(span R (MulAction.orbit G x))"
] | Submodule.comap_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Projection | {
"line": 188,
"column": 42
} | {
"line": 189,
"column": 66
} | {
"line": 191,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\np q : Submodule R E\nh : IsCompl p q\nx : E\n⊢ (p.projectionOnto q h) x = 0 ↔ x ∈ q",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Submodule",
"congrArg",
"AddCommGroup.to... | [] | by
simp [projectionOnto, prodEquivOfIsCompl_symm_apply_fst_eq_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Projection | {
"line": 775,
"column": 4
} | {
"line": 775,
"column": 46
} | {
"line": 775,
"column": 47
} | [
{
"pp": "E : Type u_1\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nf : E →ₗ[R] E\nhf : IsIdempotentElem f\nT : (E →ₗ[R] E)ˣ\n⊢ Commute f ↑((GeneralLinearGroup.generalLinearEquiv R E) T) ↔\n (map (↑((GeneralLinearGroup.generalLinearEquiv R E) T)) f.range ≤ f.range ∧\n f.... | [
"E : Type u_1\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nf : E →ₗ[R] E\nhf : IsIdempotentElem f\nT : (E →ₗ[R] E)ˣ\n⊢ Commute f ↑((GeneralLinearGroup.generalLinearEquiv R E) T) ↔\n (f.range ∈ Module.End.invtSubmodule ↑((GeneralLinearGroup.generalLinearEquiv R E) T) ∧\n f.r... | ← Module.End.mem_invtSubmodule_iff_map_le, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.RelSeries | {
"line": 71,
"column": 46
} | {
"line": 71,
"column": 63
} | {
"line": 71,
"column": 63
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\nx y : RelSeries r\nlength_eq : x.length = y.length\n⊢ x.length + 1 = y.length + 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"RelSeries.length",
"instOfNatNat... | [] | by rw [length_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Exact.Basic | {
"line": 382,
"column": 79
} | {
"line": 411,
"column": 16
} | {
"line": 413,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\nP : Type u_6\nP' : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑... | [] | by
refine
{ toFun := fun l ↦ ⟨(LinearEquiv.ofBijective (f ∘ₗ fst R M P + l.1 ∘ₗ snd R M P) ?_).symm, ?_⟩
invFun := fun e ↦ ⟨e.1.symm ∘ₗ inr R M P, ?_⟩
left_inv := ?_
right_inv := ?_ }
· have h₁ : ∀ x, g (l.1 x) = x := LinearMap.congr_fun l.2
have h₂ : ∀ x, g (f x) = 0 := congr_fun h.comp_eq_zero
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.RelSeries | {
"line": 846,
"column": 4
} | {
"line": 846,
"column": 58
} | {
"line": 846,
"column": 58
} | [
{
"pp": "α : Type u_1\nr✝ : SetRel α α\nβ : Type u_2\ns : SetRel β β\nγ : Type u_3\ninst✝¹ : Preorder γ\ninst✝ : Unique γ\nx : RelSeries {(a, b) | a < b}\nr : (RelSeries.singleton {(a, b) | a < b} default).length < x.length\n⊢ False",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Pr... | [] | exact (x.step ⟨0, by lia⟩).ne <| Subsingleton.elim _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.KrullDimension | {
"line": 501,
"column": 4
} | {
"line": 501,
"column": 63
} | {
"line": 502,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\nhfin : height x < ⊤\nm : ℕ\nhx : height x = ↑m\nh : n < m\n⊢ ∃ y < x, height y = ↑n",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"RelSeries.last",
"ENat.instNatCast",
"setOf",
"Finite... | [
"α : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\nhfin : height x < ⊤\nm : ℕ\nhx : height x = ↑m\nh : n < m\np : LTSeries α\nhp : RelSeries.last p = x\nhlen : p.length = m\n⊢ ∃ y < x, height y = ↑n"
] | obtain ⟨p, hp, hlen⟩ := exists_series_of_height_eq_coe x hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.TensorProduct.Basic | {
"line": 101,
"column": 20
} | {
"line": 101,
"column": 65
} | {
"line": 103,
"column": 0
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR' : Type u_4\nR'' : Type u_5\ninst✝³⁶ : CommSemiring R\ninst✝³⁵ : CommSemiring R₂\ninst✝³⁴ : CommSemiring R₃\ninst✝³³ : Monoid R'\ninst✝³² : Semiring R''\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nA : Type u_6\nM : Type u_7\nN : Type u_8\nP : Type u_9... | [] | dsimp; rw [LinearMap.map_smulₛₗ₂, map_smulₛₗ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.Basic | {
"line": 101,
"column": 20
} | {
"line": 101,
"column": 65
} | {
"line": 103,
"column": 0
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR' : Type u_4\nR'' : Type u_5\ninst✝³⁶ : CommSemiring R\ninst✝³⁵ : CommSemiring R₂\ninst✝³⁴ : CommSemiring R₃\ninst✝³³ : Monoid R'\ninst✝³² : Semiring R''\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nA : Type u_6\nM : Type u_7\nN : Type u_8\nP : Type u_9... | [] | dsimp; rw [LinearMap.map_smulₛₗ₂, map_smulₛₗ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.KrullDimension | {
"line": 606,
"column": 29
} | {
"line": 606,
"column": 50
} | {
"line": 606,
"column": 51
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\n⊢ krullDim α < 0 ↔ krullDim α = ⊥",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot.some",
"WithBot",
"Preorder.toLT",
"instLinearOrderENat",
"congrArg",
"Com... | [
"α : Type u_1\ninst✝ : Preorder α\n⊢ krullDim α < 0 ↔ krullDim α < ↑⊥"
] | ← WithBot.lt_coe_bot, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Defs | {
"line": 268,
"column": 74
} | {
"line": 270,
"column": 35
} | {
"line": 272,
"column": 0
} | [
{
"pp": "R : Type u_1\nR'' : Type u_5\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring R''\nM : Type u_7\nN : Type u_8\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R'' M\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : SMulCommClass R R'' M\nr s : R''\nx✝ : M ⊗[R] N\nthis : ∀ (r : R'') (m :... | [] | by
simp_rw [TensorProduct.smul_add]
rw [ihx, ihy, add_add_add_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.KrullDimension | {
"line": 849,
"column": 13
} | {
"line": 849,
"column": 49
} | {
"line": 850,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : Preorder α\nh : IsEmpty α\n⊢ krullDim α = ⨆ a, ↑(height a)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"WithBot.instSupSet",
"Eq.mpr",
"WithBot.some",
"WithBot",
"Lattice.toSemilatticeSup",
"instCompleteLinearO... | [] | rw [krullDim_eq_bot, ciSup_of_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.KrullDimension | {
"line": 849,
"column": 13
} | {
"line": 849,
"column": 49
} | {
"line": 850,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : Preorder α\nh : IsEmpty α\n⊢ krullDim α = ⨆ a, ↑(height a)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"WithBot.instSupSet",
"Eq.mpr",
"WithBot.some",
"WithBot",
"Lattice.toSemilatticeSup",
"instCompleteLinearO... | [] | rw [krullDim_eq_bot, ciSup_of_empty] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.KrullDimension | {
"line": 849,
"column": 13
} | {
"line": 849,
"column": 49
} | {
"line": 850,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : Preorder α\nh : IsEmpty α\n⊢ krullDim α = ⨆ a, ↑(height a)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"WithBot.instSupSet",
"Eq.mpr",
"WithBot.some",
"WithBot",
"Lattice.toSemilatticeSup",
"instCompleteLinearO... | [] | rw [krullDim_eq_bot, ciSup_of_empty] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.KrullDimension | {
"line": 860,
"column": 13
} | {
"line": 860,
"column": 49
} | {
"line": 861,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : Preorder α\nh : IsEmpty α\n⊢ krullDim α = ⨆ a, ↑(coheight a)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"WithBot.instSupSet",
"Eq.mpr",
"WithBot.some",
"WithBot",
"Lattice.toSemilatticeSup",
"instCompleteLinea... | [] | rw [krullDim_eq_bot, ciSup_of_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.KrullDimension | {
"line": 860,
"column": 13
} | {
"line": 860,
"column": 49
} | {
"line": 861,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : Preorder α\nh : IsEmpty α\n⊢ krullDim α = ⨆ a, ↑(coheight a)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"WithBot.instSupSet",
"Eq.mpr",
"WithBot.some",
"WithBot",
"Lattice.toSemilatticeSup",
"instCompleteLinea... | [] | rw [krullDim_eq_bot, ciSup_of_empty] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.KrullDimension | {
"line": 860,
"column": 13
} | {
"line": 860,
"column": 49
} | {
"line": 861,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : Preorder α\nh : IsEmpty α\n⊢ krullDim α = ⨆ a, ↑(coheight a)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"WithBot.instSupSet",
"Eq.mpr",
"WithBot.some",
"WithBot",
"Lattice.toSemilatticeSup",
"instCompleteLinea... | [] | rw [krullDim_eq_bot, ciSup_of_empty] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.KrullDimension | {
"line": 1067,
"column": 2
} | {
"line": 1067,
"column": 94
} | {
"line": 1068,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\n⊢ krullDim (WithTop α) = krullDim α + 1",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithBot.some",
"WithBot",
"Preorder.toLT",
"instCompleteLinearOrderENat",
"instAddMonoidWit... | [
"α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\n⊢ ↑(⨆ y, ⨆ (_ : y < ⊤), height y + 1) = ↑(⨆ a, height a) + 1"
] | rw [← height_top_eq_krullDim, krullDim_eq_iSup_height_of_nonempty, height_eq_iSup_lt_height] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.KrullDimension | {
"line": 1066,
"column": 87
} | {
"line": 1073,
"column": 6
} | {
"line": 1075,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\n⊢ krullDim (WithTop α) = krullDim α + 1",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"Eq.mpr",
"WithBot.some",
"WithBot",
"Preorder.toLT",
"instCompleteLinearO... | [] | by
rw [← height_top_eq_krullDim, krullDim_eq_iSup_height_of_nonempty, height_eq_iSup_lt_height]
norm_cast
simp_rw [WithTop.lt_top_iff_ne_top]
rw [ENat.iSup_add, iSup_subtype']
symm
apply Equiv.withTopSubtypeNe.symm.iSup_congr
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.TensorProduct.Map | {
"line": 579,
"column": 2
} | {
"line": 580,
"column": 30
} | {
"line": 582,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Type u_7\nN : Type u_8\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\nf : N →ₗ[R] N\nn : ℕ\n⊢ lTensor M f ^ n = lTensor M (f ^ n)",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"Linear... | [] | have h := TensorProduct.map_pow (id : M →ₗ[R] M) f n
rwa [Module.End.id_pow] at h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.Map | {
"line": 579,
"column": 2
} | {
"line": 580,
"column": 30
} | {
"line": 582,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Type u_7\nN : Type u_8\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\nf : N →ₗ[R] N\nn : ℕ\n⊢ lTensor M f ^ n = lTensor M (f ^ n)",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"Linear... | [] | have h := TensorProduct.map_pow (id : M →ₗ[R] M) f n
rwa [Module.End.id_pow] at h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Operations | {
"line": 329,
"column": 26
} | {
"line": 329,
"column": 45
} | {
"line": 329,
"column": 46
} | [
{
"pp": "R : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ M ^ 0 * M = M",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Submodule.pow_zero",
"Eq.mpr",
"Submodule",
"HMul.hMul",
... | [
"R : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ 1 * M = M"
] | Submodule.pow_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Operations | {
"line": 346,
"column": 18
} | {
"line": 346,
"column": 37
} | {
"line": 346,
"column": 38
} | [
{
"pp": "case succ.zero\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\nih : 0 ≠ 0 → (M ^ 0).toAddSubmonoid = M.toAddSubmonoid ^ 0\nh : 0 + 1 ≠ 0\n⊢ (M ^ 0).toAddSubmonoid * M.toAddSubmonoid = M.toAddSubmonoid ^ 0 * M.toAddS... | [
"case succ.zero\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\nih : 0 ≠ 0 → (M ^ 0).toAddSubmonoid = M.toAddSubmonoid ^ 0\nh : 0 + 1 ≠ 0\n⊢ toAddSubmonoid 1 * M.toAddSubmonoid = M.toAddSubmonoid ^ 0 * M.toAddSubmonoid"
] | Submodule.pow_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Operations | {
"line": 351,
"column": 8
} | {
"line": 351,
"column": 27
} | {
"line": 351,
"column": 28
} | [
{
"pp": "case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ M.toAddSubmonoid ^ 0 ≤ (M ^ 0).toAddSubmonoid",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Submodule.pow_zero",
"Eq.mpr... | [
"case inl\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\n⊢ M.toAddSubmonoid ^ 0 ≤ toAddSubmonoid 1"
] | Submodule.pow_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Coprime.Lemmas | {
"line": 273,
"column": 2
} | {
"line": 275,
"column": 70
} | {
"line": 276,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_2\nI : Type u_1\ninst✝² : CommMonoid α\ninst✝¹ : DecompositionMonoid α\ns : I → α\nt : Finset I\ninst✝ : DecidableEq I\nhp : Pairwise (IsRelPrime on fun i ↦ s ↑i)\ni : I\nhi : i ∈ t\nj : I\nhj : j ∈ t \\ {i}\n⊢ IsRelPrime (s i) (s j)",
"ppTerm": "?refine_1",
"assigned"... | [
"case refine_2\nα : Type u_2\nI : Type u_1\ninst✝² : CommMonoid α\ninst✝¹ : DecompositionMonoid α\ns : I → α\nt : Finset I\ninst✝ : DecidableEq I\nhp : ∀ i ∈ t, IsRelPrime (s i) (∏ j ∈ t \\ {i}, s j)\n⊢ Pairwise (IsRelPrime on fun i ↦ s ↑i)"
] | · rw [Finset.mem_sdiff, Finset.mem_singleton] at hj
obtain ⟨hj, ji⟩ := hj
exact @hp ⟨i, hi⟩ ⟨j, hj⟩ fun h ↦ ji (congrArg Subtype.val h).symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Algebra.Operations | {
"line": 612,
"column": 16
} | {
"line": 612,
"column": 50
} | {
"line": 613,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nN P : Submodule R A\nh : N * P = 1\n| ⊤",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Submodule",
"MulOne.toOne",
"Semigroup.toMul",
"HMul.hMul",
"IsScalarTo... | [
"R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nN P : Submodule R A\nh : N * P = 1\n| ⊤ * N * P"
] | rw [← mul_one ⊤, ← h, ← mul_assoc] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Algebra.Algebra.Operations | {
"line": 612,
"column": 16
} | {
"line": 612,
"column": 50
} | {
"line": 613,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nN P : Submodule R A\nh : N * P = 1\n| ⊤",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Submodule",
"MulOne.toOne",
"Semigroup.toMul",
"HMul.hMul",
"IsScalarTo... | [
"R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nN P : Submodule R A\nh : N * P = 1\n| ⊤ * N * P"
] | rw [← mul_one ⊤, ← h, ← mul_assoc] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Algebra.Algebra.Operations | {
"line": 612,
"column": 16
} | {
"line": 612,
"column": 50
} | {
"line": 613,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nN P : Submodule R A\nh : N * P = 1\n| ⊤",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Submodule",
"MulOne.toOne",
"Semigroup.toMul",
"HMul.hMul",
"IsScalarTo... | [
"R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nN P : Submodule R A\nh : N * P = 1\n| ⊤ * N * P"
] | rw [← mul_one ⊤, ← h, ← mul_assoc] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.RingTheory.Noetherian.Basic | {
"line": 292,
"column": 37
} | {
"line": 303,
"column": 90
} | {
"line": 305,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsNoetherian R M\nf : ℕ → Submodule R M\nh : ∀ (n : ℕ), Disjoint ((partialSups f) n) (f (n + 1))\n⊢ ∃ n, ∀ (m : ℕ), n ≤ m → f m = ⊥",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants":... | [] | by
-- A little off-by-one cleanup first:
suffices t : ∃ n : ℕ, ∀ m, n ≤ m → f (m + 1) = ⊥ by
obtain ⟨n, w⟩ := t
use n + 1
rintro (_ | m) p
· cases p
· apply w
exact Nat.succ_le_succ_iff.mp p
obtain ⟨n, w⟩ := monotone_stabilizes_iff_noetherian.mpr inferInstance (partialSups f)
refine ⟨n... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Maps | {
"line": 544,
"column": 57
} | {
"line": 544,
"column": 67
} | {
"line": 544,
"column": 67
} | [
{
"pp": "R : Type u\nS : Type v\nF : Type u_1\ninst✝³ : Semiring R\ninst✝² : Semiring S\ninst✝¹ : FunLike F R S\nf : F\ninst✝ : RingHomClass F R S\nhf : Function.Bijective ⇑f\nI : Ideal R\n⊢ ⊤ ≤ I ↔ I = ⊤",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toMod... | [
"R : Type u\nS : Type v\nF : Type u_1\ninst✝³ : Semiring R\ninst✝² : Semiring S\ninst✝¹ : FunLike F R S\nf : F\ninst✝ : RingHomClass F R S\nhf : Function.Bijective ⇑f\nI : Ideal R\n⊢ I = ⊤ ↔ I = ⊤"
] | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 249,
"column": 52
} | {
"line": 249,
"column": 90
} | {
"line": 249,
"column": 90
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nhxy : ¬gcd x y = 0\nz : R\nhz : x = gcd x y * z\n⊢ gcd x y * (y * z) = x * y",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"CommRing.toNonUni... | [] | rw [← mul_assoc, mul_right_comm, ← hz] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 249,
"column": 52
} | {
"line": 249,
"column": 90
} | {
"line": 249,
"column": 90
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nhxy : ¬gcd x y = 0\nz : R\nhz : x = gcd x y * z\n⊢ gcd x y * (y * z) = x * y",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"CommRing.toNonUni... | [] | rw [← mul_assoc, mul_right_comm, ← hz] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 249,
"column": 52
} | {
"line": 249,
"column": 90
} | {
"line": 249,
"column": 90
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nhxy : ¬gcd x y = 0\nz : R\nhz : x = gcd x y * z\n⊢ gcd x y * (y * z) = x * y",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"CommRing.toNonUni... | [] | rw [← mul_assoc, mul_right_comm, ← hz] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Operations | {
"line": 115,
"column": 8
} | {
"line": 115,
"column": 29
} | {
"line": 115,
"column": 30
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\nx : ↥N\n⊢ map N.subtype (I • ⊤) = I • N",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\nx : ↥N\n⊢ I • map N.subtype ⊤ = I • N"
] | Submodule.map_smul'', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 1011,
"column": 81
} | {
"line": 1012,
"column": 34
} | {
"line": 1014,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\n⊢ (span R s).annihilator = ⨅ g, (toSpanSingleton R M ↑g).ker",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Submodule",
"instHSMul",
"iInf",
"Sem... | [] | by
ext; simp [mem_annihilator_span] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Operations | {
"line": 353,
"column": 8
} | {
"line": 353,
"column": 27
} | {
"line": 353,
"column": 28
} | [
{
"pp": "case zero\nR : Type u\ninst✝ : Semiring R\nI : Ideal R\nn : ℕ\nh : 0 ≤ n\n⊢ I ^ n ≤ I ^ 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Submodule.pow_zero",
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"congrArg",
"PartialOrder.toPreorder",
... | [
"case zero\nR : Type u\ninst✝ : Semiring R\nI : Ideal R\nn : ℕ\nh : 0 ≤ n\n⊢ I ^ n ≤ 1"
] | Submodule.pow_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 365,
"column": 16
} | {
"line": 365,
"column": 35
} | {
"line": 365,
"column": 36
} | [
{
"pp": "case zero\nR : Type u\ninst✝ : Semiring R\nI J : Ideal R\ne : I ≤ J\n⊢ I ^ 0 ≤ J ^ 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Submodule.pow_zero",
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"congrArg",
"PartialOrder.toPreorder",
"... | [
"case zero\nR : Type u\ninst✝ : Semiring R\nI J : Ideal R\ne : I ≤ J\n⊢ 1 ≤ J ^ 0"
] | Submodule.pow_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 381,
"column": 12
} | {
"line": 381,
"column": 31
} | {
"line": 381,
"column": 32
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nI J K L : Ideal R\ninst✝ : I.IsTwoSided\nm n : ℕ\n⊢ (I ^ Nat.zero).IsTwoSided",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Submodule.pow_zero",
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"congrArg",
"id",
... | [
"R : Type u\ninst✝¹ : Semiring R\nI J K L : Ideal R\ninst✝ : I.IsTwoSided\nm n : ℕ\n⊢ IsTwoSided 1"
] | Submodule.pow_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 390,
"column": 18
} | {
"line": 390,
"column": 37
} | {
"line": 390,
"column": 38
} | [
{
"pp": "case inl\nR : Type u\ninst✝¹ : Semiring R\nI : Ideal R\ninst✝ : I.IsTwoSided\nm : ℕ\n⊢ I ^ m = I ^ m * I ^ 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Submodule.pow_zero",
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"HMul.hMul",
"congrArg"... | [
"case inl\nR : Type u\ninst✝¹ : Semiring R\nI : Ideal R\ninst✝ : I.IsTwoSided\nm : ℕ\n⊢ I ^ m = I ^ m * 1"
] | Submodule.pow_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Multiset | {
"line": 195,
"column": 43
} | {
"line": 195,
"column": 50
} | {
"line": 195,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns₁ s₂ : Multiset α\n⊢ gcd ?m.36 = GCDMonoid.gcd s₁.gcd s₂.gcd",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Multiset.gcd",
"congrArg",
"Multiset... | [
"α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns₁ s₂ : Multiset α\n⊢ GCDMonoid.gcd (gcd ?s₁) (gcd ?s₂) = GCDMonoid.gcd s₁.gcd s₂.gcd",
"case s₁\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns₁ s₂ : Multise... | gcd_add | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Multiset | {
"line": 200,
"column": 43
} | {
"line": 200,
"column": 50
} | {
"line": 200,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns₁ s₂ : Multiset α\n⊢ gcd ?m.36 = GCDMonoid.gcd s₁.gcd s₂.gcd",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Multiset.gcd",
"congrArg",
"Multiset... | [
"α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns₁ s₂ : Multiset α\n⊢ GCDMonoid.gcd (gcd ?s₁) (gcd ?s₂) = GCDMonoid.gcd s₁.gcd s₂.gcd",
"case s₁\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns₁ s₂ : Multise... | gcd_add | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Multiset | {
"line": 227,
"column": 6
} | {
"line": 227,
"column": 62
} | {
"line": 227,
"column": 63
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns : Multiset α\nhs : s ≠ 0\nh : ∃ x, x ∈ s ∧ x ≠ 0\nf : {a : α} → a ∈ s → α\nhf : ∀ {a : α} (h : a ∈ s), a = s.gcd * f h\n⊢ ∃ t, s = map (fun x ↦ s.gcd * x) t ∧ t.gcd = 1",
"ppTerm": "?neg✝",
"assigned": true,... | [
"case neg.refine_1\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns : Multiset α\nhs : s ≠ 0\nh : ∃ x, x ∈ s ∧ x ≠ 0\nf : {a : α} → a ∈ s → α\nhf : ∀ {a : α} (h : a ∈ s), a = s.gcd * f h\n⊢ s = map (fun x ↦ s.gcd * x) (pmap f s ⋯)",
"case neg.refine_2\nα : Type u_1\ninst✝¹ : CommMono... | refine ⟨s.pmap @f fun _ ↦ id, ?_, extract_gcd' s _ h ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.GCDMonoid.Finset | {
"line": 206,
"column": 8
} | {
"line": 208,
"column": 42
} | {
"line": 209,
"column": 4
} | [
{
"pp": "case refine_2.refine_2\nα : Type u_2\nβ : Type u_3\ninst✝² : CommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ns : Finset β\nf : β → α\ninst✝ : DecidablePred fun x ↦ f x = 0\n⊢ ∀ (a : β) (s : Finset β), a ∉ s → {x ∈ s | f x = 0}.gcd f = 0 → {x ∈ insert a s | f x = 0}.gcd f = 0",
"ppTerm": "?ref... | [] | intro a s _ h
rw [filter_insert]
split_ifs with h1 <;> simp [h, h1] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GCDMonoid.Finset | {
"line": 206,
"column": 8
} | {
"line": 208,
"column": 42
} | {
"line": 209,
"column": 4
} | [
{
"pp": "case refine_2.refine_2\nα : Type u_2\nβ : Type u_3\ninst✝² : CommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ns : Finset β\nf : β → α\ninst✝ : DecidablePred fun x ↦ f x = 0\n⊢ ∀ (a : β) (s : Finset β), a ∉ s → {x ∈ s | f x = 0}.gcd f = 0 → {x ∈ insert a s | f x = 0}.gcd f = 0",
"ppTerm": "?ref... | [] | intro a s _ h
rw [filter_insert]
split_ifs with h1 <;> simp [h, h1] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Operations | {
"line": 528,
"column": 11
} | {
"line": 528,
"column": 30
} | {
"line": 528,
"column": 31
} | [
{
"pp": "case inr\nR : Type u\ninst✝ : Semiring R\nI : Ideal R\nn : ℕ\nh : n = 0\n⊢ I ^ 0 = ⊤",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Submodule.pow_zero",
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"congrArg",
"id",
"instOfNatNat",
"... | [
"case inr\nR : Type u\ninst✝ : Semiring R\nI : Ideal R\nn : ℕ\nh : n = 0\n⊢ 1 = ⊤"
] | Submodule.pow_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 576,
"column": 4
} | {
"line": 584,
"column": 36
} | {
"line": 586,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CommMonoidWithZero α\nh : IsGCDMonoid α\nk m n : α\nH : k ∣ m * n\n⊢ ∃ a₁ a₂, a₁ ∣ m ∧ a₂ ∣ n ∧ k = a₁ * a₂",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"Dvd.dvd",
"HMul.hMul",
"GCDMonoid",
"G... | [] | cases h
by_cases h0 : gcd k m = 0
· rw [gcd_eq_zero_iff] at h0
rcases h0 with ⟨rfl, rfl⟩
exact ⟨0, n, dvd_refl 0, dvd_refl n, by simp⟩
· obtain ⟨a, ha⟩ := gcd_dvd_left k m
refine ⟨gcd k m, a, gcd_dvd_right _ _, ?_, ha⟩
rw [← mul_dvd_mul_iff_left h0, ← ha]
exact dvd_gcd_mul_of_d... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 576,
"column": 4
} | {
"line": 584,
"column": 36
} | {
"line": 586,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CommMonoidWithZero α\nh : IsGCDMonoid α\nk m n : α\nH : k ∣ m * n\n⊢ ∃ a₁ a₂, a₁ ∣ m ∧ a₂ ∣ n ∧ k = a₁ * a₂",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"Dvd.dvd",
"HMul.hMul",
"GCDMonoid",
"G... | [] | cases h
by_cases h0 : gcd k m = 0
· rw [gcd_eq_zero_iff] at h0
rcases h0 with ⟨rfl, rfl⟩
exact ⟨0, n, dvd_refl 0, dvd_refl n, by simp⟩
· obtain ⟨a, ha⟩ := gcd_dvd_left k m
refine ⟨gcd k m, a, gcd_dvd_right _ _, ?_, ha⟩
rw [← mul_dvd_mul_iff_left h0, ← ha]
exact dvd_gcd_mul_of_d... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 626,
"column": 6
} | {
"line": 626,
"column": 22
} | {
"line": 627,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c d₁ d₂ : α\nha : a ≠ 0\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nhc : c = d₁ * d₂\nhd₁ : d₁ ∣ a\n⊢ IsUnit (gcd d₁ b ^ k)",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toMono... | [
"α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c d₁ d₂ : α\nha : a ≠ 0\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nhc : c = d₁ * d₂\nhd₁ : d₁ ∣ a\n⊢ IsUnit (gcd d₁ b)"
] | apply IsUnit.pow | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.UniqueFactorizationDomain.Defs | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 97
} | {
"line": 172,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\nf : Multiset α\nhf : ∀ b ∈ f, Prime b\nu : αˣ\nha : f.prod * ↑u ≠ 0\n⊢ IsPrimal (f.prod * ↑u)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"... | [] | exact ((Submonoid.isPrimal α).multiset_prod_mem f (hf · · |>.isPrimal)).mul u.isUnit.isPrimal | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.UniqueFactorizationDomain.Defs | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 63
} | {
"line": 196,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\nane0 : a ≠ 0\n⊢ (Classical.choose ⋯).prod ~ᵤ a",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"UniqueFactorizationMonoid.exists_prime_factors",
... | [] | exact (Classical.choose_spec (exists_prime_factors a ane0)).2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.UniqueFactorizationDomain.Defs | {
"line": 197,
"column": 46
} | {
"line": 197,
"column": 63
} | {
"line": 199,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\n⊢ factors 0 = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"CommMonoidWithZero.toCommMonoid",
"UniqueFactorizationMonoid.exists_prime_factors",
"cong... | [] | by simp [factors] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 105,
"column": 31
} | {
"line": 105,
"column": 93
} | {
"line": 107,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : AddCommMonoid M\ninst✝² : Semiring R\ninst✝¹ : Module R M\nS : Submodule R M\ninst✝ : S.IsPrincipal\n⊢ S = ⊥ ↔ generator S = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
"Iff.rfl",
... | [] | by rw [← @span_singleton_eq_bot R M, span_singleton_generator] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 46
} | {
"line": 201,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nx y z : R\ninst✝ : IsPrincipal (Ideal.span {x, y})\nhx : Ideal.span {x} ≤ Ideal.span {z}\nhy : Ideal.span {y} ≤ Ideal.span {z}\n⊢ Ideal.span {gcd x y} ≤ Ideal.span {z}",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.t... | [
"R : Type u\ninst✝¹ : CommRing R\nx y z : R\ninst✝ : IsPrincipal (Ideal.span {x, y})\nhx : Ideal.span {x} ≤ Ideal.span {z}\nhy : Ideal.span {y} ≤ Ideal.span {z}\n⊢ Ideal.span {x} ≤ Ideal.span {z} ∧ Ideal.span {y} ≤ Ideal.span {z}"
] | rw [span_gcd, Ideal.span_insert, sup_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 244,
"column": 20
} | {
"line": 244,
"column": 34
} | {
"line": 244,
"column": 35
} | [
{
"pp": "case insert\nR : Type u\nα : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : NormalizedGCDMonoid R\nf : α → R\na : α\ns : Finset α\nha : a ∉ s\nx y : R\nhxy : x * f a + y * s.gcd f = IsBezout.gcd (f a) (s.gcd f)\nu : Rˣ\nhu : f a * x * ↑u + s.gcd f * y * ↑u = GCDMonoid.gcd (f a) (s.gcd f)\n... | [
"case insert\nR : Type u\nα : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : NormalizedGCDMonoid R\nf : α → R\na : α\ns : Finset α\nha : a ∉ s\nx y : R\nhxy : x * f a + y * s.gcd f = IsBezout.gcd (f a) (s.gcd f)\nu : Rˣ\nhu : f a * x * ↑u + s.gcd f * y * ↑u = GCDMonoid.gcd (f a) (s.gcd f)\ng : α → R\nh... | sum_insert ha, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1075,
"column": 4
} | {
"line": 1075,
"column": 12
} | {
"line": 1076,
"column": 4
} | [
{
"pp": "case zero\nι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\nI : Ideal R\ns : Finset ι\na b : ι\nhn : s = ∅\nh : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i)\n⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
... | [
"case zero\nι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\nI : Ideal R\na b : ι\nh : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑∅, ↑(f i)\n⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ ∅, I ≤ f i"
] | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 122,
"column": 2
} | {
"line": 125,
"column": 17
} | {
"line": 127,
"column": 0
} | [
{
"pp": "case i\nR : Type u\nM : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : RankCondition R\nι : Type u_2\ninst✝¹ : Fintype ι\nb : Basis ι R M\nw : Set M\ninst✝ : Fintype ↑w\ns : span R w = ⊤\n⊢ Surjective ⇑(↑b.repr ∘ₗ Finsupp.linearCombination R Subtype.val)",
"ppT... | [] | · apply Surjective.comp (g := b.repr.toLinearMap)
· apply LinearEquiv.surjective
rw [← LinearMap.range_eq_top, Finsupp.range_linearCombination]
simpa using s | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1107,
"column": 31
} | {
"line": 1107,
"column": 50
} | {
"line": 1107,
"column": 51
} | [
{
"pp": "ι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\nI : Ideal R\nn : ℕ\nih :\n ∀ {s : Finset ι} {a b : ι},\n (∀ i ∈ s, (f i).IsPrime) →\n s.card = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i\na b i : ι\nt : Finset ι\nhit : i ∉ t\nhn : t.card = ... | [
"ι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\nI : Ideal R\nn : ℕ\nih :\n ∀ {s : Finset ι} {a b : ι},\n (∀ i ∈ s, (f i).IsPrime) →\n s.card = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i\na b i : ι\nt : Finset ι\nhit : i ∉ t\nhn : t.card = n\nh : ↑I ⊆ ... | Set.biUnion_insert, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1119,
"column": 31
} | {
"line": 1119,
"column": 50
} | {
"line": 1119,
"column": 51
} | [
{
"pp": "ι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\nI : Ideal R\nn : ℕ\nih :\n ∀ {s : Finset ι} {a b : ι},\n (∀ i ∈ s, (f i).IsPrime) →\n s.card = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i\na b i : ι\nt : Finset ι\nhit : i ∉ t\nhn : t.card = ... | [
"ι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\nI : Ideal R\nn : ℕ\nih :\n ∀ {s : Finset ι} {a b : ι},\n (∀ i ∈ s, (f i).IsPrime) →\n s.card = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i\na b i : ι\nt : Finset ι\nhit : i ∉ t\nhn : t.card = n\nh : ↑I ⊆ ... | Set.biUnion_insert, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 520,
"column": 2
} | {
"line": 520,
"column": 32
} | {
"line": 522,
"column": 0
} | [
{
"pp": "R✝ : Type u\nS : Type u_1\nM✝ : Type v\ninst✝⁷ : Semiring R✝\ninst✝⁶ : AddCommMonoid M✝\ninst✝⁵ : Module R✝ M✝\nι : Type w\nι' : Type w'\ninst✝⁴ : StrongRankCondition R✝\nR : Type u_2\nM : Type u_3\ninst✝³ : DivisionRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ns t : Set M\ninst✝ : Module.Finit... | [] | exact Module.rank_lt_aleph0 .. | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 548,
"column": 2
} | {
"line": 548,
"column": 65
} | {
"line": 550,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : StrongRankCondition R\ninst✝ : Module.Finite R M\nN : Submodule R M\n⊢ Module.rank R ↥N < ℵ₀",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Submodule",
"lt_of_le_of_l... | [] | exact lt_of_le_of_lt (Submodule.rank_le N) (rank_lt_aleph0 R M) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Matrix.Basis | {
"line": 63,
"column": 2
} | {
"line": 65,
"column": 58
} | {
"line": 67,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type u_7\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\ni : m\nj : n\na : α\n⊢ single i j a = of (Pi.single i (Pi.single j a))",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"False",
"Equiv.instEquivLike",
"eq_fa... | [] | ext a b
unfold single
by_cases hi : i = a <;> by_cases hj : j = b <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Matrix.Basis | {
"line": 63,
"column": 2
} | {
"line": 65,
"column": 58
} | {
"line": 67,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type u_7\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\ni : m\nj : n\na : α\n⊢ single i j a = of (Pi.single i (Pi.single j a))",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"False",
"Equiv.instEquivLike",
"eq_fa... | [] | ext a b
unfold single
by_cases hi : i = a <;> by_cases hj : j = b <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Matrix.Basis | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 34
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case inl.hi\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nα : Type u_7\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq o\ninst✝ : Zero α\nf : l ≃ n\ng : m ≃ o\ni : n\nj : o\nr : α\ni' : l\nj' : m\nhi : f.symm i ≠ i'\n⊢ i ≠ f i'",
"ppTerm": "?... | [] | exact f.symm_apply_eq.not.1 hi | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1179,
"column": 50
} | {
"line": 1179,
"column": 69
} | {
"line": 1179,
"column": 70
} | [
{
"pp": "case pos\nι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\na b : ι\nI : Ideal R\nu : Finset ι\nhbu : b ∉ u\nhat : a ∉ insert b u\nhp : ∀ i ∈ insert a (insert b u), i ≠ a → i ≠ b → (f i).IsPrime\nh : ↑I ⊆ ⋃ i ∈ insert a (insert b ↑u), ↑(f i)\nhas : a ∈ insert a (insert b u)\nhbt : b ∈ inse... | [
"case pos\nι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\na b : ι\nI : Ideal R\nu : Finset ι\nhbu : b ∉ u\nhat : a ∉ insert b u\nhp : ∀ i ∈ insert a (insert b u), i ≠ a → i ≠ b → (f i).IsPrime\nh : ↑I ⊆ ↑(f a) ∪ ⋃ x ∈ insert b ↑u, ↑(f x)\nhas : a ∈ insert a (insert b u)\nhbt : b ∈ insert b u\nhp' : ... | Set.biUnion_insert, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1179,
"column": 70
} | {
"line": 1179,
"column": 89
} | {
"line": 1179,
"column": 90
} | [
{
"pp": "case pos\nι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\na b : ι\nI : Ideal R\nu : Finset ι\nhbu : b ∉ u\nhat : a ∉ insert b u\nhp : ∀ i ∈ insert a (insert b u), i ≠ a → i ≠ b → (f i).IsPrime\nh : ↑I ⊆ ↑(f a) ∪ ⋃ x ∈ insert b ↑u, ↑(f x)\nhas : a ∈ insert a (insert b u)\nhbt : b ∈ insert... | [
"case pos\nι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\na b : ι\nI : Ideal R\nu : Finset ι\nhbu : b ∉ u\nhat : a ∉ insert b u\nhp : ∀ i ∈ insert a (insert b u), i ≠ a → i ≠ b → (f i).IsPrime\nh : ↑I ⊆ ↑(f a) ∪ (↑(f b) ∪ ⋃ x ∈ ↑u, ↑(f x))\nhas : a ∈ insert a (insert b u)\nhbt : b ∈ insert b u\nhp' ... | Set.biUnion_insert, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1186,
"column": 31
} | {
"line": 1186,
"column": 50
} | {
"line": 1186,
"column": 51
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\na b : ι\nI : Ideal R\nt : Finset ι\nhat : a ∉ t\nhp : ∀ i ∈ insert a t, i ≠ a → i ≠ b → (f i).IsPrime\nh : ↑I ⊆ ⋃ i ∈ insert a ↑t, ↑(f i)\nhas : a ∈ insert a t\nhbt : b ∉ t\nhp' : ∀ j ∈ t, (f j).IsPrime\n⊢ ∃ i ∈ insert a t, I ≤ f ... | [
"case neg\nι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\na b : ι\nI : Ideal R\nt : Finset ι\nhat : a ∉ t\nhp : ∀ i ∈ insert a t, i ≠ a → i ≠ b → (f i).IsPrime\nh : ↑I ⊆ ↑(f a) ∪ ⋃ x ∈ ↑t, ↑(f x)\nhas : a ∈ insert a t\nhbt : b ∉ t\nhp' : ∀ j ∈ t, (f j).IsPrime\n⊢ ∃ i ∈ insert a t, I ≤ f i"
] | Set.biUnion_insert, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1196,
"column": 31
} | {
"line": 1196,
"column": 50
} | {
"line": 1196,
"column": 51
} | [
{
"pp": "case pos\nι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\na b : ι\nI : Ideal R\nt : Finset ι\nhbt : b ∉ t\nhp : ∀ i ∈ insert b t, i ≠ a → i ≠ b → (f i).IsPrime\nh : ↑I ⊆ ⋃ i ∈ insert b ↑t, ↑(f i)\nhas : a ∉ insert b t\nhbs : b ∈ insert b t\nhp' : ∀ j ∈ t, (f j).IsPrime\n⊢ ∃ i ∈ insert b ... | [
"case pos\nι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\na b : ι\nI : Ideal R\nt : Finset ι\nhbt : b ∉ t\nhp : ∀ i ∈ insert b t, i ≠ a → i ≠ b → (f i).IsPrime\nh : ↑I ⊆ ↑(f b) ∪ ⋃ x ∈ ↑t, ↑(f x)\nhas : a ∉ insert b t\nhbs : b ∈ insert b t\nhp' : ∀ j ∈ t, (f j).IsPrime\n⊢ ∃ i ∈ insert b t, I ≤ f i"
... | Set.biUnion_insert, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1209,
"column": 31
} | {
"line": 1209,
"column": 50
} | {
"line": 1209,
"column": 51
} | [
{
"pp": "case neg.inr\nι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\na b : ι\nI : Ideal R\ni : ι\nt : Finset ι\nleft✝ : i ∉ t\nhp : ∀ i_1 ∈ insert i t, i_1 ≠ a → i_1 ≠ b → (f i_1).IsPrime\nh : ↑I ⊆ ⋃ i_1 ∈ insert i ↑t, ↑(f i_1)\nhas : a ∉ insert i t\nhbs : b ∉ insert i t\nhis : i ∈ insert i t\n... | [
"case neg.inr\nι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\na b : ι\nI : Ideal R\ni : ι\nt : Finset ι\nleft✝ : i ∉ t\nhp : ∀ i_1 ∈ insert i t, i_1 ≠ a → i_1 ≠ b → (f i_1).IsPrime\nh : ↑I ⊆ ↑(f i) ∪ ⋃ x ∈ ↑t, ↑(f x)\nhas : a ∉ insert i t\nhbs : b ∉ insert i t\nhis : i ∈ insert i t\nhp' : ∀ j ∈ t, (... | Set.biUnion_insert, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Matrix.Mul | {
"line": 858,
"column": 2
} | {
"line": 858,
"column": 72
} | {
"line": 860,
"column": 0
} | [
{
"pp": "n : Type u_3\nR : Type u_7\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : NonUnitalNonAssocSemiring R\nv : n → R\nj : n\nx : R\ni : n\n⊢ v i * Pi.single j x i = Pi.single j (v j * x) i",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"NonUnitalNonAsso... | [] | exact Pi.apply_single (fun i x => v i * x) (fun i => mul_zero _) j x i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1468,
"column": 34
} | {
"line": 1468,
"column": 55
} | {
"line": 1468,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Semiring R\nM : Type u_2\nN : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nI : Ideal R\nf : M →ₗ[R] N\nh : Function.Surjective ⇑f\n⊢ comap f (I • ⊤) = comap f (map f (I • ⊤))",
"ppTerm": "?m.72",
"assigned": true,
... | [
"R : Type u_1\ninst✝⁴ : Semiring R\nM : Type u_2\nN : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nI : Ideal R\nf : M →ₗ[R] N\nh : Function.Surjective ⇑f\n⊢ comap f (I • ⊤) = comap f (I • map f ⊤)"
] | Submodule.map_smul'', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Matrix.Mul | {
"line": 1193,
"column": 2
} | {
"line": 1193,
"column": 48
} | {
"line": 1195,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nα : Type v\ninst✝³ : Fintype n\ninst✝² : Fintype o\ninst✝¹ : Mul α\ninst✝ : AddCommMonoid α\np : Type u_10\nM : Matrix m n α\nN : Matrix o p α\ne₁ : l → m\ne₂ : n ≃ o\n⊢ M.submatrix e₁ id * N.submatrix (⇑e₂) id = M.submatrix e₁ ⇑e₂.symm * N",
... | [] | ext; simp [mul_apply, ← e₂.bijective.sum_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Matrix.Mul | {
"line": 1193,
"column": 2
} | {
"line": 1193,
"column": 48
} | {
"line": 1195,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nα : Type v\ninst✝³ : Fintype n\ninst✝² : Fintype o\ninst✝¹ : Mul α\ninst✝ : AddCommMonoid α\np : Type u_10\nM : Matrix m n α\nN : Matrix o p α\ne₁ : l → m\ne₂ : n ≃ o\n⊢ M.submatrix e₁ id * N.submatrix (⇑e₂) id = M.submatrix e₁ ⇑e₂.symm * N",
... | [] | ext; simp [mul_apply, ← e₂.bijective.sum_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Matrix.Mul | {
"line": 1199,
"column": 2
} | {
"line": 1199,
"column": 48
} | {
"line": 1201,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nα : Type v\ninst✝³ : Fintype n\ninst✝² : Fintype o\ninst✝¹ : Mul α\ninst✝ : AddCommMonoid α\np : Type u_10\nM : Matrix m n α\nN : Matrix o p α\ne₁ : l → p\ne₂ : o ≃ n\n⊢ M.submatrix id ⇑e₂ * N.submatrix id e₁ = M * N.submatrix (⇑e₂.symm) e₁",
... | [] | ext; simp [mul_apply, ← e₂.bijective.sum_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Matrix.Mul | {
"line": 1199,
"column": 2
} | {
"line": 1199,
"column": 48
} | {
"line": 1201,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nα : Type v\ninst✝³ : Fintype n\ninst✝² : Fintype o\ninst✝¹ : Mul α\ninst✝ : AddCommMonoid α\np : Type u_10\nM : Matrix m n α\nN : Matrix o p α\ne₁ : l → p\ne₂ : o ≃ n\n⊢ M.submatrix id ⇑e₂ * N.submatrix id e₁ = M * N.submatrix (⇑e₂.symm) e₁",
... | [] | ext; simp [mul_apply, ← e₂.bijective.sum_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1479,
"column": 2
} | {
"line": 1486,
"column": 91
} | {
"line": 1487,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\ns : Set R\nI J : Ideal R\nhs : s ⊆ ↑(I ⊔ J).radical\n⊢ ∃ t, Set.range t ⊆ ↑I ∧ s ⊆ ↑(span (Set.range t) ⊔ J).radical",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ideal.subset_span",
"Submodule",
"instHSMul... | [] | replace hs : ∀ z : s, ∃ (m : ℕ) (a b : R) (ha : a ∈ I) (hb : b ∈ J), a + b = z ^ m := by
rintro ⟨z, hzs⟩
simp only [Ideal.radical, Submodule.mem_sup] at hs
obtain ⟨m, y, hyq, b, hb, hy⟩ := hs hzs
exact ⟨m, y, b, hyq, hb, hy⟩
choose m a b ha hb heq using hs
refine ⟨a, by rwa [Set.range_subset_iff], f... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1479,
"column": 2
} | {
"line": 1486,
"column": 91
} | {
"line": 1487,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\ns : Set R\nI J : Ideal R\nhs : s ⊆ ↑(I ⊔ J).radical\n⊢ ∃ t, Set.range t ⊆ ↑I ∧ s ⊆ ↑(span (Set.range t) ⊔ J).radical",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ideal.subset_span",
"Submodule",
"instHSMul... | [] | replace hs : ∀ z : s, ∃ (m : ℕ) (a b : R) (ha : a ∈ I) (hb : b ∈ J), a + b = z ^ m := by
rintro ⟨z, hzs⟩
simp only [Ideal.radical, Submodule.mem_sup] at hs
obtain ⟨m, y, hyq, b, hb, hy⟩ := hs hzs
exact ⟨m, y, b, hyq, hb, hy⟩
choose m a b ha hb heq using hs
refine ⟨a, by rwa [Set.range_subset_iff], f... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Matrix.Mul | {
"line": 1261,
"column": 2
} | {
"line": 1261,
"column": 34
} | {
"line": 1263,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\no : Type u_4\nα : Type v\nβ : Type w\ninst✝² : Fintype n\ninst✝¹ : NonAssocSemiring α\ninst✝ : NonAssocSemiring β\nM : Matrix m n α\nN : Matrix n o α\ni : m\nj : o\nf : α →+* β\n⊢ f ((M * N) i j) = (M.map ⇑f * N.map ⇑f) i j",
"ppTerm": "?m.23",
"assigned": true,
... | [] | simp [Matrix.mul_apply, map_sum] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Matrix.Mul | {
"line": 1261,
"column": 2
} | {
"line": 1261,
"column": 34
} | {
"line": 1263,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\no : Type u_4\nα : Type v\nβ : Type w\ninst✝² : Fintype n\ninst✝¹ : NonAssocSemiring α\ninst✝ : NonAssocSemiring β\nM : Matrix m n α\nN : Matrix n o α\ni : m\nj : o\nf : α →+* β\n⊢ f ((M * N) i j) = (M.map ⇑f * N.map ⇑f) i j",
"ppTerm": "?m.23",
"assigned": true,
... | [] | simp [Matrix.mul_apply, map_sum] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Matrix.Mul | {
"line": 1261,
"column": 2
} | {
"line": 1261,
"column": 34
} | {
"line": 1263,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\no : Type u_4\nα : Type v\nβ : Type w\ninst✝² : Fintype n\ninst✝¹ : NonAssocSemiring α\ninst✝ : NonAssocSemiring β\nM : Matrix m n α\nN : Matrix n o α\ni : m\nj : o\nf : α →+* β\n⊢ f ((M * N) i j) = (M.map ⇑f * N.map ⇑f) i j",
"ppTerm": "?m.23",
"assigned": true,
... | [] | simp [Matrix.mul_apply, map_sum] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.Multiset | {
"line": 74,
"column": 61
} | {
"line": 74,
"column": 78
} | {
"line": 75,
"column": 6
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nf✝ : α →₀ ℕ\ng : α → β\na : α\nn : ℕ\nf : α →₀ ℕ\na✝¹ : a ∉ f.support\na✝ : n ≠ 0\nih : Multiset.map g (toMultiset f) = toMultiset (mapDomain g f)\n⊢ Multiset.map g (toMultiset (single a n)) + toMultiset (mapDomain g f) =\n toMultiset (mapDomain g (single a... | [
"case refine_2\nα : Type u_1\nβ : Type u_2\nf✝ : α →₀ ℕ\ng : α → β\na : α\nn : ℕ\nf : α →₀ ℕ\na✝¹ : a ∉ f.support\na✝ : n ≠ 0\nih : Multiset.map g (toMultiset f) = toMultiset (mapDomain g f)\n⊢ Multiset.map g (toMultiset (single a n)) + toMultiset (mapDomain g f) = toMultiset (single (g a) n + mapDomain g f)"
] | mapDomain_single, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finsupp.Multiset | {
"line": 76,
"column": 6
} | {
"line": 76,
"column": 44
} | {
"line": 76,
"column": 44
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nf✝ : α →₀ ℕ\ng : α → β\na : α\nn : ℕ\nf : α →₀ ℕ\na✝¹ : a ∉ f.support\na✝ : n ≠ 0\nih : Multiset.map g (toMultiset f) = toMultiset (mapDomain g f)\n⊢ (Multiset.mapAddMonoidHom g) (n • {a}) + toMultiset (mapDomain g f) = n • {g a} + toMultiset (mapDomain g f)",... | [
"case refine_2\nα : Type u_1\nβ : Type u_2\nf✝ : α →₀ ℕ\ng : α → β\na : α\nn : ℕ\nf : α →₀ ℕ\na✝¹ : a ∉ f.support\na✝ : n ≠ 0\nih : Multiset.map g (toMultiset f) = toMultiset (mapDomain g f)\n⊢ n • (Multiset.mapAddMonoidHom g) {a} + toMultiset (mapDomain g f) = n • {g a} + toMultiset (mapDomain g f)"
] | (Multiset.mapAddMonoidHom g).map_nsmul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TwoSidedIdeal.Basic | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 73
} | {
"line": 63,
"column": 6
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\nt₁ t₂ : RingCon R\nh : {x | { ringCon := t₁ }.ringCon x 0} = {x | { ringCon := t₂ }.ringCon x 0}\na b : R\nH : t₁ a b\n⊢ t₂ a b",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"RingCon.ins... | [
"case refine_1\nR : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\nt₁ t₂ : RingCon R\nh : {x | { ringCon := t₁ }.ringCon x 0} = {x | { ringCon := t₂ }.ringCon x 0}\na b : R\nH : t₁ a b\nH' : a - b ∈ {x | t₁ x 0}\n⊢ t₂ a b"
] | have H' : a - b ∈ {x | t₁ x 0} := sub_self b ▸ t₁.sub H (t₁.refl b) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 940,
"column": 22
} | {
"line": 940,
"column": 35
} | {
"line": 940,
"column": 36
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁶ : CommSemiring E\ninst✝⁵ : Semiring K\ninst✝⁴ : SMul F E\ninst✝³ : Algebra E K\ninst✝² : Semiring F\ninst✝¹ : Module F K\ninst✝ : IsScalarTower F E K\nL : Submonoid K\nS : Set K\nh : ↑L = ↑(span F S)\n⊢ span E ↑(closure ↑L) = span E S",
"ppTerm": "?m... | [
"F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁶ : CommSemiring E\ninst✝⁵ : Semiring K\ninst✝⁴ : SMul F E\ninst✝³ : Algebra E K\ninst✝² : Semiring F\ninst✝¹ : Module F K\ninst✝ : IsScalarTower F E K\nL : Submonoid K\nS : Set K\nh : ↑L = ↑(span F S)\n⊢ span E ↑L = span E S"
] | L.closure_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Rat | {
"line": 32,
"column": 39
} | {
"line": 34,
"column": 55
} | {
"line": 36,
"column": 0
} | [
{
"pp": "M : Type u_1\nM₂ : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M₂\nF : Type u_3\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_4\nS : Type u_5\ninst✝³ : DivisionRing R\ninst✝² : DivisionRing S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nc : ℚ\nx : M\n⊢ f (↑c ... | [] | by
rw [Rat.cast_def, Rat.cast_def, div_eq_mul_inv, div_eq_mul_inv, mul_smul, mul_smul,
map_intCast_smul f R S, map_inv_natCast_smul f R S] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.ConjTranspose | {
"line": 250,
"column": 7
} | {
"line": 250,
"column": 33
} | {
"line": 252,
"column": 0
} | [
{
"pp": "n : Type u_3\nα : Type v\ninst✝² : DecidableEq n\ninst✝¹ : Ring α\ninst✝ : StarRing α\nM : Matrix n n α\nd : ℤ\n⊢ M = (↑d)ᴴ ↔ M = ↑d",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Matrix.instIntCastOfZero",
"Ring.toNonAssocRing",
... | [] | rw [conjTranspose_intCast] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.ConjTranspose | {
"line": 250,
"column": 7
} | {
"line": 250,
"column": 33
} | {
"line": 252,
"column": 0
} | [
{
"pp": "n : Type u_3\nα : Type v\ninst✝² : DecidableEq n\ninst✝¹ : Ring α\ninst✝ : StarRing α\nM : Matrix n n α\nd : ℤ\n⊢ M = (↑d)ᴴ ↔ M = ↑d",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Matrix.instIntCastOfZero",
"Ring.toNonAssocRing",
... | [] | rw [conjTranspose_intCast] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.ConjTranspose | {
"line": 250,
"column": 7
} | {
"line": 250,
"column": 33
} | {
"line": 252,
"column": 0
} | [
{
"pp": "n : Type u_3\nα : Type v\ninst✝² : DecidableEq n\ninst✝¹ : Ring α\ninst✝ : StarRing α\nM : Matrix n n α\nd : ℤ\n⊢ M = (↑d)ᴴ ↔ M = ↑d",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Matrix.instIntCastOfZero",
"Ring.toNonAssocRing",
... | [] | rw [conjTranspose_intCast] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Matrix.Block | {
"line": 758,
"column": 4
} | {
"line": 760,
"column": 9
} | {
"line": 762,
"column": 0
} | [
{
"pp": "case inr\no : Type u_4\nm' : o → Type u_7\nα : Type u_12\ninst✝² : Zero α\ninst✝¹ : DecidableEq o\ninst✝ : (i : o) → DecidableEq (m' i)\nd : (i : o) × m' i → α\nk : o\ni j : m' k\nhij : i ≠ j\n⊢ (diagonal d).blockDiag' k i j = diagonal (fun i ↦ d ⟨k, i⟩) i j",
"ppTerm": "?inr",
"assigned": true... | [] | · rw [blockDiag'_apply, diagonal_apply_ne _ hij, diagonal_apply_ne _ (mt (fun h => ?_) hij)]
cases h
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 399,
"column": 2
} | {
"line": 402,
"column": 21
} | {
"line": 404,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nα : Type v\ninst✝² : DecidableEq l\ninst✝¹ : Fintype m\ninst✝ : NonUnitalNonAssocSemiring α\nA : Matrix l m α\ni : l\nc v : m → α\n⊢ A.updateRow i c *ᵥ v = Function.update (A *ᵥ v) i (c ⬝ᵥ v)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"False"... | [] | ext i'
obtain rfl | hi := eq_or_ne i' i
· simp [mulVec]
· simp [mulVec, hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 399,
"column": 2
} | {
"line": 402,
"column": 21
} | {
"line": 404,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nα : Type v\ninst✝² : DecidableEq l\ninst✝¹ : Fintype m\ninst✝ : NonUnitalNonAssocSemiring α\nA : Matrix l m α\ni : l\nc v : m → α\n⊢ A.updateRow i c *ᵥ v = Function.update (A *ᵥ v) i (c ⬝ᵥ v)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"False"... | [] | ext i'
obtain rfl | hi := eq_or_ne i' i
· simp [mulVec]
· simp [mulVec, hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 24
} | {
"line": 190,
"column": 2
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsPrincipalIdealRing R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\nO : Type u_4\ninst✝¹ : AddCommGroup O\ninst✝ : Module R O\nM N : Submodule R O\nb'M : Basis ι R ↥M\nN_bot : N ≠ ⊥\nN_le_M : N ≤ M\nthis : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submo... | [
"case neg\nι : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsPrincipalIdealRing R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\nO : Type u_4\ninst✝¹ : AddCommGroup O\ninst✝ : Module R O\nM N : Submodule R O\nb'M : Basis ι R ↥M\nN_bot : N ≠ ⊥\nN_le_M : N ≤ M\nthis : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage N ... | choose c hc using hdvd | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.GroupTheory.MonoidLocalization.MonoidWithZero | {
"line": 124,
"column": 38
} | {
"line": 124,
"column": 47
} | {
"line": 124,
"column": 48
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst✝ : CommMonoidWithZero N\nf : S.LocalizationMap N\nm : M\ns : ↥S\n⊢ f.mk' m s * f ↑s = 0 * f ↑s ↔ ∃ s, ↑s * m = 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid"... | [
"M : Type u_1\ninst✝¹ : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst✝ : CommMonoidWithZero N\nf : S.LocalizationMap N\nm : M\ns : ↥S\n⊢ f m = 0 * f ↑s ↔ ∃ s, ↑s * m = 0"
] | mk'_spec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 430,
"column": 51
} | {
"line": 430,
"column": 60
} | {
"line": 430,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nn : Type u_5\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\ni✝ j✝ : n\n⊢ id (Pi.single j✝ 1) i✝ = if i✝ = j✝ then 1 else 0",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Eq.mpr",
"Pi.Function.module",
"N... | [
"R : Type u_1\ninst✝² : CommSemiring R\nn : Type u_5\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\ni✝ j✝ : n\n⊢ Pi.single j✝ 1 i✝ = if i✝ = j✝ then 1 else 0"
] | id_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 373,
"column": 4
} | {
"line": 373,
"column": 27
} | {
"line": 375,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nb✝ : ι → M\ninst✝³ : IsPrincipalIdealRing R\ninst✝² : IsDomain R\ninst✝¹ : Fintype ι\ns : ι → M\nhs : span R (range s) = ⊤\ninst✝ : IsTorsionFree R M\nthis✝¹ : ∃ s_1, LinearIndepOn R s s_1 ∧ ∀ i... | [] | exact ⟨n, b.map ψ.symm⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 15
} | {
"line": 219,
"column": 16
} | [
{
"pp": "A : Type u_4\ninst✝⁴ : CommRing A\nK : Type u_5\ninst✝³ : CommRing K\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDomain A\nx : K\nhx : x ≠ 0\n⊢ x * (mk' K ↑(sec A⁰ x).2 ⟨(sec A⁰ x).1, ⋯⟩ * (algebraMap A K) ↑⟨(sec A⁰ x).1, ⋯⟩) =\n (algebraMap A K) ↑⟨(sec A⁰ x).1, ⋯⟩",
"ppTerm": ... | [
"A : Type u_4\ninst✝⁴ : CommRing A\nK : Type u_5\ninst✝³ : CommRing K\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDomain A\nx : K\nhx : x ≠ 0\n⊢ x * (algebraMap A K) ↑(sec A⁰ x).2 = (algebraMap A K) ↑⟨(sec A⁰ x).1, ⋯⟩"
] | mk'_spec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1089,
"column": 2
} | {
"line": 1092,
"column": 95
} | {
"line": 1094,
"column": 0
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_3\nM₂ : Type u_4\nι₁ : Type u_6\nι₂ : Type u_7\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Fintype ι₁\ninst✝¹ : Fintype ι₂\ninst✝ : DecidableEq ι₁\nb₁ : Basis ι₁ R M₁\nb₂ : Basis ι₂ R M₂\n... | [] | have := Classical.decEq ι₂
rw [linearMap_apply, Matrix.stdBasis_eq_single, Matrix.toLin_self]
dsimp only [Matrix.single, of_apply]
simp_rw [ite_smul, one_smul, zero_smul, ite_and, Finset.sum_ite_eq, Finset.mem_univ, if_true] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1089,
"column": 2
} | {
"line": 1092,
"column": 95
} | {
"line": 1094,
"column": 0
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_3\nM₂ : Type u_4\nι₁ : Type u_6\nι₂ : Type u_7\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Fintype ι₁\ninst✝¹ : Fintype ι₂\ninst✝ : DecidableEq ι₁\nb₁ : Basis ι₁ R M₁\nb₂ : Basis ι₂ R M₂\n... | [] | have := Classical.decEq ι₂
rw [linearMap_apply, Matrix.stdBasis_eq_single, Matrix.toLin_self]
dsimp only [Matrix.single, of_apply]
simp_rw [ite_smul, one_smul, zero_smul, ite_and, Finset.sum_ite_eq, Finset.mem_univ, if_true] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Basic | {
"line": 564,
"column": 90
} | {
"line": 565,
"column": 35
} | {
"line": 565,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝¹⁰ : CommSemiring S\ninst✝⁹ : Algebra R S\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝⁸ : CommSemiring Rₘ\ninst✝⁷ : CommSemiring Sₘ\ninst✝⁶ : Algebra R Rₘ\ninst✝⁵ : IsLocalization M Rₘ\ninst✝⁴ : Algebra S Sₘ\ni : IsLocalization (Algebr... | [
"R : Type u_1\ninst✝¹¹ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝¹⁰ : CommSemiring S\ninst✝⁹ : Algebra R S\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝⁸ : CommSemiring Rₘ\ninst✝⁷ : CommSemiring Sₘ\ninst✝⁶ : Algebra R Rₘ\ninst✝⁵ : IsLocalization M Rₘ\ninst✝⁴ : Algebra S Sₘ\ni : IsLocalization (Algebra.algebraMap... | ←
IsScalarTower.algebraMap_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Adjoin.Basic | {
"line": 95,
"column": 90
} | {
"line": 101,
"column": 56
} | {
"line": 103,
"column": 0
} | [
{
"pp": "R : Type uR\nS : Type uS\nA : Type uA\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\ns : Set A\n⊢ Subalgebra.restrictScalars R (adjoin S s) = (IsScalarTower.toAlgHom R S A).range ⊔ ad... | [] | by
refine le_antisymm (fun _ hx ↦ adjoin_induction
(fun x hx ↦ le_sup_right (α := Subalgebra R A) (subset_adjoin hx))
(fun x ↦ le_sup_left (α := Subalgebra R A) ⟨x, rfl⟩)
(fun _ _ _ _ ↦ add_mem) (fun _ _ _ _ ↦ mul_mem) <|
(Subalgebra.mem_restrictScalars _).mp hx) (sup_le ?_ <| adjoin_le subset_adjoin)... | [anonymous] | Lean.Parser.Term.byTactic |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.