module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Order.Sublattice | {
"line": 379,
"column": 79
} | {
"line": 379,
"column": 95
} | {
"line": 381,
"column": 0
} | [
{
"pp": "κ : Type u_5\nπ : κ → Type u_6\ninst✝ : (i : κ) → Lattice (π i)\nL : (i : κ) → Sublattice (π i)\na : (i : κ) → π i\n⊢ a ∈ pi ∅ L ↔ a ∈ ⊤",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Sublattice.instTop",
"Sublattice",
"False",
"Set.mem_empty_iff_false._s... | [] | by simp [mem_pi] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Sublattice | {
"line": 382,
"column": 14
} | {
"line": 382,
"column": 30
} | {
"line": 384,
"column": 0
} | [
{
"pp": "κ : Type u_5\nπ : κ → Type u_6\ninst✝ : (i : κ) → Lattice (π i)\ns : Set κ\na : (i : κ) → π i\n⊢ (a ∈ pi s fun x ↦ ⊤) ↔ a ∈ ⊤",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Sublattice.instTop",
"Sublattice",
"congrArg",
"Membership.mem",
"Sublattice... | [] | by simp [mem_pi] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Basic | {
"line": 1066,
"column": 2
} | {
"line": 1066,
"column": 13
} | {
"line": 1067,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ p.coeffs = ∅ → p = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset",
"id",
"Ne",
"Finset.instEmptyCollection",
"Polynomial",
"Mathlib.Tactic.Contrapose.contrapose₁",
"im... | [
"R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ p ≠ 0 → p.coeffs.Nonempty"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.Module.Submodule.Invariant | {
"line": 211,
"column": 84
} | {
"line": 215,
"column": 70
} | {
"line": 217,
"column": 0
} | [
{
"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⊢ span R (MulAction.orbit G x) ∈ invtSubmodule (DistribSMul.toLinearMap R M g)",
"ppTerm": "?m.3... | [] | by
rw [mem_invtSubmodule, Submodule.span_le, Submodule.comap_coe]
intro y hy
simp only [Set.mem_preimage, DistribSMul.toLinearMap_apply, SetLike.mem_coe]
exact Submodule.subset_span <| MulAction.mem_orbit_of_mem_orbit g hy | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Projection | {
"line": 265,
"column": 15
} | {
"line": 265,
"column": 29
} | {
"line": 265,
"column": 30
} | [
{
"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\nhpq : IsCompl p q\nx : E\n⊢ x = (p.projection q hpq) x ↔ x ∈ p",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
... | [
"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\nhpq : IsCompl p q\nx : E\n⊢ x - (p.projection q hpq) x = 0 ↔ x ∈ p"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Projection | {
"line": 601,
"column": 2
} | {
"line": 606,
"column": 78
} | {
"line": 608,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\np : Submodule R E\nf : E →ₗ[R] E\nh : IsProj p f\n⊢ f = ↑(p.prodEquivOfIsCompl f.ker ⋯) ∘ₗ id.prodMap 0 ∘ₗ ↑(p.prodEquivOfIsCompl f.ker ⋯).symm",
"ppTerm": "?m.188",
"assigned": true,
"usedConstants": ... | [] | rw [← LinearMap.comp_assoc, LinearEquiv.eq_comp_toLinearMap_symm]
ext x
· simp only [coe_prodEquivOfIsCompl, comp_apply, coe_inl, coprod_apply, coe_subtype,
map_zero, add_zero, h.map_id x x.2, prodMap_apply, id_apply]
· simp only [coe_prodEquivOfIsCompl, comp_apply, coe_inr, coprod_apply, map_zero,
co... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Projection | {
"line": 601,
"column": 2
} | {
"line": 606,
"column": 78
} | {
"line": 608,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\np : Submodule R E\nf : E →ₗ[R] E\nh : IsProj p f\n⊢ f = ↑(p.prodEquivOfIsCompl f.ker ⋯) ∘ₗ id.prodMap 0 ∘ₗ ↑(p.prodEquivOfIsCompl f.ker ⋯).symm",
"ppTerm": "?m.188",
"assigned": true,
"usedConstants": ... | [] | rw [← LinearMap.comp_assoc, LinearEquiv.eq_comp_toLinearMap_symm]
ext x
· simp only [coe_prodEquivOfIsCompl, comp_apply, coe_inl, coprod_apply, coe_subtype,
map_zero, add_zero, h.map_id x x.2, prodMap_apply, id_apply]
· simp only [coe_prodEquivOfIsCompl, comp_apply, coe_inr, coprod_apply, map_zero,
co... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Projection | {
"line": 629,
"column": 2
} | {
"line": 629,
"column": 28
} | {
"line": 630,
"column": 2
} | [
{
"pp": "S : Type u_5\ninst✝² : Semiring S\nM : Type u_6\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : M →ₗ[S] M\nhf : IsProj m f\n⊢ m = ⊤ ↔ f = id",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Submodule",
"LinearMap.instFunLike"... | [
"case mp\nS : Type u_5\ninst✝² : Semiring S\nM : Type u_6\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nf : M →ₗ[S] M\nhf : IsProj ⊤ f\n⊢ f = id",
"case mpr\nS : Type u_5\ninst✝² : Semiring S\nM : Type u_6\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nhf : IsProj m id\n⊢ m = ⊤"
] | constructor <;> rintro rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.Projection | {
"line": 636,
"column": 2
} | {
"line": 636,
"column": 28
} | {
"line": 637,
"column": 2
} | [
{
"pp": "S : Type u_5\ninst✝² : Semiring S\nM : Type u_6\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nf : M →ₗ[S] M\nhf : IsProj m f\n⊢ m = ⊥ ↔ f = 0",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Submodule",
"LinearMap.instFunLike",
"Bot.bot",
... | [
"case mp\nS : Type u_5\ninst✝² : Semiring S\nM : Type u_6\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nf : M →ₗ[S] M\nhf : IsProj ⊥ f\n⊢ f = 0",
"case mpr\nS : Type u_5\ninst✝² : Semiring S\nM : Type u_6\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nhf : IsProj m 0\n⊢ m = ⊥"
] | constructor <;> rintro rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Rel | {
"line": 590,
"column": 2
} | {
"line": 590,
"column": 9
} | {
"line": 591,
"column": 2
} | [
{
"pp": "case mpr.mp\nα : Type u_1\nβ : Type u_2\nR : SetRel α β\nh : ∀ (a : α), ∃! b, (a, b) ∈ R\nf : α → β\nhf : ∀ (x : α), (x, f x) ∈ R\na : α\nb : β\n⊢ (a, b) ∈ Function.graph f → (a, b) ∈ R",
"ppTerm": "?mpr.mp",
"assigned": true,
"usedConstants": [
"SetRel",
"Function.graph",
... | [
"case mpr.mpr\nα : Type u_1\nβ : Type u_2\nR : SetRel α β\nh : ∀ (a : α), ∃! b, (a, b) ∈ R\nf : α → β\nhf : ∀ (x : α), (x, f x) ∈ R\na : α\nb : β\n⊢ (a, b) ∈ R → (a, b) ∈ Function.graph f"
] | · aesop | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Exact.Basic | {
"line": 435,
"column": 10
} | {
"line": 435,
"column": 24
} | {
"line": 435,
"column": 25
} | [
{
"pp": "case refine_1.left\nR✝ : 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✝¹³ : Semiring R✝\ninst✝¹² : AddCommGroup M✝\ninst✝¹¹ : AddCommGroup N✝\ninst✝¹⁰ : AddCommGroup P✝\ninst✝⁹ : Module R✝ M✝\ninst✝⁸ : Module R✝ N✝\ninst✝⁷ : Module R✝ P✝\nf✝ : ... | [
"case refine_1.left\nR✝ : 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✝¹³ : Semiring R✝\ninst✝¹² : AddCommGroup M✝\ninst✝¹¹ : AddCommGroup N✝\ninst✝¹⁰ : AddCommGroup P✝\ninst✝⁹ : Module R✝ M✝\ninst✝⁸ : Module R✝ N✝\ninst✝⁷ : Module R✝ P✝\nf✝ : M✝ →ₗ[R✝] N✝... | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.RelSeries | {
"line": 757,
"column": 2
} | {
"line": 757,
"column": 26
} | {
"line": 758,
"column": 2
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\np q : RelSeries r\nh : p.last = q.head\n⊢ (p.smash q h).last = q.last",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"RelSeries.last",
"id",
"RelSeries.smash",
"Eq"
],
"usedFVars": [
"α",
"r",
"p",
... | [
"α : Type u_1\nr : SetRel α α\np q : RelSeries r\nh : p.last = q.head\n⊢ Fin.addCases (p.toFun ∘ Fin.castSucc) q.toFun (Fin.last (p.length + q.length)) = q.toFun (Fin.last q.length)"
] | dsimp only [smash, last] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Exact.Basic | {
"line": 572,
"column": 30
} | {
"line": 572,
"column": 49
} | {
"line": 572,
"column": 50
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_4\nP : Type u_6\ninst✝⁶ : Ring 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 : Function.Exact ⇑f ⇑g\n⊢ Function.Surjective ⇑f ↔ g.ker = ⊤"... | [
"R : Type u_1\nM : Type u_2\nN : Type u_4\nP : Type u_6\ninst✝⁶ : Ring 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 : Function.Exact ⇑f ⇑g\n⊢ Function.Surjective ⇑f ↔ f.range = ⊤"
] | h.linearMap_ker_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.RelSeries | {
"line": 802,
"column": 2
} | {
"line": 808,
"column": 59
} | {
"line": 809,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\nr : SetRel α α\ninst✝ : Nonempty α\n⊢ (∀ (x : RelSeries r), ∃ y, x.length < y.length) → ∀ (n : ℕ), ∃ x, x.length = n",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Eq.mpr",
"Nat.recAux"... | [
"case mpr\nα : Type u_1\nr : SetRel α α\ninst✝ : Nonempty α\n⊢ (∀ (n : ℕ), ∃ x, x.length = n) → ∀ (x : RelSeries r), ∃ y, x.length < y.length"
] | · intro H n
induction n with
| zero => refine ⟨⟨0, ![_root_.Nonempty.some ‹_›], by simp⟩, by simp⟩
| succ n IH =>
obtain ⟨l, hl⟩ := IH
obtain ⟨l', hl'⟩ := H l
exact ⟨l'.take ⟨n + 1, by simpa [hl] using hl'⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Exact.Basic | {
"line": 576,
"column": 30
} | {
"line": 576,
"column": 49
} | {
"line": 576,
"column": 50
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_4\nP : Type u_6\ninst✝⁶ : Ring 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 : Function.Exact ⇑f ⇑g\n⊢ g.ker = ⊥ ↔ f = 0",
"ppTerm": "... | [
"R : Type u_1\nM : Type u_2\nN : Type u_4\nP : Type u_6\ninst✝⁶ : Ring 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 : Function.Exact ⇑f ⇑g\n⊢ f.range = ⊥ ↔ f = 0"
] | h.linearMap_ker_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.RelSeries | {
"line": 938,
"column": 4
} | {
"line": 938,
"column": 43
} | {
"line": 939,
"column": 4
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nn : ℕ\nlf : Fin n\nf : Fin (↑lf + 1) → α\nmf : StrictMono f\ng : Fin (↑lf + 1) → α\nmg : StrictMono g\ne : mk (↑lf) f ⋯ = mk (↑lf) g ⋯\n⊢ f = g",
"ppTerm": "?m.118",
"assigned": true,
"usedC... | [
"α : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nn : ℕ\nlf : Fin n\nf : Fin (↑lf + 1) → α\nmf : StrictMono f\ng : Fin (↑lf + 1) → α\nmg : StrictMono g\ne : mk (↑lf) f ⋯ = mk (↑lf) g ⋯\nfeq : ∀ (i : Fin ((mk (↑lf) f ⋯).length + 1)), (mk (↑lf) f ⋯).toFun i = (mk (↑... | have feq := fun i ↦ congr($(e).toFun i) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.KrullDimension | {
"line": 389,
"column": 4
} | {
"line": 389,
"column": 22
} | {
"line": 391,
"column": 0
} | [
{
"pp": "case h.right\nα : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ\nhne : Nonempty { p // RelSeries.last p = a }\nm : ℕ\nh : n ≤ m\nha : ⨆ x, ↑(↑x).length = ↑m\np : LTSeries α\nhlast : RelSeries.last p = a\nhlen : p.length = m\n⊢ (RelSeries.drop p ⟨m - n, ⋯⟩).length = n",
"ppTerm": "?h.right",
"assig... | [] | · simp [hlen]; lia | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.KrullDimension | {
"line": 462,
"column": 19
} | {
"line": 462,
"column": 36
} | {
"line": 462,
"column": 36
} | [
{
"pp": "case e'_2.mp\nα : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\n⊢ (∃ p, RelSeries.head p = x ∧ p.length = n) → ∃ p, RelSeries.last p = x ∧ p.length = n",
"ppTerm": "?e'_2.mp",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"RelSeries.last",
"Set.ofPred",
"Finite... | [
"case e'_2.mp\nα : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\np : LTSeries α\nhp : RelSeries.head p = x\nhl : p.length = n\n⊢ ∃ p, RelSeries.last p = x ∧ p.length = n"
] | intro ⟨p, hp, hl⟩ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Order.KrullDimension | {
"line": 462,
"column": 19
} | {
"line": 462,
"column": 36
} | {
"line": 462,
"column": 36
} | [
{
"pp": "case e'_2.mp\nα : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\n⊢ (∃ p, RelSeries.head p = x ∧ p.length = n) → ∃ p, RelSeries.last p = x ∧ p.length = n",
"ppTerm": "?e'_2.mp",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"RelSeries.last",
"Set.ofPred",
"Finite... | [
"case e'_2.mp\nα : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\np : LTSeries α\nhp : RelSeries.head p = x\nhl : p.length = n\n⊢ ∃ p, RelSeries.last p = x ∧ p.length = n"
] | intro ⟨p, hp, hl⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Order.KrullDimension | {
"line": 462,
"column": 19
} | {
"line": 462,
"column": 36
} | {
"line": 462,
"column": 36
} | [
{
"pp": "case e'_2.mpr\nα : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\n⊢ (∃ p, RelSeries.last p = x ∧ p.length = n) → ∃ p, RelSeries.head p = x ∧ p.length = n",
"ppTerm": "?e'_2.mpr",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"RelSeries.last",
"Set.ofPred",
"Fini... | [
"case e'_2.mpr\nα : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\np : LTSeries αᵒᵈ\nhp : RelSeries.last p = x\nhl : p.length = n\n⊢ ∃ p, RelSeries.head p = x ∧ p.length = n"
] | intro ⟨p, hp, hl⟩ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Order.KrullDimension | {
"line": 462,
"column": 19
} | {
"line": 462,
"column": 36
} | {
"line": 462,
"column": 36
} | [
{
"pp": "case e'_2.mpr\nα : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\n⊢ (∃ p, RelSeries.last p = x ∧ p.length = n) → ∃ p, RelSeries.head p = x ∧ p.length = n",
"ppTerm": "?e'_2.mpr",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"RelSeries.last",
"Set.ofPred",
"Fini... | [
"case e'_2.mpr\nα : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\np : LTSeries αᵒᵈ\nhp : RelSeries.last p = x\nhl : p.length = n\n⊢ ∃ p, RelSeries.head p = x ∧ p.length = n"
] | intro ⟨p, hp, hl⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Order.KrullDimension | {
"line": 607,
"column": 2
} | {
"line": 607,
"column": 13
} | {
"line": 608,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\n⊢ 0 ≤ krullDim α ↔ Nonempty α",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"Preorder.toLT",
"Mathlib.Tactic.Contrapose.contrapose_iff₁",
"instLinearOrderENat",
... | [
"α : Type u_1\ninst✝ : Preorder α\n⊢ krullDim α < 0 ↔ IsEmpty α"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Order.KrullDimension | {
"line": 637,
"column": 2
} | {
"line": 637,
"column": 13
} | {
"line": 638,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\n⊢ (∀ (i : LTSeries α), ↑i.length ≤ 1) ↔ ∀ (x : α), (∀ (b : α), ¬b < x) ∨ ∀ (b : α), ¬x < b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Eq.mpr",
"instCompleteLatticeWithBot",
"WithBo... | [
"α : Type u_1\ninst✝ : Preorder α\n⊢ (∃ i, 1 < ↑i.length) ↔ ∃ x, (∃ b, b < x) ∧ ∃ b, x < b"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Order.KrullDimension | {
"line": 646,
"column": 2
} | {
"line": 646,
"column": 13
} | {
"line": 647,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\n⊢ 0 < krullDim α ↔ ∃ x y, x < y",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"Preorder.toLT",
"Mathlib.Tactic.Contrapose.... | [
"α : Type u_1\ninst✝ : Preorder α\n⊢ krullDim α ≤ 0 ↔ ∀ (x y : α), ¬x < y"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Order.KrullDimension | {
"line": 1096,
"column": 2
} | {
"line": 1096,
"column": 38
} | {
"line": 1097,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ coheight ↑n = ⊤",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"instAddMonoidWithOneENat",
"instTopENat",
"WithTop.instPreorder",
"instAddENat",
"Order.coheight_coe_withTop",
"WithTop.some",
"AddMonoidWithOne.toOne",
"instHA... | [
"n : ℕ\n⊢ coheight n + 1 = ⊤"
] | apply (coheight_coe_withTop _).trans | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.TensorProduct.Map | {
"line": 136,
"column": 64
} | {
"line": 138,
"column": 67
} | {
"line": 140,
"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\n⊢ map LinearMap.id LinearMap.id = LinearMap.id",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"LinearMap.id",
... | [] | by
ext
simp only [mk_apply, id_coe, compr₂ₛₗ_apply, _root_.id, map_tmul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Kleene | {
"line": 158,
"column": 44
} | {
"line": 158,
"column": 78
} | {
"line": 161,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nπ : ι → Type u_4\ninst✝ : IdemSemiring α\na✝ b✝ c✝ a b c : α\nhbc : (fun x1 x2 ↦ x1 ≤ x2) b c\n⊢ (fun x1 x2 ↦ x1 * x2) a c + (fun x1 x2 ↦ x1 * x2) a b = (fun x1 x2 ↦ x1 * x2) a c",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Distr... | [] | by rw [← mul_add, hbc.add_eq_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.Submonoid.Pointwise | {
"line": 223,
"column": 6
} | {
"line": 223,
"column": 45
} | {
"line": 224,
"column": 2
} | [
{
"pp": "case refine_2\nM : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝ : NonUnitalNonAssocRing R\nx y : AddSubmonoid R\nm : R\nhm : m ∈ x\nn : R\nhn : n ∈ y\n⊢ -m * n ∈ -x * y",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddSubmonoid.involutiveNeg",
"A... | [] | exact mul_mem_mul (neg_mem_neg.2 hm) hn | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Ring.Submonoid.Pointwise | {
"line": 223,
"column": 6
} | {
"line": 223,
"column": 45
} | {
"line": 224,
"column": 2
} | [
{
"pp": "case refine_2\nM : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝ : NonUnitalNonAssocRing R\nx y : AddSubmonoid R\nm : R\nhm : m ∈ x\nn : R\nhn : n ∈ y\n⊢ -m * n ∈ -x * y",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddSubmonoid.involutiveNeg",
"A... | [] | exact mul_mem_mul (neg_mem_neg.2 hm) hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.Submonoid.Pointwise | {
"line": 223,
"column": 6
} | {
"line": 223,
"column": 45
} | {
"line": 224,
"column": 2
} | [
{
"pp": "case refine_2\nM : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝ : NonUnitalNonAssocRing R\nx y : AddSubmonoid R\nm : R\nhm : m ∈ x\nn : R\nhn : n ∈ y\n⊢ -m * n ∈ -x * y",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddSubmonoid.involutiveNeg",
"A... | [] | exact mul_mem_mul (neg_mem_neg.2 hm) hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fintype.Lattice | {
"line": 63,
"column": 2
} | {
"line": 64,
"column": 51
} | {
"line": 65,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : Finite α\ninst✝¹ : Nonempty α\ninst✝ : LinearOrder β\nf : α → β\n⊢ ∃ x₀, ∀ (x : α), f x₀ ≤ f x",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Finset.univ",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"instInh... | [] | cases nonempty_fintype α
simpa using exists_min_image univ f univ_nonempty | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fintype.Lattice | {
"line": 63,
"column": 2
} | {
"line": 64,
"column": 51
} | {
"line": 65,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : Finite α\ninst✝¹ : Nonempty α\ninst✝ : LinearOrder β\nf : α → β\n⊢ ∃ x₀, ∀ (x : α), f x₀ ≤ f x",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Finset.univ",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"instInh... | [] | cases nonempty_fintype α
simpa using exists_min_image univ f univ_nonempty | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Coprime.Basic | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 17
} | {
"line": 82,
"column": 2
} | [
{
"pp": "case h\nR : Type u_1\ninst✝ : CommRing R\na b u v : ℤ\nH : u * a + v * b = 1\n⊢ ↑u * ↑a + ↑v * ↑b = 1",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
"CommSem... | [
"case h\nR : Type u_1\ninst✝ : CommRing R\na b u v : ℤ\nH : u * a + v * b = 1\n⊢ ↑1 = 1"
] | rw_mod_cast [H] | Lean.Parser.Tactic._aux_Init_TacticsExtra___macroRules_Lean_Parser_Tactic_tacticRw_mod_cast____1 | Lean.Parser.Tactic.tacticRw_mod_cast___ |
Mathlib.Algebra.Algebra.Operations | {
"line": 329,
"column": 2
} | {
"line": 329,
"column": 64
} | {
"line": 331,
"column": 0
} | [
{
"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 ^ 1 = M",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Submodule.pow_zero",
"Eq.mpr",
"Submodule",
"HMul.hMul",
... | [] | rw [Submodule.pow_succ, Submodule.pow_zero, Submodule.one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Algebra.Operations | {
"line": 329,
"column": 2
} | {
"line": 329,
"column": 64
} | {
"line": 331,
"column": 0
} | [
{
"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 ^ 1 = M",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Submodule.pow_zero",
"Eq.mpr",
"Submodule",
"HMul.hMul",
... | [] | rw [Submodule.pow_succ, Submodule.pow_zero, Submodule.one_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Operations | {
"line": 329,
"column": 2
} | {
"line": 329,
"column": 64
} | {
"line": 331,
"column": 0
} | [
{
"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 ^ 1 = M",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Submodule.pow_zero",
"Eq.mpr",
"Submodule",
"HMul.hMul",
... | [] | rw [Submodule.pow_succ, Submodule.pow_zero, Submodule.one_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Operations | {
"line": 879,
"column": 37
} | {
"line": 879,
"column": 67
} | {
"line": 879,
"column": 67
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 ↦ x • x_1) y' = y\n⊢ y ∈ ↑I",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [] | rw [← xy'_eq_y]; exact h _ hy' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Operations | {
"line": 879,
"column": 37
} | {
"line": 879,
"column": 67
} | {
"line": 879,
"column": 67
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nx : A\nI J : Submodule R A\nh : x ∈ I / J\ny : A\nx✝ : y ∈ x • ↑J\ny' : A\nhy' : y' ∈ ↑J\nxy'_eq_y : (fun x_1 ↦ x • x_1) y' = y\n⊢ y ∈ ↑I",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [] | rw [← xy'_eq_y]; exact h _ hy' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Prod | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 44
} | {
"line": 167,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal S\nh : (⊤.prod I).IsPrime\n⊢ I.IsPrime",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Ideal.isPrime_of_isPrime_prod_top"
],
"usedFVars": [
"S",
"R",
"inst✝",
"inst✝¹"... | [
"R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal S\nh : (⊤.prod I).IsPrime\n⊢ (I.prod ⊤).IsPrime"
] | apply isPrime_of_isPrime_prod_top (S := R) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Ideal.Prod | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 13
} | {
"line": 184,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal R\nJ : Ideal S\n⊢ (I.prod J).IsPrime → I = ⊤ ∨ J = ⊤",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"id",
"Prod.instSemiring",
"Ne",
"Subm... | [
"R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal R\nJ : Ideal S\n⊢ I ≠ ⊤ ∧ J ≠ ⊤ → ¬(I.prod J).IsPrime"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.RingTheory.Ideal.Maps | {
"line": 507,
"column": 2
} | {
"line": 512,
"column": 29
} | {
"line": 514,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : Semiring R\ninst✝² : Semiring S\nE : Type u_4\ninst✝¹ : EquivLike E R S\ninst✝ : RingEquivClass E R S\ne : E\nI : Ideal R\ny : S\n⊢ y ∈ map e I ↔ ∃ x ∈ I, e x = y",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Equiv.apply_symm_apply",
... | [] | constructor
· intro h
simp_rw [show map e I = _ from map_comap_of_equiv (RingEquivClass.toRingEquiv e : R ≃+* S)] at h
exact ⟨(EquivLike.toEquiv e).symm y, h, (EquivLike.toEquiv e).apply_symm_apply y⟩
· rintro ⟨x, hx, rfl⟩
exact mem_map_of_mem e hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Maps | {
"line": 507,
"column": 2
} | {
"line": 512,
"column": 29
} | {
"line": 514,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : Semiring R\ninst✝² : Semiring S\nE : Type u_4\ninst✝¹ : EquivLike E R S\ninst✝ : RingEquivClass E R S\ne : E\nI : Ideal R\ny : S\n⊢ y ∈ map e I ↔ ∃ x ∈ I, e x = y",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Equiv.apply_symm_apply",
... | [] | constructor
· intro h
simp_rw [show map e I = _ from map_comap_of_equiv (RingEquivClass.toRingEquiv e : R ≃+* S)] at h
exact ⟨(EquivLike.toEquiv e).symm y, h, (EquivLike.toEquiv e).apply_symm_apply y⟩
· rintro ⟨x, hx, rfl⟩
exact mem_map_of_mem e hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Maps | {
"line": 673,
"column": 69
} | {
"line": 675,
"column": 16
} | {
"line": 677,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\nF : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : FunLike F R S\nrc : RingHomClass F R S\nf : F\nK : Ideal S\n⊢ comap f K.radical = (comap f K).radical",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Submodule",
"instHSMul... | [] | by
ext
simp [radical] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 299,
"column": 6
} | {
"line": 299,
"column": 44
} | {
"line": 300,
"column": 6
} | [
{
"pp": "case neg\nR : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx r s g : R\nhr : x = g * r\nhgxy : ¬g = 0\nhxy : x = 0 ∨ g * s / gcd x (g * s) = 0\nhy : g * s / g = 0\n⊢ g * s = 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
... | [
"case neg\nR : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx r s g : R\nhr : x = g * r\nhgxy : ¬g = 0\nhxy : x = 0 ∨ g * s / gcd x (g * s) = 0\nhy : s = 0\n⊢ g * s = 0"
] | rw [mul_div_cancel_left₀ _ hgxy] at hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 36
} | {
"line": 351,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\nx y z : R\nh1 : y ≠ 0\nh2 : y ∣ x\n⊢ (x - y * z) / y = x / y - z",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"EuclideanDomain.sub_mul_div_left"
],
"usedFVars": [
"R",
"inst✝",
"x",
"y",
"z",
... | [] | exact sub_mul_div_left _ _ _ h1 h2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.GCDMonoid.Multiset | {
"line": 169,
"column": 2
} | {
"line": 171,
"column": 61
} | {
"line": 173,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_2\ninst✝¹ : CommMonoidWithZero α\ninst✝ : StrongNormalizedGCDMonoid α\na : α\ns✝ : Multiset α\nb : α\ns : Multiset α\nih : (map (fun x ↦ a * x) s).gcd = normalize a * s.gcd\n⊢ (map (fun x ↦ a * x) (b ::ₘ s)).gcd = normalize a * (b ::ₘ s).gcd",
"ppTerm": "?refine_2",
"a... | [] | · simp_rw [map_cons, gcd_cons, ← gcd_mul_left]
rw [ih]
apply ((normalize_associated a).mul_right _).gcd_eq_right | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.Maps | {
"line": 1111,
"column": 8
} | {
"line": 1111,
"column": 22
} | {
"line": 1111,
"column": 23
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : FunLike F R S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ⇑f\nI : Ideal R\nH : I.IsPrime\nhk : RingHom.ker f ≤ I\nx y : S\na : R\nha : f a = x\nb : R\nhb : f b = y\nc : R\nhc : c ∈ I\nhc' : ... | [
"case refine_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : FunLike F R S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ⇑f\nI : Ideal R\nH : I.IsPrime\nhk : RingHom.ker f ≤ I\nx y : S\na : R\nha : f a = x\nb : R\nhb : f b = y\nc : R\nhc : c ∈ I\nhc' : f c - f (a *... | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Nat | {
"line": 160,
"column": 2
} | {
"line": 161,
"column": 67
} | {
"line": 163,
"column": 0
} | [
{
"pp": "case refine_2\nn : ℕ\n⊢ (fun x ↦ x.out.natAbs) ((fun x ↦ Associates.mk ↑x) n) = n",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Associates.mk",
"Int.strongNormalizationMonoid",
"abs",
"congrArg",
"Int.instStrongNormalizedGCDMono... | [] | · dsimp only [Associates.out_mk]
rw [← Int.abs_eq_normalize, Int.natAbs_abs, Int.natAbs_natCast] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.Maps | {
"line": 1171,
"column": 23
} | {
"line": 1171,
"column": 37
} | {
"line": 1171,
"column": 38
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : Ring A\ninst✝¹ : Ring B\ninst✝ : Ring C\nf : A →+* B\nf_inv : B → A\nhf : Function.RightInverse f_inv ⇑f\ng : A →+* C\nhg : ker f ≤ ker g\n⊢ g (f_inv 1) = g 1",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtr... | [
"A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : Ring A\ninst✝¹ : Ring B\ninst✝ : Ring C\nf : A →+* B\nf_inv : B → A\nhf : Function.RightInverse f_inv ⇑f\ng : A →+* C\nhg : ker f ≤ ker g\n⊢ g (f_inv 1) - g 1 = 0"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 1177,
"column": 23
} | {
"line": 1177,
"column": 37
} | {
"line": 1177,
"column": 38
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : Ring A\ninst✝¹ : Ring B\ninst✝ : Ring C\nf : A →+* B\nf_inv : B → A\nhf : Function.RightInverse f_inv ⇑f\ng : A →+* C\nhg : ker f ≤ ker g\nx y : B\n⊢ g (f_inv (x * y)) = g (f_inv x * f_inv y)",
"ppTerm": "?m.139",
"assigned": true,
"usedCon... | [
"A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : Ring A\ninst✝¹ : Ring B\ninst✝ : Ring C\nf : A →+* B\nf_inv : B → A\nhf : Function.RightInverse f_inv ⇑f\ng : A →+* C\nhg : ker f ≤ ker g\nx y : B\n⊢ g (f_inv (x * y)) - g (f_inv x * f_inv y) = 0"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Finset | {
"line": 237,
"column": 2
} | {
"line": 241,
"column": 30
} | {
"line": 243,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns : Finset β\nf g : β → α\nhs : ∃ x ∈ s, f x ≠ 0\nhg : ∀ b ∈ s, f b = s.gcd f * g b\n⊢ s.gcd g = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_... | [] | rw [← normalize_gcd, normalize_eq_one, ← associated_one_iff_isUnit]
refine .of_mul_left (.symm <| .trans ?_ (gcd_mul_left' ..)) .rfl (a := s.gcd f) ?_
· simp [← gcd_congr rfl hg]
contrapose! hs
exact s.gcd_eq_zero_iff.1 hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GCDMonoid.Finset | {
"line": 237,
"column": 2
} | {
"line": 241,
"column": 30
} | {
"line": 243,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns : Finset β\nf g : β → α\nhs : ∃ x ∈ s, f x ≠ 0\nhg : ∀ b ∈ s, f b = s.gcd f * g b\n⊢ s.gcd g = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_... | [] | rw [← normalize_gcd, normalize_eq_one, ← associated_one_iff_isUnit]
refine .of_mul_left (.symm <| .trans ?_ (gcd_mul_left' ..)) .rfl (a := s.gcd f) ?_
· simp [← gcd_congr rfl hg]
contrapose! hs
exact s.gcd_eq_zero_iff.1 hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GCDMonoid.Finset | {
"line": 260,
"column": 2
} | {
"line": 263,
"column": 58
} | {
"line": 265,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝³ : CommMonoidWithZero α\ninst✝² : NormalizedGCDMonoid α\ninst✝¹ : Div α\ninst✝ : MulDivCancelClass α\nf : ι → α\ns : Finset ι\ni : ι\nhis : i ∈ s\nhfi : f i ≠ 0\n⊢ (s.gcd fun j ↦ f j / s.gcd f) = 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
... | [] | obtain ⟨g, he, hg⟩ := Finset.extract_gcd f ⟨i, his⟩
refine (Finset.gcd_congr rfl fun a ha ↦ ?_).trans hg
rw [he a ha, mul_div_cancel_left₀]
exact mt Finset.gcd_eq_zero_iff.1 fun h ↦ hfi <| h i his | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GCDMonoid.Finset | {
"line": 260,
"column": 2
} | {
"line": 263,
"column": 58
} | {
"line": 265,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝³ : CommMonoidWithZero α\ninst✝² : NormalizedGCDMonoid α\ninst✝¹ : Div α\ninst✝ : MulDivCancelClass α\nf : ι → α\ns : Finset ι\ni : ι\nhis : i ∈ s\nhfi : f i ≠ 0\n⊢ (s.gcd fun j ↦ f j / s.gcd f) = 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
... | [] | obtain ⟨g, he, hg⟩ := Finset.extract_gcd f ⟨i, his⟩
refine (Finset.gcd_congr rfl fun a ha ↦ ?_).trans hg
rw [he a ha, mul_div_cancel_left₀]
exact mt Finset.gcd_eq_zero_iff.1 fun h ↦ hfi <| h i his | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 562,
"column": 2
} | {
"line": 562,
"column": 22
} | {
"line": 563,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\nm n k : α\nH : k ∣ m * n\n⊢ k ∣ m * gcd k n",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"CommMonoid.toCommSemigr... | [
"α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\nm n k : α\nH : k ∣ n * m\n⊢ k ∣ gcd k n * m"
] | rw [mul_comm] at H ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1010,
"column": 2
} | {
"line": 1013,
"column": 9
} | {
"line": 1015,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nI P : Ideal R\nhP : P.IsPrime\nn : ℕ\nhn : n ≠ 0\n⊢ I ^ n ≤ P ↔ I ≤ P",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Multiset.prod_replicate",
"False",
"Semiring.toModule",
"eq_false",
"congrArg",
"CommSemir... | [] | have h : (Multiset.replicate n I).prod ≤ P ↔ _ := hP.multiset_prod_le
simp_rw [Multiset.prod_replicate, Multiset.mem_replicate, ne_eq, hn, not_false_eq_true,
true_and, exists_eq_left] at h
exact h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1010,
"column": 2
} | {
"line": 1013,
"column": 9
} | {
"line": 1015,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nI P : Ideal R\nhP : P.IsPrime\nn : ℕ\nhn : n ≠ 0\n⊢ I ^ n ≤ P ↔ I ≤ P",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Multiset.prod_replicate",
"False",
"Semiring.toModule",
"eq_false",
"congrArg",
"CommSemir... | [] | have h : (Multiset.replicate n I).prod ≤ P ↔ _ := hP.multiset_prod_le
simp_rw [Multiset.prod_replicate, Multiset.mem_replicate, ne_eq, hn, not_false_eq_true,
true_and, exists_eq_left] at h
exact h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1027,
"column": 2
} | {
"line": 1029,
"column": 33
} | {
"line": 1030,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommRing α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nh : a ∣ b - c\n⊢ gcd a b = gcd a c",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"GCDMonoid.toIsCancelMulZero",
"MulZeroClass.toMul",
"congrArg",
... | [
"case hab\nα : Type u_1\ninst✝¹ : CommRing α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nh : a ∣ b - c\n⊢ gcd a b ∣ c",
"case hba\nα : Type u_1\ninst✝¹ : CommRing α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nh : a ∣ b - c\n⊢ gcd a c ∣ b"
] | apply dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _) <;>
rw [dvd_gcd_iff] <;>
refine ⟨gcd_dvd_left _ _, ?_⟩ | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1101,
"column": 6
} | {
"line": 1101,
"column": 23
} | {
"line": 1102,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\na b : α\n⊢ Associated (gcd a b * if a = 0 then 0 else... | [
"case pos\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\na b : α\na0 : a = 0\n⊢ Associated (gcd a b * 0) (a * b)... | split_ifs with a0 | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1106,
"column": 6
} | {
"line": 1106,
"column": 23
} | {
"line": 1107,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\na : α\n⊢ (if a = 0 then 0 else Classical.choose ⋯) = ... | [
"case pos\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\na : α\na0 : a = 0\n⊢ 0 = 0",
"case neg\nα : Type u_1\... | split_ifs with a0 | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1134,
"column": 6
} | {
"line": 1134,
"column": 23
} | {
"line": 1135,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : IsCancelMulZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\nnormalize_gcd : ∀ (a ... | [
"case pos\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : IsCancelMulZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\nnormalize_gcd : ∀ (a b ... | split_ifs with a0 | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1140,
"column": 6
} | {
"line": 1140,
"column": 23
} | {
"line": 1141,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : IsCancelMulZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\nnormalize_gcd : ∀ (a ... | [
"case pos\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : IsCancelMulZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\nnormalize_gcd : ∀ (a b ... | split_ifs with a0 | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 329,
"column": 6
} | {
"line": 329,
"column": 88
} | {
"line": 330,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\ninst✝ : IsPrincipalIdealRing R\na b : R\nhp : Irreducible (a * b)\nhI : R ∙ a * b < R ∙ a\n⊢ IsUnit a",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"Semiring.toModule",
"HMul.hMul",
"CommSemiring.toN... | [
"R : Type u\ninst✝¹ : CommSemiring R\ninst✝ : IsPrincipalIdealRing R\na b : R\nhp : Irreducible (a * b)\nhI : R ∙ a * b < R ∙ a\nhb : IsUnit b\n⊢ R ∙ a ≤ R ∙ a * b"
] | refine (of_irreducible_mul hp).resolve_right (mt (fun hb => ?_) (not_le_of_gt hI)) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 443,
"column": 37
} | {
"line": 443,
"column": 64
} | {
"line": 443,
"column": 65
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsBezout R\np n : R\nhp : Irreducible p\n⊢ IsCoprime p n ↔ ¬p ∣ n",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"CommRing.toNonUnitalCommRing",
"congrArg",
"CommSemiring.toSemiring",
... | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsBezout R\np n : R\nhp : Irreducible p\n⊢ IsRelPrime p n ↔ ¬p ∣ n"
] | ← isRelPrime_iff_isCoprime, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 472,
"column": 35
} | {
"line": 472,
"column": 48
} | {
"line": 472,
"column": 49
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : IsBezout R\ninst✝¹ : IsDomain R\ninst✝ : GCDMonoid R\na b z : R\n⊢ gcd a b ∣ z ↔ ∃ x y, z = x * a + y * b",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
... | [
"R : Type u\ninst✝³ : CommRing R\ninst✝² : IsBezout R\ninst✝¹ : IsDomain R\ninst✝ : GCDMonoid R\na b z : R\n⊢ gcd a b ∣ z ↔ ∃ x y, x * a + y * b = z"
] | @eq_comm _ z, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 546,
"column": 15
} | {
"line": 546,
"column": 47
} | {
"line": 546,
"column": 47
} | [
{
"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⊢ ↑(toNat (Module.rank R ↥N)) = Module.rank R ↥N",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"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 = Module.rank R ↥N",
"R : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : S... | Cardinal.cast_toNat_of_lt_aleph0 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1222,
"column": 2
} | {
"line": 1224,
"column": 73
} | {
"line": 1225,
"column": 2
} | [
{
"pp": "R : Type u_2\nι : Type u_3\ninst✝ : CommRing R\ns : Set ι\nhs : s.Finite\nf : ι → Ideal R\na b : ι\nhp : ∀ i ∈ s, i ≠ a → i ≠ b → (f i).IsPrime\nI : Ideal R\nt : Finset ι\nht : ∀ (a : ι), a ∈ t ↔ a ∈ s\n⊢ ↑I ⊆ ⋃ i ∈ s, ↑(f i) ↔ ∃ i ∈ s, I ≤ f i",
"ppTerm": "?m.44",
"assigned": true,
"usedCo... | [
"R : Type u_2\nι : Type u_3\ninst✝ : CommRing R\ns : Set ι\nhs : s.Finite\nf : ι → Ideal R\na b : ι\nhp : ∀ i ∈ s, i ≠ a → i ≠ b → (f i).IsPrime\nI : Ideal R\nt : Finset ι\nht : ∀ (a : ι), a ∈ t ↔ a ∈ s\nheq : ⋃ i ∈ s, ↑(f i) = ⋃ i ∈ t, ↑(f i)\n⊢ ↑I ⊆ ⋃ i ∈ s, ↑(f i) ↔ ∃ i ∈ s, I ≤ f i"
] | have heq : ⋃ i ∈ s, f i = ⋃ i ∈ t, (f i : Set R) := by
ext
simpa using exists_congr (fun i ↦ (and_congr_left fun a ↦ ht i).symm) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Matrix.Mul | {
"line": 460,
"column": 2
} | {
"line": 461,
"column": 44
} | {
"line": 463,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type v\ninst✝² : NonAssocSemiring α\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nM : Matrix m n α\n⊢ M * 1 = M",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
... | [] | ext
rw [← diagonal_one, mul_diagonal, mul_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Matrix.Mul | {
"line": 460,
"column": 2
} | {
"line": 461,
"column": 44
} | {
"line": 463,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type v\ninst✝² : NonAssocSemiring α\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nM : Matrix m n α\n⊢ M * 1 = M",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
... | [] | ext
rw [← diagonal_one, mul_diagonal, mul_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1382,
"column": 52
} | {
"line": 1382,
"column": 79
} | {
"line": 1382,
"column": 79
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nr : R\n⊢ ¬Ideal.span {r} = ⊥ ↔ r ≠ 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"Ideal.span_singleton_eq_bot",
"congrArg",
"CommSemiring.toSemiring",
"Set.instSingletonSet... | [
"R : Type u_1\ninst✝ : CommSemiring R\nr : R\n⊢ ¬r = 0 ↔ r ≠ 0"
] | Ideal.span_singleton_eq_bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FreeModule.Finite.Basic | {
"line": 49,
"column": 40
} | {
"line": 52,
"column": 67
} | {
"line": 54,
"column": 0
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nM : Type u_4\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Free R M\ninst✝² : Module.Finite R M\ninst✝¹ : Finite ι₁\ninst✝ : Finite ι₂\n⊢ Module.Finite R (Matrix ι₁ ι₂ M)",
"ppTerm": "?m.12",
"assigned": true,
"used... | [] | by
cases nonempty_fintype ι₁
cases nonempty_fintype ι₂
exact Module.Finite.of_basis <| (Free.chooseBasis _ _).matrix _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.AlgebraTower | {
"line": 164,
"column": 18
} | {
"line": 164,
"column": 32
} | {
"line": 164,
"column": 33
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\nι : Type u_5\nι' : Type u_6\nb : Basis ι R S\nc : Basis ι' S A\nij : ι' × ι\n⊢ ((b.smulTower c).reind... | [
"R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\nι : Type u_5\nι' : Type u_6\nb : Basis ι R S\nc : Basis ι' S A\nij : ι' × ι\n⊢ (b.smulTower c) ((Equiv.prodComm ι... | reindex_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Matrix.Mul | {
"line": 1248,
"column": 8
} | {
"line": 1248,
"column": 22
} | {
"line": 1248,
"column": 23
} | [
{
"pp": "m : Type u_10\nn : Type u_11\nR : Type u_12\ninst✝⁶ : Fintype m\ninst✝⁵ : DecidableEq m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : MulOne R\ninst✝¹ : AddCommMonoid R\ninst✝ : IsStablyFiniteRing R\nA : Matrix m n R\nB : Matrix n m R\ne : m ≃ n\n⊢ (reindex e e) (A * B) = (reindex e e) 1 ↔ B * ... | [
"m : Type u_10\nn : Type u_11\nR : Type u_12\ninst✝⁶ : Fintype m\ninst✝⁵ : DecidableEq m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : MulOne R\ninst✝¹ : AddCommMonoid R\ninst✝ : IsStablyFiniteRing R\nA : Matrix m n R\nB : Matrix n m R\ne : m ≃ n\n⊢ (A * B).submatrix ⇑e.symm ⇑e.symm = (reindex e e) 1 ↔ B * ... | reindex_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Matrix.Mul | {
"line": 1248,
"column": 23
} | {
"line": 1248,
"column": 37
} | {
"line": 1248,
"column": 38
} | [
{
"pp": "m : Type u_10\nn : Type u_11\nR : Type u_12\ninst✝⁶ : Fintype m\ninst✝⁵ : DecidableEq m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : MulOne R\ninst✝¹ : AddCommMonoid R\ninst✝ : IsStablyFiniteRing R\nA : Matrix m n R\nB : Matrix n m R\ne : m ≃ n\n⊢ (A * B).submatrix ⇑e.symm ⇑e.symm = (reindex e... | [
"m : Type u_10\nn : Type u_11\nR : Type u_12\ninst✝⁶ : Fintype m\ninst✝⁵ : DecidableEq m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : MulOne R\ninst✝¹ : AddCommMonoid R\ninst✝ : IsStablyFiniteRing R\nA : Matrix m n R\nB : Matrix n m R\ne : m ≃ n\n⊢ (A * B).submatrix ⇑e.symm ⇑e.symm = submatrix 1 ⇑e.symm ⇑e... | reindex_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finsupp.Multiset | {
"line": 71,
"column": 2
} | {
"line": 77,
"column": 7
} | {
"line": 79,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α →₀ ℕ\ng : α → β\n⊢ Multiset.map g (toMultiset f) = toMultiset (mapDomain g f)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"instHSMul",
"Finsupp.toMulti... | [] | refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.map_zero, mapDomain_zero, toMultiset_zero]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
(Multiset.mapAddMo... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Multiset | {
"line": 71,
"column": 2
} | {
"line": 77,
"column": 7
} | {
"line": 79,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α →₀ ℕ\ng : α → β\n⊢ Multiset.map g (toMultiset f) = toMultiset (mapDomain g f)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"instHSMul",
"Finsupp.toMulti... | [] | refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.map_zero, mapDomain_zero, toMultiset_zero]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
(Multiset.mapAddMo... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.Multiset | {
"line": 93,
"column": 2
} | {
"line": 97,
"column": 40
} | {
"line": 99,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : DecidableEq α\nf : α →₀ ℕ\n⊢ ∀ (a : α) (b : ℕ) (f : α →₀ ℕ),\n a ∉ f.support →\n b ≠ 0 → (toMultiset f).toFinset = f.support → (toMultiset (single a b + f)).toFinset = (single a b + f).support",
"ppTerm": "?refine_2",
"assigned": true,
"usedConst... | [] | · intro a n f ha hn ih
rw [toMultiset_add, Multiset.toFinset_add, ih, toMultiset_single, support_add_eq,
support_single _ hn, Multiset.toFinset_nsmul _ _ hn, Multiset.toFinset_singleton]
refine Disjoint.mono_left support_single_subset ?_
rwa [Finset.disjoint_singleton_left] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Matrix.Basic | {
"line": 203,
"column": 70
} | {
"line": 204,
"column": 75
} | {
"line": 206,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type u_11\ninst✝⁴ : Semiring α\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\nr : α\nM : Matrix m n α\n⊢ (scalar m) r * M = M * (scalar n) r ↔ r • M = MulOpposite.op r • M",
"ppTerm": "?m.30",
"assigned": true,
"usedCo... | [] | by
simp_rw [scalar_apply, ← smul_eq_diagonal_mul, ← op_smul_eq_mul_diagonal] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finsupp.Multiset | {
"line": 161,
"column": 28
} | {
"line": 161,
"column": 44
} | {
"line": 161,
"column": 45
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\na a✝ : α\n⊢ count a✝ {a} = (Finsupp.single a 1) a✝",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"Multiset.count",
"Multiset",
"id"... | [
"α : Type u_1\ninst✝ : DecidableEq α\na a✝ : α\n⊢ (if a✝ = a then 1 else 0) = (Finsupp.single a 1) a✝"
] | count_singleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.Free | {
"line": 49,
"column": 80
} | {
"line": 49,
"column": 95
} | {
"line": 50,
"column": 4
} | [
{
"pp": "F : Type u\nK : Type v\nA : Type w\ninst✝¹⁰ : Semiring F\ninst✝⁹ : Semiring K\ninst✝⁸ : AddCommMonoid A\ninst✝⁷ : Module F K\ninst✝⁶ : Module K A\ninst✝⁵ : Module F A\ninst✝⁴ : IsScalarTower F K A\ninst✝³ : StrongRankCondition F\ninst✝² : StrongRankCondition K\ninst✝¹ : Free F K\ninst✝ : Free K A\nb : ... | [
"F : Type u\nK : Type v\nA : Type w\ninst✝¹⁰ : Semiring F\ninst✝⁹ : Semiring K\ninst✝⁸ : AddCommMonoid A\ninst✝⁷ : Module F K\ninst✝⁶ : Module K A\ninst✝⁵ : Module F A\ninst✝⁴ : IsScalarTower F K A\ninst✝³ : StrongRankCondition F\ninst✝² : StrongRankCondition K\ninst✝¹ : Free F K\ninst✝ : Free K A\nb : Basis (Free.... | ← c.mk_eq_rank, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Subalgebra.Basic | {
"line": 669,
"column": 30
} | {
"line": 669,
"column": 87
} | {
"line": 669,
"column": 87
} | [
{
"pp": "R : Type u\nA : Type v\nB✝ : Type w\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B✝\ninst✝¹ : Algebra R B✝\nS T U : Subalgebra R A\ninst✝ : Subsingleton A\nB C : Subalgebra R A\nx : A\n⊢ x ∈ B ↔ x ∈ C",
"ppTerm": "?m.28",
"assigned": true,
"usedCons... | [] | simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Algebra.Subalgebra.Basic | {
"line": 669,
"column": 30
} | {
"line": 669,
"column": 87
} | {
"line": 669,
"column": 87
} | [
{
"pp": "R : Type u\nA : Type v\nB✝ : Type w\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B✝\ninst✝¹ : Algebra R B✝\nS T U : Subalgebra R A\ninst✝ : Subsingleton A\nB C : Subalgebra R A\nx : A\n⊢ x ∈ B ↔ x ∈ C",
"ppTerm": "?m.28",
"assigned": true,
"usedCons... | [] | simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Subalgebra.Basic | {
"line": 669,
"column": 30
} | {
"line": 669,
"column": 87
} | {
"line": 669,
"column": 87
} | [
{
"pp": "R : Type u\nA : Type v\nB✝ : Type w\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B✝\ninst✝¹ : Algebra R B✝\nS T U : Subalgebra R A\ninst✝ : Subsingleton A\nB C : Subalgebra R A\nx : A\n⊢ x ∈ B ↔ x ∈ C",
"ppTerm": "?m.28",
"assigned": true,
"usedCons... | [] | simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.NonUnitalSubalgebra | {
"line": 885,
"column": 30
} | {
"line": 885,
"column": 87
} | {
"line": 885,
"column": 87
} | [
{
"pp": "R : Type u\nA : Type v\nB✝ : Type w\ninst✝³ : CommSemiring R\ninst✝² : NonUnitalNonAssocSemiring A\ninst✝¹ : Module R A\nS : NonUnitalSubalgebra R A\ninst✝ : Subsingleton A\nB C : NonUnitalSubalgebra R A\nx : A\n⊢ x ∈ B ↔ x ∈ C",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
... | [] | simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Algebra.NonUnitalSubalgebra | {
"line": 885,
"column": 30
} | {
"line": 885,
"column": 87
} | {
"line": 885,
"column": 87
} | [
{
"pp": "R : Type u\nA : Type v\nB✝ : Type w\ninst✝³ : CommSemiring R\ninst✝² : NonUnitalNonAssocSemiring A\ninst✝¹ : Module R A\nS : NonUnitalSubalgebra R A\ninst✝ : Subsingleton A\nB C : NonUnitalSubalgebra R A\nx : A\n⊢ x ∈ B ↔ x ∈ C",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
... | [] | simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.NonUnitalSubalgebra | {
"line": 885,
"column": 30
} | {
"line": 885,
"column": 87
} | {
"line": 885,
"column": 87
} | [
{
"pp": "R : Type u\nA : Type v\nB✝ : Type w\ninst✝³ : CommSemiring R\ninst✝² : NonUnitalNonAssocSemiring A\ninst✝¹ : Module R A\nS : NonUnitalSubalgebra R A\ninst✝ : Subsingleton A\nB C : NonUnitalSubalgebra R A\nx : A\n⊢ x ∈ B ↔ x ∈ C",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
... | [] | simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Matrix.Composition | {
"line": 63,
"column": 2
} | {
"line": 64,
"column": 81
} | {
"line": 65,
"column": 2
} | [
{
"pp": "case inr.inl\nI : Type u_1\nJ : Type u_2\nK : Type u_3\nL : Type u_4\nR : Type u_5\ninst✝⁴ : DecidableEq I\ninst✝³ : DecidableEq J\ninst✝² : DecidableEq K\ninst✝¹ : DecidableEq L\ninst✝ : Zero R\ni : I\nj : J\nk : K\nl : L\nr : R\nk' : K\nj' : J\nl' : L\nhj : j ≠ j'\n⊢ single i j (single k l r) i j' k'... | [
"case inr.inr\nI : Type u_1\nJ : Type u_2\nK : Type u_3\nL : Type u_4\nR : Type u_5\ninst✝⁴ : DecidableEq I\ninst✝³ : DecidableEq J\ninst✝² : DecidableEq K\ninst✝¹ : DecidableEq L\ninst✝ : Zero R\ni : I\nj : J\nk : K\nl : L\nr : R\nk' : K\nl' : L\n⊢ single i j (single k l r) i j k' l' = single (i, k) (j, l) r (i, k... | · rw [single_apply_of_col_ne _ _ hj,
single_apply_of_col_ne _ _ (ne_of_apply_ne Prod.fst hj), Matrix.zero_apply] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Matrix.Composition | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 24
} | {
"line": 66,
"column": 2
} | [
{
"pp": "case inr.inr\nI : Type u_1\nJ : Type u_2\nK : Type u_3\nL : Type u_4\nR : Type u_5\ninst✝⁴ : DecidableEq I\ninst✝³ : DecidableEq J\ninst✝² : DecidableEq K\ninst✝¹ : DecidableEq L\ninst✝ : Zero R\ni : I\nj : J\nk : K\nl : L\nr : R\nk' : K\nl' : L\n⊢ single i j (single k l r) i j k' l' = single (i, k) (j... | [
"case inr.inr\nI : Type u_1\nJ : Type u_2\nK : Type u_3\nL : Type u_4\nR : Type u_5\ninst✝⁴ : DecidableEq I\ninst✝³ : DecidableEq J\ninst✝² : DecidableEq K\ninst✝¹ : DecidableEq L\ninst✝ : Zero R\ni : I\nj : J\nk : K\nl : L\nr : R\nk' : K\nl' : L\n⊢ single k l r k' l' = single (i, k) (j, l) r (i, k') (j, l')"
] | rw [single_apply_same] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Star.Basic | {
"line": 498,
"column": 8
} | {
"line": 498,
"column": 26
} | {
"line": 498,
"column": 27
} | [
{
"pp": "case pos\nR : Type u\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nu : Rˣ\n⊢ (star ↑u)⁻¹ʳ = star (↑u)⁻¹ʳ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"congrArg",
"Units",
"id",
"Ring.inverse",
"StarAddMonoid.toInvolut... | [
"case pos\nR : Type u\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nu : Rˣ\n⊢ (star ↑u)⁻¹ʳ = star ↑u⁻¹"
] | Ring.inverse_unit, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Star.Basic | {
"line": 498,
"column": 45
} | {
"line": 498,
"column": 63
} | {
"line": 498,
"column": 64
} | [
{
"pp": "case pos\nR : Type u\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nu : Rˣ\n⊢ (↑(star u))⁻¹ʳ = star ↑u⁻¹",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"congrArg",
"Units",
"id",
"Ring.inverse",
"StarAddMonoid.toInvoluti... | [
"case pos\nR : Type u\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nu : Rˣ\n⊢ ↑(star u)⁻¹ = star ↑u⁻¹"
] | Ring.inverse_unit, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Star.SelfAdjoint | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 54
} | {
"line": 344,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : Monoid R\ninst✝³ : StarMul R\ninst✝² : Star A\ninst✝¹ : MulAction R A\ninst✝ : StarModule R A\nx : A\nr : Rˣ\nhr : IsSelfAdjoint ↑r\nhrx : IsSelfAdjoint (↑r • x)\n⊢ IsSelfAdjoint (r⁻¹ • r • x)",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\nA : Type u_2\ninst✝⁴ : Monoid R\ninst✝³ : StarMul R\ninst✝² : Star A\ninst✝¹ : MulAction R A\ninst✝ : StarModule R A\nx : A\nr : Rˣ\nhrx : IsSelfAdjoint (↑r • x)\nhr : IsSelfAdjoint r\n⊢ IsSelfAdjoint (r⁻¹ • r • x)"
] | replace hr : IsSelfAdjoint r := Units.ext hr.star_eq | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Algebra.Star.SelfAdjoint | {
"line": 654,
"column": 2
} | {
"line": 654,
"column": 56
} | {
"line": 655,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : StarRing R\na b : R\nhab : Commute a (star b)\nha : Commute (star a) a\nhb : Commute (star b) b\nthis : Commute (star a) b\n⊢ Commute (star (a + b)) (a + b)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"add_mul",... | [
"R : Type u_1\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : StarRing R\na b : R\nhab : Commute a (star b)\nha : Commute (star a) a\nhb : Commute (star b) b\nthis : Commute (star a) b\n⊢ star a * a + star b * a + (star a * b + star b * b) = a * star a + b * star a + (a * star b + b * star b)"
] | simp only [star_add, commute_iff_eq, mul_add, add_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 61
} | {
"line": 261,
"column": 2
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type v\nβ : Type w\nM : Matrix m n α\nj : n\nc : m → α\ninst✝ : DecidableEq n\nf : α → β\ni✝ : m\nj✝ : n\n⊢ (M.updateCol j c).map f i✝ j✝ = (M.map f).updateCol j (f ∘ c) i✝ j✝",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"co... | [
"m : Type u_2\nn : Type u_3\nα : Type v\nβ : Type w\nM : Matrix m n α\nj : n\nc : m → α\ninst✝ : DecidableEq n\nf : α → β\ni✝ : m\nj✝ : n\n⊢ f (if j✝ = j then c i✝ else M i✝ j✝) = if j✝ = j then (f ∘ c) i✝ else f (M i✝ j✝)"
] | rw [updateCol_apply, map_apply, map_apply, updateCol_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Matrix.Block | {
"line": 682,
"column": 2
} | {
"line": 683,
"column": 22
} | {
"line": 684,
"column": 2
} | [
{
"pp": "o : Type u_4\nm' : o → Type u_7\nn' : o → Type u_8\np' : o → Type u_9\nα : Type u_12\ninst✝³ : DecidableEq o\ninst✝² : NonUnitalNonAssocSemiring α\ninst✝¹ : (i : o) → Fintype (n' i)\ninst✝ : Fintype o\nM : (i : o) → Matrix (m' i) (n' i) α\nN : (i : o) → Matrix (n' i) (p' i) α\nk : o\ni : m' k\nk' : o\n... | [
"o : Type u_4\nm' : o → Type u_7\nn' : o → Type u_8\np' : o → Type u_9\nα : Type u_12\ninst✝³ : DecidableEq o\ninst✝² : NonUnitalNonAssocSemiring α\ninst✝¹ : (i : o) → Fintype (n' i)\ninst✝ : Fintype o\nM : (i : o) → Matrix (m' i) (n' i) α\nN : (i : o) → Matrix (n' i) (p' i) α\nk : o\ni : m' k\nk' : o\nj : p' k'\n⊢... | · simp only [dif_pos]
split_ifs <;> simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.TensorProduct.Tower | {
"line": 787,
"column": 4
} | {
"line": 787,
"column": 59
} | {
"line": 788,
"column": 2
} | [
{
"pp": "case succ\nR : Type u_1\nA : Type u_2\nM : Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : M ≃ₗ[R] M\nn : ℕ\nh : LinearEquiv.baseChange R A M M (f ^ ↑n) = LinearEquiv.baseChange R A M M f ^ ↑n\n⊢ LinearEquiv.baseChange R A ... | [] | simp only [zpow_add_one, LinearEquiv.baseChange_mul, h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.TensorProduct.Tower | {
"line": 787,
"column": 4
} | {
"line": 787,
"column": 59
} | {
"line": 788,
"column": 2
} | [
{
"pp": "case succ\nR : Type u_1\nA : Type u_2\nM : Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : M ≃ₗ[R] M\nn : ℕ\nh : LinearEquiv.baseChange R A M M (f ^ ↑n) = LinearEquiv.baseChange R A M M f ^ ↑n\n⊢ LinearEquiv.baseChange R A ... | [] | simp only [zpow_add_one, LinearEquiv.baseChange_mul, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.Tower | {
"line": 787,
"column": 4
} | {
"line": 787,
"column": 59
} | {
"line": 788,
"column": 2
} | [
{
"pp": "case succ\nR : Type u_1\nA : Type u_2\nM : Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : M ≃ₗ[R] M\nn : ℕ\nh : LinearEquiv.baseChange R A M M (f ^ ↑n) = LinearEquiv.baseChange R A M M f ^ ↑n\n⊢ LinearEquiv.baseChange R A ... | [] | simp only [zpow_add_one, LinearEquiv.baseChange_mul, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.OreLocalization.Basic | {
"line": 228,
"column": 6
} | {
"line": 228,
"column": 38
} | {
"line": 228,
"column": 38
} | [
{
"pp": "case c.c.c\nR : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type u_2\ninst✝¹ : AddMonoid X\ninst✝ : DistribMulAction R X\nr₁ : X\ns₁ : ↥S\nr₂ : X\ns₂ : ↥S\nr₃ : R\ns₃ : ↥S\nra : R\nsa : ↥S\nha : ↑(sa * s₁) = ra * ↑s₂\n⊢ (r₃ /ₒ s₃) • ((sa • r₁ + ra • r₂) /ₒ (sa * s₁)) = (r₃ /ₒ s... | [
"case c.c.c\nR : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type u_2\ninst✝¹ : AddMonoid X\ninst✝ : DistribMulAction R X\nr₁ : X\ns₁ : ↥S\nr₂ : X\ns₂ : ↥S\nr₃ : R\ns₃ : ↥S\nra : R\nsa : ↥S\nha : ↑(sa * s₁) = ra * ↑s₂\n⊢ (r₃ /ₒ s₃) • ((sa • r₁ + ra • r₂) /ₒ (sa * s₁)) = (r₃ /ₒ s₃) • (sa • r... | OreLocalization.expand' r₁ s₁ sa | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 265,
"column": 31
} | {
"line": 265,
"column": 54
} | {
"line": 265,
"column": 54
} | [
{
"pp": "case neg.intro.refine_2.refine_2\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... | [
"case neg.intro.refine_2.refine_2\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] ... | Basis.coe_mkFinConsOfLE | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.OreLocalization.Ring | {
"line": 199,
"column": 12
} | {
"line": 199,
"column": 26
} | {
"line": 199,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\nS : Submonoid R\ninst✝ : OreSet S\nhS : S ≤ nonZeroDivisorsLeft R\nr₁ r₂ : R\nu : ↥S\nv : R\nh₁ : ↑u * r₂ = ↑u * r₁\nh₂ : ↑u = v\n⊢ r₁ = r₂",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"HMul.hMul",
... | [
"R : Type u_1\ninst✝¹ : Ring R\nS : Submonoid R\ninst✝ : OreSet S\nhS : S ≤ nonZeroDivisorsLeft R\nr₁ r₂ : R\nu : ↥S\nv : R\nh₁ : ↑u * r₂ - ↑u * r₁ = 0\nh₂ : ↑u = v\n⊢ r₁ = r₂"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
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