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.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 598,
"column": 36
} | {
"line": 598,
"column": 47
} | {
"line": 598,
"column": 48
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.IsIrreducible\n... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.IsIrreducible\ni : ι\n⊢ i =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 598,
"column": 33
} | {
"line": 598,
"column": 64
} | {
"line": 600,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.IsIrreducible\n... | [] | by simpa using mem_allRoots P i | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 616,
"column": 50
} | {
"line": 616,
"column": 66
} | {
"line": 616,
"column": 66
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsG2\nb : P.Base\ninst✝² : Finite ι\ninst✝¹ : CharZero R\ninst✝ : IsDomain R\n_i✝ : P.EmbeddedG2\n... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsG2\nb : P.Base\ninst✝² : Finite ι\ninst✝¹ : CharZero R\ninst✝ : IsDomain R\n_i✝ : P.EmbeddedG2\n_i : Nonempt... | Fintype.card_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 288,
"column": 6
} | {
"line": 288,
"column": 65
} | {
"line": 288,
"column": 66
} | [
{
"pp": "case inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.... | [
"case inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j k ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 320,
"column": 4
} | {
"line": 320,
"column": 37
} | {
"line": 320,
"column": 38
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j k l m : ι\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 319,
"column": 4
} | {
"line": 319,
"column": 37
} | {
"line": 320,
"column": 2
} | [
{
"pp": "case h₀\nι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : CharZero K\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : Fintype ι\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module K M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module K N\nP : RootPairing ι K M N\ninst✝² : P.IsCrystallographic\nb : P.... | [] | exact h fun k ↦ by simp [hωu, hU] | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 321,
"column": 10
} | {
"line": 321,
"column": 43
} | {
"line": 321,
"column": 44
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j k l m : ι\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 322,
"column": 10
} | {
"line": 322,
"column": 43
} | {
"line": 322,
"column": 44
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j k l m : ι\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 327,
"column": 59
} | {
"line": 327,
"column": 92
} | {
"line": 327,
"column": 93
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j k l m : ι\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 342,
"column": 40
} | {
"line": 342,
"column": 72
} | {
"line": 342,
"column": 73
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module K M\ninst✝² : AddCommGroup N\ninst✝¹ : Module K N\nP : RootPairing ι K M N\ninst✝ : P.IsCrystallographic\nb : P.Base\nw : ↥... | [
"ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module K M\ninst✝² : AddCommGroup N\ninst✝¹ : Module K N\nP : RootPairing ι K M N\ninst✝ : P.IsCrystallographic\nb : P.Base\nw : ↥b.support ⊕ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SModEq.Pow | {
"line": 27,
"column": 34
} | {
"line": 27,
"column": 45
} | {
"line": 27,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI J : Ideal R\np : ℕ\nhpI : ↑p ∈ I\nx y : R\nh : x - y ∈ J\nhJI : J ≤ I\nh₁ : (Ideal.Quotient.mk I) x = (Ideal.Quotient.mk I) y\n⊢ ↑p = 0",
"ppTerm": "?m.97",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\nI J : Ideal R\np : ℕ\nhpI : ↑p ∈ I\nx y : R\nh : x - y ∈ J\nhJI : J ≤ I\nh₁ : (Ideal.Quotient.mk I) x = (Ideal.Quotient.mk I) y\n⊢ ↑p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 348,
"column": 4
} | {
"line": 349,
"column": 46
} | {
"line": 349,
"column": 47
} | [
{
"pp": "case refine_1\nι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module K M\ninst✝² : AddCommGroup N\ninst✝¹ : Module K N\nP : RootPairing ι K M N\ninst✝ : P.IsCrystallographic\nb ... | [
"case refine_1\nι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module K M\ninst✝² : AddCommGroup N\ninst✝¹ : Module K N\nP : RootPairing ι K M N\ninst✝ : P.IsCrystallographic\nb : P.Base\nw ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 354,
"column": 32
} | {
"line": 354,
"column": 43
} | {
"line": 354,
"column": 44
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module K M\ninst✝² : AddCommGroup N\ninst✝¹ : Module K N\nP : RootPairing ι K M N\ninst✝ : P.IsCrystallographic\nb : P.Base\ni : ↥... | [
"ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module K M\ninst✝² : AddCommGroup N\ninst✝¹ : Module K N\nP : RootPairing ι K M N\ninst✝ : P.IsCrystallographic\nb : P.Base\ni : ↥b.support\nx... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 349,
"column": 42
} | {
"line": 349,
"column": 70
} | {
"line": 349,
"column": 71
} | [
{
"pp": "case e'_5\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P... | [
"case e'_5\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j k... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SesquilinearForm.Star | {
"line": 62,
"column": 2
} | {
"line": 69,
"column": 13
} | {
"line": 70,
"column": 0
} | [
{
"pp": "M : Type u_2\nn : Type u_3\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq n\nR : Type u_4\ninst✝³ : CommRing R\ninst✝² : StarRing R\ninst✝¹ : PartialOrder R\ninst✝ : Module R M\nB : M →ₗ⋆[R] M →ₗ[R] R\nb : Basis n R M\n⊢ B.IsPosSemidef ↔ ((toMatrix₂ b b) B).PosSemidef",
"ppTerm... | [] | rw [isPosSemidef_def, Matrix.posSemidef_iff_dotProduct_mulVec]
apply and_congr (B.isSymm_iff_isHermitian_toMatrix b)
rw [isNonneg_def]
refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩
· rw [star_dotProduct_toMatrix₂_mulVec]
exact h _
· rw [apply_eq_star_dotProduct_toMatrix₂_mulVec b]
exact h _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SesquilinearForm.Star | {
"line": 62,
"column": 2
} | {
"line": 69,
"column": 13
} | {
"line": 70,
"column": 0
} | [
{
"pp": "M : Type u_2\nn : Type u_3\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq n\nR : Type u_4\ninst✝³ : CommRing R\ninst✝² : StarRing R\ninst✝¹ : PartialOrder R\ninst✝ : Module R M\nB : M →ₗ⋆[R] M →ₗ[R] R\nb : Basis n R M\n⊢ B.IsPosSemidef ↔ ((toMatrix₂ b b) B).PosSemidef",
"ppTerm... | [] | rw [isPosSemidef_def, Matrix.posSemidef_iff_dotProduct_mulVec]
apply and_congr (B.isSymm_iff_isHermitian_toMatrix b)
rw [isNonneg_def]
refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩
· rw [star_dotProduct_toMatrix₂_mulVec]
exact h _
· rw [apply_eq_star_dotProduct_toMatrix₂_mulVec b]
exact h _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 375,
"column": 4
} | {
"line": 375,
"column": 70
} | {
"line": 375,
"column": 71
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrysta... | [
"ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrystallographic\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 377,
"column": 70
} | {
"line": 377,
"column": 81
} | {
"line": 377,
"column": 82
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrysta... | [
"ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrystallographic\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 364,
"column": 4
} | {
"line": 364,
"column": 30
} | {
"line": 364,
"column": 31
} | [
{
"pp": "case inr.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb ... | [
"case inr.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 388,
"column": 34
} | {
"line": 388,
"column": 49
} | {
"line": 388,
"column": 50
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrysta... | [
"ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrystallographic\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 369,
"column": 37
} | {
"line": 369,
"column": 48
} | {
"line": 369,
"column": 49
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j k l m : ι\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 371,
"column": 4
} | {
"line": 371,
"column": 15
} | {
"line": 371,
"column": 16
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j k l m : ι\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 71,
"column": 11
} | {
"line": 71,
"column": 20
} | {
"line": 72,
"column": 2
} | [
{
"pp": "case ι\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nmotive : SymmetricAlgebra R M → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R (SymmetricAlgebra R M)) r)\nι : ∀ (x : M), motive ((SymmetricAlgebra.ι R M) x)\nmul : ∀ (a b : Symmetric... | [] | exact ι x | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 71,
"column": 11
} | {
"line": 71,
"column": 20
} | {
"line": 72,
"column": 2
} | [
{
"pp": "case ι\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nmotive : SymmetricAlgebra R M → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R (SymmetricAlgebra R M)) r)\nι : ∀ (x : M), motive ((SymmetricAlgebra.ι R M) x)\nmul : ∀ (a b : Symmetric... | [] | exact ι x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 71,
"column": 11
} | {
"line": 71,
"column": 20
} | {
"line": 72,
"column": 2
} | [
{
"pp": "case ι\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nmotive : SymmetricAlgebra R M → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R (SymmetricAlgebra R M)) r)\nι : ∀ (x : M), motive ((SymmetricAlgebra.ι R M) x)\nmul : ∀ (a b : Symmetric... | [] | exact ι x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorAlgebra.Grading | {
"line": 32,
"column": 36
} | {
"line": 32,
"column": 62
} | {
"line": 32,
"column": 63
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\n⊢ (TensorAlgebra.ι R) m ∈ (TensorAlgebra.ι R).range ^ 1",
"ppTerm": "?m.126",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.ids",
... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\n⊢ (TensorAlgebra.ι R) m ∈ (TensorAlgebra.ι R).range"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorAlgebra.Grading | {
"line": 37,
"column": 33
} | {
"line": 37,
"column": 59
} | {
"line": 37,
"column": 60
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\n⊢ (TensorAlgebra.ι R) m ∈ (TensorAlgebra.ι R).range ^ 1",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.ids",
... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\n⊢ (TensorAlgebra.ι R) m ∈ (TensorAlgebra.ι R).range"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 22
} | {
"line": 144,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : R\n⊢ algebraMapInv ((algebraMap R (SymmetricAlgebra R M)) x) = x",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Equiv.instEquivLike",
"Te... | [] | simp [algebraMapInv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 22
} | {
"line": 144,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : R\n⊢ algebraMapInv ((algebraMap R (SymmetricAlgebra R M)) x) = x",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Equiv.instEquivLike",
"Te... | [] | simp [algebraMapInv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 22
} | {
"line": 144,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : R\n⊢ algebraMapInv ((algebraMap R (SymmetricAlgebra R M)) x) = x",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Equiv.instEquivLike",
"Te... | [] | simp [algebraMapInv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 13
} | {
"line": 207,
"column": 14
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\ne : SymmetricAlgebra R M ≃ₐ[R] A\nhe : ↑↑e ∘ₗ SymmetricAlgebra.ι R M = f\nx : M\n⊢ (↑↑e ∘ₗ SymmetricAlgebra.ι R M) x = (↑(Symmet... | [
"R : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\ne : SymmetricAlgebra R M ≃ₐ[R] A\nhe : ↑↑e ∘ₗ SymmetricAlgebra.ι R M = f\nx : M\n⊢ e ((SymmetricAlgebra.ι R M) x) = f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 13
} | {
"line": 234,
"column": 14
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nA : Type u_3\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nf : M →ₗ[R] A\nA' : Type u_4\ninst✝¹ : CommSemiring A'\ninst✝ : Algebra R A'\nh : IsSymmetricAlgebra f\nF G : A →ₐ[R] A'\nhFG : ↑F ∘ₗ f = ↑G ∘... | [
"R : Type u_1\nM : Type u_2\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nA : Type u_3\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nf : M →ₗ[R] A\nA' : Type u_4\ninst✝¹ : CommSemiring A'\ninst✝ : Algebra R A'\nh : IsSymmetricAlgebra f\nF G : A →ₐ[R] A'\nhFG : ↑F ∘ₗ f = ↑G ∘ₗ f\nx : M\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 253,
"column": 20
} | {
"line": 253,
"column": 31
} | {
"line": 253,
"column": 32
} | [
{
"pp": "case algebraMap\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nh : IsSymmetricAlgebra f\nmotive : A → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R A) r)\nι : ∀... | [
"case algebraMap\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nh : IsSymmetricAlgebra f\nmotive : A → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R A) r)\nι : ∀ (x : M), mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 254,
"column": 11
} | {
"line": 254,
"column": 22
} | {
"line": 254,
"column": 23
} | [
{
"pp": "case ι\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nh : IsSymmetricAlgebra f\nmotive : A → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R A) r)\nι : ∀ (x : M),... | [
"case ι\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nh : IsSymmetricAlgebra f\nmotive : A → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R A) r)\nι : ∀ (x : M), motive (f x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 255,
"column": 21
} | {
"line": 255,
"column": 32
} | {
"line": 255,
"column": 33
} | [
{
"pp": "case mul\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nh : IsSymmetricAlgebra f\nmotive : A → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R A) r)\nι : ∀ (x : M... | [
"case mul\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nh : IsSymmetricAlgebra f\nmotive : A → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R A) r)\nι : ∀ (x : M), motive (f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 256,
"column": 21
} | {
"line": 256,
"column": 32
} | {
"line": 256,
"column": 33
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nh : IsSymmetricAlgebra f\nmotive : A → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R A) r)\nι : ∀ (x : M... | [
"case add\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nh : IsSymmetricAlgebra f\nmotive : A → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R A) r)\nι : ∀ (x : M), motive (f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Logic.Godel.GodelBetaFunction | {
"line": 55,
"column": 6
} | {
"line": 55,
"column": 66
} | {
"line": 56,
"column": 8
} | [
{
"pp": "n m a : ℕ\nha : m - n ∣ a\np : ℕ\npp : Prime p\nhn : p ∣ n * a + 1\nhm : p ∣ m * a + 1\n⊢ p ∣ (m - n) * a",
"ppTerm": "?m.50",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n m a : ℕ\nha : m - n ∣ a\np : ℕ\npp : Prime p\nhn : p ∣ n * a + 1\nhm : p ∣ m * a + 1\n⊢ p ∣ (m - n) * a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Logic.Godel.GodelBetaFunction | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 41
} | {
"line": 62,
"column": 42
} | [
{
"pp": "n m a : ℕ\nha : m - n ∣ a\np : ℕ\npp : Prime p\nhn : p ∣ n * a + 1\nhm : p ∣ m * a + 1\nthis✝ : p ∣ (m - n) * a\nthis : p ∣ a\n⊢ p = 1",
"ppTerm": "?m.93",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n m a : ℕ\nha : m - n ∣ a\np : ℕ\npp : Prime p\nhn : p ∣ n * a + 1\nhm : p ∣ m * a + 1\nthis✝ : p ∣ (m - n) * a\nthis : p ∣ a\n⊢ p = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Logic.Godel.GodelBetaFunction | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 29
} | {
"line": 80,
"column": 30
} | [
{
"pp": "m : ℕ\na : Fin m → ℕ\ni : Fin m\nh₁ : a i < supOfSeq a\nh₂ : supOfSeq a ≤ (↑i + 1) * (supOfSeq a)! + 1\n⊢ a i < coprimes a i",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"id",
"Nat",
"LT.lt",
"_private.Mathlib.Logic.Godel.GodelBetaFunction.0.Nat.coprimes... | [
"m : ℕ\na : Fin m → ℕ\ni : Fin m\nh₁ : a i < supOfSeq a\nh₂ : supOfSeq a ≤ (↑i + 1) * (supOfSeq a)! + 1\n⊢ a i < (↑i + 1) * (supOfSeq a)! + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Logic.Godel.GodelBetaFunction | {
"line": 91,
"column": 10
} | {
"line": 91,
"column": 46
} | {
"line": 91,
"column": 47
} | [
{
"pp": "m✝ m : ℕ\na : Fin m → ℕ\ni j : Fin m\nhij : i ≠ j\nltij : i < j\nhja : ↑j < supOfSeq a\n⊢ ↑j + 1 - (↑i + 1) ≤ supOfSeq a",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"HSub.hSub",
"id",
"instSubNat",
"instOfNatNat",
... | [
"m✝ m : ℕ\na : Fin m → ℕ\ni j : Fin m\nhij : i ≠ j\nltij : i < j\nhja : ↑j < supOfSeq a\n⊢ ↑j - ↑i ≤ supOfSeq a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Logic.Godel.GodelBetaFunction | {
"line": 104,
"column": 8
} | {
"line": 104,
"column": 19
} | {
"line": 104,
"column": 20
} | [
{
"pp": "m : ℕ\nl : List ℕ\n⊢ (↑Finset.univ).Pairwise (Function.onFun Coprime (coprimes fun x ↦ l[x]))",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Coprime",
"Finset.univ",
"Finset.coe_univ",
"Function.onFun",
"congrArg",
"Finset"... | [
"m : ℕ\nl : List ℕ\n⊢ Set.univ.Pairwise (Function.onFun Coprime (coprimes fun x ↦ l[↑x]))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Logic.Godel.GodelBetaFunction | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 38
} | {
"line": 109,
"column": 39
} | [
{
"pp": "l : List ℕ\ni : Fin l.length\n⊢ (unbeta l).beta ↑i = l[i]",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Finset.univ",
"congrArg",
"Nat.unpair",
"Finset",
"Nat.beta",
"Nat.unbeta",
"Membership.mem",
... | [
"l : List ℕ\ni : Fin l.length\n⊢ ↑(chineseRemainderOfFinset (fun x ↦ l[↑x]) (coprimes fun x ↦ l[↑x]) Finset.univ ⋯ ⋯) %\n ((↑i + 1) * (supOfSeq fun x ↦ l[↑x])! + 1) =\n l[↑i]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Logic.Godel.GodelBetaFunction | {
"line": 112,
"column": 10
} | {
"line": 112,
"column": 21
} | {
"line": 112,
"column": 22
} | [
{
"pp": "l : List ℕ\ni : Fin l.length\n⊢ (↑Finset.univ).Pairwise (Function.onFun Coprime (coprimes fun x ↦ l[x]))",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Coprime",
"Finset.univ",
"Finset.coe_univ",
"Function.onFun",
"congrArg",
... | [
"l : List ℕ\ni : Fin l.length\n⊢ Set.univ.Pairwise (Function.onFun Coprime (coprimes fun x ↦ l[↑x]))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Logic.Hydra | {
"line": 69,
"column": 4
} | {
"line": 70,
"column": 11
} | {
"line": 70,
"column": 12
} | [
{
"pp": "case refine_1\nα : Type u_1\nr : α → α → Prop\ninst✝¹ : DecidableEq α\ninst✝ : Std.Irrefl r\ns t u : Multiset α\na : α\nhr : ∀ (a' : α), ¬r a' a → a' ∉ u\nb : α\nh : (rᶜ ⊓ fun x1 x2 ↦ x1 ≠ x2) b a\nhe : count b (s + {a}) = count b (t + u)\n⊢ count b s = count b t",
"ppTerm": "?refine_1",
"assig... | [
"case refine_1\nα : Type u_1\nr : α → α → Prop\ninst✝¹ : DecidableEq α\ninst✝ : Std.Irrefl r\ns t u : Multiset α\na : α\nhr : ∀ (a' : α), ¬r a' a → a' ∉ u\nb : α\nh : (rᶜ ⊓ fun x1 x2 ↦ x1 ≠ x2) b a\nhe : count b (s + {a}) = count b (t + u)\n⊢ count b s = count b t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Logic.Hydra | {
"line": 80,
"column": 35
} | {
"line": 80,
"column": 58
} | {
"line": 82,
"column": 0
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\nx' x : α\nh✝ : r x' x\na : α\nh : a ∈ {x'}\n⊢ r a x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"Multiset",
"id",
"Multiset.instSingleton",
"Multiset.instMembershi... | [] | rwa [mem_singleton.1 h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Logic.Hydra | {
"line": 80,
"column": 35
} | {
"line": 80,
"column": 58
} | {
"line": 82,
"column": 0
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\nx' x : α\nh✝ : r x' x\na : α\nh : a ∈ {x'}\n⊢ r a x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"Multiset",
"id",
"Multiset.instSingleton",
"Multiset.instMembershi... | [] | rwa [mem_singleton.1 h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Hydra | {
"line": 80,
"column": 35
} | {
"line": 80,
"column": 58
} | {
"line": 82,
"column": 0
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\nx' x : α\nh✝ : r x' x\na : α\nh : a ∈ {x'}\n⊢ r a x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"Multiset",
"id",
"Multiset.instSingleton",
"Multiset.instMembershi... | [] | rwa [mem_singleton.1 h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Hydra | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 20
} | {
"line": 110,
"column": 2
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ninst✝ : Std.Irrefl r\ns : Multiset α\n⊢ ¬CutExpand r s 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Classical.propDecidable",
"Membership.mem",
"Exists",
"Relation.CutExpand",
"Mul... | [
"α : Type u_1\nr : α → α → Prop\ninst✝ : Std.Irrefl r\ns : Multiset α\n⊢ ¬∃ t a, (∀ a' ∈ t, r a' a) ∧ a ∈ 0 ∧ s = erase 0 a + t"
] | rw [cutExpand_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Transvection.Generation | {
"line": 109,
"column": 14
} | {
"line": 109,
"column": 49
} | {
"line": 110,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne f : V ≃ₗ[K] V\nhf : f ∈ dilatransvections K V\n| 1 + finrank K ↥(↑e).fixedSubmodule",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"mul_inv_cancel... | [
"K : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne f : V ≃ₗ[K] V\nhf : f ∈ dilatransvections K V\n| 1 + finrank K ↥(↑(e * f * f⁻¹)).fixedSubmodule"
] | rw [show e = (e * f) * f⁻¹ by simp] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.LinearAlgebra.Transvection.Generation | {
"line": 109,
"column": 14
} | {
"line": 109,
"column": 49
} | {
"line": 110,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne f : V ≃ₗ[K] V\nhf : f ∈ dilatransvections K V\n| 1 + finrank K ↥(↑e).fixedSubmodule",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"mul_inv_cancel... | [
"K : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne f : V ≃ₗ[K] V\nhf : f ∈ dilatransvections K V\n| 1 + finrank K ↥(↑(e * f * f⁻¹)).fixedSubmodule"
] | rw [show e = (e * f) * f⁻¹ by simp] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.LinearAlgebra.Transvection.Generation | {
"line": 109,
"column": 14
} | {
"line": 109,
"column": 49
} | {
"line": 110,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne f : V ≃ₗ[K] V\nhf : f ∈ dilatransvections K V\n| 1 + finrank K ↥(↑e).fixedSubmodule",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"mul_inv_cancel... | [
"K : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne f : V ≃ₗ[K] V\nhf : f ∈ dilatransvections K V\n| 1 + finrank K ↥(↑(e * f * f⁻¹)).fixedSubmodule"
] | rw [show e = (e * f) * f⁻¹ by simp] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Logic.Hydra | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 20
} | {
"line": 136,
"column": 2
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ninst✝ : Std.Irrefl r\np : α → Prop\nh : ∀ {a' a : α}, r a' a → p a → p a'\ns' s : Multiset α\n⊢ CutExpand r s' s → (∀ a ∈ s, p a) → ∀ a ∈ s', p a",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Classical.prop... | [
"α : Type u_1\nr : α → α → Prop\ninst✝ : Std.Irrefl r\np : α → Prop\nh : ∀ {a' a : α}, r a' a → p a → p a'\ns' s : Multiset α\n⊢ (∃ t a, (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t) → (∀ a ∈ s, p a) → ∀ a ∈ s', p a"
] | rw [cutExpand_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Logic.Hydra | {
"line": 163,
"column": 8
} | {
"line": 163,
"column": 23
} | {
"line": 163,
"column": 23
} | [
{
"pp": "case cons\nα : Type u_1\nr : α → α → Prop\ninst✝ : Std.Irrefl r\na : α\ns : Multiset α\nihs : (∀ a ∈ s, Acc (CutExpand r) {a}) → Acc (CutExpand r) s\nhs : ∀ a_1 ∈ a ::ₘ s, Acc (CutExpand r) {a_1}\n⊢ Acc (CutExpand r) ({a} + s)",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
... | [
"case cons\nα : Type u_1\nr : α → α → Prop\ninst✝ : Std.Irrefl r\na : α\ns : Multiset α\nihs : (∀ a ∈ s, Acc (CutExpand r) {a}) → Acc (CutExpand r) s\nhs : Acc (CutExpand r) {a} ∧ ∀ x ∈ s, Acc (CutExpand r) {x}\n⊢ Acc (CutExpand r) ({a} + s)"
] | forall_mem_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 49
} | {
"line": 85,
"column": 2
} | [
{
"pp": "ι : Type u_2\nα : ι → Type u_1\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), IsPiSystem (C i)\nhC_univ : ∀ (i : ι), univ ∈ C i\ns₁ : Finset ι\nt₁ : (i : ι) → Set (α i)\nh₁ : t₁ ∈ univ.pi C\ns₂ : Finset ι\nt₂ : (i : ι) → Set (α i)\nh₂ : t₂ ∈ univ.pi C\nhst_nonempty : ((↑s₁).pi t₁ ∩ (↑s₂).pi t₂).Nonemp... | [
"case refine_1\nι : Type u_2\nα : ι → Type u_1\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), IsPiSystem (C i)\nhC_univ : ∀ (i : ι), univ ∈ C i\ns₁ : Finset ι\nt₁ : (i : ι) → Set (α i)\nh₁ : t₁ ∈ univ.pi C\ns₂ : Finset ι\nt₂ : (i : ι) → Set (α i)\nh₂ : t₂ ∈ univ.pi C\nhst_nonempty : ((↑s₁).pi t₁ ∩ (↑s₂).pi t₂).Non... | refine ⟨s₁ ∪ s₂, fun i ↦ t₁' i ∩ t₂' i, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Constructions.Projective | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 35
} | {
"line": 58,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\nP : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)\nh : IsEmpty ((i : ι) → α i)\nhP : IsProjectiveMeasureFamily P\nI : Finset ι\n⊢ P I = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"MeasureTheory.Meas... | [
"ι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\nP : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)\nh : IsEmpty ((i : ι) → α i)\nhP : IsProjectiveMeasureFamily P\nI : Finset ι\ni : ι\nhi : IsEmpty (α i)\n⊢ P I = 0"
] | obtain ⟨i, hi⟩ := isEmpty_pi.mp h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Constructions.Projective | {
"line": 85,
"column": 4
} | {
"line": 87,
"column": 17
} | {
"line": 88,
"column": 4
} | [
{
"pp": "case inl\nι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\nP : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)\nI J : Finset ι\nhP : IsProjectiveMeasureFamily P\nS : Set ((i : ↥I) → α ↑i)\nT : Set ((i : ↥J) → α ↑i)\nhT : MeasurableSet T\nh_eq : cylinder I S = cylinder J T\nhJI : J ⊆ ... | [
"case inl\nι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\nP : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)\nI J : Finset ι\nhP : IsProjectiveMeasureFamily P\nS : Set ((i : ↥I) → α ↑i)\nT : Set ((i : ↥J) → α ↑i)\nhT : MeasurableSet T\nh_eq : cylinder I S = cylinder J T\nhJI : J ⊆ I\nh : IsEmp... | suffices ∀ I, P I univ = 0 by
simp only [Measure.measure_univ_eq_zero] at this
simp [this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.MeasureTheory.Constructions.Projective | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 15
} | {
"line": 88,
"column": 16
} | [
{
"pp": "case inl\nι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\nP : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)\nI J : Finset ι\nhP : IsProjectiveMeasureFamily P\nS : Set ((i : ↥I) → α ↑i)\nT : Set ((i : ↥J) → α ↑i)\nhT : MeasurableSet T\nh_eq : cylinder I S = cylinder J T\nhJI : J ⊆ ... | [
"case inl\nι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\nP : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)\nI J : Finset ι\nhP : IsProjectiveMeasureFamily P\nS : Set ((i : ↥I) → α ↑i)\nT : Set ((i : ↥J) → α ↑i)\nhT : MeasurableSet T\nh_eq : cylinder I S = cylinder J T\nhJI : J ⊆ I\nh : IsEmp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Transvection.Generation | {
"line": 259,
"column": 8
} | {
"line": 259,
"column": 24
} | {
"line": 259,
"column": 25
} | [
{
"pp": "K : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nn✝ : ℕ\ne : V ≃ₗ[K] V\nhe : e.fixedReduce = 1\nn : ℕ\nh : finrank K (V ⧸ (↑e).fixedSubmodule) = n + 2\nhind :\n ∀ {e : V ≃ₗ[K] V},\n e.fixedReduce = 1 → finrank K (V ⧸ (↑e).... | [
"K : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nn✝ : ℕ\ne : V ≃ₗ[K] V\nhe : e.fixedReduce = 1\nn : ℕ\nh : finrank K (V ⧸ (↑e).fixedSubmodule) = n + 2\nhind :\n ∀ {e : V ≃ₗ[K] V},\n e.fixedReduce = 1 → finrank K (V ⧸ (↑e).fixedSubmodu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.SetSemiring | {
"line": 106,
"column": 17
} | {
"line": 106,
"column": 28
} | {
"line": 106,
"column": 29
} | [
{
"pp": "case singleton\nα : Type u_1\nC : Set (Set α)\nι : Type u_2\nhC : IsSetRing C\ns : ι → Set α\nS : Finset ι\na✝ : ι\nhs : ∀ n ∈ {a✝}, s n ∈ C\n⊢ ⋂ i ∈ {a✝}, s i ∈ C",
"ppTerm": "?singleton",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Iff.of_eq",
"congrArg",
"Set.... | [
"case singleton\nα : Type u_1\nC : Set (Set α)\nι : Type u_2\nhC : IsSetRing C\ns : ι → Set α\nS : Finset ι\na✝ : ι\nhs : ∀ n ∈ {a✝}, s n ∈ C\n⊢ s a✝ ∈ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.SetSemiring | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 13
} | {
"line": 116,
"column": 14
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nhC : IsSetRing C\nι : Type u_2\ns : ι → Set α\nt : Finset ι\nhs : ∀ i ∈ t, s i ∈ C\n⊢ t.sup s ∈ C",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"CompleteBooleanAlgebra.toCompleteDistribLattice... | [
"α : Type u_1\nC : Set (Set α)\nhC : IsSetRing C\nι : Type u_2\ns : ι → Set α\nt : Finset ι\nhs : ∀ i ∈ t, s i ∈ C\n⊢ ⋃ x ∈ t, s x ∈ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.SetSemiring | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 51
} | {
"line": 121,
"column": 52
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nι : Type u_2\ninst✝¹ : Preorder ι\ninst✝ : LocallyFiniteOrderBot ι\nhC : IsSetRing C\ns : ι → Set α\nhs : ∀ (n : ι), s n ∈ C\nn : ι\n⊢ (partialSups s) n ∈ C",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeS... | [
"α : Type u_1\nC : Set (Set α)\nι : Type u_2\ninst✝¹ : Preorder ι\ninst✝ : LocallyFiniteOrderBot ι\nhC : IsSetRing C\ns : ι → Set α\nhs : ∀ (n : ι), s n ∈ C\nn : ι\n⊢ (Finset.Iic n).sup s ∈ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 373,
"column": 4
} | {
"line": 373,
"column": 19
} | {
"line": 374,
"column": 4
} | [
{
"pp": "case a\nι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\ni : ι\nx : Set ((i : ι) → α i)\n⊢ ∀ (x_1 : (i : ι) → Set (α i)),\n (∀ (i : ι), MeasurableSet (x_1 i)) → eval i ⁻¹' x_1 i = x → ∃ s S, MeasurableSet S ∧ x = cylinder s S",
"ppTerm": "?a✝",
"assigned": true,
"... | [
"case a\nι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\ni : ι\nt : (i : ι) → Set (α i)\nht : ∀ (i : ι), MeasurableSet (t i)\n⊢ ∃ s S, MeasurableSet S ∧ eval i ⁻¹' t i = cylinder s S"
] | rintro t ht rfl | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 414,
"column": 2
} | {
"line": 414,
"column": 13
} | {
"line": 414,
"column": 14
} | [
{
"pp": "ι : Type u_2\nX : ι → Type u_3\nm : (i : ι) → MeasurableSpace (X i)\nΔ : Set ι\n⊢ cylinderEvents Δ ≤ MeasurableSpace.pi",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_2\nX : ι → Type u_3\nm : (i : ι) → MeasurableSpace (X i)\nΔ : Set ι\n⊢ cylinderEvents Δ ≤ MeasurableSpace.pi"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Transvection.Generation | {
"line": 341,
"column": 2
} | {
"line": 341,
"column": 74
} | {
"line": 341,
"column": 75
} | [
{
"pp": "K : Type u_1\ninst✝² : DivisionRing K\nV : Type u_2\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ne : V ≃ₗ[K] V\nhe : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x\nh : 1 < finrank K (V ⧸ (↑e).fixedSubmodule)\nu : V\nf : Dual K V\nhu : LinearIndependent K ![(f u)⁻¹ • (↑e).fixedSubmodule.mkQ u, (f u)⁻¹ • e.fi... | [
"K : Type u_1\ninst✝² : DivisionRing K\nV : Type u_2\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ne : V ≃ₗ[K] V\nhe : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x\nh : 1 < finrank K (V ⧸ (↑e).fixedSubmodule)\nu : V\nf : Dual K V\nhu : LinearIndependent K ![(f u)⁻¹ • (↑e).fixedSubmodule.mkQ u, (f u)⁻¹ • e.fixedReduce ((... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 57
} | {
"line": 126,
"column": 58
} | [
{
"pp": "case convert_2\nα : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝¹ : AddCommMonoid G\nm : AddContent G C\nι : Type u_3\ninst✝ : Fintype ι\nf : ι → Set α\nhf : ∀ (i : ι), f i ∈ C\nh_dis : Pairwise (Disjoint on f)\nh_mem : ⋃ i, f i ∈ C\n⊢ (↑Finset.univ).PairwiseDisjoint f",
"ppTerm": "?convert_2",
... | [
"case convert_2\nα : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝¹ : AddCommMonoid G\nm : AddContent G C\nι : Type u_3\ninst✝ : Fintype ι\nf : ι → Set α\nhf : ∀ (i : ι), f i ∈ C\nh_dis : Pairwise (Disjoint on f)\nh_mem : ⋃ i, f i ∈ C\n⊢ Pairwise (Disjoint on f)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.WithTop | {
"line": 111,
"column": 12
} | {
"line": 111,
"column": 23
} | {
"line": 111,
"column": 24
} | [
{
"pp": "case a.inl.coe.top.inr.refine_1\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c... | [
"case a.inl.coe.top.inr.refine_1\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Transvection.Generation | {
"line": 385,
"column": 6
} | {
"line": 385,
"column": 22
} | {
"line": 385,
"column": 23
} | [
{
"pp": "K : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nf g : Dual K V\nv : V\na b : K\nhv : ∀ (s t : K), s • (↑e).fixedSubmodule.mkQ v + t • e.fixedReduce ((↑e).fixedSubmodule.mkQ v) = 0 → s = 0 ∧ t = 0\nhf : (↑e).fix... | [
"K : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nf g : Dual K V\nv : V\na b : K\nhv : ∀ (s t : K), s • (↑e).fixedSubmodule.mkQ v + t • e.fixedReduce ((↑e).fixedSubmodule.mkQ v) = 0 → s = 0 ∧ t = 0\nhf : (↑e).fixedSubmodule ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.WithTop | {
"line": 120,
"column": 62
} | {
"line": 120,
"column": 73
} | {
"line": 120,
"column": 74
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι := if h : ∃ x, Ioi x = ... | [
"ι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι := if h : ∃ x, Ioi x = ∅ then h.cho... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.WithTop | {
"line": 127,
"column": 12
} | {
"line": 127,
"column": 23
} | {
"line": 127,
"column": 24
} | [
{
"pp": "case a.inl.coe.coe.refine_1.coe\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c... | [
"case a.inl.coe.coe.refine_1.coe\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.SetAlgebra | {
"line": 229,
"column": 6
} | {
"line": 229,
"column": 17
} | {
"line": 229,
"column": 18
} | [
{
"pp": "α : Type u_1\n𝒜 : Set (Set α)\nh : 𝒜.Countable\nℬ : Set (Set α) := {s | s ∈ 𝒜} ∪ {s | sᶜ ∈ 𝒜}\ns : Set α\n⊢ s ∈ compl '' 𝒜 ↔ s ∈ {s | sᶜ ∈ 𝒜}",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.ofPred",
"Compl.compl",
"Set... | [
"α : Type u_1\n𝒜 : Set (Set α)\nh : 𝒜.Countable\nℬ : Set (Set α) := {s | s ∈ 𝒜} ∪ {s | sᶜ ∈ 𝒜}\ns : Set α\n⊢ (∃ x ∈ 𝒜, xᶜ = s) ↔ sᶜ ∈ 𝒜"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 253,
"column": 35
} | {
"line": 253,
"column": 50
} | {
"line": 253,
"column": 51
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\ns t : Set α\nI✝ : Finset (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm✝ m' m : AddContent G C\nhC : IsSetSemiring C\nI : Finset (Set α)\nhI : ↑I ⊆ _root_.supClosure C\nh'I : (↑I).PairwiseDisjoint id\nhh'I : ⋃₀ ↑I ∈ _root_.supClosure C\nJ : (s : Set α) → Finpartition s... | [
"α : Type u_1\nC : Set (Set α)\ns t : Set α\nI✝ : Finset (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm✝ m' m : AddContent G C\nhC : IsSetSemiring C\nI : Finset (Set α)\nhI : ↑I ⊆ _root_.supClosure C\nh'I : (↑I).PairwiseDisjoint id\nhh'I : ⋃₀ ↑I ∈ _root_.supClosure C\nJ : (s : Set α) → Finpartition s\nhJC : ∀ s ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 293,
"column": 4
} | {
"line": 293,
"column": 36
} | {
"line": 293,
"column": 37
} | [
{
"pp": "case refine_4\nα : Type u_1\nC : Set (Set α)\ns t : Set α\nG : Type u_2\ninst✝² : AddCommMonoid G\nm : AddContent G C\ninst✝¹ : PartialOrder G\ninst✝ : CanonicallyOrderedAdd G\nhC : IsSetSemiring C\nhs : s ∈ C\nht : t ∈ C\nhst : s ⊆ t\nh : ∑ u ∈ {s}, m u ≤ m t\n⊢ m s ≤ m t",
"ppTerm": "?refine_4",
... | [
"case refine_4\nα : Type u_1\nC : Set (Set α)\ns t : Set α\nG : Type u_2\ninst✝² : AddCommMonoid G\nm : AddContent G C\ninst✝¹ : PartialOrder G\ninst✝ : CanonicallyOrderedAdd G\nhC : IsSetSemiring C\nhs : s ∈ C\nht : t ∈ C\nhst : s ⊆ t\nh : ∑ u ∈ {s}, m u ≤ m t\n⊢ m s ≤ m t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.WithTop | {
"line": 160,
"column": 72
} | {
"line": 160,
"column": 83
} | {
"line": 160,
"column": 84
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι := if h : ∃ x, Ioi x = ... | [
"ι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι := if h : ∃ x, Ioi x = ∅ then h.cho... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 76,
"column": 60
} | {
"line": 76,
"column": 75
} | {
"line": 76,
"column": 76
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nhs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace... | [
"α : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nhs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologic... | iInf_eq_iInter, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.WithTop | {
"line": 167,
"column": 12
} | {
"line": 167,
"column": 23
} | {
"line": 167,
"column": 24
} | [
{
"pp": "case a.inr.coe.coe.refine_1.coe\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c... | [
"case a.inr.coe.coe.refine_1.coe\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Transvection.Generation | {
"line": 409,
"column": 4
} | {
"line": 409,
"column": 48
} | {
"line": 409,
"column": 49
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nf g : Dual K V\nv : V\na b : K\nhv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]\nhf : (↑e).fixedSubmodu... | [
"case inr\nK : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nf g : Dual K V\nv : V\na b : K\nhv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]\nhf : (↑e).fixedSubmodule ⊔ K ∙ (e ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 423,
"column": 48
} | {
"line": 423,
"column": 59
} | {
"line": 423,
"column": 60
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\ns t : Set α\nI✝ : Finset (Set α)\nG✝ : Type u_2\ninst✝² : AddCommMonoid G✝\nm m' : AddContent G✝ C\ninst✝¹ : LinearOrder α\nG : Type u_3\ninst✝ : AddCommGroup G\nf : α → G\nn : ℕ\nih :\n ∀ (I : Finset (Set α)),\n ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →\n (↑I)... | [
"α : Type u_1\nC : Set (Set α)\ns t : Set α\nI✝ : Finset (Set α)\nG✝ : Type u_2\ninst✝² : AddCommMonoid G✝\nm m' : AddContent G✝ C\ninst✝¹ : LinearOrder α\nG : Type u_3\ninst✝ : AddCommGroup G\nf : α → G\nn : ℕ\nih :\n ∀ (I : Finset (Set α)),\n ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →\n (↑I).PairwiseDis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Transvection.Generation | {
"line": 437,
"column": 8
} | {
"line": 437,
"column": 51
} | {
"line": 437,
"column": 52
} | [
{
"pp": "K : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nn✝ : ℕ\ne : V ≃ₗ[K] V\nhe : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x\nn : ℕ\nhind :\n ∀ {e : V ≃ₗ[K] V},\n (∀ (a : K), ∃ x, ¬e.fixedReduce x = a • x) →\n finrank K (V ⧸ ... | [
"K : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nn✝ : ℕ\ne : V ≃ₗ[K] V\nhe : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x\nn : ℕ\nhind :\n ∀ {e : V ≃ₗ[K] V},\n (∀ (a : K), ∃ x, ¬e.fixedReduce x = a • x) →\n finrank K (V ⧸ (↑e).fixedSu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 102,
"column": 6
} | {
"line": 102,
"column": 17
} | {
"line": 102,
"column": 18
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nhs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUn... | [
"case inr\nα : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nhs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 47
} | {
"line": 113,
"column": 8
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nhs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace... | [
"α : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nhs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologic... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.WithTop | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 15
} | {
"line": 281,
"column": 16
} | [
{
"pp": "case inl\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nα : Type u_2\nf : Filter α\nx : α → WithTop ι\nh : IsEmpty ι\n⊢ Tendsto x f (𝓝 ⊤) ↔ ∀ (i : ι), ∀ᶠ (a : α) in f, ↑i < x a",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Pur... | [
"case inl\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nα : Type u_2\nf : Filter α\nx : α → WithTop ι\nh : IsEmpty ι\n⊢ ∀ᶠ (x_1 : α) in f, x x_1 = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.WithTop | {
"line": 289,
"column": 4
} | {
"line": 289,
"column": 15
} | {
"line": 289,
"column": 16
} | [
{
"pp": "case inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : OrderTopology ι\ninst✝ : NoMaxOrder ι\nh : IsEmpty ι\n⊢ Tendsto some atTop (𝓝 ⊤)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"False",
"WithTop.i... | [
"case inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : OrderTopology ι\ninst✝ : NoMaxOrder ι\nh : IsEmpty ι\n⊢ atTop = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 601,
"column": 2
} | {
"line": 608,
"column": 34
} | {
"line": 609,
"column": 2
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nhC : IsSetRing C\nm : AddContent ℝ≥0∞ C\nhm_ne_top : ∀ s ∈ C, m s ≠ ∞\nhm_tendsto : ∀ ⦃s : ℕ → Set α⦄, (∀ (n : ℕ), s n ∈ C) → Antitone s → ⋂ n, s n = ∅ → Tendsto (fun n ↦ m (s n)) atTop (𝓝 0)\nf : ℕ → Set α\nhf : ∀ (i : ℕ), f i ∈ C\nhUf : ⋃ i, f i ∈ C\nh_disj : Pairwise ... | [
"α : Type u_1\nC : Set (Set α)\nhC : IsSetRing C\nm : AddContent ℝ≥0∞ C\nhm_ne_top : ∀ s ∈ C, m s ≠ ∞\nhm_tendsto : ∀ ⦃s : ℕ → Set α⦄, (∀ (n : ℕ), s n ∈ C) → Antitone s → ⋂ n, s n = ∅ → Tendsto (fun n ↦ m (s n)) atTop (𝓝 0)\nf : ℕ → Set α\nhf : ∀ (i : ℕ), f i ∈ C\nhUf : ⋃ i, f i ∈ C\nh_disj : Pairwise (Disjoint on... | have h_tendsto : Tendsto (fun n ↦ m (s n)) atTop (𝓝 0) := by
refine hm_tendsto hCs ?_ ?_
· intro i j hij x hxj
rw [Set.mem_sdiff] at hxj ⊢
exact ⟨hxj.1, fun hxi ↦ hxj.2 (Set.monotone_accumulate hij hxi)⟩
· simp_rw [s, Set.sdiff_eq]
rw [Set.iInter_inter_distrib, Set.iInter_const, ← Set.com... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Transvection.Generation | {
"line": 475,
"column": 6
} | {
"line": 477,
"column": 69
} | {
"line": 479,
"column": 6
} | [
{
"pp": "K : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nn✝ : ℕ\ne : V ≃ₗ[K] V\nhe : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x\nn : ℕ\nhind :\n ∀ {e : V ≃ₗ[K] V},\n (∀ (a : K), ∃ x, ¬e.fixedReduce x = a • x) →\n finrank K (V ⧸ ... | [
"K : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nn✝ : ℕ\ne : V ≃ₗ[K] V\nhe : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x\nn : ℕ\nhind :\n ∀ {e : V ≃ₗ[K] V},\n (∀ (a : K), ∃ x, ¬e.fixedReduce x = a • x) →\n finrank K (V ⧸ (↑e).fixedSu... | obtain ⟨a, ha⟩ : ∃ a : K, ∀ x, (auxTransvection hf * e).fixedReduce x = a • x := by
contrapose! he'
exact hind he' (by rw [auxTransvection_mul_fixed hfv, hrank]) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 158,
"column": 91
} | {
"line": 159,
"column": 75
} | {
"line": 160,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nE : Set α\nε : ℝ\nhε : 0 < ε\nδ : ℝ\nhδ_pos : 0 < δ\nhδε :\n ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.in... | [] | by
rw [eLpNorm_indicator_eq_eLpNorm_restrict hs, μ.restrict_restrict hs] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Transvection.Generation | {
"line": 493,
"column": 4
} | {
"line": 493,
"column": 86
} | {
"line": 493,
"column": 87
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nhe : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1\n⊢ e ∈ transvections K V ^ (finrank K (V ⧸ (↑e).fixedSubmodule) - 1) * dilatransvections K V",
"ppTerm"... | [
"case inl\nK : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nhe : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1\n⊢ finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 35
} | {
"line": 181,
"column": 36
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : T2Space M\nv : VectorMeasure α M\nA : Set α\nhA : MeasurableSet A\n⊢ v Aᶜ = v univ - v A",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg... | [
"α : Type u_1\nm : MeasurableSpace α\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : T2Space M\nv : VectorMeasure α M\nA : Set α\nhA : MeasurableSet A\n⊢ v (univ \\ A) = v univ - v A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 199,
"column": 50
} | {
"line": 199,
"column": 65
} | {
"line": 199,
"column": 66
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv : VectorMeasure α M\ninst✝ : T2Space M\nA B : Set α\nhA : MeasurableSet A\nhB : MeasurableSet B\nh' : v (B \\ A) = 0\n⊢ v (A \\ B) + v (B \\ A ∪ A ∩ B) = v (A \\ B) + v B",
"ppTerm": "?m.163"... | [
"α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv : VectorMeasure α M\ninst✝ : T2Space M\nA B : Set α\nhA : MeasurableSet A\nhB : MeasurableSet B\nh' : v (B \\ A) = 0\n⊢ v (A \\ B) + v (B \\ A ∪ B ∩ A) = v (A \\ B) + v B"
] | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 252,
"column": 65
} | {
"line": 252,
"column": 76
} | {
"line": 252,
"column": 77
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : T2Space M\ninst✝ : ContinuousSub M\nv : VectorMeasure α M\ns : ℕ → Set α\nhm : Antitone s\nhs : ∀ (i : ℕ), MeasurableSet (s i)\nI : ∀ (n : ℕ), v (s n) = v univ - v (s n)ᶜ\nJ : v (⋂ n, s n) ... | [
"α : Type u_1\nm : MeasurableSpace α\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : T2Space M\ninst✝ : ContinuousSub M\nv : VectorMeasure α M\ns : ℕ → Set α\nhm : Antitone s\nhs : ∀ (i : ℕ), MeasurableSet (s i)\nI : ∀ (n : ℕ), v (s n) = v univ - v (s n)ᶜ\nJ : v (⋂ n, s n) = v univ - v... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 438,
"column": 40
} | {
"line": 438,
"column": 51
} | {
"line": 438,
"column": 52
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : MeasurableSpace β\nx✝ : β\nv✝ : M\ns : Set β\nx : β\nv : M\nf : ℕ → Set β\nf_meas : ∀ (i : ℕ), MeasurableSet (f i)\nf_disj : Pairwise (Disjoint on f)\nhx : x ∈ ⋃ i, f i\nthis ... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : MeasurableSpace β\nx✝ : β\nv✝ : M\ns : Set β\nx : β\nv : M\nf : ℕ → Set β\nf_meas : ∀ (i : ℕ), MeasurableSet (f i)\nf_disj : Pairwise (Disjoint on f)\nhx : x ∈ ⋃ i, f i\nthis : Measurable... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 17
} | {
"line": 166,
"column": 18
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\ninst✝¹ : CompleteSpace G\nB : F →L[... | [
"Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\ninst✝¹ : CompleteSpace G\nB : F →L[ℝ] E →L[ℝ] G... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn | {
"line": 225,
"column": 8
} | {
"line": 225,
"column": 23
} | {
"line": 225,
"column": 24
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni : Set α\nn m : ℕ\nh✝ : n ≠ m\nh : m < n\n⊢ s.restrictNonposSeq i n ∩ s.restrictNonposSeq i m = ∅",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Set.instIn... | [
"case inr\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni : Set α\nn m : ℕ\nh✝ : n ≠ m\nh : m < n\n⊢ s.restrictNonposSeq i m ∩ s.restrictNonposSeq i n = ∅"
] | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 377,
"column": 8
} | {
"line": 377,
"column": 19
} | {
"line": 377,
"column": 20
} | [
{
"pp": "case neg.h\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nhf : MemLp f p μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhMpos : 0 < M\nhM : eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ EN... | [
"case neg.h\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nhf : MemLp f p μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhMpos : 0 < M\nhM : eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 377,
"column": 8
} | {
"line": 377,
"column": 19
} | {
"line": 377,
"column": 20
} | [
{
"pp": "case neg.h\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nhf : MemLp f p μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhMpos : 0 < M\nhM : eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ EN... | [
"case neg.h\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nhf : MemLp f p μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhMpos : 0 < M\nhM : eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 413,
"column": 2
} | {
"line": 419,
"column": 73
} | {
"line": 421,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), MemLp (f i) p μ\n⊢ UnifIntegrable f p μ",
"ppTerm": "?m.20",
"assigned": true,
"use... | [] | intro ε hε
by_cases hι : Nonempty ι
· obtain ⟨i⟩ := hι
obtain ⟨δ, hδpos, hδ⟩ := (hf i).eLpNorm_indicator_le hp_one hp_top hε
refine ⟨δ, hδpos, fun j s hs hμs => ?_⟩
convert! hδ s hs hμs
· exact ⟨1, zero_lt_one, fun i => False.elim <| hι <| Nonempty.intro i⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 413,
"column": 2
} | {
"line": 419,
"column": 73
} | {
"line": 421,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : Subsingleton ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), MemLp (f i) p μ\n⊢ UnifIntegrable f p μ",
"ppTerm": "?m.20",
"assigned": true,
"use... | [] | intro ε hε
by_cases hι : Nonempty ι
· obtain ⟨i⟩ := hι
obtain ⟨δ, hδpos, hδ⟩ := (hf i).eLpNorm_indicator_le hp_one hp_top hε
refine ⟨δ, hδpos, fun j s hs hμs => ?_⟩
convert! hδ s hs hμs
· exact ⟨1, zero_lt_one, fun i => False.elim <| hι <| Nonempty.intro i⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 17
} | {
"line": 451,
"column": 18
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), MemLp (f i) p μ\nn : ℕ\nhn : Nonempty (ι ≃ Fin n)\nε : ℝ\nhε : 0 < ε\ng : Fin n → α → β := f ∘ ⇑hn.so... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), MemLp (f i) p μ\nn : ℕ\nhn : Nonempty (ι ≃ Fin n)\nε : ℝ\nhε : 0 < ε\ng : Fin n → α → β := f ∘ ⇑hn.some.symm\nhg ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 85,
"column": 2
} | {
"line": 91,
"column": 25
} | {
"line": 93,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : α → E\nhf : Integrable f μ\nhg : Integrable g μ\n⊢ μ.withDensityᵥ (f + g) = μ.withDensityᵥ f + μ.withDensityᵥ g",
"ppTerm": "?m.38",
"assigned": true,
"usedConstan... | [] | ext1 i hi
rw [withDensityᵥ_apply (hf.add hg) hi, _root_.add_apply, withDensityᵥ_apply hf hi,
withDensityᵥ_apply hg hi]
simp_rw [Pi.add_apply]
rw [integral_add]
· exact hf.integrableOn
· exact hg.integrableOn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 85,
"column": 2
} | {
"line": 91,
"column": 25
} | {
"line": 93,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : α → E\nhf : Integrable f μ\nhg : Integrable g μ\n⊢ μ.withDensityᵥ (f + g) = μ.withDensityᵥ f + μ.withDensityᵥ g",
"ppTerm": "?m.38",
"assigned": true,
"usedConstan... | [] | ext1 i hi
rw [withDensityᵥ_apply (hf.add hg) hi, _root_.add_apply, withDensityᵥ_apply hf hi,
withDensityᵥ_apply hg hi]
simp_rw [Pi.add_apply]
rw [integral_add]
· exact hf.integrableOn
· exact hg.integrableOn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn | {
"line": 398,
"column": 4
} | {
"line": 400,
"column": 12
} | {
"line": 401,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nf : ℕ → ℝ\nleft✝ : Antitone f\nhf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))\nB : ℕ → Set α\nhB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ s (B n) = f n\nhB₁ : ∀ (n : ℕ), MeasurableSet (B n)\nhB₂ : ∀ (n : ℕ), s ≤[B n] 0... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nf : ℕ → ℝ\nleft✝ : Antitone f\nhf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))\nB : ℕ → Set α\nhB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ s (B n) = f n\nhB₁ : ∀ (n : ℕ), MeasurableSet (B n)\nhB₂ : ∀ (n : ℕ), s ≤[B n] 0\nA : Set α ... | rw [← hA₃,
of_union (Set.disjoint_of_subset_right (Set.Subset.trans hD hC₁) disjoint_compl_right) hA₁
hD₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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