module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 415,
"column": 4
} | {
"line": 415,
"column": 64
} | {
"line": 415,
"column": 65
} | [
{
"pp": "A : Type u\ninst✝ : Ring A\nrelations : Relations A\n⊢ Function.Bijective ⇑(ofQuotient relations).fromQuotient",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Eq.mpr",
"Module.Relations.Solution.ofQuotient",
"Module.Relations.Quotient",
... | [
"A : Type u\ninst✝ : Ring A\nrelations : Relations A\n⊢ Function.Bijective id"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Nondegenerate | {
"line": 55,
"column": 2
} | {
"line": 56,
"column": 13
} | {
"line": 58,
"column": 0
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : CommSemiring R\nM : Matrix m n R\ninst✝¹ : Fintype m\ninst✝ : Fintype n\n⊢ M.SeparatingRight ↔ ∀ (w : n → R), (∀ (v : m → R), v ⬝ᵥ M *ᵥ w = 0) → w = 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.... | [] | refine forall_congr' fun w ↦ ⟨fun hM hw ↦ hM ?_, fun hM hw ↦ hM ?_⟩ <;>
convert! hw | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.Matrix.Nondegenerate | {
"line": 55,
"column": 2
} | {
"line": 56,
"column": 13
} | {
"line": 58,
"column": 0
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : CommSemiring R\nM : Matrix m n R\ninst✝¹ : Fintype m\ninst✝ : Fintype n\n⊢ M.SeparatingRight ↔ ∀ (w : n → R), (∀ (v : m → R), v ⬝ᵥ M *ᵥ w = 0) → w = 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.... | [] | refine forall_congr' fun w ↦ ⟨fun hM hw ↦ hM ?_, fun hM hw ↦ hM ?_⟩ <;>
convert! hw | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Nondegenerate | {
"line": 55,
"column": 2
} | {
"line": 56,
"column": 13
} | {
"line": 58,
"column": 0
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : CommSemiring R\nM : Matrix m n R\ninst✝¹ : Fintype m\ninst✝ : Fintype n\n⊢ M.SeparatingRight ↔ ∀ (w : n → R), (∀ (v : m → R), v ⬝ᵥ M *ᵥ w = 0) → w = 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.... | [] | refine forall_congr' fun w ↦ ⟨fun hM hw ↦ hM ?_, fun hM hw ↦ hM ?_⟩ <;>
convert! hw | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Nondegenerate | {
"line": 82,
"column": 14
} | {
"line": 82,
"column": 25
} | {
"line": 82,
"column": 26
} | [
{
"pp": "case refine_2\nm : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : CommSemiring R\nM : Matrix m n R\ninst✝¹ : Finite m\ninst✝ : Fintype n\nthis : Fintype m\nh : ∀ (v : n → R), M *ᵥ v = 0 → v = 0\nw : n → R\nhw : ∀ (v : m → R), v ⬝ᵥ M *ᵥ w = 0\ni : m\n⊢ (M *ᵥ w) i = 0 i",
"ppTerm": "?refine_2",
"... | [
"case refine_2\nm : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : CommSemiring R\nM : Matrix m n R\ninst✝¹ : Finite m\ninst✝ : Fintype n\nthis : Fintype m\nh : ∀ (v : n → R), M *ᵥ v = 0 → v = 0\nw : n → R\nhw : ∀ (v : m → R), v ⬝ᵥ M *ᵥ w = 0\ni : m\n⊢ (M *ᵥ w) i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Nondegenerate | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 13
} | {
"line": 152,
"column": 14
} | [
{
"pp": "m : Type u_1\nR : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nM : Matrix m m R\nhM : M.det ∈ R⁰\nv : m → R\nh : ∀ (w : m → R), v ⬝ᵥ M *ᵥ w = 0\ni : m\n⊢ M.det * v i = 0",
"ppTerm": "?m.39",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedG... | [
"m : Type u_1\nR : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nM : Matrix m m R\nhM : M.det ∈ R⁰\nv : m → R\nh : ∀ (w : m → R), v ⬝ᵥ M *ᵥ w = 0\ni : m\n⊢ M.det * v i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Nondegenerate | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 42
} | {
"line": 201,
"column": 43
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nR : Type u_3\nM : Type u_4\ninst✝⁴ : Fintype ι\ninst✝³ : Finite κ\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nA : Matrix κ ι R\nhA : A.Nondegenerate\nthis : Fintype κ\nw : κ → R\nhw : ∑ x, (∑ i, w i • A i x) • v x = 0\nhv : w ᵥ* A = 0\nw' : ... | [
"ι : Type u_1\nκ : Type u_2\nR : Type u_3\nM : Type u_4\ninst✝⁴ : Fintype ι\ninst✝³ : Finite κ\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nA : Matrix κ ι R\nhA : A.Nondegenerate\nthis : Fintype κ\nw : κ → R\nhw : ∑ x, (∑ i, w i • A i x) • v x = 0\nhv : w ᵥ* A = 0\nw' : ι → R\n⊢ w ᵥ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralDomain | {
"line": 194,
"column": 6
} | {
"line": 194,
"column": 54
} | {
"line": 194,
"column": 55
} | [
{
"pp": "R : Type u_1\nG : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : Group G\ninst✝ : Fintype G\nf : G →* R\nx : ↥f.toHomUnits.range\nhx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x\nhf : ↑↑x = 1\ng : G\nn : ℕ\nhn : (fun x_1 ↦ x ^ x_1) n = ⟨f.toHomUnits g, ⋯⟩\n⊢ f g = 1 g",
"ppTe... | [
"R : Type u_1\nG : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : Group G\ninst✝ : Fintype G\nf : G →* R\nx : ↥f.toHomUnits.range\nhx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x\nhf : ↑↑x = 1\ng : G\nn : ℕ\nhn : (fun x_1 ↦ x ^ x_1) n = ⟨f.toHomUnits g, ⋯⟩\n⊢ f g = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralDomain | {
"line": 210,
"column": 8
} | {
"line": 210,
"column": 71
} | {
"line": 210,
"column": 72
} | [
{
"pp": "case calc_1.hx\nR : Type u_1\nG : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : Group G\ninst✝ : Fintype G\nf : G →* R\nhf : f ≠ 1\nx : ↥f.toHomUnits.range\nhx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x\nhx1 : ↑↑x - 1 ≠ 0\nc : ℕ := ⋯\nu : Rˣ\nhu : u ∈ image (⇑f.toHomUnits) uni... | [
"case calc_1.hx\nR : Type u_1\nG : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : Group G\ninst✝ : Fintype G\nf : G →* R\nhf : f ≠ 1\nx : ↥f.toHomUnits.range\nhx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x\nhx1 : ↑↑x - 1 ≠ 0\nc : ℕ := #{g | f.toHomUnits g = 1}\nu : Rˣ\nhu : u ∈ image (⇑f.toH... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralDomain | {
"line": 218,
"column": 16
} | {
"line": 218,
"column": 27
} | {
"line": 218,
"column": 28
} | [
{
"pp": "R : Type u_1\nG : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : Group G\ninst✝ : Fintype G\nf : G →* R\nhf : f ≠ 1\nx : ↥f.toHomUnits.range\nhx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x\nhx1 : ↑↑x - 1 ≠ 0\nc : ℕ := #{g | f.toHomUnits g = 1}\n⊢ Set.InjOn (fun x_1 ↦ x ^ x_1) ↑(... | [
"R : Type u_1\nG : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : Group G\ninst✝ : Fintype G\nf : G →* R\nhf : f ≠ 1\nx : ↥f.toHomUnits.range\nhx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x\nhx1 : ↑↑x - 1 ≠ 0\nc : ℕ := #{g | f.toHomUnits g = 1}\n⊢ Set.InjOn (fun x_1 ↦ x ^ x_1) (Set.Iio (orde... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 30
} | {
"line": 302,
"column": 31
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nL M N : C\ng : M ⟶ N\nhg : Epi g\nf : N ⟶ L\nhx : addEquiv₀.symm f ∈ ((mk₀ g).precomp L ⋯).ker\n⊢ addEquiv₀.symm f = 0",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPrea... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nL M N : C\ng : M ⟶ N\nhg : Epi g\nf : N ⟶ L\nhx : addEquiv₀.symm f ∈ ((mk₀ g).precomp L ⋯).ker\n⊢ g ≫ f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ToLinearEquiv | {
"line": 120,
"column": 6
} | {
"line": 120,
"column": 83
} | {
"line": 120,
"column": 84
} | [
{
"pp": "n : Type u_1\ninst✝² : Fintype n\nK : Type u_4\ninst✝¹ : DecidableEq n\ninst✝ : Field K\nM : Matrix n n K\nh : ∀ (v : n → K), v ≠ 0 → M *ᵥ v ≠ 0\n⊢ Function.Injective ⇑(toLin' M)",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Sub... | [
"n : Type u_1\ninst✝² : Fintype n\nK : Type u_4\ninst✝¹ : DecidableEq n\ninst✝ : Field K\nM : Matrix n n K\nh : ∀ (v : n → K), v ≠ 0 → M *ᵥ v ≠ 0\n⊢ ∀ (v : n → K), M *ᵥ v = 0 → v = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ToLinearEquiv | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 58
} | {
"line": 167,
"column": 59
} | [
{
"pp": "n : Type u_1\ninst✝³ : Fintype n\nA : Type u_4\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\nM : Matrix n n A\ninst✝ : DecidableEq n\n⊢ (∃ v, v ≠ 0 ∧ v ᵥ* M = 0) ↔ M.det = 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCo... | [
"n : Type u_1\ninst✝³ : Fintype n\nA : Type u_4\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\nM : Matrix n n A\ninst✝ : DecidableEq n\n⊢ (∃ v, v ≠ 0 ∧ Mᵀ *ᵥ v = 0) ↔ Mᵀ.det = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ToLinearEquiv | {
"line": 181,
"column": 2
} | {
"line": 183,
"column": 68
} | {
"line": 185,
"column": 0
} | [
{
"pp": "n : Type u_1\nA : Type u_4\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\nM : Matrix n n A\ninst✝ : Finite n\n⊢ M.Nondegenerate ↔ M.SeparatingLeft",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.ofFinite",
"NonUnitalCommRing.toNonUnitalNonAssocComm... | [] | classical
have := Fintype.ofFinite n
rw [nondegenerate_iff_det_ne_zero, separatingLeft_iff_det_ne_zero] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.LinearAlgebra.Matrix.ToLinearEquiv | {
"line": 181,
"column": 2
} | {
"line": 183,
"column": 68
} | {
"line": 185,
"column": 0
} | [
{
"pp": "n : Type u_1\nA : Type u_4\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\nM : Matrix n n A\ninst✝ : Finite n\n⊢ M.Nondegenerate ↔ M.SeparatingLeft",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.ofFinite",
"NonUnitalCommRing.toNonUnitalNonAssocComm... | [] | classical
have := Fintype.ofFinite n
rw [nondegenerate_iff_det_ne_zero, separatingLeft_iff_det_ne_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.ToLinearEquiv | {
"line": 181,
"column": 2
} | {
"line": 183,
"column": 68
} | {
"line": 185,
"column": 0
} | [
{
"pp": "n : Type u_1\nA : Type u_4\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\nM : Matrix n n A\ninst✝ : Finite n\n⊢ M.Nondegenerate ↔ M.SeparatingLeft",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.ofFinite",
"NonUnitalCommRing.toNonUnitalNonAssocComm... | [] | classical
have := Fintype.ofFinite n
rw [nondegenerate_iff_det_ne_zero, separatingLeft_iff_det_ne_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 353,
"column": 4
} | {
"line": 354,
"column": 11
} | {
"line": 354,
"column": 12
} | [
{
"pp": "n : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR : Type v\ninst✝² : CommRing R\nS : Type u_1\ninst✝¹ : CommRing S\ninst✝ : Fact (Even (Fintype.card n))\ng : SpecialLinearGroup n R\n⊢ (-↑g).det = 1",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"n : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR : Type v\ninst✝² : CommRing R\nS : Type u_1\ninst✝¹ : CommRing S\ninst✝ : Fact (Even (Fintype.card n))\ng : SpecialLinearGroup n R\n⊢ (-↑g).det = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 54
} | {
"line": 375,
"column": 55
} | [
{
"pp": "R : Type v\ninst✝ : CommRing R\nA : SL(2, R)\n⊢ det ![![↑A 1 1, -↑A 0 1], ![-↑A 1 0, ↑A 0 0]] = 1",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonUnitalCommRing... | [
"R : Type v\ninst✝ : CommRing R\nA : SL(2, R)\n⊢ ↑A 0 0 * ↑A 1 1 - ↑A 0 1 * ↑A 1 0 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Basic | {
"line": 206,
"column": 4
} | {
"line": 210,
"column": 11
} | {
"line": 210,
"column": 12
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nC : CliffordAlgebra Q → Prop\nalgebraMap : ∀ (r : R), C ((Algebra.algebraMap R (CliffordAlgebra Q)) r)\nι : ∀ (x : M), C ((CliffordAlgebra.ι Q) x)\nmul : ∀ (a b : CliffordAlgebra Q), C a... | [
"R : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nC : CliffordAlgebra Q → Prop\nalgebraMap : ∀ (r : R), C ((Algebra.algebraMap R (CliffordAlgebra Q)) r)\nι : ∀ (x : M), C ((CliffordAlgebra.ι Q) x)\nmul : ∀ (a b : CliffordAlgebra Q), C a → C b → C (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 395,
"column": 27
} | {
"line": 395,
"column": 38
} | {
"line": 395,
"column": 39
} | [
{
"pp": "R : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * c = 1\nhg : ↑⟨!![a, b; c, d], ⋯⟩ 1 0 = 0\n⊢ c = 0",
"ppTerm": "?m.77",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * c = 1\nhg : ↑⟨!![a, b; c, d], ⋯⟩ 1 0 = 0\n⊢ c = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GradedMonoid | {
"line": 428,
"column": 24
} | {
"line": 428,
"column": 59
} | {
"line": 428,
"column": 59
} | [
{
"pp": "ι : Type u_1\nA : ι → Type u_3\ninst✝¹ : AddMonoid ι\ninst✝ : GMonoid A\nn : ℕ\nf : Fin n → GradedMonoid A\n⊢ (List.map f (List.finRange n)).prod =\n mk ((List.finRange n).dProdIndex fun i ↦ (f i).fst) ((List.finRange n).dProd (fun i ↦ (f i).fst) fun i ↦ (f i).snd)",
"ppTerm": "?m.28",
"assi... | [
"ι : Type u_1\nA : ι → Type u_3\ninst✝¹ : AddMonoid ι\ninst✝ : GMonoid A\nn : ℕ\nf : Fin n → GradedMonoid A\n⊢ mk ((List.finRange n).dProdIndex fun i ↦ (f i).fst) ((List.finRange n).dProd (fun i ↦ (f i).fst) fun i ↦ (f i).snd) =\n mk ((List.finRange n).dProdIndex fun i ↦ (f i).fst) ((List.finRange n).dProd (fun ... | GradedMonoid.list_prod_map_eq_dProd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 548,
"column": 4
} | {
"line": 548,
"column": 39
} | {
"line": 548,
"column": 40
} | [
{
"pp": "case refine_1\nι : Type u_1\nF : Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\ninst✝ : CommRing F\ni j : ι\nhij : i ≠ j\nb : F\nh : transvection hij b ∈ Subgroup.center (SpecialLinearGroup ι F)\nr : F\nleft✝ : r ^ Fintype.card ι = 1\nhr : (scalar ι) r = ↑(transvection hij b)\n⊢ b = 0",
"ppT... | [
"case refine_1\nι : Type u_1\nF : Type u_2\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\ninst✝ : CommRing F\ni j : ι\nhij : i ≠ j\nb : F\nh : transvection hij b ∈ Subgroup.center (SpecialLinearGroup ι F)\nr : F\nleft✝ : r ^ Fintype.card ι = 1\nhr : (scalar ι) r = ↑(transvection hij b)\n⊢ b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 617,
"column": 25
} | {
"line": 617,
"column": 77
} | {
"line": 619,
"column": 0
} | [
{
"pp": "case «0»\nF : Type u_1\ninst✝ : Field F\na : F\nha : a ≠ 0\n⊢ (diag2 a ha • Pi.single 0 1) ((fun i ↦ i) ⟨0, ⋯⟩) = (a • Pi.single 0 1) ((fun i ↦ i) ⟨0, ⋯⟩)",
"ppTerm": "?«0»",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"instHSMul",
"Matrix.SpecialLinearGrou... | [] | simp [Matrix.SpecialLinearGroup.smul_def, diag2_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 617,
"column": 25
} | {
"line": 617,
"column": 77
} | {
"line": 619,
"column": 0
} | [
{
"pp": "case «1»\nF : Type u_1\ninst✝ : Field F\na : F\nha : a ≠ 0\n⊢ (diag2 a ha • Pi.single 0 1) ((fun i ↦ i) ⟨1, ⋯⟩) = (a • Pi.single 0 1) ((fun i ↦ i) ⟨1, ⋯⟩)",
"ppTerm": "?«1»",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"MulOne.toOne",
"False",
"instHS... | [] | simp [Matrix.SpecialLinearGroup.smul_def, diag2_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 621,
"column": 25
} | {
"line": 621,
"column": 77
} | {
"line": 623,
"column": 0
} | [
{
"pp": "case «0»\nF : Type u_1\ninst✝ : Field F\na : F\nha : a ≠ 0\n⊢ (diag2 a ha • Pi.single 1 1) ((fun i ↦ i) ⟨0, ⋯⟩) = (a⁻¹ • Pi.single 1 1) ((fun i ↦ i) ⟨0, ⋯⟩)",
"ppTerm": "?«0»",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"MulOne.toOne",
"False",
"inst... | [] | simp [Matrix.SpecialLinearGroup.smul_def, diag2_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 621,
"column": 25
} | {
"line": 621,
"column": 77
} | {
"line": 623,
"column": 0
} | [
{
"pp": "case «1»\nF : Type u_1\ninst✝ : Field F\na : F\nha : a ≠ 0\n⊢ (diag2 a ha • Pi.single 1 1) ((fun i ↦ i) ⟨1, ⋯⟩) = (a⁻¹ • Pi.single 1 1) ((fun i ↦ i) ⟨1, ⋯⟩)",
"ppTerm": "?«1»",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"instHSMul",
"Matrix.SpecialLinearGr... | [] | simp [Matrix.SpecialLinearGroup.smul_def, diag2_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 656,
"column": 4
} | {
"line": 656,
"column": 50
} | {
"line": 656,
"column": 51
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ni₀ : ι\nD : ι → F\nhD : D i₀ = ∏ x with x ≠ i₀, (D x)⁻¹\nx : ι\nhx : x = i₀\n⊢ (∏ i with i ≠ i₀, fun k ↦ if k = i then D i else if k = i₀ then (D i)⁻¹ else 1) x = D x",
"ppTerm": "?pos✝",
"assigne... | [
"case pos\nF : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ni₀ : ι\nD : ι → F\nhD : D i₀ = ∏ x with x ≠ i₀, (D x)⁻¹\nx : ι\nhx : x = i₀\n⊢ (∏ x with ¬x = i₀, if i₀ = x then D x else (D x)⁻¹) = ∏ x with ¬x = i₀, (D x)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.DirectSum.Ring | {
"line": 241,
"column": 6
} | {
"line": 241,
"column": 82
} | {
"line": 242,
"column": 8
} | [
{
"pp": "ι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\na b c : ⨁ (i : ι), A i\nthis : AddMonoidHom.mulLeft₃ = AddMonoidHom.mulRight₃\n⊢ a * b * c = a * (b * c)",
"ppTerm": "?m.38",
"assigned": false,
"usedCo... | [
"ι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\na b c : ⨁ (i : ι), A i\nthis : AddMonoidHom.mulLeft₃ = AddMonoidHom.mulRight₃\n⊢ a * b * c = a * (b * c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.DirectSum.Ring | {
"line": 317,
"column": 20
} | {
"line": 317,
"column": 31
} | {
"line": 317,
"column": 31
} | [
{
"pp": "ι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddCommMonoid ι\ninst✝ : GCommSemiring A\na b : ⨁ (i : ι), A i\n⊢ ∀ (i : ι) (y : A i), (mulHom A) ((of A i) y) = (mulHom A).flip ((of A i) y)",
"ppTerm": "?m.78",
"assigned": true,
"used... | [
"ι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddCommMonoid ι\ninst✝ : GCommSemiring A\na b : ⨁ (i : ι), A i\nai : ι\nax : A ai\n⊢ (mulHom A) ((of A ai) ax) = (mulHom A).flip ((of A ai) ax)"
] | intro ai ax | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 698,
"column": 23
} | {
"line": 698,
"column": 39
} | {
"line": 698,
"column": 40
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : SpecialLinearGroup ι F\nL L' : List (TransvectionStruct ι F)\nD : ι → F\nhM : ↑M = (List.map TransvectionStruct.toMatrix L).prod * diagonal D * (List.map TransvectionStruct.toMatrix L').prod\n⊢ (diagonal D).det... | [
"F : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : SpecialLinearGroup ι F\nL L' : List (TransvectionStruct ι F)\nD : ι → F\nhM : ↑M = (List.map TransvectionStruct.toMatrix L).prod * diagonal D * (List.map TransvectionStruct.toMatrix L').prod\n⊢ ∏ i, D i = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 716,
"column": 13
} | {
"line": 716,
"column": 24
} | {
"line": 716,
"column": 25
} | [
{
"pp": "case nil\nF : Type u_1\ninst✝³ : Field F\nι : Type u_2\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial ι\nP : SpecialLinearGroup ι F → Prop\nM : SpecialLinearGroup ι F\nhdiag : ∀ (i j : ι) (hij : i ≠ j) {c : F} (hc : c ≠ 0), P (diag2n hij c hc)\nhtransvec : ∀ (i j : ι) (hij : i ≠ j) (a ... | [
"case nil\nF : Type u_1\ninst✝³ : Field F\nι : Type u_2\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial ι\nP : SpecialLinearGroup ι F → Prop\nM : SpecialLinearGroup ι F\nhdiag : ∀ (i j : ι) (hij : i ≠ j) {c : F} (hc : c ≠ 0), P (diag2n hij c hc)\nhtransvec : ∀ (i j : ι) (hij : i ≠ j) (a : F), P (tra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 854,
"column": 2
} | {
"line": 854,
"column": 13
} | {
"line": 854,
"column": 14
} | [
{
"pp": "g : SL(2, ℤ)\n⊢ ↑(T * g) 1 = ↑g 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Matrix.SpecialLinearGroup",
"HMul.hMul",
"Matrix",
"instDecidableEqFin",
"AddGroupWithOne.toAddMonoidWithOne",
"Matrix.SpecialLinearGroup.hasMul",
"ModularG... | [
"g : SL(2, ℤ)\n⊢ (↑T * ↑g) 1 = ↑g 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 858,
"column": 2
} | {
"line": 858,
"column": 13
} | {
"line": 858,
"column": 14
} | [
{
"pp": "g : SL(2, ℤ)\n⊢ ↑(T⁻¹ * g) 1 = ↑g 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Matrix.SpecialLinearGroup",
"HMul.hMul",
"Matrix",
"instDecidableEqFin",
"AddGroupWithOne.toAddMonoidWithOne",
"Matrix.SpecialLinearGroup.hasMul",
"Modula... | [
"g : SL(2, ℤ)\n⊢ ((↑T).adjugate * ↑g) 1 = ↑g 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 678,
"column": 2
} | {
"line": 678,
"column": 72
} | {
"line": 678,
"column": 73
} | [
{
"pp": "R : Type u_1\nn : Type u_11\nm : Type u_12\ninst✝⁴ : CommRing R\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix m n R\nh : M.SeparatingLeft\n⊢ ((Matrix.toLinearMap₂' R) M).SeparatingLeft",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants... | [
"R : Type u_1\nn : Type u_11\nm : Type u_12\ninst✝⁴ : CommRing R\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix m n R\nh : M.SeparatingLeft\n⊢ ∀ (x : m → R), (∀ (y : n → R), x ⬝ᵥ M *ᵥ y = 0) → x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 682,
"column": 2
} | {
"line": 682,
"column": 74
} | {
"line": 682,
"column": 75
} | [
{
"pp": "R : Type u_1\nn : Type u_11\nm : Type u_12\ninst✝⁴ : CommRing R\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix m n R\nh : M.SeparatingRight\n⊢ ((Matrix.toLinearMap₂' R) M).SeparatingRight",
"ppTerm": "?m.61",
"assigned": true,
"usedConstan... | [
"R : Type u_1\nn : Type u_11\nm : Type u_12\ninst✝⁴ : CommRing R\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix m n R\nh : M.SeparatingRight\n⊢ ∀ (y : n → R), (∀ (x : m → R), x ⬝ᵥ M *ᵥ y = 0) → y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 835,
"column": 2
} | {
"line": 835,
"column": 13
} | {
"line": 835,
"column": 14
} | [
{
"pp": "R : Type u_1\nn : Type u_11\ninst✝³ : CommRing R\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : IsDomain R\nM : Matrix n n R\n⊢ ((toLinearMap₂' R) M).SeparatingLeft ↔ M.det ≠ 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
... | [
"R : Type u_1\nn : Type u_11\ninst✝³ : CommRing R\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : IsDomain R\nM : Matrix n n R\n⊢ M.SeparatingLeft ↔ ¬M.det = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 839,
"column": 2
} | {
"line": 839,
"column": 13
} | {
"line": 839,
"column": 14
} | [
{
"pp": "R : Type u_1\nn : Type u_11\ninst✝³ : CommRing R\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : IsDomain R\nM : Matrix n n R\n⊢ ((toLinearMap₂' R) M).SeparatingRight ↔ M.det ≠ 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
... | [
"R : Type u_1\nn : Type u_11\ninst✝³ : CommRing R\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : IsDomain R\nM : Matrix n n R\n⊢ M.SeparatingRight ↔ ¬M.det = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.DirectSum.Internal | {
"line": 473,
"column": 4
} | {
"line": 473,
"column": 46
} | {
"line": 474,
"column": 4
} | [
{
"pp": "case cons\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : CanonicallyOrdered... | [
"case cons\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : CanonicallyOrderedAdd ι\nm : ι... | refine mul_apply_eq_zero hl.1 (ih hl.2) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.DirectSum.Internal | {
"line": 474,
"column": 4
} | {
"line": 474,
"column": 38
} | {
"line": 474,
"column": 39
} | [
{
"pp": "case cons\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : CanonicallyOrdered... | [
"case cons\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : CanonicallyOrderedAdd ι\nm : ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.DirectSum.Internal | {
"line": 490,
"column": 54
} | {
"line": 490,
"column": 65
} | {
"line": 490,
"column": 66
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CanonicallyOrderedAdd ι\ninst✝³ : CommSemiring R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : SetLike.GradedMonoid A\n... | [
"ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CanonicallyOrderedAdd ι\ninst✝³ : CommSemiring R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : SetLike.GradedMonoid A\ns : Multiset... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.DirectSum.Internal | {
"line": 497,
"column": 53
} | {
"line": 497,
"column": 64
} | {
"line": 497,
"column": 65
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CanonicallyOrderedAdd ι\ninst✝³ : CommSemiring R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : SetLike.GradedMonoid A\n... | [
"ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CanonicallyOrderedAdd ι\ninst✝³ : CommSemiring R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : SetLike.GradedMonoid A\ns : Multiset... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.DirectSum.Internal | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 13
} | {
"line": 506,
"column": 14
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CanonicallyOrderedAdd ι\ninst✝³ : CommSemiring R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : SetLike.GradedMonoid A\n... | [
"ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CanonicallyOrderedAdd ι\ninst✝³ : CommSemiring R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : SetLike.GradedMonoid A\ns : Finset (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.DirectSum.Internal | {
"line": 506,
"column": 58
} | {
"line": 506,
"column": 69
} | {
"line": 506,
"column": 70
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CanonicallyOrderedAdd ι\ninst✝³ : CommSemiring R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : SetLike.GradedMonoid A\n... | [
"ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CanonicallyOrderedAdd ι\ninst✝³ : CommSemiring R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : SetLike.GradedMonoid A\ns : Finset (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.DirectSum.Internal | {
"line": 511,
"column": 2
} | {
"line": 511,
"column": 13
} | {
"line": 511,
"column": 14
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CanonicallyOrderedAdd ι\ninst✝³ : CommSemiring R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : SetLike.GradedMonoid A\n... | [
"ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CanonicallyOrderedAdd ι\ninst✝³ : CommSemiring R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : SetLike.GradedMonoid A\ns : Finset (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.DirectSum.Internal | {
"line": 511,
"column": 57
} | {
"line": 511,
"column": 68
} | {
"line": 511,
"column": 69
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CanonicallyOrderedAdd ι\ninst✝³ : CommSemiring R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : SetLike.GradedMonoid A\n... | [
"ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CanonicallyOrderedAdd ι\ninst✝³ : CommSemiring R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : SetLike.GradedMonoid A\ns : Finset (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Fold | {
"line": 143,
"column": 12
} | {
"line": 143,
"column": 39
} | {
"line": 143,
"column": 40
} | [
{
"pp": "case zero\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nP : CliffordAlgebra Q → Prop\nalgebraMap : ∀ (r : R), P ((Algebra.algebraMap R (CliffordAlgebra Q)) r)\nadd : ∀ (x y : CliffordAlgebra Q), P x → P y → P (x + y)\nmul_ι : ∀ (m ... | [
"case zero\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nP : CliffordAlgebra Q → Prop\nalgebraMap : ∀ (r : R), P ((Algebra.algebraMap R (CliffordAlgebra Q)) r)\nadd : ∀ (x y : CliffordAlgebra Q), P x → P y → P (x + y)\nmul_ι : ∀ (m : M) (x : Cl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Fold | {
"line": 153,
"column": 20
} | {
"line": 153,
"column": 55
} | {
"line": 153,
"column": 56
} | [
{
"pp": "case algebraMap\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nP : CliffordAlgebra Q → Prop\nalgebraMap : ∀ (r : R), P ((Algebra.algebraMap R (CliffordAlgebra Q)) r)\nadd : ∀ (x y : CliffordAlgebra Q), P x → P y → P (x + y)\nι_mul :... | [
"case algebraMap\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nP : CliffordAlgebra Q → Prop\nalgebraMap : ∀ (r : R), P ((Algebra.algebraMap R (CliffordAlgebra Q)) r)\nadd : ∀ (x y : CliffordAlgebra Q), P x → P y → P (x + y)\nι_mul : ∀ (x : Clif... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Fold | {
"line": 154,
"column": 21
} | {
"line": 154,
"column": 47
} | {
"line": 154,
"column": 48
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nP : CliffordAlgebra Q → Prop\nalgebraMap : ∀ (r : R), P ((Algebra.algebraMap R (CliffordAlgebra Q)) r)\nadd : ∀ (x y : CliffordAlgebra Q), P x → P y → P (x + y)\nι_mul : ∀ (x :... | [
"case add\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nP : CliffordAlgebra Q → Prop\nalgebraMap : ∀ (r : R), P ((Algebra.algebraMap R (CliffordAlgebra Q)) r)\nadd : ∀ (x y : CliffordAlgebra Q), P x → P y → P (x + y)\nι_mul : ∀ (x : CliffordAlg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Fold | {
"line": 155,
"column": 20
} | {
"line": 155,
"column": 65
} | {
"line": 155,
"column": 66
} | [
{
"pp": "case mul_ι\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nP : CliffordAlgebra Q → Prop\nalgebraMap : ∀ (r : R), P ((Algebra.algebraMap R (CliffordAlgebra Q)) r)\nadd : ∀ (x y : CliffordAlgebra Q), P x → P y → P (x + y)\nι_mul : ∀ (x... | [
"case mul_ι\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nP : CliffordAlgebra Q → Prop\nalgebraMap : ∀ (r : R), P ((Algebra.algebraMap R (CliffordAlgebra Q)) r)\nadd : ∀ (x y : CliffordAlgebra Q), P x → P y → P (x + y)\nι_mul : ∀ (x : CliffordA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.PowersetCard | {
"line": 83,
"column": 9
} | {
"line": 83,
"column": 20
} | {
"line": 83,
"column": 21
} | [
{
"pp": "α : Type u_1\nn : ℕ\nhn : 1 ≤ n\nhα : ↑n < ENat.card α\na b : α\nhab : a ≠ b\nha' : ↑n ≤ {b}ᶜ.encard\ns : Set α\nhas : {a} ⊆ s\nhas' : s ⊆ {b}ᶜ\nhs : s.encard = ↑n\nthis : s.Finite\n⊢ a ∈ ⟨this.toFinset, ⋯⟩",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Se... | [
"α : Type u_1\nn : ℕ\nhn : 1 ≤ n\nhα : ↑n < ENat.card α\na b : α\nhab : a ≠ b\nha' : ↑n ≤ {b}ᶜ.encard\ns : Set α\nhas : {a} ⊆ s\nhas' : s ⊆ {b}ᶜ\nhs : s.encard = ↑n\nthis : s.Finite\n⊢ a ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.PowersetCard | {
"line": 83,
"column": 29
} | {
"line": 83,
"column": 40
} | {
"line": 83,
"column": 41
} | [
{
"pp": "α : Type u_1\nn : ℕ\nhn : 1 ≤ n\nhα : ↑n < ENat.card α\na b : α\nhab : a ≠ b\nha' : ↑n ≤ {b}ᶜ.encard\ns : Set α\nhas : {a} ⊆ s\nhas' : s ⊆ {b}ᶜ\nhs : s.encard = ↑n\nthis : s.Finite\n⊢ b ∉ ⟨this.toFinset, ⋯⟩",
"ppTerm": "?m.121",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Se... | [
"α : Type u_1\nn : ℕ\nhn : 1 ≤ n\nhα : ↑n < ENat.card α\na b : α\nhab : a ≠ b\nha' : ↑n ≤ {b}ᶜ.encard\ns : Set α\nhas : {a} ⊆ s\nhas' : s ⊆ {b}ᶜ\nhs : s.encard = ↑n\nthis : s.Finite\n⊢ b ∉ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Grading | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 15
} | {
"line": 138,
"column": 16
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nthis : Finset.univ = {0, 1}\n⊢ Set.univ = {0, 1}",
"ppTerm": "?m.68",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nthis : Finset.univ = {0, 1}\n⊢ Set.univ = {0, 1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.PowersetCard | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 26
} | {
"line": 219,
"column": 27
} | [
{
"pp": "α : Type u_1\nn : ℕ\ninst✝ : Finite α\nthis : Fintype α\n⊢ Finite ↑(powersetCard α n)",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"congrArg",
"Finset.subset_univ._simp_1",
"Finset",
"Finite",
"PartialOrder.toPre... | [
"α : Type u_1\nn : ℕ\ninst✝ : Finite α\nthis : Fintype α\n⊢ Finite { x // #x = n }"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.PowersetCard | {
"line": 226,
"column": 31
} | {
"line": 226,
"column": 42
} | {
"line": 226,
"column": 43
} | [
{
"pp": "α : Type u_1\nn : ℕ\nh : 0 < n\ninst✝ : Infinite α\na : α\ns : Finset α\na_mem : {a} ⊆ s\ns_card : #s = n\n⊢ a ∈ s",
"ppTerm": "?m.53",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nn : ℕ\nh : 0 < n\ninst✝ : Infinite α\na : α\ns : Finset α\na_mem : {a} ⊆ s\ns_card : #s = n\n⊢ a ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 130,
"column": 22
} | {
"line": 130,
"column": 49
} | {
"line": 130,
"column": 50
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n.succ) → M... | [
"case last\nR : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n.succ) → Mo... | cases i using Fin.lastCases | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 131,
"column": 22
} | {
"line": 131,
"column": 49
} | {
"line": 131,
"column": 50
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n.succ) → M... | [
"case last\nR : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n.succ) → Mo... | cases i using Fin.lastCases | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 22
} | {
"line": 157,
"column": 0
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : R\n⊢ algebraMapInv ((algebraMap R (ExteriorAlgebra R M)) x) = x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Equiv.instEquivLike",
"HMul.hMul"... | [] | simp [algebraMapInv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 22
} | {
"line": 157,
"column": 0
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : R\n⊢ algebraMapInv ((algebraMap R (ExteriorAlgebra R M)) x) = x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Equiv.instEquivLike",
"HMul.hMul"... | [] | simp [algebraMapInv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 22
} | {
"line": 157,
"column": 0
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : R\n⊢ algebraMapInv ((algebraMap R (ExteriorAlgebra R M)) x) = x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Equiv.instEquivLike",
"HMul.hMul"... | [] | simp [algebraMapInv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | {
"line": 470,
"column": 2
} | {
"line": 470,
"column": 13
} | {
"line": 470,
"column": 14
} | [
{
"pp": "R : Type u1\ninst✝⁴ : CommRing R\nM : Type u2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u4\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\ng : N →ₗ[R] M\nh : LeftInverse ⇑(map g) ⇑(map f)\nx : M\n⊢ g (f x) = x",
"ppTerm": "?m.54",
"assigned": false,
"usedConstants... | [
"R : Type u1\ninst✝⁴ : CommRing R\nM : Type u2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u4\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\ng : N →ₗ[R] M\nh : LeftInverse ⇑(map g) ⇑(map f)\nx : M\n⊢ g (f x) = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 1420,
"column": 4
} | {
"line": 1420,
"column": 61
} | {
"line": 1421,
"column": 2
} | [
{
"pp": "case succ.inr.refine_2.refine_1\nK : Type v\ninst✝³ : Field K\nhK : Invertible 2\nd : ℕ\nih :\n ∀ {V : Type u} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] {B : BilinForm K V},\n IsSymm B → finrank K V = d → ∃ v, IsOrthoᵢ B ⇑v\nV : Type u\ninst✝² : AddCommGroup V\ninst✝¹ : ... | [] | exact (v' i).prop _ (Submodule.mem_span_singleton_self x) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 1420,
"column": 4
} | {
"line": 1420,
"column": 61
} | {
"line": 1421,
"column": 2
} | [
{
"pp": "case succ.inr.refine_2.refine_1\nK : Type v\ninst✝³ : Field K\nhK : Invertible 2\nd : ℕ\nih :\n ∀ {V : Type u} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] {B : BilinForm K V},\n IsSymm B → finrank K V = d → ∃ v, IsOrthoᵢ B ⇑v\nV : Type u\ninst✝² : AddCommGroup V\ninst✝¹ : ... | [] | exact (v' i).prop _ (Submodule.mem_span_singleton_self x) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 1420,
"column": 4
} | {
"line": 1420,
"column": 61
} | {
"line": 1421,
"column": 2
} | [
{
"pp": "case succ.inr.refine_2.refine_1\nK : Type v\ninst✝³ : Field K\nhK : Invertible 2\nd : ℕ\nih :\n ∀ {V : Type u} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] {B : BilinForm K V},\n IsSymm B → finrank K V = d → ∃ v, IsOrthoᵢ B ⇑v\nV : Type u\ninst✝² : AddCommGroup V\ninst✝¹ : ... | [] | exact (v' i).prop _ (Submodule.mem_span_singleton_self x) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 51
} | {
"line": 121,
"column": 52
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nn : ℕ\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : Set M\nhs : span R s = ⊤\n⊢ Submodule.map (⋀[R]^n M).subtype (span R (⇑(ιMulti R n) '' {a | range a ⊆ s})) = Submodule.map (⋀[R]^n M).subtype ⊤",
"ppTerm": "?m.58",
"assigned": true,
"used... | [
"R : Type u\ninst✝² : CommRing R\nn : ℕ\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : Set M\nhs : span R s = ⊤\n⊢ span R ((fun a ↦ (ExteriorAlgebra.ιMulti R n) a) '' {a | range a ⊆ s}) = ⋀[R]^n M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 173,
"column": 11
} | {
"line": 173,
"column": 22
} | {
"line": 173,
"column": 23
} | [
{
"pp": "R : Type u\ninst✝⁹ : CommRing R\nn : ℕ\nM✝ : Type u_1\nN : Type u_2\nN' : Type u_3\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R M✝\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\ninst✝⁴ : AddCommGroup N'\ninst✝³ : Module R N'\nι : Type u_4\ninst✝² : DecidableEq ι\nM : Type u_5\ninst✝¹ : AddCommGroup M\n... | [
"R : Type u\ninst✝⁹ : CommRing R\nn : ℕ\nM✝ : Type u_1\nN : Type u_2\nN' : Type u_3\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R M✝\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\ninst✝⁴ : AddCommGroup N'\ninst✝³ : Module R N'\nι : Type u_4\ninst✝² : DecidableEq ι\nM : Type u_5\ninst✝¹ : AddCommGroup M\ninst✝ : Modu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 180,
"column": 10
} | {
"line": 180,
"column": 21
} | {
"line": 180,
"column": 22
} | [
{
"pp": "case alt\nR : Type u\ninst✝⁹ : CommRing R\nn : ℕ\nM✝ : Type u_1\nN : Type u_2\nN' : Type u_3\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R M✝\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\ninst✝⁴ : AddCommGroup N'\ninst✝³ : Module R N'\nι : Type u_4\ninst✝² : DecidableEq ι\nM : Type u_5\ninst✝¹ : AddCom... | [
"case alt\nR : Type u\ninst✝⁹ : CommRing R\nn : ℕ\nM✝ : Type u_1\nN : Type u_2\nN' : Type u_3\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R M✝\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\ninst✝⁴ : AddCommGroup N'\ninst✝³ : Module R N'\nι : Type u_4\ninst✝² : DecidableEq ι\nM : Type u_5\ninst✝¹ : AddCommGroup M\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 13
} | {
"line": 387,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nn : ℕ\nM : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_4\ninst✝ : LinearOrder I\nv : I → M\nhv : Submodule.span R (range v) = ⊤\n⊢ Submodule.span R ((fun x ↦ (⋀[R]^n M).subtype (ιMulti_family R n v x)) '' univ) = Submodule.map (⋀[R]^n M).subtype ⊤... | [
"R : Type u\ninst✝³ : CommRing R\nn : ℕ\nM : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_4\ninst✝ : LinearOrder I\nv : I → M\nhv : Submodule.span R (range v) = ⊤\n⊢ Submodule.span R (range fun a ↦ ExteriorAlgebra.ιMulti_family R n v a) = ⋀[R]^n M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Colimits | {
"line": 111,
"column": 8
} | {
"line": 111,
"column": 19
} | {
"line": 111,
"column": 20
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nJ : Type u₂\ninst✝² : Category.{v₂, u₂} J\nF : J ⥤ PresheafOfModules R\ninst✝¹ :\n ∀ {X Y : Cᵒᵖ} (f : X ⟶ Y),\n PreservesColimit (F ⋙ evaluation R Y) (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f)))\ninst✝ : ∀ (X : Cᵒᵖ), HasColimi... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nJ : Type u₂\ninst✝² : Category.{v₂, u₂} J\nF : J ⥤ PresheafOfModules R\ninst✝¹ :\n ∀ {X Y : Cᵒᵖ} (f : X ⟶ Y),\n PreservesColimit (F ⋙ evaluation R Y) (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f)))\ninst✝ : ∀ (X : Cᵒᵖ), HasColimit (F ⋙ evalu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits | {
"line": 109,
"column": 8
} | {
"line": 109,
"column": 19
} | {
"line": 109,
"column": 20
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\nF : J ⥤ PresheafOfModules R\ninst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections\nj j' : J\nf : j ⟶ j'\nX : Cᵒᵖ\n⊢ (((Functor.const J).obj (limi... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\nF : J ⥤ PresheafOfModules R\ninst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections\nj j' : J\nf : j ⟶ j'\nX : Cᵒᵖ\n⊢ limit.π (F ⋙ evaluation R X) j' = limit.π... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.FunctorCategory | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 30
} | {
"line": 66,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nF✝ G F' G' : C ⥤ D\nα : F✝ ⟶ G\nβ✝ : F' ⟶ G'\nF : C ⥤ D\nβ : F' ⟶ G'\nX Y : C\nf : X ⟶ Y\n⊢ (tensorObj F F').map f ≫ F.obj Y ◁ β.app Y = F.obj X ◁ β.app X ≫ (tensorObj F G').map f",
"pp... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nF✝ G F' G' : C ⥤ D\nα : F✝ ⟶ G\nβ✝ : F' ⟶ G'\nF : C ⥤ D\nβ : F' ⟶ G'\nX Y : C\nf : X ⟶ Y\n⊢ (tensorObj F F').map f ≫ (𝟙 (F.obj Y) ⊗ₘ β.app Y) = (𝟙 (F.obj X) ⊗ₘ β.app X) ≫ (tensorObj F G').map f"
] | simp only [← id_tensorHom] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Sifted | {
"line": 76,
"column": 40
} | {
"line": 76,
"column": 94
} | {
"line": 76,
"column": 94
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : IsSifted C\nD : Type u₁\ninst✝ : Category.{v₁, u₁} D\ne : D ≌ C\nthis : D × D ≌ C × C := e.prod e\nc : C\n⊢ (e.functor.obj (e.inverse.obj c), e.functor.obj (e.inverse.obj c)) ≅ (c, c)",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
... | [] | exact Iso.prod (e.counitIso.app c) (e.counitIso.app c) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Sifted | {
"line": 98,
"column": 12
} | {
"line": 98,
"column": 23
} | {
"line": 98,
"column": 24
} | [
{
"pp": "case h.left.h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : IsSifted C\nc₁ c₂ : C\nX : StructuredArrow (c₁, c₂) (Functor.diag C)\n⊢ List.IsChain Zag [X.right, c₂]",
"ppTerm": "?h.left.h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"and_true",
"CategoryTheory.Functor... | [
"case h.left.h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : IsSifted C\nc₁ c₂ : C\nX : StructuredArrow (c₁, c₂) (Functor.diag C)\n⊢ Zag X.right c₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 76,
"column": 70
} | {
"line": 76,
"column": 81
} | {
"line": 76,
"column": 82
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nd : D\nh₁ : ∃ c, Nonempty (d ⟶ F.obj c)\nh₂ : ∀ {c : C} (s s' : d ⟶ F.obj c), ∃ c' t, s ≫ F.map t = s' ≫ F.map t\nthis : Nonempty (StructuredArrow d F)\nf g : StructuredArrow d ... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nd : D\nh₁ : ∃ c, Nonempty (d ⟶ F.obj c)\nh₂ : ∀ {c : C} (s s' : d ⟶ F.obj c), ∃ c' t, s ≫ F.map t = s' ≫ F.map t\nthis : Nonempty (StructuredArrow d F)\nf g : StructuredArrow d F\nc : C\nt ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 15
} | {
"line": 79,
"column": 16
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nd : D\nh₁ : ∃ c, Nonempty (d ⟶ F.obj c)\nh₂ : ∀ {c : C} (s s' : d ⟶ F.obj c), ∃ c' t, s ≫ F.map t = s' ≫ F.map t\nthis : Nonempty (StructuredArrow d F)\nf g : Str... | [
"case refine_2\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nd : D\nh₁ : ∃ c, Nonempty (d ⟶ F.obj c)\nh₂ : ∀ {c : C} (s s' : d ⟶ F.obj c), ∃ c' t, s ≫ F.map t = s' ≫ F.map t\nthis : Nonempty (StructuredArrow d F)\nf g : StructuredArrow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 29
} | {
"line": 162,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝² : IsCofilteredOrEmpty D\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nh : ∀ (d : D), ∃ c, Nonempty (F.obj c ⟶ d)\nd : Dᵒᵖ\n⊢ ∃ c, Nonempty (d ⟶ F.op.obj c)",
"ppTerm": "?m.40",
"assigned": true,
... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝² : IsCofilteredOrEmpty D\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nh : ∀ (d : D), ∃ c, Nonempty (F.obj c ⟶ d)\nd : Dᵒᵖ\nc : C\nf : F.obj c ⟶ unop d\n⊢ ∃ c, Nonempty (d ⟶ F.op.obj c)"
] | obtain ⟨c, ⟨f⟩⟩ := h d.unop | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 29
} | {
"line": 201,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝² : IsCofilteredOrEmpty D\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nh : ∀ (d : D), ∃ c, Nonempty (F.obj c ⟶ d)\nd : Dᵒᵖ\n⊢ ∃ c, Nonempty (d ⟶ F.op.obj c)",
"ppTerm": "?m.53",
"assigned": true,
... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝² : IsCofilteredOrEmpty D\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nh : ∀ (d : D), ∃ c, Nonempty (F.obj c ⟶ d)\nd : Dᵒᵖ\nc : C\nf : F.obj c ⟶ unop d\n⊢ ∃ c, Nonempty (d ⟶ F.op.obj c)"
] | obtain ⟨c, ⟨f⟩⟩ := h d.unop | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 110,
"column": 18
} | {
"line": 110,
"column": 29
} | {
"line": 110,
"column": 30
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝ : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\nD : DiagramOfCones F\nQ : (j : J) → IsLimit (D.obj j)\nc : Cone (uncurry.obj F)\nj : J\nk k' : K\nf : k ⟶ k'\n⊢ ((const K).obj c.pt).map... | [
"J : Type u_1\nK : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝ : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\nD : DiagramOfCones F\nQ : (j : J) → IsLimit (D.obj j)\nc : Cone (uncurry.obj F)\nj : J\nk k' : K\nf : k ⟶ k'\n⊢ 𝟙 c.pt ≫ c.π.app (j, k') = c.π.app ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 113,
"column": 23
} | {
"line": 113,
"column": 34
} | {
"line": 113,
"column": 35
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝ : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\nD : DiagramOfCones F\nQ : (j : J) → IsLimit (D.obj j)\nc : Cone (uncurry.obj F)\nj j' : J\nf : j ⟶ j'\nk : K\n⊢ (((const J).obj c.pt).ma... | [
"J : Type u_1\nK : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝ : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\nD : DiagramOfCones F\nQ : (j : J) → IsLimit (D.obj j)\nc : Cone (uncurry.obj F)\nj j' : J\nf : j ⟶ j'\nk : K\n⊢ 𝟙 c.pt ≫ c.π.app (j', k) = c.π.app ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.FinallySmall | {
"line": 148,
"column": 35
} | {
"line": 152,
"column": 57
} | {
"line": 154,
"column": 0
} | [
{
"pp": "J : Type u\ninst✝³ : Category.{v, u} J\nK : Type u₁\ninst✝² : Category.{v₁, u₁} K\ninst✝¹ : LocallySmall.{w, v, u} J\ninst✝ : InitiallySmall J\nX : J\n⊢ InitiallySmall (Over X)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"CategoryTheory.InitiallySmall",
"CategoryThe... | [] | by
have : InitiallySmall.{w} (CostructuredArrow (fromInitialModel.{w} J) X) :=
initiallySmall_of_essentiallySmall _
exact initiallySmall_of_initial_of_initiallySmall
(CostructuredArrow.toOver (fromInitialModel.{w} J) X) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.FinallySmall | {
"line": 209,
"column": 2
} | {
"line": 210,
"column": 76
} | {
"line": 211,
"column": 2
} | [
{
"pp": "J : Type u\ninst✝² : Category.{v, u} J\ninst✝¹ : IsCofilteredOrEmpty J\ns : Set J\ninst✝ : Small.{v, u} ↑s\nhs : ∀ (i : J), ∃ j ∈ s, Nonempty (j ⟶ i)\n⊢ InitiallySmall J",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CategoryTheory.ObjectProperty.ι",
"CategoryTheory.... | [
"J : Type u\ninst✝² : Category.{v, u} J\ninst✝¹ : IsCofilteredOrEmpty J\ns : Set J\ninst✝ : Small.{v, u} ↑s\nhs : ∀ (i : J), ∃ j ∈ s, Nonempty (j ⟶ i)\n⊢ (ObjectProperty.ι fun x ↦ x ∈ s).Initial"
] | suffices Functor.Initial (ObjectProperty.ι (· ∈ s)) from
initiallySmall_of_initial_of_essentiallySmall (ObjectProperty.ι (· ∈ s)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 431,
"column": 70
} | {
"line": 431,
"column": 81
} | {
"line": 431,
"column": 82
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nα : Type u₁\nI : α → Type u₂\ninst✝¹ : (s : α) → Category.{v₂, u₂} (I s)\ninst✝ : ∀ (s : α), IsFiltered (I s)\ns : α\ni : I s\n⊢ i ⟶ (Pi.eval I s).obj (Function.update (fun t ↦ ⋯.some) s i)",
"ppTerm": ... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nα : Type u₁\nI : α → Type u₂\ninst✝¹ : (s : α) → Category.{v₂, u₂} (I s)\ninst✝ : ∀ (s : α), IsFiltered (I s)\ns : α\ni : I s\n⊢ i ⟶ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 438,
"column": 4
} | {
"line": 438,
"column": 15
} | {
"line": 438,
"column": 16
} | [
{
"pp": "case h₂\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nα : Type u₁\nI : α → Type u₂\ninst✝¹ : (s : α) → Category.{v₂, u₂} (I s)\ninst✝ : ∀ (s : α), IsFiltered (I s)\ns : α\nd : I s\nc : (i : α) → I i\nf g : d ⟶ (Pi.eval I s).obj c\nc't : (s : α) → (c' ... | [
"case h₂\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nα : Type u₁\nI : α → Type u₂\ninst✝¹ : (s : α) → Category.{v₂, u₂} (I s)\ninst✝ : ∀ (s : α), IsFiltered (I s)\ns : α\nd : I s\nc : (i : α) → I i\nf g : d ⟶ (Pi.eval I s).obj c\nc't : (s : α) → (c' : I s) × (c ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 445,
"column": 70
} | {
"line": 445,
"column": 81
} | {
"line": 445,
"column": 82
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nα : Type u₁\nI : α → Type u₂\ninst✝¹ : (s : α) → Category.{v₂, u₂} (I s)\ninst✝ : ∀ (s : α), IsCofiltered (I s)\ns : α\ni : I s\n⊢ (Pi.eval I s).obj (Function.update (fun t ↦ ⋯.some) s i) ⟶ i",
"ppTerm"... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nα : Type u₁\nI : α → Type u₂\ninst✝¹ : (s : α) → Category.{v₂, u₂} (I s)\ninst✝ : ∀ (s : α), IsCofiltered (I s)\ns : α\ni : I s\n⊢ i ⟶ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 452,
"column": 4
} | {
"line": 452,
"column": 15
} | {
"line": 452,
"column": 16
} | [
{
"pp": "case h₂\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nα : Type u₁\nI : α → Type u₂\ninst✝¹ : (s : α) → Category.{v₂, u₂} (I s)\ninst✝ : ∀ (s : α), IsCofiltered (I s)\ns : α\nd : I s\nc : (i : α) → I i\nf g : (Pi.eval I s).obj c ⟶ d\nc't : (s : α) → (c... | [
"case h₂\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nα : Type u₁\nI : α → Type u₂\ninst✝¹ : (s : α) → Category.{v₂, u₂} (I s)\ninst✝ : ∀ (s : α), IsCofiltered (I s)\ns : α\nd : I s\nc : (i : α) → I i\nf g : (Pi.eval I s).obj c ⟶ d\nc't : (s : α) → (c' : I s) × (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 237,
"column": 23
} | {
"line": 237,
"column": 38
} | {
"line": 237,
"column": 39
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝ : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\nD : DiagramOfCones F\nQ : (j : J) → IsLimit (D.obj j)\nc : Cone (uncurry.obj F)\nP : IsLimit (coneOfConeUncurry Q c)\nE : (j : J) → Prod... | [
"J : Type u_1\nK : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝ : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\nD : DiagramOfCones F\nQ : (j : J) → IsLimit (D.obj j)\nc : Cone (uncurry.obj F)\nP : IsLimit (coneOfConeUncurry Q c)\nE : (j : J) → Prod.sectR j K ⋙... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 255,
"column": 42
} | {
"line": 255,
"column": 60
} | {
"line": 255,
"column": 61
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝ : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\nD : DiagramOfCones F\nQ : (j : J) → IsLimit (D.obj j)\nc : Cone (uncurry.obj F)\nP : IsLimit (coneOfConeUncurry Q c)\nE : (j : J) → Prod... | [
"J : Type u_1\nK : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝ : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\nD : DiagramOfCones F\nQ : (j : J) → IsLimit (D.obj j)\nc : Cone (uncurry.obj F)\nP : IsLimit (coneOfConeUncurry Q c)\nE : (j : J) → Prod.sectR j K ⋙... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 313,
"column": 23
} | {
"line": 313,
"column": 38
} | {
"line": 313,
"column": 39
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝ : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\nD : DiagramOfCocones F\nQ : (j : J) → IsColimit (D.obj j)\nc : Cocone (uncurry.obj F)\nP : IsColimit (coconeOfCoconeUncurry Q c)\nE : (j... | [
"J : Type u_1\nK : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝ : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\nD : DiagramOfCocones F\nQ : (j : J) → IsColimit (D.obj j)\nc : Cocone (uncurry.obj F)\nP : IsColimit (coconeOfCoconeUncurry Q c)\nE : (j : J) → Prod... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 331,
"column": 42
} | {
"line": 331,
"column": 60
} | {
"line": 331,
"column": 61
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝ : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\nD : DiagramOfCocones F\nQ : (j : J) → IsColimit (D.obj j)\nc : Cocone (uncurry.obj F)\nP : IsColimit (coconeOfCoconeUncurry Q c)\nE : (j... | [
"J : Type u_1\nK : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝ : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\nD : DiagramOfCocones F\nQ : (j : J) → IsColimit (D.obj j)\nc : Cocone (uncurry.obj F)\nP : IsColimit (coconeOfCoconeUncurry Q c)\nE : (j : J) → Prod... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 37
} | {
"line": 184,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\ninst✝¹ : IsCofiltered C\ninst✝ : InitiallySmall C\nR : Cᵒᵖ ⥤ RingCat\ncR : Cocone R\nhcR : IsColimit cR\nM : PresheafOfModules R\ncM : Cocone M.presheaf\nhcM : IsColimit cM\nM' : PresheafOfModules R\ncM' : Cocone M'.presheaf\nhc... | [] | exact ⟨U, a, x₁, x₂, rfl, rfl, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 15
} | {
"line": 190,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\ninst✝¹ : IsCofiltered C\ninst✝ : InitiallySmall C\nR : Cᵒᵖ ⥤ RingCat\ncR : Cocone R\nhcR : IsColimit cR\nM : PresheafOfModules R\ncM : Cocone M.presheaf\nhcM : IsColimit cM\nM' : PresheafOfModules R\ncM' : Cocone M'.presheaf\nhc... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\ninst✝¹ : IsCofiltered C\ninst✝ : InitiallySmall C\nR : Cᵒᵖ ⥤ RingCat\ncR : Cocone R\nhcR : IsColimit cR\nM : PresheafOfModules R\ncM : Cocone M.presheaf\nhcM : IsColimit cM\nM' : PresheafOfModules R\ncM' : Cocone M'.presheaf\nhcM' : IsColim... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 15
} | {
"line": 196,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\ninst✝¹ : IsCofiltered C\ninst✝ : InitiallySmall C\nR : Cᵒᵖ ⥤ RingCat\ncR : Cocone R\nhcR : IsColimit cR\nM : PresheafOfModules R\ncM : Cocone M.presheaf\nhcM : IsColimit cM\nM' : PresheafOfModules R\ncM' : Cocone M'.presheaf\nhc... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\ninst✝¹ : IsCofiltered C\ninst✝ : InitiallySmall C\nR : Cᵒᵖ ⥤ RingCat\ncR : Cocone R\nhcR : IsColimit cR\nM : PresheafOfModules R\ncM : Cocone M.presheaf\nhcM : IsColimit cM\nM' : PresheafOfModules R\ncM' : Cocone M'.presheaf\nhcM' : IsColim... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor | {
"line": 201,
"column": 4
} | {
"line": 202,
"column": 44
} | {
"line": 204,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\ninst✝¹ : IsCofiltered C\ninst✝ : InitiallySmall C\nR : Cᵒᵖ ⥤ RingCat\ncR : Cocone R\nhcR : IsColimit cR\nM : PresheafOfModules R\ncM : Cocone M.presheaf\nhcM : IsColimit cM\nM' : PresheafOfModules R\ncM' : Cocone M'.presheaf\nhc... | [] | obtain ⟨U, r₁, r₂, m, rfl, rfl, rfl⟩ := jointly_surjective₃ r₁ r₂ m
simp only [smul_eq, ← map_add, add_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor | {
"line": 201,
"column": 4
} | {
"line": 202,
"column": 44
} | {
"line": 204,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\ninst✝¹ : IsCofiltered C\ninst✝ : InitiallySmall C\nR : Cᵒᵖ ⥤ RingCat\ncR : Cocone R\nhcR : IsColimit cR\nM : PresheafOfModules R\ncM : Cocone M.presheaf\nhcM : IsColimit cM\nM' : PresheafOfModules R\ncM' : Cocone M'.presheaf\nhc... | [] | obtain ⟨U, r₁, r₂, m, rfl, rfl, rfl⟩ := jointly_surjective₃ r₁ r₂ m
simp only [smul_eq, ← map_add, add_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 15
} | {
"line": 193,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\ninst✝¹ : IsCofiltered C\ninst✝ : InitiallySmall C\nR : Cᵒᵖ ⥤ RingCat\ncR : Cocone R\nhcR : IsColimit cR\nM : PresheafOfModules R\ncM : Cocone M.presheaf\nhcM : IsColimit cM\nM' : PresheafOfModules R\ncM' : Cocone M'.presheaf\nhc... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\ninst✝¹ : IsCofiltered C\ninst✝ : InitiallySmall C\nR : Cᵒᵖ ⥤ RingCat\ncR : Cocone R\nhcR : IsColimit cR\nM : PresheafOfModules R\ncM : Cocone M.presheaf\nhcM : IsColimit cM\nM' : PresheafOfModules R\ncM' : Cocone M'.presheaf\nhcM' : IsColim... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Adjunction.PartialAdjoint | {
"line": 139,
"column": 2
} | {
"line": 141,
"column": 79
} | {
"line": 143,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : D ⥤ C\nh : ∀ (X : C), (F ⋙ coyoneda.obj (op X)).IsCorepresentable\n⊢ F.IsRightAdjoint",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"Opposite",
"C... | [] | exact (Adjunction.adjunctionOfEquivLeft
(fun X Y ↦ (F ⋙ coyoneda.obj (op X)).corepresentableBy.homEquiv)
(fun X Y Y' g f ↦ by apply CorepresentableBy.homEquiv_comp)).isRightAdjoint | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Adjunction.PartialAdjoint | {
"line": 148,
"column": 2
} | {
"line": 149,
"column": 9
} | {
"line": 149,
"column": 10
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : D ⥤ C\nh : F.IsRightAdjoint\nX : C\n⊢ F.leftAdjointObjIsDefined X ↔ ⊤ X",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iff_true",
"CategoryTheory.Functor.leftAdjo... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : D ⥤ C\nh : F.IsRightAdjoint\nX : C\n⊢ F.leftAdjointObjIsDefined X"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Monoidal | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 19
} | {
"line": 74,
"column": 19
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nR : Cᵒᵖ ⥤ CommRingCat\nM₁ M₂ M₃ M₄ : PresheafOfModules (R ⋙ forget₂ CommRingCat RingCat)\nX✝ Y✝ Z✝ : Cᵒᵖ\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nm₁ : ↑(M₁.obj X✝)\nm₂ : ↑(M₂.obj X✝)\n⊢ (ConcreteCategory.hom (M₁.map (f ≫ g))) m₁ ⊗ₜ[↑(R.obj Z✝)] (ConcreteCategory.hom (M₂.m... | [] | simp +instances | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Adjunction.PartialAdjoint | {
"line": 298,
"column": 2
} | {
"line": 299,
"column": 9
} | {
"line": 299,
"column": 10
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nh : F.IsLeftAdjoint\nX : D\n⊢ F.rightAdjointObjIsDefined X ↔ ⊤ X",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iff_true",
"id",
"Iff",
"Boolea... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nh : F.IsLeftAdjoint\nX : D\n⊢ F.rightAdjointObjIsDefined X"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Adjunction.CompositionIso | {
"line": 182,
"column": 8
} | {
"line": 182,
"column": 19
} | {
"line": 182,
"column": 20
} | [
{
"pp": "C₀ : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\ninst✝³ : Category.{v_1, u_1} C₀\ninst✝² : Category.{v_2, u_2} C₁\ninst✝¹ : Category.{v_3, u_3} C₂\ninst✝ : Category.{v_4, u_4} C₃\nF₀₁ : C₀ ⥤ C₁\nF₁₂ : C₁ ⥤ C₂\nF₂₃ : C₂ ⥤ C₃\nF₀₂ : C₀ ⥤ C₂\nF₁₃ : C₁ ⥤ C₃\nF₀₃ : C₀ ⥤ C₃\nG₁₀ : C₁ ⥤ C₀\nG₂₁ : C... | [
"C₀ : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\ninst✝³ : Category.{v_1, u_1} C₀\ninst✝² : Category.{v_2, u_2} C₁\ninst✝¹ : Category.{v_3, u_3} C₂\ninst✝ : Category.{v_4, u_4} C₃\nF₀₁ : C₀ ⥤ C₁\nF₁₂ : C₁ ⥤ C₂\nF₂₃ : C₂ ⥤ C₃\nF₀₂ : C₀ ⥤ C₂\nF₁₃ : C₁ ⥤ C₃\nF₀₃ : C₀ ⥤ C₃\nG₁₀ : C₁ ⥤ C₀\nG₂₁ : C₂ ⥤ C₁\nG₃₂ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subfunctor.Basic | {
"line": 56,
"column": 17
} | {
"line": 59,
"column": 34
} | {
"line": 59,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF✝ F' F'' : C ⥤ Type w\nG✝ G' F G : Subfunctor F✝\nU✝ V✝ : C\nx✝¹ : U✝ ⟶ V✝\nx✝ : F✝.obj U✝\n⊢ x✝ ∈ F.obj U✝ ⊔ G.obj U✝ → x✝ ∈ ⇑(hom (F✝.map x✝¹)) ⁻¹' (F.obj V✝ ⊔ G.obj V✝)",
"ppTerm": "?m.227",
"assigned": true,
"usedConstants": [
"Lattice.toSem... | [] | by
rintro (h | h)
· exact Or.inl (F.map _ h)
· exact Or.inr (G.map _ h) | [anonymous] | Lean.Parser.Term.byTactic |
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