module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Span.Defs | {
"line": 709,
"column": 15
} | {
"line": 709,
"column": 20
} | {
"line": 710,
"column": 2
} | [
{
"pp": "case mem\nR : Type u_8\nA : Type u_9\ninst✝⁴ : Semiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nx : A\nh : ∀ y ∈ s, Commute y x\ny x✝ : A\nh✝ : x✝ ∈ s\n⊢ Commute x✝ x",
"ppTerm": "?mem",
"assigned": true... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SupIndep | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 53
} | {
"line": 226,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nι : Type u_3\ninst✝² : Lattice α\ninst✝¹ : IsModularLattice α\ninst✝ : OrderBot α\nβ : ι → Type u_5\ns : Finset ι\ng : (i : ι) → Finset (β i)\nf : Sigma β → α\nh : (s.sigma g).SupIndep f\nt : Finset ι\nx✝² : t ⊆ s\ni : ι\nx✝¹ : i ∈ s\nx✝ : i ∉ t\n⊢ Disjoint ((fun i ↦ (g i).... | [
"case refine_1\nα : Type u_1\nι : Type u_3\ninst✝² : Lattice α\ninst✝¹ : IsModularLattice α\ninst✝ : OrderBot α\nβ : ι → Type u_5\ns : Finset ι\ng : (i : ι) → Finset (β i)\nf : Sigma β → α\nh : (s.sigma g).SupIndep f\nt : Finset ι\nx✝² : t ⊆ s\ni : ι\nx✝¹ : i ∈ s\nx✝ : i ∉ t\nu : Finset ((x : ι) × β x) := map (Func... | let u := (g i).map (Function.Embedding.sigmaMk i) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Order.SupIndep | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 7
} | {
"line": 352,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝ : CompleteLattice α\nt : ι → α\n⊢ (∀ (i : ι), Disjoint (t i) (sSup (t '' {j | j ≠ i}))) ↔ ∀ (i : ι), Disjoint (t i) (sSup {a | ∃ j, j ≠ i ∧ t j = a})",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Iff.rfl",
"PartialOrder.toPreorder",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.SupIndep | {
"line": 523,
"column": 4
} | {
"line": 523,
"column": 9
} | {
"line": 524,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝¹ : CompleteLattice α\ninst✝ : IsModularLattice α\nf : ι → α\ns t : Set ι\nhf : iSupIndep f\nhst : Disjoint s t\nhs : s.Finite\nthis : Disjoint (↑hs.toFinset) t → Disjoint (⨆ i ∈ hs.toFinset, f i) (⨆ i ∈ t, f i)\n⊢ Disjoint (⨆ i ∈ s, f i) (⨆ i ∈ t, f i)",
"ppTerm": ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.SupIndep | {
"line": 530,
"column": 27
} | {
"line": 530,
"column": 32
} | {
"line": 531,
"column": 4
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝¹ : CompleteLattice α\ninst✝ : IsModularLattice α\nf : ι → α\nhf : iSupIndep f\nj : ι\ns₀ : Finset ι\nhj : j ∉ s₀\nih : ∀ {t : Set ι}, Disjoint (↑s₀) t → Disjoint (⨆ i ∈ s₀, f i) (⨆ i ∈ t, f i)\nt : Set ι\nhst : Disjoint (↑(insert j s₀)) t\n⊢ j ∉ t",
"ppTerm": "?m.1... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.SupIndep | {
"line": 531,
"column": 50
} | {
"line": 531,
"column": 55
} | {
"line": 532,
"column": 4
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝¹ : CompleteLattice α\ninst✝ : IsModularLattice α\nf : ι → α\nhf : iSupIndep f\nj : ι\ns₀ : Finset ι\nhj : j ∉ s₀\nih : ∀ {t : Set ι}, Disjoint (↑s₀) t → Disjoint (⨆ i ∈ s₀, f i) (⨆ i ∈ t, f i)\nt : Set ι\nhst : Disjoint (↑(insert j s₀)) t\nhjt : j ∉ t\n⊢ Disjoint (↑s₀)... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.SupIndep | {
"line": 537,
"column": 36
} | {
"line": 537,
"column": 41
} | {
"line": 538,
"column": 4
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝¹ : CompleteLattice α\ninst✝ : IsModularLattice α\nf : ι → α\nhf : iSupIndep f\nj : ι\ns₀ : Finset ι\nhj : j ∉ s₀\nt : Set ι\nhjt : j ∉ t\nhst : Disjoint (↑s₀) (insert j t)\nih : Disjoint (⨆ i ∈ s₀, f i) (f j ⊔ ⨆ i ∈ t, f i)\n⊢ j ∉ t ∪ ↑s₀",
"ppTerm": "?m.214",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.SupIndep | {
"line": 537,
"column": 36
} | {
"line": 537,
"column": 41
} | {
"line": 538,
"column": 4
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝¹ : CompleteLattice α\ninst✝ : IsModularLattice α\nf : ι → α\nhf : iSupIndep f\nj : ι\ns₀ : Finset ι\nhj : j ∉ s₀\nt : Set ι\nhjt : j ∉ t\nhst : Disjoint (↑s₀) (insert j t)\nih : Disjoint (⨆ i ∈ s₀, f i) (f j ⊔ ⨆ i ∈ t, f i)\n⊢ j ∉ t ∪ ↑s₀",
"ppTerm": "?m.214",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SupIndep | {
"line": 537,
"column": 36
} | {
"line": 537,
"column": 41
} | {
"line": 538,
"column": 4
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝¹ : CompleteLattice α\ninst✝ : IsModularLattice α\nf : ι → α\nhf : iSupIndep f\nj : ι\ns₀ : Finset ι\nhj : j ∉ s₀\nt : Set ι\nhjt : j ∉ t\nhst : Disjoint (↑s₀) (insert j t)\nih : Disjoint (⨆ i ∈ s₀, f i) (f j ⊔ ⨆ i ∈ t, f i)\n⊢ j ∉ t ∪ ↑s₀",
"ppTerm": "?m.214",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SupIndep | {
"line": 545,
"column": 86
} | {
"line": 545,
"column": 91
} | {
"line": 546,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝ : CompleteLattice α\nf : ι → α\ns : Set ι\nh₁ : iSupIndep f\nh₂ : ⨆ i ∈ s, f i = ⊤\ni : ι\nhi : f i ≠ ⊥\ncontra : i ∉ s\n⊢ ∀ i_1 ∈ s, i_1 ≠ i",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"False",
"Lattice.toSemilatticeSup",
"e... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.SupIndep | {
"line": 545,
"column": 86
} | {
"line": 545,
"column": 91
} | {
"line": 546,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝ : CompleteLattice α\nf : ι → α\ns : Set ι\nh₁ : iSupIndep f\nh₂ : ⨆ i ∈ s, f i = ⊤\ni : ι\nhi : f i ≠ ⊥\ncontra : i ∉ s\n⊢ ∀ i_1 ∈ s, i_1 ≠ i",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"False",
"Lattice.toSemilatticeSup",
"e... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SupIndep | {
"line": 545,
"column": 86
} | {
"line": 545,
"column": 91
} | {
"line": 546,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝ : CompleteLattice α\nf : ι → α\ns : Set ι\nh₁ : iSupIndep f\nh₂ : ⨆ i ∈ s, f i = ⊤\ni : ι\nhi : f i ≠ ⊥\ncontra : i ∉ s\n⊢ ∀ i_1 ∈ s, i_1 ≠ i",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"False",
"Lattice.toSemilatticeSup",
"e... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SupIndep | {
"line": 546,
"column": 2
} | {
"line": 546,
"column": 7
} | {
"line": 548,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝ : CompleteLattice α\nf : ι → α\ns : Set ι\nh₂ : ⨆ i ∈ s, f i = ⊤\ni : ι\nhi : f i ≠ ⊥\ncontra : i ∉ s\nh₁ : Disjoint (f i) (⨆ i ∈ s, f i)\n⊢ False",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"False",
"Lattice.toSemilatticeSup",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Notation.Indicator | {
"line": 100,
"column": 91
} | {
"line": 102,
"column": 16
} | {
"line": 104,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_3\ninst✝ : One M\ns : Set α\nf : α → M\n⊢ (s.mulIndicator f = fun x ↦ 1) ↔ Disjoint (mulSupport f) s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"congrArg",
"PartialOrder.toPreorder",
"Set.mulIndicator",
"Set.mulIndicator_apply_... | [] | by
simp only [funext_iff, mulIndicator_apply_eq_one, Set.disjoint_left, mem_mulSupport,
not_imp_not] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 470,
"column": 15
} | {
"line": 470,
"column": 20
} | {
"line": 470,
"column": 20
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\nι : Type u_3\nf : ι → α\nh : ∀ (s : Finset ι), s.SupIndep f\ni : ι\ns : Finset α\nhs : ↑s ⊆ f '' {j | j ≠ i}\nt : Finset α := s.erase ⊥\nhf : InjOn f (f ⁻¹' ↑t)\na : α\n⊢ (∃ a_1, (¬a_1 = i ∧ ¬f a_1 = ⊥ ∧ f a_1 ∈ s) ∧ f a_1 = a) →... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 470,
"column": 15
} | {
"line": 470,
"column": 20
} | {
"line": 470,
"column": 20
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\nι : Type u_3\nf : ι → α\nh : ∀ (s : Finset ι), s.SupIndep f\ni : ι\ns : Finset α\nhs : ↑s ⊆ f '' {j | j ≠ i}\nt : Finset α := s.erase ⊥\nhf : InjOn f (f ⁻¹' ↑t)\na : α\n⊢ (∃ a_1, (¬a_1 = i ∧ ¬f a_1 = ⊥ ∧ f a_1 ∈ s) ∧ f a_1 = a) →... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 470,
"column": 15
} | {
"line": 470,
"column": 20
} | {
"line": 470,
"column": 20
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\nι : Type u_3\nf : ι → α\nh : ∀ (s : Finset ι), s.SupIndep f\ni : ι\ns : Finset α\nhs : ↑s ⊆ f '' {j | j ≠ i}\nt : Finset α := s.erase ⊥\nhf : InjOn f (f ⁻¹' ↑t)\na : α\n⊢ (∃ a_1, (¬a_1 = i ∧ ¬f a_1 = ⊥ ∧ f a_1 ∈ s) ∧ f a_1 = a) →... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.GroupWithZero.Finset | {
"line": 36,
"column": 40
} | {
"line": 36,
"column": 53
} | {
"line": 37,
"column": 2
} | [
{
"pp": "ι : Type u_1\nM₀ : Type u_4\ninst✝¹ : CommMonoidWithZero M₀\np : ι → Prop\ninst✝ : DecidablePred p\nf : ι → M₀\ns : Finset ι\nh : ∀ i ∈ s, p i\ni : ι\nhi : i ∈ s\n⊢ (if p i then f i else 0) = f i",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"congrArg",
"CommMonoidWi... | [] | simp [h i hi] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.GroupWithZero.Finset | {
"line": 36,
"column": 40
} | {
"line": 36,
"column": 53
} | {
"line": 37,
"column": 2
} | [
{
"pp": "ι : Type u_1\nM₀ : Type u_4\ninst✝¹ : CommMonoidWithZero M₀\np : ι → Prop\ninst✝ : DecidablePred p\nf : ι → M₀\ns : Finset ι\nh : ∀ i ∈ s, p i\ni : ι\nhi : i ∈ s\n⊢ (if p i then f i else 0) = f i",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"congrArg",
"CommMonoidWi... | [] | simp [h i hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.GroupWithZero.Finset | {
"line": 36,
"column": 40
} | {
"line": 36,
"column": 53
} | {
"line": 37,
"column": 2
} | [
{
"pp": "ι : Type u_1\nM₀ : Type u_4\ninst✝¹ : CommMonoidWithZero M₀\np : ι → Prop\ninst✝ : DecidablePred p\nf : ι → M₀\ns : Finset ι\nh : ∀ i ∈ s, p i\ni : ι\nhi : i ∈ s\n⊢ (if p i then f i else 0) = f i",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"congrArg",
"CommMonoidWi... | [] | simp [h i hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 494,
"column": 15
} | {
"line": 494,
"column": 20
} | {
"line": 494,
"column": 20
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\nι : Type u_3\nf : ι → α\nh : ∀ (s : Finset ι), s.SupIndep f\ni : ι\ns : Finset α\nhs : ↑s ⊆ f '' {j | j ≠ i}\nt : Finset α := s.erase ⊥\nhf : InjOn f (f ⁻¹' ↑t)\na : α\n⊢ (∃ a_1, (¬a_1 = i ∧ ¬f a_1 = ⊥ ∧ f a_1 ∈ s) ∧ f a_1 = a) →... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 494,
"column": 15
} | {
"line": 494,
"column": 20
} | {
"line": 494,
"column": 20
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\nι : Type u_3\nf : ι → α\nh : ∀ (s : Finset ι), s.SupIndep f\ni : ι\ns : Finset α\nhs : ↑s ⊆ f '' {j | j ≠ i}\nt : Finset α := s.erase ⊥\nhf : InjOn f (f ⁻¹' ↑t)\na : α\n⊢ (∃ a_1, (¬a_1 = i ∧ ¬f a_1 = ⊥ ∧ f a_1 ∈ s) ∧ f a_1 = a) →... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 494,
"column": 15
} | {
"line": 494,
"column": 20
} | {
"line": 494,
"column": 20
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\nι : Type u_3\nf : ι → α\nh : ∀ (s : Finset ι), s.SupIndep f\ni : ι\ns : Finset α\nhs : ↑s ⊆ f '' {j | j ≠ i}\nt : Finset α := s.erase ⊥\nhf : InjOn f (f ⁻¹' ↑t)\na : α\n⊢ (∃ a_1, (¬a_1 = i ∧ ¬f a_1 = ⊥ ∧ f a_1 ∈ s) ∧ f a_1 = a) →... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Ring.List | {
"line": 72,
"column": 14
} | {
"line": 72,
"column": 81
} | {
"line": 74,
"column": 0
} | [
{
"pp": "M₀ : Type u_4\ninst✝² : MonoidWithZero M₀\ninst✝¹ : Nontrivial M₀\ninst✝ : NoZeroDivisors M₀\na : M₀\nl : List M₀\n⊢ (a :: l).prod = 0 ↔ 0 ∈ a :: l",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"MulZeroClass.toMul",
... | [] | by rw [prod_cons, mul_eq_zero, prod_eq_zero_iff, mem_cons, eq_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.Pi | {
"line": 64,
"column": 32
} | {
"line": 64,
"column": 63
} | {
"line": 64,
"column": 63
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\ns : Finset ι\nf : ι → M\ng : ι → N\nthis : DecidableEq ι\n⊢ ∀ (a : ι) (s : Finset ι),\n a ∉ s →\n (∏ x ∈ s, f x, ∏ x ∈ s, g x) = ∏ x ∈ s, (f x, g x) →\n (∏ x ∈ insert a s, f x, ∏ x ∈ insert a s, g x) =... | [] | simp +contextual [Prod.ext_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.Pi | {
"line": 64,
"column": 32
} | {
"line": 64,
"column": 63
} | {
"line": 64,
"column": 63
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\ns : Finset ι\nf : ι → M\ng : ι → N\nthis : DecidableEq ι\n⊢ ∀ (a : ι) (s : Finset ι),\n a ∉ s →\n (∏ x ∈ s, f x, ∏ x ∈ s, g x) = ∏ x ∈ s, (f x, g x) →\n (∏ x ∈ insert a s, f x, ∏ x ∈ insert a s, g x) =... | [] | simp +contextual [Prod.ext_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Pi | {
"line": 64,
"column": 32
} | {
"line": 64,
"column": 63
} | {
"line": 64,
"column": 63
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\ns : Finset ι\nf : ι → M\ng : ι → N\nthis : DecidableEq ι\n⊢ ∀ (a : ι) (s : Finset ι),\n a ∉ s →\n (∏ x ∈ s, f x, ∏ x ∈ s, g x) = ∏ x ∈ s, (f x, g x) →\n (∏ x ∈ insert a s, f x, ∏ x ∈ insert a s, g x) =... | [] | simp +contextual [Prod.ext_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Pi | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 7
} | {
"line": 78,
"column": 0
} | [
{
"pp": "case a\nι : Type u_7\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\nR : Type u_8\ninst✝ : NonAssocSemiring R\nx : ι → R\nx✝¹ : ι\na✝ : x✝¹ ∈ Finset.univ\nx✝ : ι\n⊢ Pi.single x✝¹ 1 x✝ = if x✝¹ = x✝ then 1 else 0",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Non... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Pi | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 33
} | {
"line": 209,
"column": 2
} | [
{
"pp": "case intro\nι : Type u_1\ninst✝² : Finite ι\ninst✝¹ : DecidableEq ι\nM : ι → Type u_7\ninst✝ : (i : ι) → AddCommMonoid (M i)\np : ((i : ι) → M i) → Prop\nf : (i : ι) → M i\nzero : p 0\nadd : ∀ (f g : (i : ι) → M i), p f → p g → p (f + g)\nsingle : ∀ (i : ι) (m : M i), p (Pi.single i m)\nval✝ : Fintype ... | [
"case intro\nι : Type u_1\ninst✝² : Finite ι\ninst✝¹ : DecidableEq ι\nM : ι → Type u_7\ninst✝ : (i : ι) → AddCommMonoid (M i)\np : ((i : ι) → M i) → Prop\nf : (i : ι) → M i\nzero : p 0\nadd : ∀ (f g : (i : ι) → M i), p f → p g → p (f + g)\nsingle : ∀ (i : ι) (m : M i), p (Pi.single i m)\nval✝ : Fintype ι\n⊢ p (∑ i,... | rw [← Finset.univ_sum_single f] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 604,
"column": 2
} | {
"line": 604,
"column": 76
} | {
"line": 605,
"column": 2
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nhf : Surjective ⇑f\np q : Submodule R₂ M₂... | [
"R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nhf : Surjective ⇑f\np q : Submodule R₂ M₂\nh : p ⋖ q\... | rwa [← comap_lt_comap_iff_of_surjective hf, comap_map_eq, sup_eq_left.mpr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Group.Indicator | {
"line": 210,
"column": 92
} | {
"line": 210,
"column": 97
} | {
"line": 211,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : One β\ns : γ → Set α\nf : α → β\nhs : Pairwise (Disjoint on fun j ↦ s j ∩ mulSupport f)\ni : α\nj : γ\nhj : i ∈ s j\nthis✝ : ∀ (j' : γ), j' ≠ j → {i} ⊆ s j → {i} ⊆ s j' → {i} ⊆ mulSupport f → False\nthis : ¬(mulSupport fun d ↦ (s d).mulIndicator f i) ⊆ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 809,
"column": 2
} | {
"line": 809,
"column": 7
} | {
"line": 811,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\nx : M\nhx : x ∈ N\ny : ↥N\n⊢ (∃ a, a • x = ↑y) ↔ ∃ a, a • ⟨x, hx⟩ = y",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"Submodule",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Group.Finset.Preimage | {
"line": 40,
"column": 12
} | {
"line": 40,
"column": 91
} | {
"line": 42,
"column": 0
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nβ : Type u_3\ninst✝ : CommMonoid β\nf : ι → κ\ns : Finset κ\nhf : Set.InjOn f (f ⁻¹' ↑s)\ng : κ → β\nhg : ∀ x ∈ s, x ∉ Set.range f → g x = 1\n⊢ ∏ x ∈ s.preimage f hf, g (f x) = ∏ x ∈ s, g x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | rw [prod_preimage', prod_filter_of_ne]; exact fun x hx ↦ Not.imp_symm (hg x hx) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Group.Finset.Preimage | {
"line": 40,
"column": 12
} | {
"line": 40,
"column": 91
} | {
"line": 42,
"column": 0
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nβ : Type u_3\ninst✝ : CommMonoid β\nf : ι → κ\ns : Finset κ\nhf : Set.InjOn f (f ⁻¹' ↑s)\ng : κ → β\nhg : ∀ x ∈ s, x ∉ Set.range f → g x = 1\n⊢ ∏ x ∈ s.preimage f hf, g (f x) = ∏ x ∈ s, g x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | rw [prod_preimage', prod_filter_of_ne]; exact fun x hx ↦ Not.imp_symm (hg x hx) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Finsupp.Basic | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 31
} | {
"line": 62,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝¹ : Zero M\ninst✝ : CommMonoid N\nf : α →₀ M\ns : Finset α\nhs : f.support ⊆ s\ng : α → M → N\nh : ∀ i ∈ s, g i 0 = 1\nx : α\nhxs : x ∈ s\nhx : x ∉ f.support\n⊢ f x = 0",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Finsupp.i... | [] | exact notMem_support_iff.1 hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.BigOperators.Finsupp.Basic | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 27
} | {
"line": 159,
"column": 27
} | [
{
"pp": "α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝¹ : Zero M\ninst✝ : CommMonoid N\nf : α →₀ M\ny : α\ng : α → M → N\nhyf : y ∈ f.support\n⊢ g y (f y) * ∏ a ∈ f.support.erase y, g a ((erase y f) a) = g y (f y) * ∏ x ∈ f.support.erase y, g x (f x)",
"ppTerm": "?m.61",
"assigned": true,
"usedCo... | [
"α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝¹ : Zero M\ninst✝ : CommMonoid N\nf : α →₀ M\ny : α\ng : α → M → N\nhyf : y ∈ f.support\n⊢ g y (f y) * (f.support.erase y).prod ?m.68 = g y (f y) * ∏ x ∈ f.support.erase y, g x (f x)",
"α : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝¹ : Zero M\ninst✝ : CommMonoid ... | Finset.prod_congr rfl | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.AbsoluteValue.Basic | {
"line": 125,
"column": 7
} | {
"line": 125,
"column": 30
} | {
"line": 127,
"column": 0
} | [
{
"pp": "R : Type u_5\nS : Type u_6\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : PartialOrder S\nabv : AbsoluteValue R S\nh : IsLeftRegular (abv 1)\n⊢ (fun x ↦ abv 1 * x) (abv 1) = (fun x ↦ abv 1 * x) 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddComm... | [] | by simp [← abv.map_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 465,
"column": 41
} | {
"line": 465,
"column": 46
} | {
"line": 465,
"column": 46
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ninst✝³ : CommMonoid M\ninst✝² : Preorder M\ninst✝¹ : IsOrderedCancelMonoid M\nf g : ι → M\ns : Finset ι\ninst✝ : MulLeftStrictMono M\nhs : s.Nonempty\nhlt : ∀ i ∈ s, f i < g i\n⊢ s.val ≠ ∅",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 465,
"column": 41
} | {
"line": 465,
"column": 46
} | {
"line": 465,
"column": 46
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ninst✝³ : CommMonoid M\ninst✝² : Preorder M\ninst✝¹ : IsOrderedCancelMonoid M\nf g : ι → M\ns : Finset ι\ninst✝ : MulLeftStrictMono M\nhs : s.Nonempty\nhlt : ∀ i ∈ s, f i < g i\n⊢ s.val ≠ ∅",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 465,
"column": 41
} | {
"line": 465,
"column": 46
} | {
"line": 465,
"column": 46
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ninst✝³ : CommMonoid M\ninst✝² : Preorder M\ninst✝¹ : IsOrderedCancelMonoid M\nf g : ι → M\ns : Finset ι\ninst✝ : MulLeftStrictMono M\nhs : s.Nonempty\nhlt : ∀ i ∈ s, f i < g i\n⊢ s.val ≠ ∅",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.Basic | {
"line": 483,
"column": 10
} | {
"line": 483,
"column": 26
} | {
"line": 484,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : AddCommMonoid M\nS : Set α\nf : α → β\nhf : Set.InjOn f S\nv₁ : α →₀ M\nhv₁ : v₁ ∈ {w | ↑w.support ⊆ S}\nv₂ : α →₀ M\nhv₂ : v₂ ∈ {w | ↑w.support ⊆ S}\neq : mapDomain f v₁ = mapDomain f v₂\na : α\nh : a ∈ v₁.support ∪ v₂.support\n⊢ a ∈ ↑v₁.support ∪ ↑v₂.... | [] | exact mod_cast h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finsupp.Basic | {
"line": 483,
"column": 10
} | {
"line": 483,
"column": 26
} | {
"line": 484,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : AddCommMonoid M\nS : Set α\nf : α → β\nhf : Set.InjOn f S\nv₁ : α →₀ M\nhv₁ : v₁ ∈ {w | ↑w.support ⊆ S}\nv₂ : α →₀ M\nhv₂ : v₂ ∈ {w | ↑w.support ⊆ S}\neq : mapDomain f v₁ = mapDomain f v₂\na : α\nh : a ∈ v₁.support ∪ v₂.support\n⊢ a ∈ ↑v₁.support ∪ ↑v₂.... | [] | exact mod_cast h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finsupp.Basic | {
"line": 548,
"column": 4
} | {
"line": 548,
"column": 65
} | {
"line": 549,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : Zero M\nf : α ↪ β\ng : β →₀ M\nhg : ↑g.support ⊆ Set.range ⇑f\nb : β\nhb : b ∉ Set.range ⇑f\n⊢ (embDomain f (comapDomain (⇑f) g ⋯)) b = g b",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
... | [
"case neg\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : Zero M\nf : α ↪ β\ng : β →₀ M\nb : β\nhb : b ∉ Set.range ⇑f\nhg : g b = 0\n⊢ (embDomain f (comapDomain (⇑f) g ⋯)) b = g b"
] | replace hg : g b = 0 := notMem_support_iff.mp <| mt (hg ·) hb | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Algebra.BigOperators.GroupWithZero.Action | {
"line": 88,
"column": 45
} | {
"line": 90,
"column": 91
} | {
"line": 92,
"column": 0
} | [
{
"pp": "N : Type u_2\ninst✝³ : CommMonoid N\nG : Type u_4\ninst✝² : Group G\ninst✝¹ : MulDistribMulAction G N\ninst✝ : Fintype G\nb : N\ng : G\n⊢ g • ∏ h, h • b = ∏ h, h • b",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"HMul.hMul",
"Finset... | [] | by
simp only [smul_prod', smul_smul]
exact Finset.prod_bijective (g * ·) (Group.mulLeft_bijective g) (by simp) (fun _ _ ↦ rfl) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finsupp.Basic | {
"line": 816,
"column": 55
} | {
"line": 816,
"column": 60
} | {
"line": 818,
"column": 0
} | [
{
"pp": "case refine_1\nα : Type u_1\nM : Type u_5\nN : Type u_6\ninst✝¹ : Zero M\np : α → Prop\ninst✝ : CommMonoid N\nv : α →₀ M\nh : α → M → N\nhp : ∀ x ∈ v.support, p x\n⊢ ∀ a ∈ (subtypeDomain p v).support, ↑a ∈ v.support",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Finsup... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Finsupp.Basic | {
"line": 816,
"column": 55
} | {
"line": 816,
"column": 60
} | {
"line": 818,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nM : Type u_5\nN : Type u_6\ninst✝¹ : Zero M\np : α → Prop\ninst✝ : CommMonoid N\nv : α →₀ M\nh : α → M → N\nhp : ∀ x ∈ v.support, p x\n⊢ ∀ a₁ ∈ (subtypeDomain p v).support, ∀ a₂ ∈ (subtypeDomain p v).support, ↑a₁ = ↑a₂ → a₁ = a₂",
"ppTerm": "?refine_2",
"assigned": ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Finsupp.Basic | {
"line": 816,
"column": 55
} | {
"line": 816,
"column": 60
} | {
"line": 818,
"column": 0
} | [
{
"pp": "case refine_3\nα : Type u_1\nM : Type u_5\nN : Type u_6\ninst✝¹ : Zero M\np : α → Prop\ninst✝ : CommMonoid N\nv : α →₀ M\nh : α → M → N\nhp : ∀ x ∈ v.support, p x\n⊢ ∀ b ∈ v.support, ∃ a, ∃ (_ : a ∈ (subtypeDomain p v).support), ↑a = b",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstan... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Finsupp.Basic | {
"line": 816,
"column": 55
} | {
"line": 816,
"column": 60
} | {
"line": 818,
"column": 0
} | [
{
"pp": "case refine_4\nα : Type u_1\nM : Type u_5\nN : Type u_6\ninst✝¹ : Zero M\np : α → Prop\ninst✝ : CommMonoid N\nv : α →₀ M\nh : α → M → N\nhp : ∀ x ∈ v.support, p x\n⊢ ∀ a ∈ (subtypeDomain p v).support, (fun a b ↦ h (↑a) b) a ((subtypeDomain p v) a) = h (↑a) (v ↑a)",
"ppTerm": "?refine_4",
"assig... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 58,
"column": 25
} | {
"line": 58,
"column": 30
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case ha\nι : Type u_1\nR : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\ns : Finset ι\ni : ι\nf g h : ι → R\nhi : i ∈ s\nh2i : g i + h i ≤ f i\nhgf : ∀ j ∈ s, j ≠ i → g j ≤ f j\nhhf : ∀ j ∈ s, j ≠ i → h j ≤ f j\nhg : ∀ i ∈ s, 0 ≤ g i\nhh : ∀ i ∈ s, 0 ≤ h i\n⊢ 0 ≤ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 58,
"column": 25
} | {
"line": 58,
"column": 30
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case h0\nι : Type u_1\nR : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\ns : Finset ι\ni : ι\nf g h : ι → R\nhi : i ∈ s\nh2i : g i + h i ≤ f i\nhgf : ∀ j ∈ s, j ≠ i → g j ≤ f j\nhhf : ∀ j ∈ s, j ≠ i → h j ≤ f j\nhg : ∀ i ∈ s, 0 ≤ g i\nhh : ∀ i ∈ s, 0 ≤ h i\n⊢ ∀ i_... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 58,
"column": 25
} | {
"line": 58,
"column": 30
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case h1\nι : Type u_1\nR : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\ns : Finset ι\ni : ι\nf g h : ι → R\nhi : i ∈ s\nh2i : g i + h i ≤ f i\nhgf : ∀ j ∈ s, j ≠ i → g j ≤ f j\nhhf : ∀ j ∈ s, j ≠ i → h j ≤ f j\nhg : ∀ i ∈ s, 0 ≤ g i\nhh : ∀ i ∈ s, 0 ≤ h i\nj : ι\... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 58,
"column": 25
} | {
"line": 58,
"column": 30
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case ha\nι : Type u_1\nR : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\ns : Finset ι\ni : ι\nf g h : ι → R\nhi : i ∈ s\nh2i : g i + h i ≤ f i\nhgf : ∀ j ∈ s, j ≠ i → g j ≤ f j\nhhf : ∀ j ∈ s, j ≠ i → h j ≤ f j\nhg : ∀ i ∈ s, 0 ≤ g i\nhh : ∀ i ∈ s, 0 ≤ h i\n⊢ 0 ≤ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 58,
"column": 25
} | {
"line": 58,
"column": 30
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case h0\nι : Type u_1\nR : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\ns : Finset ι\ni : ι\nf g h : ι → R\nhi : i ∈ s\nh2i : g i + h i ≤ f i\nhgf : ∀ j ∈ s, j ≠ i → g j ≤ f j\nhhf : ∀ j ∈ s, j ≠ i → h j ≤ f j\nhg : ∀ i ∈ s, 0 ≤ g i\nhh : ∀ i ∈ s, 0 ≤ h i\n⊢ ∀ i_... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 58,
"column": 25
} | {
"line": 58,
"column": 30
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case h1\nι : Type u_1\nR : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\ns : Finset ι\ni✝ : ι\nf g h : ι → R\nhi : i✝ ∈ s\nh2i : g i✝ + h i✝ ≤ f i✝\nhgf : ∀ j ∈ s, j ≠ i✝ → g j ≤ f j\nhhf : ∀ j ∈ s, j ≠ i✝ → h j ≤ f j\nhg : ∀ i ∈ s, 0 ≤ g i\nhh : ∀ i ∈ s, 0 ≤ h i\... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Finsupp.Basic | {
"line": 934,
"column": 26
} | {
"line": 934,
"column": 31
} | {
"line": 936,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nN : Type u_6\nP : Type u_7\nG : Type u_8\nH : Type u_9\nR : Type u_10\nS : Type u_11\ninst✝ : Zero M\nf : α →₀ β →₀ M\n⊢ ∀ (a : α × β), a ∈ f.support.disjiUnion (fun a ↦ map (Embedding.sectR a β) (f a).support) ⋯ ↔ (f a.1) a.2 ≠ 0",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Finsupp.Basic | {
"line": 934,
"column": 26
} | {
"line": 934,
"column": 31
} | {
"line": 936,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nN : Type u_6\nP : Type u_7\nG : Type u_8\nH : Type u_9\nR : Type u_10\nS : Type u_11\ninst✝ : Zero M\nf : α →₀ β →₀ M\n⊢ ∀ (a : α × β), a ∈ f.support.disjiUnion (fun a ↦ map (Embedding.sectR a β) (f a).support) ⋯ ↔ (f a.1) a.2 ≠ 0",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Basic | {
"line": 934,
"column": 26
} | {
"line": 934,
"column": 31
} | {
"line": 936,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nN : Type u_6\nP : Type u_7\nG : Type u_8\nH : Type u_9\nR : Type u_10\nS : Type u_11\ninst✝ : Zero M\nf : α →₀ β →₀ M\n⊢ ∀ (a : α × β), a ∈ f.support.disjiUnion (fun a ↦ map (Embedding.sectR a β) (f a).support) ⋯ ↔ (f a.1) a.2 ≠ 0",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.Basic | {
"line": 1407,
"column": 2
} | {
"line": 1407,
"column": 50
} | {
"line": 1408,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : AddCommMonoid M\nf : α ↪ β\nx : β →₀ M\n⊢ x ∈ Set.range (embDomain f) ↔ ↑x.support ⊆ Set.range ⇑f",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Finsupp.mem_range_mapDomain_iff",... | [
"case e'_1.e'_4\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : AddCommMonoid M\nf : α ↪ β\nx : β →₀ M\n⊢ embDomain f = mapDomain ⇑f",
"case e'_2\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : AddCommMonoid M\nf : α ↪ β\nx : β →₀ M\n⊢ ↑x.support ⊆ Set.range ⇑f ↔ ∀ b ∉ Set.range ⇑f, x b = 0"
] | convert! mem_range_mapDomain_iff _ f.injective _ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.LinearAlgebra.Basis.Defs | {
"line": 152,
"column": 6
} | {
"line": 152,
"column": 23
} | {
"line": 152,
"column": 23
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nM : Type u_6\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nb : Basis ι R M\nv : ι →₀ R\n⊢ b.repr ((linearCombination R ⇑b) v) = v",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearEquiv.symm",
"Semi... | [
"ι : Type u_1\nR : Type u_3\nM : Type u_6\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nb : Basis ι R M\nv : ι →₀ R\n⊢ b.repr (↑b.repr.symm v) = v"
] | ← b.coe_repr_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Basis.Defs | {
"line": 157,
"column": 6
} | {
"line": 157,
"column": 23
} | {
"line": 157,
"column": 23
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nM : Type u_6\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nb : Basis ι R M\nx : M\n⊢ (linearCombination R ⇑b) (b.repr x) = x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearEquiv.symm",
"Semiring.... | [
"ι : Type u_1\nR : Type u_3\nM : Type u_6\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nb : Basis ι R M\nx : M\n⊢ ↑b.repr.symm (b.repr x) = x"
] | ← b.coe_repr_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 84,
"column": 48
} | {
"line": 84,
"column": 65
} | {
"line": 84,
"column": 65
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nc : M\na : α\nf : α →₀ R\ninst✝ : DecidableEq α\n⊢ f a • Pi.single a c a = f a • c",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.m... | [
"α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nc : M\na : α\nf : α →₀ R\ninst✝ : DecidableEq α\n⊢ f a • c = f a • c",
"case h₀\nα : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nc : M\na : α... | Pi.single_eq_same | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 91,
"column": 64
} | {
"line": 93,
"column": 32
} | {
"line": 95,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nM' : Type u_8\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nv : α → M\nf : M →ₗ[R] M'\n⊢ linearCombination R (⇑f ∘ v) = f ∘ₗ linearCombination R v",
"ppTerm": "?m.55",
"assigned": tr... | [] | by
ext
simp [linearCombination_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 216,
"column": 2
} | {
"line": 221,
"column": 73
} | {
"line": 223,
"column": 0
} | [
{
"pp": "M : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nα : Type u_9\nβ : Type u_10\nA : α → M\nB : β → α →₀ R\nf : β →₀ R\n⊢ (linearCombination R A) ((linearCombination R B) f) = (linearCombination R fun b ↦ (linearCombination R A) (B b)) f",
"ppTerm": "?m.37... | [] | classical
simp only [linearCombination_apply]
induction f using induction_linear with
| zero => simp only [sum_zero_index]
| add f₁ f₂ h₁ h₂ => simp [sum_add_index, h₁, h₂, add_smul]
| single => simp [sum_single_index, sum_smul_index, smul_sum, mul_smul] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 216,
"column": 2
} | {
"line": 221,
"column": 73
} | {
"line": 223,
"column": 0
} | [
{
"pp": "M : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nα : Type u_9\nβ : Type u_10\nA : α → M\nB : β → α →₀ R\nf : β →₀ R\n⊢ (linearCombination R A) ((linearCombination R B) f) = (linearCombination R fun b ↦ (linearCombination R A) (B b)) f",
"ppTerm": "?m.37... | [] | classical
simp only [linearCombination_apply]
induction f using induction_linear with
| zero => simp only [sum_zero_index]
| add f₁ f₂ h₁ h₂ => simp [sum_add_index, h₁, h₂, add_smul]
| single => simp [sum_single_index, sum_smul_index, smul_sum, mul_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 216,
"column": 2
} | {
"line": 221,
"column": 73
} | {
"line": 223,
"column": 0
} | [
{
"pp": "M : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nα : Type u_9\nβ : Type u_10\nA : α → M\nB : β → α →₀ R\nf : β →₀ R\n⊢ (linearCombination R A) ((linearCombination R B) f) = (linearCombination R fun b ↦ (linearCombination R A) (B b)) f",
"ppTerm": "?m.37... | [] | classical
simp only [linearCombination_apply]
induction f using induction_linear with
| zero => simp only [sum_zero_index]
| add f₁ f₂ h₁ h₂ => simp [sum_add_index, h₁, h₂, add_smul]
| single => simp [sum_single_index, sum_smul_index, smul_sum, mul_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 259,
"column": 85
} | {
"line": 261,
"column": 32
} | {
"line": 263,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nα' : Type u_7\nv : α → M\nf : α' → α\n⊢ linearCombination R (v ∘ f) = linearCombination R v ∘ₗ lmapDomain R R f",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"NonAs... | [] | by
ext
simp [linearCombination_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 405,
"column": 71
} | {
"line": 405,
"column": 76
} | {
"line": 407,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : α → M\nx : M\ns : Set α\nx✝¹ : ∃ t, ↑t ⊆ s ∧ ∃ c, ∑ i, c i • v ↑i = x\nt : Finset α\nht : ↑t ⊆ s\nc : ↥t → R\nhx : ∑ i, c i • v ↑i = x\na : ↥t\nx✝ : a ∈ Finset.univ\n⊢ v ↑a ∈ v '' s",
"p... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 405,
"column": 71
} | {
"line": 405,
"column": 76
} | {
"line": 407,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : α → M\nx : M\ns : Set α\nx✝¹ : ∃ t, ↑t ⊆ s ∧ ∃ c, ∑ i, c i • v ↑i = x\nt : Finset α\nht : ↑t ⊆ s\nc : ↥t → R\nhx : ∑ i, c i • v ↑i = x\na : ↥t\nx✝ : a ∈ Finset.univ\n⊢ v ↑a ∈ v '' s",
"p... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 405,
"column": 71
} | {
"line": 405,
"column": 76
} | {
"line": 407,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : α → M\nx : M\ns : Set α\nx✝¹ : ∃ t, ↑t ⊆ s ∧ ∃ c, ∑ i, c i • v ↑i = x\nt : Finset α\nht : ↑t ⊆ s\nc : ↥t → R\nhx : ∑ i, c i • v ↑i = x\na : ↥t\nx✝ : a ∈ Finset.univ\n⊢ v ↑a ∈ v '' s",
"p... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 589,
"column": 6
} | {
"line": 589,
"column": 11
} | {
"line": 590,
"column": 2
} | [
{
"pp": "case hs\nι : Type u_3\nM : Type u_7\ninst✝² : CommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : IsOrderedCancelMonoid M\nf : ι → M\np : ι → Prop\nh : ∀ (i : ι), p i → 1 ≤ f i\nh' : ∃ i, p i ∧ 1 < f i\nhf : (mulSupport f ∩ {i | p i}).Finite\n⊢ ∃ i ∈ hf.toFinset, 1 < f i",
"ppTerm": "?hs",
"assigned... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 589,
"column": 6
} | {
"line": 589,
"column": 11
} | {
"line": 590,
"column": 2
} | [
{
"pp": "case hs\nι : Type u_3\nM : Type u_7\ninst✝² : CommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : IsOrderedCancelMonoid M\nf : ι → M\np : ι → Prop\nh : ∀ (i : ι), p i → 1 ≤ f i\nh' : ∃ i, p i ∧ 1 < f i\nhf : (mulSupport f ∩ {i | p i}).Finite\n⊢ ∃ i ∈ hf.toFinset, 1 < f i",
"ppTerm": "?hs",
"assigned... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 589,
"column": 6
} | {
"line": 589,
"column": 11
} | {
"line": 590,
"column": 2
} | [
{
"pp": "case hs\nι : Type u_3\nM : Type u_7\ninst✝² : CommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : IsOrderedCancelMonoid M\nf : ι → M\np : ι → Prop\nh : ∀ (i : ι), p i → 1 ≤ f i\nh' : ∃ i, p i ∧ 1 < f i\nhf : (mulSupport f ∩ {i | p i}).Finite\n⊢ ∃ i ∈ hf.toFinset, 1 < f i",
"ppTerm": "?hs",
"assigned... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 700,
"column": 90
} | {
"line": 702,
"column": 38
} | {
"line": 704,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝ : CommMonoid M\nf : α → M\ns : Set α\nh : ∏ᶠ (i : α) (_ : i ∈ s), f i ≠ 1\n⊢ ∃ x ∈ s, f x ≠ 1",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Mathlib.Tactic.Push.not_and_eq",
"MulOne.toO... | [] | by
by_contra! h'
exact h (finprod_mem_of_eqOn_one h') | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Finset.Fin | {
"line": 885,
"column": 54
} | {
"line": 885,
"column": 72
} | {
"line": 885,
"column": 72
} | [
{
"pp": "n : ℕ\na : Fin n\n⊢ n - ↑a - 1 = n - 1 - ↑a",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"HSub.hSub",
"id",
"instSubNat",
"instOfNatNat",
"Fin.val",
"instHSub",
"Nat",
"Nat.sub_right_comm",
... | [
"n : ℕ\na : Fin n\n⊢ n - 1 - ↑a = n - 1 - ↑a"
] | Nat.sub_right_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1247,
"column": 45
} | {
"line": 1247,
"column": 50
} | {
"line": 1247,
"column": 51
} | [
{
"pp": "α : Type u_1\nR : Type u_7\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nf : α → R\nr : R\nhr : ¬r = 0\nh : ¬(support f).Finite\n⊢ (support fun x ↦ r * f x) = support f",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"HMul.h... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1247,
"column": 45
} | {
"line": 1247,
"column": 50
} | {
"line": 1247,
"column": 51
} | [
{
"pp": "α : Type u_1\nR : Type u_7\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nf : α → R\nr : R\nhr : ¬r = 0\nh : ¬(support f).Finite\n⊢ (support fun x ↦ r * f x) = support f",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"HMul.h... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1247,
"column": 45
} | {
"line": 1247,
"column": 50
} | {
"line": 1247,
"column": 51
} | [
{
"pp": "α : Type u_1\nR : Type u_7\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nf : α → R\nr : R\nhr : ¬r = 0\nh : ¬(support f).Finite\n⊢ (support fun x ↦ r * f x) = support f",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"HMul.h... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1273,
"column": 45
} | {
"line": 1273,
"column": 50
} | {
"line": 1273,
"column": 51
} | [
{
"pp": "α : Type u_1\nR : Type u_7\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nf : α → R\nr : R\nhr : ¬r = 0\nh : ¬(support f).Finite\n⊢ (support fun x ↦ f x * r) = support f",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"HMul.h... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1273,
"column": 45
} | {
"line": 1273,
"column": 50
} | {
"line": 1273,
"column": 51
} | [
{
"pp": "α : Type u_1\nR : Type u_7\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nf : α → R\nr : R\nhr : ¬r = 0\nh : ¬(support f).Finite\n⊢ (support fun x ↦ f x * r) = support f",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"HMul.h... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1273,
"column": 45
} | {
"line": 1273,
"column": 50
} | {
"line": 1273,
"column": 51
} | [
{
"pp": "α : Type u_1\nR : Type u_7\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nf : α → R\nr : R\nhr : ¬r = 0\nh : ¬(support f).Finite\n⊢ (support fun x ↦ f x * r) = support f",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"HMul.h... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1294,
"column": 31
} | {
"line": 1294,
"column": 36
} | {
"line": 1296,
"column": 0
} | [
{
"pp": "case h\nN : Type u_6\ninst✝ : CommMonoid N\nα : Type u_7\nι : Type u_8\nf : ι → α → N\nhf : HasFiniteMulSupport f\na : α\nhf' : HasFiniteMulSupport fun i ↦ f i a\n⊢ Finite.toFinset ⋯ ⊆ Finite.toFinset ⋯",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toO... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1294,
"column": 31
} | {
"line": 1294,
"column": 36
} | {
"line": 1296,
"column": 0
} | [
{
"pp": "case hf\nN : Type u_6\ninst✝ : CommMonoid N\nα : Type u_7\nι : Type u_8\nf : ι → α → N\nhf : HasFiniteMulSupport f\na : α\nhf' : HasFiniteMulSupport fun i ↦ f i a\n⊢ ∀ x ∈ Finite.toFinset ⋯, x ∉ Finite.toFinset ⋯ → f x a = 1",
"ppTerm": "?hf",
"assigned": true,
"usedConstants": [
"Mul... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1319,
"column": 74
} | {
"line": 1319,
"column": 79
} | {
"line": 1320,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ns : Finset (α × β)\nf : α × β → M\na : α\n⊢ ∀ a_1 ∈ Finset.image Prod.snd ({ab ∈ s | ab.1 = a}), (a, a_1) ∈ {x ∈ s | x.1 = a}",
"ppTerm": "?refine_1",
"assigned": true,... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1319,
"column": 74
} | {
"line": 1319,
"column": 79
} | {
"line": 1320,
"column": 2
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ns : Finset (α × β)\nf : α × β → M\na : α\n⊢ ∀ a_1 ∈ {x ∈ s | x.1 = a}, a_1.2 ∈ Finset.image Prod.snd ({ab ∈ s | ab.1 = a})",
"ppTerm": "?refine_2",
"assigned": true,
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1319,
"column": 74
} | {
"line": 1319,
"column": 79
} | {
"line": 1320,
"column": 2
} | [
{
"pp": "case refine_3\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ns : Finset (α × β)\nf : α × β → M\na : α\n⊢ ∀ a_1 ∈ Finset.image Prod.snd ({ab ∈ s | ab.1 = a}), (a, a_1).2 = a_1",
"ppTerm": "?refine_3",
"assigned": true,
"usedCo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1319,
"column": 74
} | {
"line": 1319,
"column": 79
} | {
"line": 1320,
"column": 2
} | [
{
"pp": "case refine_4\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ns : Finset (α × β)\nf : α × β → M\na : α\n⊢ ∀ a_1 ∈ {x ∈ s | x.1 = a}, (a, a_1.2) = a_1",
"ppTerm": "?refine_4",
"assigned": true,
"usedConstants": [
"Finset.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1319,
"column": 74
} | {
"line": 1319,
"column": 79
} | {
"line": 1320,
"column": 2
} | [
{
"pp": "case refine_5\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ns : Finset (α × β)\nf : α × β → M\na : α\n⊢ ∀ a_1 ∈ Finset.image Prod.snd ({ab ∈ s | ab.1 = a}), f (a, a_1) = f (a, a_1)",
"ppTerm": "?refine_5",
"assigned": true,
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Basis.Basic | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 47
} | {
"line": 161,
"column": 2
} | [
{
"pp": "case inl\nι : Type u_1\nR : Type u_3\nM : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nv : ι → M\nhli : LinearIndependent R v\nhsp : ⊤ ≤ span R (range v)\ninst✝ : DecidableEq ι\ni j : ι\nh : j = i\n⊢ ((Basis.mk hli hsp).coord i) (v j) = if j = i then 1 else 0",
"ppT... | [] | simp only [h, if_true, mk_coord_apply_eq i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Basis.Basic | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 47
} | {
"line": 161,
"column": 2
} | [
{
"pp": "case inl\nι : Type u_1\nR : Type u_3\nM : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nv : ι → M\nhli : LinearIndependent R v\nhsp : ⊤ ≤ span R (range v)\ninst✝ : DecidableEq ι\ni j : ι\nh : j = i\n⊢ ((Basis.mk hli hsp).coord i) (v j) = if j = i then 1 else 0",
"ppT... | [] | simp only [h, if_true, mk_coord_apply_eq i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Basis.Basic | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 47
} | {
"line": 161,
"column": 2
} | [
{
"pp": "case inl\nι : Type u_1\nR : Type u_3\nM : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nv : ι → M\nhli : LinearIndependent R v\nhsp : ⊤ ≤ span R (range v)\ninst✝ : DecidableEq ι\ni j : ι\nh : j = i\n⊢ ((Basis.mk hli hsp).coord i) (v j) = if j = i then 1 else 0",
"ppT... | [] | simp only [h, if_true, mk_coord_apply_eq i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 151,
"column": 40
} | {
"line": 151,
"column": 85
} | {
"line": 151,
"column": 85
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nhv : LinearIndependent R v\ni : ι\n⊢ Injective fun r ↦ r • v i",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Finsupp.single_injective",
"Eq.mpr",
... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nhv : LinearIndependent R v\ni : ι\nx✝ : R\n⊢ x✝ • v i = (⇑(Finsupp.linearCombination R v) ∘ Finsupp.single i) x✝"
] | convert! hv.comp (Finsupp.single_injective i) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Data.ENat.Pow | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 99
} | {
"line": 116,
"column": 2
} | [
{
"pp": "x y : ℕ∞\n⊢ x ^ y = 1 ↔ x = 1 ∨ y = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"instAddMonoidWithOneENat",
"Classical.or_iff_not_imp_right",
"congrArg",
"CommSemiring.toSemiring",
"Or.casesOn",
"AddMonoidWithOne.toOne",
... | [
"x y : ℕ∞\nh : x ^ y = 1\ny_0 : ¬y = 0\n⊢ x = 1"
] | refine ⟨fun h ↦ or_iff_not_imp_right.2 fun y_0 ↦ ?_, fun h ↦ by rcases h with h | h <;> simp [h]⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 232,
"column": 64
} | {
"line": 232,
"column": 92
} | {
"line": 232,
"column": 92
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nhv : ∀ (l₁ l₂ : ι →₀ R), (Finsupp.linearCombination R v) l₁ = (Finsupp.linearCombination R v) l₂ → l₁ = l₂\ns : Finset ι\nf✝ g : ι → R\neq : ∑ i ∈ s, f✝ i • v i = ∑ i ∈ s, g i • v i\ni... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nhv : ∀ (l₁ l₂ : ι →₀ R), (Finsupp.linearCombination R v) l₁ = (Finsupp.linearCombination R v) l₂ → l₁ = l₂\ns : Finset ι\nf✝ g : ι → R\neq : ∑ i ∈ s, f✝ i • v i = ∑ i ∈ s, g i • v i\ni : ι\nhis : ... | Finsupp.single_eq_of_ne' hji | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 257,
"column": 11
} | {
"line": 257,
"column": 74
} | {
"line": 257,
"column": 74
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nH :\n ∀ (s : Finset ι) (f g : ι → R), (∀ i ∉ s, f i = g i) → ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i → ∀ (i : ι), f i = g i\ns : Finset ι\nf g : ι → R\neq : ∑ i ∈ s, f i • v i = ∑ i ∈... | [] | by simp_rw [ite_smul, zero_smul, Finset.sum_extend_by_zero, eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 307,
"column": 71
} | {
"line": 307,
"column": 76
} | {
"line": 309,
"column": 0
} | [
{
"pp": "ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nhv : ∀ (t : Finset ι), ↑t ⊆ s → LinearIndepOn R v ↑t\nt : Finset ↑s\nx : ↥t\n⊢ ∀ ⦃a₂ : ↥t⦄, (fun x ↦ ⟨↑↑x, ⋯⟩) x = (fun x ↦ ⟨↑↑x, ⋯⟩) a₂ → x = a₂",
"ppTerm": "?m.78",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 307,
"column": 71
} | {
"line": 307,
"column": 76
} | {
"line": 309,
"column": 0
} | [
{
"pp": "ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nhv : ∀ (t : Finset ι), ↑t ⊆ s → LinearIndepOn R v ↑t\nt : Finset ↑s\nx : ↥t\n⊢ ∀ ⦃a₂ : ↥t⦄, (fun x ↦ ⟨↑↑x, ⋯⟩) x = (fun x ↦ ⟨↑↑x, ⋯⟩) a₂ → x = a₂",
"ppTerm": "?m.78",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 307,
"column": 71
} | {
"line": 307,
"column": 76
} | {
"line": 309,
"column": 0
} | [
{
"pp": "ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nhv : ∀ (t : Finset ι), ↑t ⊆ s → LinearIndepOn R v ↑t\nt : Finset ↑s\nx : ↥t\n⊢ ∀ ⦃a₂ : ↥t⦄, (fun x ↦ ⟨↑↑x, ⋯⟩) x = (fun x ↦ ⟨↑↑x, ⋯⟩) a₂ → x = a₂",
"ppTerm": "?m.78",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 313,
"column": 25
} | {
"line": 313,
"column": 63
} | {
"line": 313,
"column": 64
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nM' : Type u_5\nv : ι → M\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nhfv : Injective ⇑(Finsupp.linearCombination R (⇑f ∘ v))\n⊢ LinearIndependent R v",
"ppTerm": "?m.39"... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\nM' : Type u_5\nv : ι → M\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nhfv : Injective ⇑(f ∘ₗ Finsupp.linearCombination R v)\n⊢ LinearIndependent R v"
] | Finsupp.linearCombination_linear_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 332,
"column": 30
} | {
"line": 332,
"column": 68
} | {
"line": 332,
"column": 69
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nM' : Type u_5\nv : ι → M\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nhf_inj : InjOn ⇑f ↑(span R (Set.range v))\n⊢ Injective ⇑(Finsupp.linearCombination R (⇑f ∘ v)) ↔ Injecti... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\nM' : Type u_5\nv : ι → M\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nhf_inj : InjOn ⇑f ↑(span R (Set.range v))\n⊢ Injective ⇑(f ∘ₗ Finsupp.linearCombination R v) ↔ Injective ⇑(Finsupp.l... | Finsupp.linearCombination_linear_comp, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Group.ModEq | {
"line": 211,
"column": 4
} | {
"line": 211,
"column": 20
} | {
"line": 212,
"column": 2
} | [
{
"pp": "case h\nG : Type u_1\ninst✝ : AddCommGroup G\np a b : G\nm n : ℕ\nh : m • p + a = n • p + b\n⊢ ↑m • p + a = ↑n • p + b",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"instHSMul",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"AddMonoid.toAddZeroClass",
"A... | [] | exact mod_cast h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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