module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 224,
"column": 37
} | {
"line": 224,
"column": 64
} | {
"line": 225,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_3\nm m₀ : MeasurableSpace α\nμ : Measure α\nf : α → E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhm : ¬m ≤ m₀\n⊢ Integrable 0 μ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"PseudoMetricSpace.toUni... | [] | exact integrable_zero _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 226,
"column": 50
} | {
"line": 226,
"column": 77
} | {
"line": 227,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_3\nm m₀ : MeasurableSpace α\nμ : Measure α\nf : α → E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhm : m ≤ m₀\nhμm : ¬SigmaFinite (μ.trim hm)\n⊢ Integrable 0 μ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [... | [] | exact integrable_zero _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 101,
"column": 10
} | {
"line": 101,
"column": 54
} | {
"line": 102,
"column": 6
} | [
{
"pp": "case refine_2\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator... | [] | rw [Set.indicator_indicator, Set.inter_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 101,
"column": 10
} | {
"line": 101,
"column": 54
} | {
"line": 102,
"column": 6
} | [
{
"pp": "case refine_2\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator... | [] | rw [Set.indicator_indicator, Set.inter_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 101,
"column": 10
} | {
"line": 101,
"column": 54
} | {
"line": 102,
"column": 6
} | [
{
"pp": "case refine_2\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator... | [] | rw [Set.indicator_indicator, Set.inter_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 104,
"column": 6
} | {
"line": 104,
"column": 49
} | {
"line": 105,
"column": 6
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝¹ : SigmaFinite (μ.trim hm)\nthis✝ : s.indicator μ[... | [
"case neg\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝¹ : SigmaFinite (μ.trim hm)\nthis✝ : s.indicator μ[f | m] =ᵐ[μ]... | · simp only [hx, hxs, Set.indicator_of_mem] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 111,
"column": 8
} | {
"line": 111,
"column": 52
} | {
"line": 113,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator... | [] | rw [Set.indicator_indicator, Set.inter_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 111,
"column": 8
} | {
"line": 111,
"column": 52
} | {
"line": 113,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator... | [] | rw [Set.indicator_indicator, Set.inter_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 111,
"column": 8
} | {
"line": 111,
"column": 52
} | {
"line": 113,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator... | [] | rw [Set.indicator_indicator, Set.inter_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 121,
"column": 71
} | {
"line": 121,
"column": 86
} | {
"line": 121,
"column": 87
} | [
{
"pp": "case refine_1\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nhs_m : MeasurableSet s\nhf_int : Integrable f μ\nthis : SigmaFinite ... | [
"case refine_1\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nhs_m : MeasurableSet s\nhf_int : Integrable f μ\nthis : SigmaFinite ((μ.restrict... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 57
} | {
"line": 127,
"column": 2
} | [
{
"pp": "case pos\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX : Ω → ℝ≥0∞\nhm : mΩ ≤ mΩ₀\n⊢ Measurable P⁻[X | mΩ]",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"le_refl",
"MeasurableSpace.instPartialOrder",
"PartialOrder.toPreorder",
"ENNReal.measur... | [] | exact (measurable_condLExp _ _ _).mono hm (le_refl _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 57
} | {
"line": 127,
"column": 2
} | [
{
"pp": "case pos\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX : Ω → ℝ≥0∞\nhm : mΩ ≤ mΩ₀\n⊢ Measurable P⁻[X | mΩ]",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"le_refl",
"MeasurableSpace.instPartialOrder",
"PartialOrder.toPreorder",
"ENNReal.measur... | [] | exact (measurable_condLExp _ _ _).mono hm (le_refl _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 57
} | {
"line": 127,
"column": 2
} | [
{
"pp": "case pos\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX : Ω → ℝ≥0∞\nhm : mΩ ≤ mΩ₀\n⊢ Measurable P⁻[X | mΩ]",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"le_refl",
"MeasurableSpace.instPartialOrder",
"PartialOrder.toPreorder",
"ENNReal.measur... | [] | exact (measurable_condLExp _ _ _).mono hm (le_refl _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 35
} | {
"line": 151,
"column": 36
} | [
{
"pp": "Ω : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nhm : mΩ ≤ mΩ₀\nP : Measure Ω\nhσ : SigmaFinite (P.trim hm)\nX : Ω → ℝ≥0∞\n⊢ ∫⁻ (ω : Ω), P⁻[X | mΩ] ω ∂P = ∫⁻ (ω : Ω), X ω ∂P",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
... | [
"Ω : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nhm : mΩ ≤ mΩ₀\nP : Measure Ω\nhσ : SigmaFinite (P.trim hm)\nX : Ω → ℝ≥0∞\n⊢ ∫⁻ (x : Ω) in Set.univ, P⁻[X | mΩ] x ∂P = ∫⁻ (x : Ω) in Set.univ, X x ∂P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 166,
"column": 10
} | {
"line": 166,
"column": 25
} | {
"line": 166,
"column": 26
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nf : α → E\ns : Set α\nm m₂ m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nhm₂ : m₂ ≤ m0\ninst✝¹ : SigmaFinite (μ.trim hm)\ninst✝ : SigmaFinite (μ.trim hm₂)\nhs_m : MeasurableSet s\nhs : ∀... | [
"α : Type u_1\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nf : α → E\ns : Set α\nm m₂ m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nhm₂ : m₂ ≤ m0\ninst✝¹ : SigmaFinite (μ.trim hm)\ninst✝ : SigmaFinite (μ.trim hm₂)\nhs_m : MeasurableSet s\nhs : ∀ (t : Set α)... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 51
} | {
"line": 251,
"column": 2
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nα : Type u_2\nm mα : MeasurableSpace α\nμ : Measure α\nf : α → E\nhm : m ≤ mα\nhμm : ¬SigmaFinite (μ.trim hm)\n⊢ (fun x ↦ ‖μ[f | m] x‖) ≤ᵐ[μ] μ[fun x ↦ ‖f x‖ | m]",
"ppTerm": "?pos✝",
"assi... | [] | simp [condExp_of_not_sigmaFinite hm hμm]; aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 51
} | {
"line": 251,
"column": 2
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nα : Type u_2\nm mα : MeasurableSpace α\nμ : Measure α\nf : α → E\nhm : m ≤ mα\nhμm : ¬SigmaFinite (μ.trim hm)\n⊢ (fun x ↦ ‖μ[f | m] x‖) ≤ᵐ[μ] μ[fun x ↦ ‖f x‖ | m]",
"ppTerm": "?pos✝",
"assi... | [] | simp [condExp_of_not_sigmaFinite hm hμm]; aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Matrix | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 64
} | {
"line": 72,
"column": 65
} | [
{
"pp": "case intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : AffineSpace V P\ninst✝⁴ : Ring k\ninst✝³ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝² : Fintype ι\ninst✝¹ : Finite ι'\ninst✝ : DecidableEq ι'\np : ι' → P\nA : Matrix ι ι' k\nhA : b.toMatrix p... | [
"case intro\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : AffineSpace V P\ninst✝⁴ : Ring k\ninst✝³ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝² : Fintype ι\ninst✝¹ : Finite ι'\ninst✝ : DecidableEq ι'\np : ι' → P\nA : Matrix ι ι' k\nhA : b.toMatrix p * A = 1\nva... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift | {
"line": 130,
"column": 25
} | {
"line": 130,
"column": 40
} | {
"line": 130,
"column": 41
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : AddTorsor V P\ninst✝² : Module k V\nι : Type u_4\ninst✝¹ : Fintype ι\ninst✝ : Nontrivial ι\np : ι → P\nh✝ : AffineIndependent k p\ni : ι\nw : ι → k\nhw : ∑ i, w i = 1\nc : k\nh : w i = 1 - c\n⊢ (affineCombinati... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : AddTorsor V P\ninst✝² : Module k V\nι : Type u_4\ninst✝¹ : Fintype ι\ninst✝ : Nontrivial ι\np : ι → P\nh✝ : AffineIndependent k p\ni : ι\nw : ι → k\nhw : ∑ i, w i = 1\nc : k\nh : w i = 1 - c\n⊢ (affineCombination k univ p)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift | {
"line": 137,
"column": 74
} | {
"line": 137,
"column": 85
} | {
"line": 137,
"column": 86
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : AddTorsor V P\ninst✝² : Module k V\nι : Type u_4\ninst✝¹ : Fintype ι\ninst✝ : Nontrivial ι\np : ι → P\nh : AffineIndependent k p\ni : ι\nw : ι → k\nhw : ∑ i, w i = 1\nc : k\nj : ι\nhj : j ≠ i\nt : Finset ι\nw' ... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : AddTorsor V P\ninst✝² : Module k V\nι : Type u_4\ninst✝¹ : Fintype ι\ninst✝ : Nontrivial ι\np : ι → P\nh : AffineIndependent k p\ni : ι\nw : ι → k\nhw : ∑ i, w i = 1\nc : k\nj : ι\nhj : j ≠ i\nt : Finset ι\nw' : ι → k\nht ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 29
} | {
"line": 142,
"column": 30
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : AddTorsor V P\ninst✝² : Module k V\nι : Type u_4\ninst✝¹ : Fintype ι\ninst✝ : Nontrivial ι\np : ι → P\nh : AffineIndependent k p\ni : ι\nw : ι → k\nhw : ∑ i, w i = 1\nc : k\nj : ι\nhj : j ≠ i\nt : Finset ι\nw' ... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : AddTorsor V P\ninst✝² : Module k V\nι : Type u_4\ninst✝¹ : Fintype ι\ninst✝ : Nontrivial ι\np : ι → P\nh : AffineIndependent k p\ni : ι\nw : ι → k\nhw : ∑ i, w i = 1\nc : k\nj : ι\nhj : j ≠ i\nt : Finset ι\nw' : ι → k\nht✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 106,
"column": 18
} | {
"line": 106,
"column": 29
} | {
"line": 106,
"column": 30
} | [
{
"pp": "ιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' ... | [
"ιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 107,
"column": 18
} | {
"line": 107,
"column": 29
} | {
"line": 107,
"column": 30
} | [
{
"pp": "ιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' ... | [
"ιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift | {
"line": 201,
"column": 17
} | {
"line": 201,
"column": 28
} | {
"line": 201,
"column": 29
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁷ : Ring k\ninst✝⁶ : PartialOrder k\ninst✝⁵ : IsOrderedAddMonoid k\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Module k V\nn : ℕ\ninst✝¹ : NeZero n\ns : Simplex k P n\ni : Fin (n + 1)\ninst✝ : ZeroLEOneClass k\nw : Fin (n + 1) → k\nhw : ∀ ... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁷ : Ring k\ninst✝⁶ : PartialOrder k\ninst✝⁵ : IsOrderedAddMonoid k\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Module k V\nn : ℕ\ninst✝¹ : NeZero n\ns : Simplex k P n\ni : Fin (n + 1)\ninst✝ : ZeroLEOneClass k\nw : Fin (n + 1) → k\nhw : ∀ i_1 ∈ univ.e... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TensorProduct.IsBaseChangeHom | {
"line": 80,
"column": 36
} | {
"line": 80,
"column": 52
} | {
"line": 80,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝¹² : CommSemiring R\nS : Type u_2\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Algebra R S\nM : Type u_3\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nN : Type u_4\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nP : Type u_5\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R P\ninst✝³ : Module S... | [
"R : Type u_1\ninst✝¹² : CommSemiring R\nS : Type u_2\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Algebra R S\nM : Type u_3\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nN : Type u_4\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nP : Type u_5\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R P\ninst✝³ : Module S P\ninst✝² :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift | {
"line": 225,
"column": 25
} | {
"line": 225,
"column": 36
} | {
"line": 225,
"column": 37
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\ninst✝¹ : LinearOrder k\ninst✝ : IsOrderedRing k\nn : ℕ\ni : Fin n\nx : k\nhxpos : 0 < x\nhx1 : x ≤ 1\nw : Fin n → k\nhw : ∑ i, w i = 1\nhi : w i = 1 - x\nhj : ∀ (j : Fin n), w j ∈ Set.Icc 0 1\nj : Fin n\nhji : j ≠ i\n⊢ 0 ≤ x⁻¹",
"ppTerm": "?m.144",
"assigned": tr... | [
"k : Type u_1\ninst✝² : Field k\ninst✝¹ : LinearOrder k\ninst✝ : IsOrderedRing k\nn : ℕ\ni : Fin n\nx : k\nhxpos : 0 < x\nhx1 : x ≤ 1\nw : Fin n → k\nhw : ∑ i, w i = 1\nhi : w i = 1 - x\nhj : ∀ (j : Fin n), w j ∈ Set.Icc 0 1\nj : Fin n\nhji : j ≠ i\n⊢ 0 ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TensorProduct.IsBaseChangeHom | {
"line": 84,
"column": 2
} | {
"line": 86,
"column": 57
} | {
"line": 87,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹² : CommSemiring R\nS : Type u_2\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Algebra R S\nM : Type u_3\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nN : Type u_4\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nP : Type u_5\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R P\ninst✝³ : Module S... | [
"R : Type u_1\ninst✝¹² : CommSemiring R\nS : Type u_2\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Algebra R S\nM : Type u_3\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nN : Type u_4\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nP : Type u_5\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R P\ninst✝³ : Module S P\ninst✝² :... | suffices f.toLinearMap.comp (linearMapRightBaseChangeHom S M ε) =
(finitePow ι ibc).equiv.toLinearMap.comp e'.toLinearMap by
simp [h', this, ← LinearEquiv.trans_assoc e'.symm e'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift | {
"line": 241,
"column": 4
} | {
"line": 241,
"column": 51
} | {
"line": 241,
"column": 52
} | [
{
"pp": "case inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Field k\ninst✝⁵ : LinearOrder k\ninst✝⁴ : IsOrderedRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex k P n\ni : Fin (n + 1)\nx : k\nhx : x ∈ Set.Icc 0 1\nhx0 : 0 = x\n⊢ s.clos... | [
"case inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Field k\ninst✝⁵ : LinearOrder k\ninst✝⁴ : IsOrderedRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex k P n\ni : Fin (n + 1)\nx : k\nhx : x ∈ Set.Icc 0 1\nhx0 : 0 = x\n⊢ s.closedInterior ∩... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FixedSubmodule | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 35
} | {
"line": 156,
"column": 36
} | [
{
"pp": "R : Type u_4\nV : Type u_5\ninst✝² : Ring R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\ne : V ≃ₗ[R] V\n⊢ e.fixedReduce = refl R (V ⧸ (↑e).fixedSubmodule) ↔ ∀ (v : V), e v - v ∈ (↑e).fixedSubmodule",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",... | [
"R : Type u_4\nV : Type u_5\ninst✝² : Ring R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\ne : V ≃ₗ[R] V\n⊢ (∀ (x : V ⧸ (↑e).fixedSubmodule), e.fixedReduce x = x) ↔ ∀ (v : V), e (e v) - e v = e v - v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 46
} | {
"line": 81,
"column": 47
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nhcomm : ∀ (i j : ι) (r : R), i ≠ j → f * (b.coord i).transvection (r • b j) = (b.coord i).transvection (r • b j) * f\ni j : ι... | [
"R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nhcomm : ∀ (i j : ι) (r : R), i ≠ j → f * (b.coord i).transvection (r • b j) = (b.coord i).transvection (r • b j) * f\ni j : ι\nhij : i ≠ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 15
} | {
"line": 85,
"column": 16
} | [
{
"pp": "case neg\nR : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (r : R), r • f (b j) = (b.coord i) (f (b i)) • r • b j\ni j : ι\nhij : ¬j = i\n⊢ (b.... | [
"case neg\nR : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (r : R), r • f (b j) = (b.coord i) (f (b i)) • r • b j\ni j : ι\nhij : ¬j = i\n⊢ (b.repr (f (b i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 29
} | {
"line": 88,
"column": 30
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (r : R), r • f (b j) = (b.coord i) (f (b i)) • r • b j\nh_allEq : ∀ (i j : ι), (b.coord i) (f ... | [
"R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (r : R), r • f (b j) = (b.coord i) (f (b i)) • r • b j\nh_allEq : ∀ (i j : ι), (b.coord i) (f (b i)) = (b.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 90,
"column": 42
} | {
"line": 90,
"column": 53
} | {
"line": 90,
"column": 54
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nh_allEq : ∀ (i j : ι), (b.coord i) (f (b i)) = (b.coord j) (f (b j))\nhcomm : ∀ (i : ι) (r : R), r • f (b i) = (b.coord i) (f... | [
"R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nh_allEq : ∀ (i j : ι), (b.coord i) (f (b i)) = (b.coord j) (f (b j))\nhcomm : ∀ (i : ι) (r : R), r • f (b i) = (b.coord i) (f (b i)) • r ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 130,
"column": 8
} | {
"line": 130,
"column": 41
} | {
"line": 130,
"column": 42
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\ni j : ι\nh✝ : ∀ (v : V), ∃ x x_1, ∃ (_ : x • v + x_1 • f v = 0), ¬(x = 0 ∧ x_1 = 0)\ns t : R\nh : ∀ (i_1 : ι), (b.repr (s •... | [
"R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\ni j : ι\nh✝ : ∀ (v : V), ∃ x x_1, ∃ (_ : x • v + x_1 • f v = 0), ¬(x = 0 ∧ x_1 = 0)\ns t : R\nh : ∀ (i_1 : ι), (b.repr (s • b i + t • f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 137,
"column": 6
} | {
"line": 137,
"column": 17
} | {
"line": 137,
"column": 18
} | [
{
"pp": "case neg\nR : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\ni j : ι\nh✝ : ∀ (v : V), ∃ x x_1, ∃ (_ : x • v + x_1 • f v = 0), ¬(x = 0 ∧ x_1 = 0)\ns t : R\nthis : t = 0 ∨ (b.r... | [
"case neg\nR : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\ni j : ι\nh✝ : ∀ (v : V), ∃ x x_1, ∃ (_ : x • v + x_1 • f v = 0), ¬(x = 0 ∧ x_1 = 0)\ns t : R\nthis : t = 0 ∨ (b.repr (f (b i)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 156,
"column": 6
} | {
"line": 156,
"column": 43
} | {
"line": 156,
"column": 44
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nfeq : ∀ (i : ι), f (b i) = (b.coord i) (f (b i)) • b i\ni j : ι\nhij : i ≠ j\nr : R\nx : V := b.repr.symm ((Finsupp.single ... | [
"R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nfeq : ∀ (i : ι), f (b i) = (b.coord i) (f (b i)) • b i\ni j : ι\nhij : i ≠ j\nr : R\nx : V := b.repr.symm ((Finsupp.single i 1).update ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 299,
"column": 8
} | {
"line": 299,
"column": 19
} | {
"line": 299,
"column": 20
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\ne : V ≃ₗ[R] V\nthis : ∀ e ∈ dilatransvections R V, e.symm ∈ dilatransvections R V\n⊢ e.symm ∈ dilatransvections R V → e ∈ dilatransvections R V",
"ppTerm": "?m.72",
"assigned": false,
"usedConstants": ... | [
"R : Type u_1\nV : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\ne : V ≃ₗ[R] V\nthis : ∀ e ∈ dilatransvections R V, e.symm ∈ dilatransvections R V\n⊢ e.symm ∈ dilatransvections R V → e ∈ dilatransvections R V"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 162,
"column": 6
} | {
"line": 162,
"column": 60
} | {
"line": 162,
"column": 61
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nfeq : ∀ (i : ι), f (b i) = (b.coord i) (f (b i)) • b i\ni j : ι\nhij : i ≠ j\nr : R\nx : V := b.repr.symm ((Finsupp.single ... | [
"R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nfeq : ∀ (i : ι), f (b i) = (b.coord i) (f (b i)) • b i\ni j : ι\nhij : i ≠ j\nr : R\nx : V := b.repr.symm ((Finsupp.single i 1).update ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 65
} | {
"line": 182,
"column": 66
} | [
{
"pp": "case refine_1\nR : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nh : ∀ (v : V), ¬LinearIndependent R ![v, f v]\nh' : ∀ (i j : ι), i ≠ j → ∀ (r : R), (b.coord i) (f (b i)) * ... | [
"case refine_1\nR : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nh : ∀ (v : V), ¬LinearIndependent R ![v, f v]\nh' : ∀ (i j : ι), i ≠ j → ∀ (r : R), (b.coord i) (f (b i)) * r = r * (b.c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Center | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 64
} | {
"line": 184,
"column": 65
} | [
{
"pp": "case refine_2\nR : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nh : ∀ (v : V), ¬LinearIndependent R ![v, f v]\nh' : ∀ (i j : ι), i ≠ j → ∀ (r : R), (b.coord i) (f (b i)) * ... | [
"case refine_2\nR : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nh : ∀ (v : V), ¬LinearIndependent R ![v, f v]\nh' : ∀ (i j : ι), i ≠ j → ∀ (r : R), (b.coord i) (f (b i)) * r = r * (b.c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 388,
"column": 6
} | {
"line": 388,
"column": 37
} | {
"line": 388,
"column": 38
} | [
{
"pp": "case h\nV : Type u_2\ninst✝² : AddCommGroup V\nK : Type u_3\ninst✝¹ : DivisionRing K\ninst✝ : Module K V\ne : V ≃ₗ[K] V\nu : V →ₗ[K] V := ↑e - LinearMap.id\nhe : Module.rank K ↥u.range ≤ 1\nhu : u + LinearMap.id = ↑e\nhr : Subsingleton ↥u.range\nx : V\n⊢ u x = 0",
"ppTerm": "?h",
"assigned": fa... | [
"case h\nV : Type u_2\ninst✝² : AddCommGroup V\nK : Type u_3\ninst✝¹ : DivisionRing K\ninst✝ : Module K V\ne : V ≃ₗ[K] V\nu : V →ₗ[K] V := ↑e - LinearMap.id\nhe : Module.rank K ↥u.range ≤ 1\nhu : u + LinearMap.id = ↑e\nhr : Subsingleton ↥u.range\nx : V\n⊢ u x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.QuadraticForm.TensorProduct | {
"line": 163,
"column": 8
} | {
"line": 163,
"column": 20
} | {
"line": 163,
"column": 21
} | [
{
"pp": "R : Type uR\nA : Type uA\nM₂ : Type uM₂\nN₁ : Type uN₁\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : AddCommGroup M₂\ninst✝⁵ : AddCommGroup N₁\ninst✝⁴ : Algebra R A\ninst✝³ : Module R N₁\ninst✝² : Module A N₁\ninst✝¹ : IsScalarTower R A N₁\ninst✝ : Module R M₂\nQ₁ Q₂ : QuadraticMap A (A ⊗[R] M₂) ... | [
"R : Type uR\nA : Type uA\nM₂ : Type uM₂\nN₁ : Type uN₁\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : AddCommGroup M₂\ninst✝⁵ : AddCommGroup N₁\ninst✝⁴ : Algebra R A\ninst✝³ : Module R N₁\ninst✝² : Module A N₁\ninst✝¹ : IsScalarTower R A N₁\ninst✝ : Module R M₂\nQ₁ Q₂ : QuadraticMap A (A ⊗[R] M₂) N₁\nh : ∀ (m... | ← mul_one a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.BaseChange | {
"line": 55,
"column": 4
} | {
"line": 56,
"column": 39
} | {
"line": 58,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nV : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : AddCommGroup V\ninst✝² : Algebra R A\ninst✝¹ : Module R V\ninst✝ : Invertible 2\nQ : QuadraticForm R V\nv : V\n⊢ (algebraMap A (CliffordAlgebra (QuadraticForm.baseChange A Q))) ((QuadraticForm.baseChange A Q) (... | [] | rw [QuadraticForm.baseChange_tmul, one_mul, ← Algebra.algebraMap_eq_smul_one,
← IsScalarTower.algebraMap_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.CliffordAlgebra.BaseChange | {
"line": 102,
"column": 43
} | {
"line": 102,
"column": 79
} | {
"line": 102,
"column": 80
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nV : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : AddCommGroup V\ninst✝² : Algebra R A\ninst✝¹ : Module R V\ninst✝ : Invertible 2\nQ : QuadraticForm R V\nthis✝ : Invertible 2 := (Invertible.map (algebraMap R A) 2).copy 2 ⋯\nthis : Invertible 2 := (Invertible.m... | [
"R : Type u_1\nA : Type u_2\nV : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : AddCommGroup V\ninst✝² : Algebra R A\ninst✝¹ : Module R V\ninst✝ : Invertible 2\nQ : QuadraticForm R V\nthis✝ : Invertible 2 := (Invertible.map (algebraMap R A) 2).copy 2 ⋯\nthis : Invertible 2 := (Invertible.map (algebraM... | LinearMap.BilinForm.baseChange_tmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 478,
"column": 10
} | {
"line": 478,
"column": 71
} | {
"line": 478,
"column": 72
} | [
{
"pp": "V : Type u_2\ninst✝³ : AddCommGroup V\nK : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nx✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1\nhe : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1\nhe' : e.fixedReduce = 1\nhe_one : ¬e = 1\nhefixed_ne_top : (↑e).... | [
"V : Type u_2\ninst✝³ : AddCommGroup V\nK : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nx✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1\nhe : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1\nhe' : e.fixedReduce = 1\nhe_one : ¬e = 1\nhefixed_ne_top : (↑e).fixedSubmodu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 23
} | {
"line": 134,
"column": 24
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\na : M\nb : CliffordAlgebra Q\n⊢ (contractRight (b * (ι Q) a)) d = d a • b - (contractRight b) d * (ι Q) a",
"ppTerm": "?m.57",
"assigned": true,
"usedConst... | [
"R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\na : M\nb : CliffordAlgebra Q\n⊢ reverse ((contractLeft d) (reverse (b * (ι Q) a))) = d a • b - (contractRight b) d * (ι Q) a"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 165,
"column": 6
} | {
"line": 165,
"column": 23
} | {
"line": 165,
"column": 24
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nx : M\n⊢ (contractRight ((ι Q) x)) d = (algebraMap R (CliffordAlgebra Q)) (d x)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"CliffordAl... | [
"R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nx : M\n⊢ reverse ((contractLeft d) (reverse ((ι Q) x))) = (algebraMap R (CliffordAlgebra Q)) (d x)"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 175,
"column": 6
} | {
"line": 175,
"column": 23
} | {
"line": 175,
"column": 24
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nr : R\n⊢ (contractRight ((algebraMap R (CliffordAlgebra Q)) r)) d = 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"CliffordAlgebra.cont... | [
"R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nr : R\n⊢ reverse ((contractLeft d) (reverse ((algebraMap R (CliffordAlgebra Q)) r))) = 0"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 28
} | {
"line": 179,
"column": 29
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\n⊢ (contractLeft d) 1 = 0",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\n⊢ (contractLeft d) 1 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 28
} | {
"line": 183,
"column": 29
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\n⊢ (contractRight 1) d = 0",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\n⊢ (contractRight 1) d = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 197,
"column": 6
} | {
"line": 197,
"column": 23
} | {
"line": 197,
"column": 24
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ (contractRight ((contractRight x) d)) d = 0",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"CliffordAlgebra.contr... | [
"R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ reverse ((contractLeft d) (reverse ((contractRight x) d))) = 0"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 197,
"column": 24
} | {
"line": 197,
"column": 41
} | {
"line": 197,
"column": 42
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ reverse ((contractLeft d) (reverse ((contractRight x) d))) = 0",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Cl... | [
"R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ reverse ((contractLeft d) (reverse (reverse ((contractLeft d) (reverse x))))) = 0"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Equivs | {
"line": 299,
"column": 14
} | {
"line": 299,
"column": 25
} | {
"line": 301,
"column": 0
} | [
{
"pp": "case hl\nR : Type u_1\ninst✝ : CommRing R\nc₁ c₂ : R\n⊢ ((ofQuaternion.comp toQuaternion).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inl R R R =\n ((AlgHom.id R (CliffordAlgebra (Q c₁ c₂))).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inl R R R",
"ppTerm": "?hl",
"assigned": true,
"usedConstan... | [] | (ext; simp) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.LinearAlgebra.CliffordAlgebra.Equivs | {
"line": 299,
"column": 14
} | {
"line": 299,
"column": 25
} | {
"line": 301,
"column": 0
} | [
{
"pp": "case hr\nR : Type u_1\ninst✝ : CommRing R\nc₁ c₂ : R\n⊢ ((ofQuaternion.comp toQuaternion).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inr R R R =\n ((AlgHom.id R (CliffordAlgebra (Q c₁ c₂))).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inr R R R",
"ppTerm": "?hr",
"assigned": true,
"usedConstan... | [] | (ext; simp) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 210,
"column": 6
} | {
"line": 210,
"column": 23
} | {
"line": 210,
"column": 24
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ (contractRight ((contractRight x) d)) d' = -(contractRight ((contractRight x) d')) d",
"ppTerm": "?m.40",
"assigned": true,
"us... | [
"R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ reverse ((contractLeft d') (reverse ((contractRight x) d))) = -(contractRight ((contractRight x) d')) d"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 210,
"column": 24
} | {
"line": 210,
"column": 41
} | {
"line": 210,
"column": 42
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ reverse ((contractLeft d') (reverse ((contractRight x) d))) = -(contractRight ((contractRight x) d')) d",
"ppTerm": "?m.49",
"assig... | [
"R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ reverse ((contractLeft d') (reverse (reverse ((contractLeft d) (reverse x))))) =\n -(contractRight ((contractRight x) d')) d"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 210,
"column": 42
} | {
"line": 210,
"column": 59
} | {
"line": 210,
"column": 60
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ reverse ((contractLeft d') (reverse (reverse ((contractLeft d) (reverse x))))) =\n -(contractRight ((contractRight x) d')) d",
"ppTe... | [
"R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ reverse ((contractLeft d') (reverse (reverse ((contractLeft d) (reverse x))))) =\n -reverse ((contractLeft d) (reverse ((contractRight x) d')))"
] | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 210,
"column": 60
} | {
"line": 210,
"column": 77
} | {
"line": 210,
"column": 78
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ reverse ((contractLeft d') (reverse (reverse ((contractLeft d) (reverse x))))) =\n -reverse ((contractLeft d) (reverse ((contractRight x... | [
"R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n⊢ reverse ((contractLeft d') (reverse (reverse ((contractLeft d) (reverse x))))) =\n -reverse ((contractLeft d) (reverse (reverse ((contractLeft d') (... | contractRight_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 13
} | {
"line": 274,
"column": 14
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' : QuadraticForm R M\nB : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\n⊢ (changeForm h) 1 = 1",
"ppTerm": "?m.63",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' : QuadraticForm R M\nB : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\n⊢ (changeForm h) 1 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 631,
"column": 2
} | {
"line": 631,
"column": 56
} | {
"line": 631,
"column": 57
} | [
{
"pp": "R : Type u_3\nV : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module R V\ninst✝² : Free R V\ninst✝¹ : Module.Finite R V\ninst✝ : IsDomain R\nf : Dual R V\nv : V\nK : Type u_3 := FractionRing R\nthis✝ : Field K := inferInstance\nthis : (algebraMap R K) (LinearMap.det (transvection f... | [
"R : Type u_3\nV : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module R V\ninst✝² : Free R V\ninst✝¹ : Module.Finite R V\ninst✝ : IsDomain R\nf : Dual R V\nv : V\nK : Type u_3 := FractionRing R\nthis✝ : Field K := inferInstance\nthis : (algebraMap R K) (LinearMap.det (transvection f v)) = ↑1 + ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.Internal | {
"line": 132,
"column": 47
} | {
"line": 134,
"column": 59
} | {
"line": 136,
"column": 0
} | [
{
"pp": "R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁸ : CommSemiring ι\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : CommRing R\ninst✝⁵ : Ring A\ninst✝⁴ : Ring B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝¹ : GradedAlgebra 𝒜\ninst✝ : GradedAlgebra ℬ\n... | [] | by
rw [← of_one, Algebra.TensorProduct.one_def, auxEquiv_tmul 𝒜 ℬ, DirectSum.decompose_one,
DirectSum.decompose_one, Algebra.TensorProduct.one_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 34
} | {
"line": 90,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\nx : (CliffordAlgebra Q)ˣ\ninst✝ : Invertible 2\ny z : (CliffordAlgebra Q)ˣ\nhx✝ : y ∈ Subgroup.closure (Units.val ⁻¹' Set.range ⇑(ι Q))\nhy✝ : z ∈ Subgroup.closure (Units.val ⁻¹' Set.ra... | [
"R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\nx : (CliffordAlgebra Q)ˣ\ninst✝ : Invertible 2\ny z : (CliffordAlgebra Q)ˣ\nhx✝ : y ∈ Subgroup.closure (Units.val ⁻¹' Set.range ⇑(ι Q))\nhy✝ : z ∈ Subgroup.closure (Units.val ⁻¹' Set.range ⇑(ι Q))\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup | {
"line": 121,
"column": 6
} | {
"line": 121,
"column": 34
} | {
"line": 121,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\ninst✝ : Invertible 2\nx y z : (CliffordAlgebra Q)ˣ\nhx✝ : y ∈ Subgroup.closure (Units.val ⁻¹' Set.range ⇑(ι Q))\nhy✝ : z ∈ Subgroup.closure (Units.val ⁻¹' Set.range ⇑(ι Q))\nhy : ∀ (b :... | [
"R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\ninst✝ : Invertible 2\nx y z : (CliffordAlgebra Q)ˣ\nhx✝ : y ∈ Subgroup.closure (Units.val ⁻¹' Set.range ⇑(ι Q))\nhy✝ : z ∈ Subgroup.closure (Units.val ⁻¹' Set.range ⇑(ι Q))\nhy : ∀ (b : M), involut... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Torsion.Finite | {
"line": 22,
"column": 28
} | {
"line": 22,
"column": 58
} | {
"line": 22,
"column": 59
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nh : IsTorsion R M\nh' : Module.rank R M ≠ 0\n⊢ ?m.22",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Cardinal.instOne",
"Cardinal",
"congrArg",
... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nh : IsTorsion R M\nh'✝ : Module.rank R M ≠ 0\nh' : 1 ≤ Module.rank R M\n⊢ ?m.22"
] | ← Cardinal.one_le_iff_ne_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.Torsion.Finite | {
"line": 23,
"column": 2
} | {
"line": 23,
"column": 59
} | {
"line": 23,
"column": 60
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nh : IsTorsion R M\nh' : Module.rank R M ≠ 0\nf : R →ₗ[R] M\nhf : Function.Injective ⇑f\n⊢ False",
"ppTerm": "?m.46",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nh : IsTorsion R M\nh' : Module.rank R M ≠ 0\nf : R →ₗ[R] M\nhf : Function.Injective ⇑f\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basis | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 45
} | {
"line": 56,
"column": 46
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nm n : ℕ\nI : Type u_3\ninst✝³ : LinearOrder I\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nb : Basis I R M\ns : ↑(powersetCard I m)\nt : ↑(powersetCard I n)\nh : ¬Disjoint ↑s ↑t\n⊢ b.ExteriorAlgebra ↑s * b.ExteriorAlgebra ↑t = 0",
"ppTerm": "?m.38",... | [
"R : Type u_1\nM : Type u_2\nm n : ℕ\nI : Type u_3\ninst✝³ : LinearOrder I\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nb : Basis I R M\ns : ↑(powersetCard I m)\nt : ↑(powersetCard I n)\nh : ¬Disjoint ↑s ↑t\n⊢ ιMulti_family R m (⇑b) s * ιMulti_family R n (⇑b) t = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basis | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 45
} | {
"line": 61,
"column": 46
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nm n : ℕ\nI : Type u_3\ninst✝³ : LinearOrder I\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nb : Basis I R M\ns : ↑(powersetCard I m)\nt : ↑(powersetCard I n)\nh : Disjoint ↑s ↑t\n⊢ b.ExteriorAlgebra ↑s * b.ExteriorAlgebra ↑t = Equiv.Perm.sign (permOfDisj... | [
"R : Type u_1\nM : Type u_2\nm n : ℕ\nI : Type u_3\ninst✝³ : LinearOrder I\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nb : Basis I R M\ns : ↑(powersetCard I m)\nt : ↑(powersetCard I n)\nh : Disjoint ↑s ↑t\n⊢ ιMulti_family R m (⇑b) s * ιMulti_family R n (⇑b) t =\n Equiv.Perm.sign (permOfDis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorAlgebra.Grading | {
"line": 34,
"column": 36
} | {
"line": 34,
"column": 62
} | {
"line": 34,
"column": 63
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\n⊢ (ι R) m ∈ ⋀[R]^1 M",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"QuadraticMap.instZero",
"ExteriorAlg... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\n⊢ (ι R) m ∈ (ι R).range"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.ExteriorAlgebra.Grading | {
"line": 39,
"column": 19
} | {
"line": 39,
"column": 45
} | {
"line": 39,
"column": 46
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\n⊢ (ι R) m ∈ ⋀[R]^1 M",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"QuadraticMap.instZero",
"ExteriorAlg... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\n⊢ (ι R) m ∈ (ι R).range"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.ModN | {
"line": 43,
"column": 65
} | {
"line": 43,
"column": 76
} | {
"line": 43,
"column": 77
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nM : Type u_3\ninst✝¹ : AddCommGroup G\nn : ℕ\ninst✝ : AddMonoid M\nφ : { φ // ∀ (g : G), n • φ g = 0 }\ng : G\n⊢ ((LinearMap.lsmul ℤ G) ↑n) g ∈ (↑φ).ker",
"ppTerm": "?m.133",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"ModN._pro... | [
"G : Type u_1\nH : Type u_2\nM : Type u_3\ninst✝¹ : AddCommGroup G\nn : ℕ\ninst✝ : AddMonoid M\nφ : { φ // ∀ (g : G), n • φ g = 0 }\ng : G\n⊢ n • ↑φ g = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeProduct.Basic | {
"line": 215,
"column": 2
} | {
"line": 216,
"column": 33
} | {
"line": 218,
"column": 0
} | [
{
"pp": "I : Type u\ninst✝⁵ : DecidableEq I\ni : I\nR : Type v\ninst✝⁴ : CommSemiring R\nA : I → Type w\ninst✝³ : (i : I) → Semiring (A i)\ninst✝² : (i : I) → Algebra R (A i)\nB : Type w'\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nmaps : {i : I} → A i →ₐ[R] B\n⊢ ((lift R A) fun {i} ↦ maps) ∘ₐ ι R A i = maps",
... | [] | ext a
simp [lift_apply, ι, ← ι_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.FreeProduct.Basic | {
"line": 215,
"column": 2
} | {
"line": 216,
"column": 33
} | {
"line": 218,
"column": 0
} | [
{
"pp": "I : Type u\ninst✝⁵ : DecidableEq I\ni : I\nR : Type v\ninst✝⁴ : CommSemiring R\nA : I → Type w\ninst✝³ : (i : I) → Semiring (A i)\ninst✝² : (i : I) → Algebra R (A i)\nB : Type w'\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nmaps : {i : I} → A i →ₐ[R] B\n⊢ ((lift R A) fun {i} ↦ maps) ∘ₐ ι R A i = maps",
... | [] | ext a
simp [lift_apply, ι, ← ι_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Goursat | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 73
} | {
"line": 114,
"column": 6
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nL : Submodule R (M × N)\nM' : Submodule R M := map (LinearMap.fst R M N) L\nN' : Submodule R N := map (LinearMap.snd R M N) L\nP : ↥L →ₗ[R] ↥M' := (Linea... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nL : Submodule R (M × N)\nM' : Submodule R M := map (LinearMap.fst R M N) L\nN' : Submodule R N := map (LinearMap.snd R M N) L\nP : ↥L →ₗ[R] ↥M' := (LinearMap.fst R M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Goursat | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 73
} | {
"line": 118,
"column": 6
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nL : Submodule R (M × N)\nM' : Submodule R M := map (LinearMap.fst R M N) L\nN' : Submodule R N := map (LinearMap.snd R M N) L\nP : ↥L →ₗ[R] ↥M' := (Linea... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nL : Submodule R (M × N)\nM' : Submodule R M := map (LinearMap.fst R M N) L\nN' : Submodule R N := map (LinearMap.snd R M N) L\nP : ↥L →ₗ[R] ↥M' := (LinearMap.fst R M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.CharP | {
"line": 27,
"column": 73
} | {
"line": 27,
"column": 98
} | {
"line": 28,
"column": 4
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝³ : AddMonoidWithOne R\ninst✝² : DecidableEq n\ninst✝¹ : Nonempty n\np : ℕ\ninst✝ : CharP R p\nk : ℕ\n⊢ ((diagonal fun x ↦ ↑k) = diagonal fun x ↦ 0) ↔ p ∣ k",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Lin... | [
"n : Type u_1\nR : Type u_2\ninst✝³ : AddMonoidWithOne R\ninst✝² : DecidableEq n\ninst✝¹ : Nonempty n\np : ℕ\ninst✝ : CharP R p\nk : ℕ\n⊢ (∀ (i : n), ↑k = 0) ↔ p ∣ k"
] | diagonal_eq_diagonal_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.LinearIndependent.BaseChange | {
"line": 56,
"column": 25
} | {
"line": 56,
"column": 52
} | {
"line": 56,
"column": 53
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\ninst✝⁵ : Finite ι'\nR : Type u_3\nS : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FaithfulSMul R S\ninst✝ : IsDomain S\nv✝ : ι → ι' → R\nh : LinearIndependent R v✝\nthis : IsDomain R\nK : Type u_3 := FractionRing R\nL : Type u_4 := Fra... | [
"ι : Type u_1\nι' : Type u_2\ninst✝⁵ : Finite ι'\nR : Type u_3\nS : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FaithfulSMul R S\ninst✝ : IsDomain S\nv✝ : ι → ι' → R\nh : LinearIndependent R v✝\nthis : IsDomain R\nK : Type u_3 := FractionRing R\nL : Type u_4 := FractionRing S\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 97,
"column": 21
} | {
"line": 105,
"column": 25
} | {
"line": 106,
"column": 6
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Fintype ι\ninst✝ : Module R M\nN : Submodule R M\nbM : Basis ι R M\nbN : Basis (Fin n) R ↥N\nf : Fin n ↪ ι\na : Fin n → R\nsnf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)\nN' : Submodule R (ι → R) := S... | [] | by
simp only [hj.choose_spec, ↓reduceIte]
rw [mul_comm]
conv_rhs =>
rw [← hj.choose_spec, (h (f hj.choose)).choose_spec]
simp only [EmbeddingLike.apply_eq_iff_eq, exists_eq, ↓reduceDIte, Classical.choose_eq]
congr!
· exa... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 63
} | {
"line": 69,
"column": 2
} | [
{
"pp": "case mp\nm : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝² : CommRing R\nA : Matrix m n R\nι : Type w\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : ι → m\ng : ι → n\nhA : (A.submatrix (f ∘ ⇑(Fintype.equivFin ι).symm) (g ∘ ⇑(Fintype.equivFin ι).symm)).det ∈ Set.range SignType.cast\n⊢ (A.submatrix f g).... | [] | rwa [← submatrix_submatrix, det_submatrix_equiv_self] at hA | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 63
} | {
"line": 73,
"column": 0
} | [
{
"pp": "case mpr\nm : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝ : CommRing R\nA : Matrix m n R\nk : ℕ\nf : Fin k → m\ng : Fin k → n\nhA : (A.submatrix (f ∘ ⇑Equiv.ulift) (g ∘ ⇑Equiv.ulift)).det ∈ Set.range SignType.cast\n⊢ (A.submatrix f g).det ∈ Set.range SignType.cast",
"ppTerm": "?mpr",
"assigned"... | [] | rwa [← submatrix_submatrix, det_submatrix_equiv_self] at hA | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 13
} | {
"line": 76,
"column": 14
} | [
{
"pp": "m : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝ : CommRing R\nA : Matrix m n R\nhA : ∀ (k : ℕ) (f : Fin k → m) (g : Fin k → n), (A.submatrix f g).det ∈ Set.range SignType.cast\ni : m\nj : n\n⊢ A i j ∈ Set.range SignType.cast",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"S... | [
"m : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝ : CommRing R\nA : Matrix m n R\nhA : ∀ (k : ℕ) (f : Fin k → m) (g : Fin k → n), (A.submatrix f g).det ∈ Set.range SignType.cast\ni : m\nj : n\n⊢ ∃ y, ↑y = A i j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 102,
"column": 16
} | {
"line": 102,
"column": 49
} | {
"line": 102,
"column": 50
} | [
{
"pp": "m : Type u_1\nm' : Type u_2\nn : Type u_3\nn' : Type u_4\nR : Type u_5\ninst✝ : CommRing R\nA : Matrix m n R\nem : m ≃ m'\nen : n ≃ n'\nhA : ((reindex em en) A).IsTotallyUnimodular\n⊢ A.IsTotallyUnimodular",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"m : Type u_1\nm' : Type u_2\nn : Type u_3\nn' : Type u_4\nR : Type u_5\ninst✝ : CommRing R\nA : Matrix m n R\nem : m ≃ m'\nen : n ≃ n'\nhA : ((reindex em en) A).IsTotallyUnimodular\n⊢ A.IsTotallyUnimodular"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Echelon.Pivot | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 28
} | {
"line": 67,
"column": 29
} | [
{
"pp": "case top\nm : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : Zero R\nA : Matrix m n R\nl : m → WithTop n\ninst✝¹ : LT m\ninst✝ : LT n\ni : m\nhA : A.IsPivotedBy l\nhc : l i = ⊤\nh : ∀ (j : n), ↑j < ⊤ → A i j = 0\n⊢ ⊤ = ⊤ ↔ A i = 0",
"ppTerm": "?top",
"assigned": true,
"usedConstants": [
... | [
"case top\nm : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : Zero R\nA : Matrix m n R\nl : m → WithTop n\ninst✝¹ : LT m\ninst✝ : LT n\ni : m\nhA : A.IsPivotedBy l\nhc : l i = ⊤\nh : ∀ (j : n), ↑j < ⊤ → A i j = 0\n⊢ ∀ (x : n), A i x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Echelon.Pivot | {
"line": 68,
"column": 13
} | {
"line": 68,
"column": 24
} | {
"line": 68,
"column": 25
} | [
{
"pp": "case coe\nm : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : Zero R\nA : Matrix m n R\nl : m → WithTop n\ninst✝¹ : LT m\ninst✝ : LT n\ni : m\nhA : A.IsPivotedBy l\nc : n\nhc : l i = ↑c\n⊢ ↑c = ⊤ ↔ A i = 0",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False"... | [
"case coe\nm : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : Zero R\nA : Matrix m n R\nl : m → WithTop n\ninst✝¹ : LT m\ninst✝ : LT n\ni : m\nhA : A.IsPivotedBy l\nc : n\nhc : l i = ↑c\n⊢ ¬A i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 158,
"column": 24
} | {
"line": 158,
"column": 35
} | {
"line": 158,
"column": 36
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nn : ℕ\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Fintype ι\ninst✝¹ : Infinite R\ninst✝ : Module R M\nN : Submodule R M\nsnf : SmithNormalForm N ι n\nh : ¬n = Fintype.card ι\n⊢ n ≤ Fintype.card ι",
"ppTerm": "?m.82",
"assigned": false,
"... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nn : ℕ\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Fintype ι\ninst✝¹ : Infinite R\ninst✝ : Module R M\nN : Submodule R M\nsnf : SmithNormalForm N ι n\nh : ¬n = Fintype.card ι\n⊢ n ≤ Fintype.card ι"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 20
} | {
"line": 173,
"column": 21
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\nn : ℕ\ninst✝¹ : AddCommGroup M\ninst✝ : Fintype ι\nN : Submodule ℤ M\nbM : Basis ι ℤ M\nbN : Basis (Fin n) ℤ ↥N\nf : Fin n ↪ ι\na : Fin n → ℤ\ni : Fin n\nsnf : ↑(bN i) = a i • bM (f i)\nhi : a i = 0\n⊢ bN i = 0",
"ppTerm": "?m.67",
"assigned": false,
"usedConstan... | [
"ι : Type u_1\nM : Type u_3\nn : ℕ\ninst✝¹ : AddCommGroup M\ninst✝ : Fintype ι\nN : Submodule ℤ M\nbM : Basis ι ℤ M\nbN : Basis (Fin n) ℤ ↥N\nf : Fin n ↪ ι\na : Fin n → ℤ\ni : Fin n\nsnf : ↑(bN i) = a i • bM (f i)\nhi : a i = 0\n⊢ bN i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 175,
"column": 2
} | {
"line": 176,
"column": 34
} | {
"line": 176,
"column": 35
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\nn : ℕ\ninst✝¹ : AddCommGroup M\ninst✝ : Fintype ι\nN : Submodule ℤ M\nbM : Basis ι ℤ M\nbN : Basis (Fin n) ℤ ↥N\nf : Fin n ↪ ι\na : Fin n → ℤ\nsnf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)\nha : ∀ (i : Fin n), a i ≠ 0\nh : n = Fintype.card ι\n⊢ ¬∏ x, (Submodule.toAddSubgroup... | [
"ι : Type u_1\nM : Type u_3\nn : ℕ\ninst✝¹ : AddCommGroup M\ninst✝ : Fintype ι\nN : Submodule ℤ M\nbM : Basis ι ℤ M\nbN : Basis (Fin n) ℤ ↥N\nf : Fin n ↪ ι\na : Fin n → ℤ\nsnf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)\nha : ∀ (i : Fin n), a i ≠ 0\nh : n = Fintype.card ι\n⊢ ∀ (x : Fin n), ¬a x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Determinant.Misc | {
"line": 95,
"column": 8
} | {
"line": 95,
"column": 19
} | {
"line": 95,
"column": 20
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nM : Matrix (Fin (n + 1)) (Fin (n + 1)) R\ni₀ j₀ : Fin (n + 1)\nhv : ∀ (i : Fin (n + 1)), i ≠ i₀ → ∑ j, M i j = 0\n⊢ ∀ (j : Fin (n + 1)), j ≠ i₀ → ∑ i, Mᵀ i j = 0",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"Finset.univ",
"Co... | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nM : Matrix (Fin (n + 1)) (Fin (n + 1)) R\ni₀ j₀ : Fin (n + 1)\nhv : ∀ (i : Fin (n + 1)), i ≠ i₀ → ∑ j, M i j = 0\n⊢ ∀ (j : Fin (n + 1)), ¬j = i₀ → ∑ i, M j i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 194,
"column": 4
} | {
"line": 194,
"column": 15
} | {
"line": 194,
"column": 16
} | [
{
"pp": "case refine_2\nι : Type u_1\ninst✝ : Finite ι\nN : Submodule ℤ (ι → ℤ)\nn : ℕ\nthis : Fintype ι\nbN : Module.Basis (Fin n) ℤ ↥N\nx✝ : Nonempty (↥N ≃ₗ[ℤ] ι → ℤ)\ne : ↥N ≃ₗ[ℤ] ι → ℤ\nhc : Fintype.card (Fin n) = Fintype.card ι\n⊢ n = Fintype.card ι",
"ppTerm": "?refine_2",
"assigned": false,
"... | [
"case refine_2\nι : Type u_1\ninst✝ : Finite ι\nN : Submodule ℤ (ι → ℤ)\nn : ℕ\nthis : Fintype ι\nbN : Module.Basis (Fin n) ℤ ↥N\nx✝ : Nonempty (↥N ≃ₗ[ℤ] ι → ℤ)\ne : ↥N ≃ₗ[ℤ] ι → ℤ\nhc : Fintype.card (Fin n) = Fintype.card ι\n⊢ n = Fintype.card ι"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 158,
"column": 10
} | {
"line": 158,
"column": 28
} | {
"line": 158,
"column": 29
} | [
{
"pp": "case h\nm : Type u_1\nm' : Type u_2\nn : Type u_3\nR : Type u_5\ninst✝¹ : CommRing R\ninst✝ : DecidableEq n\nA : Matrix m n R\nB : Matrix m' n R\nhA : A.IsTotallyUnimodular\nhB : ∀ (i : m'), ∃ j s, B i = Pi.single j ↑s\nk : ℕ\nih :\n ∀ (f : Fin k → m ⊕ m') (g : Fin k → n),\n Function.Injective f → ... | [
"case h\nm : Type u_1\nm' : Type u_2\nn : Type u_3\nR : Type u_5\ninst✝¹ : CommRing R\ninst✝ : DecidableEq n\nA : Matrix m n R\nB : Matrix m' n R\nhA : A.IsTotallyUnimodular\nhB : ∀ (i : m'), ∃ j s, B i = Pi.single j ↑s\nk : ℕ\nih :\n ∀ (f : Fin k → m ⊕ m') (g : Fin k → n),\n Function.Injective f → Function.Inj... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Card | {
"line": 47,
"column": 42
} | {
"line": 47,
"column": 77
} | {
"line": 47,
"column": 78
} | [
{
"pp": "case zero\nK : Type u_1\nV : Type u_2\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Fintype K\ninst✝ : Finite V\nhk : 0 ≤ n\nthis : Unique { s // ⊤ = ⊥ }\n⊢ card { s // (Finsupp.linearCombination K s).ker = ⊥ } = ∏ i, (q ^ n - q ^ ↑i)",
"ppTerm": "?zero",
"ass... | [
"case zero\nK : Type u_1\nV : Type u_2\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Fintype K\ninst✝ : Finite V\nhk : 0 ≤ n\nthis : Unique { s // ⊤ = ⊥ }\n⊢ card { s // ker 0 = ⊥ } = ∏ i, (q ^ n - q ^ ↑i)"
] | Finsupp.linearCombination_fin_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 56
} | {
"line": 131,
"column": 57
} | [
{
"pp": "m : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nha : ↑A 1 0 = 0\n⊢ ↑A 0 0 * ↑A 1 1 = m",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nha : ↑A 1 0 = 0\n⊢ ↑A 0 0 * ↑A 1 1 = m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 99,
"column": 30
} | {
"line": 99,
"column": 55
} | {
"line": 99,
"column": 56
} | [
{
"pp": "m : Type u_1\nn : Type u_2\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nA : Matrix m n ℚ\nx✝ : ℕ\n⊢ Aᵀ.den ∣ x✝ ↔ A.den ∣ x✝",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"Rat",
"semigroupDvd",
"_private.Mathlib.Lin... | [
"m : Type u_1\nn : Type u_2\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nA : Matrix m n ℚ\nx✝ : ℕ\n⊢ (∀ (i : n) (j : m), (A j i).den ∣ x✝) ↔ ∀ (i : m) (j : n), (A i j).den ∣ x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 34
} | {
"line": 130,
"column": 35
} | [
{
"pp": "m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℕ\n⊢ (↑a).den = 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Matrix",
"Rat",
"id",
"Matrix.instNatCastOfZero",
"instOfNatNat",
"Nat.cast",
"Nat",
"Matrix.den",
"... | [
"m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℕ\n⊢ (diagonal fun x ↦ ↑a).den = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 34
} | {
"line": 134,
"column": 35
} | [
{
"pp": "m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℕ\n⊢ (↑a).num = ↑a",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Matrix",
"Rat",
"id",
"Matrix.instNatCastOfZero",
"Int",
"Nat.cast",
"Matrix.num",
"instNatCastInt",
... | [
"m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℕ\n⊢ (diagonal fun x ↦ ↑a).num = diagonal fun x ↦ ↑a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 34
} | {
"line": 148,
"column": 35
} | [
{
"pp": "m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℤ\n⊢ (↑a).den = 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Int.cast",
"Matrix.instIntCastOfZero",
"Matrix",
"Rat",
"Rat.instIntCast",
"id",
"instOfNatNat",
"Nat",
... | [
"m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℤ\n⊢ (diagonal fun x ↦ ↑a).den = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 34
} | {
"line": 152,
"column": 35
} | [
{
"pp": "m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℤ\n⊢ (↑a).num = ↑a",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Int.cast",
"Matrix.instIntCastOfZero",
"Matrix",
"Rat",
"Rat.instIntCast",
"id",
"Int",
"Matrix.num",
... | [
"m : Type u_1\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\na : ℤ\n⊢ (diagonal fun x ↦ ↑a).num = diagonal fun x ↦ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 37
} | {
"line": 166,
"column": 38
} | [
{
"pp": "n : Type u_1\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type u_2\ninst✝ : CommRing R\nm k : ℤ\nH : Finset ℤ := Finset.Icc (-|k|) |k|\nH4 : Type := Fin 2 → Fin 2 → ↥H\nM N : ↑(reps k)\nh : (fun M i j ↦ ⟨↑↑M i j, ⋯⟩) M = (fun M i j ↦ ⟨↑↑M i j, ⋯⟩) N\ni j : Fin 2\n⊢ ↑↑M i j = ↑↑N i j",
"ppTerm":... | [
"n : Type u_1\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type u_2\ninst✝ : CommRing R\nm k : ℤ\nH : Finset ℤ := Finset.Icc (-|k|) |k|\nH4 : Type := Fin 2 → Fin 2 → ↥H\nM N : ↑(reps k)\nh : (fun M i j ↦ ⟨↑↑M i j, ⋯⟩) M = (fun M i j ↦ ⟨↑↑M i j, ⋯⟩) N\ni j : Fin 2\n⊢ ↑↑M i j = ↑↑N i j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 180,
"column": 20
} | {
"line": 180,
"column": 59
} | {
"line": 180,
"column": 60
} | [
{
"pp": "case step\nm : ℤ\nhm : m ≠ 0\nA : FixedDetMatrix (Fin 2) ℤ m\nh1 : ↑A 1 0 ≠ 0\nh2 : reduce (reduceStep A) ∈ reps m\n⊢ reduce A ∈ reps m",
"ppTerm": "?step",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case step\nm : ℤ\nhm : m ≠ 0\nA : FixedDetMatrix (Fin 2) ℤ m\nh1 : ↑A 1 0 ≠ 0\nh2 : reduce (reduceStep A) ∈ reps m\n⊢ reduce A ∈ reps m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 180,
"column": 20
} | {
"line": 180,
"column": 62
} | {
"line": 181,
"column": 2
} | [
{
"pp": "case step\nm : ℤ\nhm : m ≠ 0\nA : FixedDetMatrix (Fin 2) ℤ m\nh1 : ↑A 1 0 ≠ 0\nh2 : reduce (reduceStep A) ∈ reps m\n⊢ reduce A ∈ reps m",
"ppTerm": "?step",
"assigned": true,
"usedConstants": [
"FixedDetMatrices.reduce_reduceStep",
"congrArg",
"instDecidableEqFin",
"... | [] | simpa only [reduce_reduceStep h1] using h2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 180,
"column": 20
} | {
"line": 180,
"column": 62
} | {
"line": 181,
"column": 2
} | [
{
"pp": "case step\nm : ℤ\nhm : m ≠ 0\nA : FixedDetMatrix (Fin 2) ℤ m\nh1 : ↑A 1 0 ≠ 0\nh2 : reduce (reduceStep A) ∈ reps m\n⊢ reduce A ∈ reps m",
"ppTerm": "?step",
"assigned": true,
"usedConstants": [
"FixedDetMatrices.reduce_reduceStep",
"congrArg",
"instDecidableEqFin",
"... | [] | simpa only [reduce_reduceStep h1] using h2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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