module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 58
} | {
"line": 209,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\n⊢ ν.singularPart μ ≪ ν",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.singularPart_le",
"LE.le.absolutelyContinuous",
"MeasureTheory.Measu... | [] | exact (Measure.singularPart_le _ _).absolutelyContinuous | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Dynamics.OmegaLimit | {
"line": 121,
"column": 4
} | {
"line": 123,
"column": 47
} | {
"line": 124,
"column": 2
} | [
{
"pp": "case mp\nτ : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\ns : Set α\ny : β\n⊢ (∀ i ∈ f, ∀ t ∈ 𝓝 y, (t ∩ image2 ϕ i s).Nonempty) →\n ∀ n ∈ 𝓝 y, ∀ {U : Set τ}, U ∈ f → ∃ x ∈ U, (s ∩ ϕ x ⁻¹' n).Nonempty",
"ppTerm": "?mp",
"assigned": true,
... | [] | intro h _ hn _ hu
rcases h _ hu _ hn with ⟨_, _, _, ht, _, hx, rfl⟩
exact ⟨_, ht, _, hx, by rwa [mem_preimage]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Dynamics.OmegaLimit | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 23
} | {
"line": 170,
"column": 23
} | [
{
"pp": "τ : Type u_1\nα : Type u_2\nβ : Type u_3\nι : Type u_4\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\np : ι → Set α\n⊢ ⋃ i, ω f ϕ (p i) ⊆ ω f ϕ (⋃ i, p i)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"omegaLimit",
"Eq.mpr",
"Set.iUnion_subset_iff",
... | [
"τ : Type u_1\nα : Type u_2\nβ : Type u_3\nι : Type u_4\ninst✝ : TopologicalSpace β\nf : Filter τ\nϕ : τ → α → β\np : ι → Set α\n⊢ ∀ (i : ι), ω f ϕ (p i) ⊆ ω f ϕ (⋃ i, p i)"
] | iUnion_subset_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Dynamics.TopologicalEntropy.NetEntropy | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 50
} | {
"line": 145,
"column": 4
} | [
{
"pp": "case mp\nX : Type u_1\nT : X → X\nF : Set X\nU : SetRel X X\nn : ℕ\nh : netMaxcard T F U n < ⊤\nk : ℕ\nk_max : ↑k = ↑(sSup (Finset.card '' {s | IsDynNetIn T F U n ↑s}))\nthis : netMaxcard T F U n = sSup (WithTop.some '' Finset.card '' {s | IsDynNetIn T F U n ↑s})\nh_bdda : BddAbove (Finset.card '' {s |... | [
"case mp\nX : Type u_1\nT : X → X\nF : Set X\nU : SetRel X X\nn : ℕ\nh : netMaxcard T F U n < ⊤\nk : ℕ\nk_max : ↑k = ↑(sSup (Finset.card '' {s | IsDynNetIn T F U n ↑s}))\nthis : netMaxcard T F U n = sSup (WithTop.some '' Finset.card '' {s | IsDynNetIn T F U n ↑s})\nh_bdda : BddAbove (Finset.card '' {s | IsDynNetIn ... | rw [← Nat.cast_inj.mp k_max, mem_image] at key | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 79
} | {
"line": 429,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : μ.HaveLebesgueDecomposition ν\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nh : μ.rnDeriv ν ≤ᵐ[ν] 1\ns : Set α\n⊢ μ s ≤ ν s",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.withDensit... | [
"α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : μ.HaveLebesgueDecomposition ν\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nh : μ.rnDeriv ν ≤ᵐ[ν] 1\ns : Set α\n⊢ ∫⁻ (a : α) in s, μ.rnDeriv ν a ∂ν ≤ ∫⁻ (x : α) in s, 1 ∂ν"
] | rw [← withDensity_rnDeriv_eq _ _ hμν, withDensity_apply', ← setLIntegral_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 460,
"column": 7
} | {
"line": 460,
"column": 64
} | {
"line": 460,
"column": 64
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nh_add : (μ + ν).rnDeriv (μ + ν) =ᵐ[ν] μ.rnDeriv (μ + ν) + ν.rnDeriv (μ + ν)\nh_one_add : ∀ᵐ (x : α) ∂ν, (μ + ν).rnDeriv (μ + ν) x = (fun x ↦ 1) x\nthis : μ.rnDeriv (μ + ν) =ᵐ[ν] fun x ↦ 1 - (μ.rnDeriv ν... | [
"α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nh_add : (μ + ν).rnDeriv (μ + ν) =ᵐ[ν] μ.rnDeriv (μ + ν) + ν.rnDeriv (μ + ν)\nh_one_add : ∀ᵐ (x : α) ∂ν, (μ + ν).rnDeriv (μ + ν) x = (fun x ↦ 1) x\nthis : μ.rnDeriv (μ + ν) =ᵐ[ν] fun x ↦ 1 - (μ.rnDeriv ν x + 1)⁻¹\na... | ENNReal.mul_inv_cancel (by simp) (by simp [ha_lt_top.ne]) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Dynamics.TopologicalEntropy.CoverEntropy | {
"line": 419,
"column": 29
} | {
"line": 419,
"column": 44
} | {
"line": 419,
"column": 45
} | [
{
"pp": "X : Type u_1\nT : X → X\nU : SetRel X X\nF : Set X\ninst✝ : UniformSpace X\nF_comp : IsCompact F\nF_inv : MapsTo T F F\nU_uni : U ∈ 𝓤 X\nV : Set (X × X)\nV_uni : V ∈ 𝓤 X\nV_symm : SetRel.IsSymm V\nV_U : V ○ V ⊆ U\ns : Finset X\ns_cover : IsDynCoverOf T F V 1 ↑s\n⊢ (↑(#s)).log < ⊤",
"ppTerm": "?m.... | [
"X : Type u_1\nT : X → X\nU : SetRel X X\nF : Set X\ninst✝ : UniformSpace X\nF_comp : IsCompact F\nF_inv : MapsTo T F F\nU_uni : U ∈ 𝓤 X\nV : Set (X × X)\nV_uni : V ∈ 𝓤 X\nV_symm : SetRel.IsSymm V\nV_U : V ○ V ⊆ U\ns : Finset X\ns_cover : IsDynCoverOf T F V 1 ↑s\n⊢ ↑(#s) < ∞"
] | log_lt_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.AbelRuffini | {
"line": 53,
"column": 2
} | {
"line": 57,
"column": 42
} | {
"line": 59,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\ns : Multiset F[X]\nhs : ∀ p ∈ s, IsSolvable p.Gal\n⊢ IsSolvable s.prod.Gal",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsSolvable",
"HMul.hMul",
"Monoid.toMulOneClass",
"Polynomial.Gal",
"congrArg",
... | [] | apply Multiset.induction_on' s
· exact gal_one_isSolvable
· intro p t hps _ ht
rw [Multiset.insert_eq_cons, Multiset.prod_cons]
exact gal_mul_isSolvable (hs p hps) ht | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.AbelRuffini | {
"line": 53,
"column": 2
} | {
"line": 57,
"column": 42
} | {
"line": 59,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\ns : Multiset F[X]\nhs : ∀ p ∈ s, IsSolvable p.Gal\n⊢ IsSolvable s.prod.Gal",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsSolvable",
"HMul.hMul",
"Monoid.toMulOneClass",
"Polynomial.Gal",
"congrArg",
... | [] | apply Multiset.induction_on' s
· exact gal_one_isSolvable
· intro p t hps _ ht
rw [Multiset.insert_eq_cons, Multiset.prod_cons]
exact gal_mul_isSolvable (hs p hps) ht | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IsAlgClosed.Classification | {
"line": 97,
"column": 54
} | {
"line": 99,
"column": 90
} | {
"line": 100,
"column": 4
} | [
{
"pp": "R : Type u\nK : Type v\ninst✝³ : CommRing R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsAlgClosed K\nι : Type w\nv : ι → K\nhv : IsTranscendenceBasis R v\n⊢ Cardinal.lift.{max u w, v} #K ≤ Cardinal.lift.{max u w, v} (max #↥(Algebra.adjoin R (Set.range v)) ℵ₀)",
"ppTerm": "?m.28",
"assign... | [] | by
letI := isAlgClosure_of_transcendence_basis v hv
simpa using Algebra.IsAlgebraic.cardinalMk_le_max (Algebra.adjoin R (Set.range v)) K | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.LanguageMap | {
"line": 519,
"column": 2
} | {
"line": 519,
"column": 32
} | {
"line": 520,
"column": 2
} | [
{
"pp": "L : Language\nL' : Language\nM : Type w\ninst✝² : L.Structure M\nα : Type u_1\ninst✝¹ : (constantsOn α).Structure M\nA B : Set M\nh : A ⊆ B\nN : Type w'\ninst✝ : L.Structure N\nf : M ↪[L] N\n⊢ M ↪[L[[↑A]]] f.withConstants A",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Fi... | [
"case refine_1\nL : Language\nL' : Language\nM : Type w\ninst✝² : L.Structure M\nα : Type u_1\ninst✝¹ : (constantsOn α).Structure M\nA B : Set M\nh : A ⊆ B\nN : Type w'\ninst✝ : L.Structure N\nf : M ↪[L] N\n⊢ ∀ {n : ℕ} (f_1 : L[[↑A]].Functions n) (x : Fin n → M), f.toFun (funMap f_1 x) = funMap f_1 (f.toFun ∘ x)",
... | refine ⟨f.toEmbedding, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.ModelTheory.LanguageMap | {
"line": 521,
"column": 4
} | {
"line": 521,
"column": 16
} | {
"line": 522,
"column": 4
} | [
{
"pp": "case refine_1\nL : Language\nL' : Language\nM : Type w\ninst✝² : L.Structure M\nα : Type u_1\ninst✝¹ : (constantsOn α).Structure M\nA B : Set M\nh : A ⊆ B\nN : Type w'\ninst✝ : L.Structure N\nf : M ↪[L] N\nn : ℕ\ng : L[[↑A]].Functions n\nx : Fin n → M\n⊢ f.toFun (funMap g x) = funMap g (f.toFun ∘ x)",
... | [] | cases g with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.FieldTheory.AbelRuffini | {
"line": 179,
"column": 8
} | {
"line": 179,
"column": 11
} | {
"line": 179,
"column": 12
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝ : Field F\nn : ℕ\nx : F\nhx : x = 0\n⊢ IsSolvable (X ^ n - C x).Gal",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"IsSolvable",
"Polynomial.Gal",
"congrArg",
"HSub.hSub",
"RingHom",... | [
"case pos\nF : Type u_1\ninst✝ : Field F\nn : ℕ\nx : F\nhx : x = 0\n⊢ IsSolvable (X ^ n - C 0).Gal"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.AbelRuffini | {
"line": 298,
"column": 2
} | {
"line": 300,
"column": 82
} | {
"line": 301,
"column": 2
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nx y z : E\nhx : x ∈ solvableByRad F E\nhy : y ∈ solvableByRad F E\nhz : z ∈ solvableByRad F E\nhx' : IsSolvable (minpoly F x).Gal\nhy' : IsSolvable (minpoly F y).Gal\nhz' : z ∈ F⟮x, y⟯\np : F[X] := minpoly F x\nq : F[X... | [
"F : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nx y z : E\nhx : x ∈ solvableByRad F E\nhy : y ∈ solvableByRad F E\nhz : z ∈ solvableByRad F E\nhx' : IsSolvable (minpoly F x).Gal\nhy' : IsSolvable (minpoly F y).Gal\nhz' : z ∈ F⟮x, y⟯\np : F[X] := minpoly F x\nq : F[X] := minpoly... | rw [Polynomial.map_mul,
splits_mul (map_ne_zero (minpoly.ne_zero (isIntegral_of_mem_solvableByRad hx)))
(map_ne_zero (minpoly.ne_zero (isIntegral_of_mem_solvableByRad hy)))] at hpq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.Syntax | {
"line": 700,
"column": 15
} | {
"line": 702,
"column": 31
} | {
"line": 704,
"column": 0
} | [
{
"pp": "L : Language\nL' : Language\nM : Type w\nα : Type u'\nβ : Type v'\nγ : Type u_1\nn : ℕ\nφ : L ≃ᴸ L'\n⊢ Function.RightInverse φ.invLHom.onBoundedFormula φ.toLHom.onBoundedFormula",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FirstOrder.Language.LEquiv.right... | [] | by
rw [Function.rightInverse_iff_comp, ← LHom.comp_onBoundedFormula, φ.right_inv,
LHom.id_onBoundedFormula] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.AbelRuffini | {
"line": 313,
"column": 6
} | {
"line": 313,
"column": 9
} | {
"line": 313,
"column": 9
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nx y z : E\nhx : x ∈ solvableByRad F E\nhy : y ∈ solvableByRad F E\nhz : z ∈ solvableByRad F E\nhx' : IsSolvable (minpoly F x).Gal\nhy' : IsSolvable (minpoly F y).Gal\nhz' : z ∈ F⟮x, y⟯\np : F[X] := minpoly F x\nq : F[X... | [
"F : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nx y z : E\nhx : x ∈ solvableByRad F E\nhy : y ∈ solvableByRad F E\nhz : z ∈ solvableByRad F E\nhx' : IsSolvable (minpoly F x).Gal\nhy' : IsSolvable (minpoly F y).Gal\nhz' : z ∈ F⟮x, y⟯\np : F[X] := minpoly F x\nq : F[X] := minpoly... | key | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.AbelRuffini | {
"line": 322,
"column": 4
} | {
"line": 322,
"column": 100
} | {
"line": 323,
"column": 2
} | [
{
"pp": "case add\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nx y z : E\nhy : y ∈ solvableByRad F E\nhz : z ∈ solvableByRad F E\nhy' : IsSolvable (minpoly F y).Gal\nhz' : IsSolvable (minpoly F z).Gal\n⊢ IsSolvable (minpoly F (y + z)).Gal",
"ppTerm": "?add",
"ass... | [] | apply induction_step hy hz (add_mem hy hz) hy' hz' (add_mem ..) <;> apply subset_adjoin <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.FieldTheory.AbelRuffini | {
"line": 322,
"column": 4
} | {
"line": 322,
"column": 100
} | {
"line": 323,
"column": 2
} | [
{
"pp": "case add\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nx y z : E\nhy : y ∈ solvableByRad F E\nhz : z ∈ solvableByRad F E\nhy' : IsSolvable (minpoly F y).Gal\nhz' : IsSolvable (minpoly F z).Gal\n⊢ IsSolvable (minpoly F (y + z)).Gal",
"ppTerm": "?add",
"ass... | [] | apply induction_step hy hz (add_mem hy hz) hy' hz' (add_mem ..) <;> apply subset_adjoin <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.AbelRuffini | {
"line": 322,
"column": 4
} | {
"line": 322,
"column": 100
} | {
"line": 323,
"column": 2
} | [
{
"pp": "case add\nF : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nx y z : E\nhy : y ∈ solvableByRad F E\nhz : z ∈ solvableByRad F E\nhy' : IsSolvable (minpoly F y).Gal\nhz' : IsSolvable (minpoly F z).Gal\n⊢ IsSolvable (minpoly F (y + z)).Gal",
"ppTerm": "?add",
"ass... | [] | apply induction_step hy hz (add_mem hy hz) hy' hz' (add_mem ..) <;> apply subset_adjoin <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.ElementaryMaps | {
"line": 185,
"column": 27
} | {
"line": 185,
"column": 53
} | {
"line": 187,
"column": 0
} | [
{
"pp": "L : Language\nM : Type u_1\nN : Type u_2\nP : Type u_3\nQ : Type u_4\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\ninst✝¹ : L.Structure P\ninst✝ : L.Structure Q\nhnp : N ↪ₑ[L] P\nhmn : M ↪ₑ[L] N\nn : ℕ\nφ : L.Formula (Fin n)\nx : Fin n → M\n⊢ φ.Realize ((⇑hnp ∘ ⇑hmn) ∘ x) ↔ φ.Realize x",
"ppTerm... | [] | simp [Function.comp_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.ElementaryMaps | {
"line": 185,
"column": 27
} | {
"line": 185,
"column": 53
} | {
"line": 187,
"column": 0
} | [
{
"pp": "L : Language\nM : Type u_1\nN : Type u_2\nP : Type u_3\nQ : Type u_4\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\ninst✝¹ : L.Structure P\ninst✝ : L.Structure Q\nhnp : N ↪ₑ[L] P\nhmn : M ↪ₑ[L] N\nn : ℕ\nφ : L.Formula (Fin n)\nx : Fin n → M\n⊢ φ.Realize ((⇑hnp ∘ ⇑hmn) ∘ x) ↔ φ.Realize x",
"ppTerm... | [] | simp [Function.comp_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.ElementaryMaps | {
"line": 185,
"column": 27
} | {
"line": 185,
"column": 53
} | {
"line": 187,
"column": 0
} | [
{
"pp": "L : Language\nM : Type u_1\nN : Type u_2\nP : Type u_3\nQ : Type u_4\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\ninst✝¹ : L.Structure P\ninst✝ : L.Structure Q\nhnp : N ↪ₑ[L] P\nhmn : M ↪ₑ[L] N\nn : ℕ\nφ : L.Formula (Fin n)\nx : Fin n → M\n⊢ φ.Realize ((⇑hnp ∘ ⇑hmn) ∘ x) ↔ φ.Realize x",
"ppTerm... | [] | simp [Function.comp_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Semantics | {
"line": 596,
"column": 73
} | {
"line": 598,
"column": 14
} | {
"line": 600,
"column": 0
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nn : ℕ\nf : L.Functions n\nx : Fin n → M\ny : M\n⊢ (graph f).Realize (cons y x) ↔ funMap f x = y",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fin.cons_succ",
"Fin.succ",
"congrArg",
"FirstO... | [] | by
simp only [Formula.graph, Term.realize, realize_equal, Fin.cons_zero, Fin.cons_succ]
rw [eq_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Encoding | {
"line": 233,
"column": 4
} | {
"line": 261,
"column": 36
} | {
"line": 262,
"column": 2
} | [
{
"pp": "case rel\nL : Language\nα : Type u'\nl : List ((n : ℕ) × L.BoundedFormula α n)\nn φ_n φ_l : ℕ\nφ_R : L.Relations φ_l\nts : Fin φ_l → L.Term (α ⊕ Fin φ_n)\n⊢ ∀ (l' : List ((k : ℕ) × L.Term (α ⊕ Fin k) ⊕ (n : ℕ) × L.Relations n ⊕ ℕ)),\n listDecode (⟨φ_n, rel φ_R ts⟩.snd.listEncode ++ l') = ⟨φ_n, rel φ... | [] | intro l
rw [listEncode, cons_append, cons_append, singleton_append, cons_append, listDecode]
have h : ∀ i : Fin φ_l, ((List.map Sum.getLeft? (List.map (fun i : Fin φ_l =>
Sum.inl (⟨(⟨φ_n, rel φ_R ts⟩ : Σ n, L.BoundedFormula α n).fst, ts i⟩ :
Σ n, L.Term (α ⊕ (Fin n)))) (finRange φ_l) ++ l))[↑i]?).... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Encoding | {
"line": 233,
"column": 4
} | {
"line": 261,
"column": 36
} | {
"line": 262,
"column": 2
} | [
{
"pp": "case rel\nL : Language\nα : Type u'\nl : List ((n : ℕ) × L.BoundedFormula α n)\nn φ_n φ_l : ℕ\nφ_R : L.Relations φ_l\nts : Fin φ_l → L.Term (α ⊕ Fin φ_n)\n⊢ ∀ (l' : List ((k : ℕ) × L.Term (α ⊕ Fin k) ⊕ (n : ℕ) × L.Relations n ⊕ ℕ)),\n listDecode (⟨φ_n, rel φ_R ts⟩.snd.listEncode ++ l') = ⟨φ_n, rel φ... | [] | intro l
rw [listEncode, cons_append, cons_append, singleton_append, cons_append, listDecode]
have h : ∀ i : Fin φ_l, ((List.map Sum.getLeft? (List.map (fun i : Fin φ_l =>
Sum.inl (⟨(⟨φ_n, rel φ_R ts⟩ : Σ n, L.BoundedFormula α n).fst, ts i⟩ :
Σ n, L.Term (α ⊕ (Fin n)))) (finRange φ_l) ++ l))[↑i]?).... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Substructures | {
"line": 836,
"column": 6
} | {
"line": 836,
"column": 38
} | {
"line": 837,
"column": 6
} | [
{
"pp": "L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nS : L.Substructure M\nf g : M →[L] N\nn : ℕ\nfn : L.Functions n\nx : Fin n → M\nhx : ∀ (i : Fin n), x i ∈ {x | f x = g x}\nx✝ : Fin n\n⊢ (⇑f ∘ x) x✝ = (⇑g ∘ x) x✝",
"ppTerm":... | [
"L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nS : L.Substructure M\nf g : M →[L] N\nn : ℕ\nfn : L.Functions n\nx : Fin n → M\nhx : ∀ (i : Fin n), x i ∈ {x | f x = g x}\nx✝ : Fin n\n⊢ f (x x✝) = g (x x✝)"
] | repeat' rw [Function.comp_apply] | Lean.Elab.Tactic.evalRepeat' | Lean.Parser.Tactic.repeat' |
Mathlib.ModelTheory.Definability | {
"line": 85,
"column": 2
} | {
"line": 86,
"column": 6
} | {
"line": 88,
"column": 0
} | [
{
"pp": "M : Type w\nL : Language\ninst✝ : L.Structure M\nα : Type u₁\ns : Set (α → M)\n⊢ ∅.Definable L s ↔ ∃ φ, s = setOf φ.Realize",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FirstOrder.Language.LEquiv.symm",
"FirstOrder.Language.LHom.setOf_realize_onForm... | [] | rw [Definable, Equiv.exists_congr_left (LEquiv.addEmptyConstants L (∅ : Set M)).onFormula]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Definability | {
"line": 85,
"column": 2
} | {
"line": 86,
"column": 6
} | {
"line": 88,
"column": 0
} | [
{
"pp": "M : Type w\nL : Language\ninst✝ : L.Structure M\nα : Type u₁\ns : Set (α → M)\n⊢ ∅.Definable L s ↔ ∃ φ, s = setOf φ.Realize",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FirstOrder.Language.LEquiv.symm",
"FirstOrder.Language.LHom.setOf_realize_onForm... | [] | rw [Definable, Equiv.exists_congr_left (LEquiv.addEmptyConstants L (∅ : Set M)).onFormula]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Definability | {
"line": 127,
"column": 48
} | {
"line": 131,
"column": 24
} | {
"line": 133,
"column": 0
} | [
{
"pp": "M : Type w\nA : Set M\nL : Language\ninst✝ : L.Structure M\nα : Type u₁\nι : Type u_2\nf : ι → Set (α → M)\nhf : ∀ (i : ι), A.Definable L (f i)\ns : Finset ι\n⊢ A.Definable L (s.inf f)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Definable.inter",
... | [] | by
classical
refine Finset.induction definable_univ (fun i s _ h => ?_) s
rw [Finset.inf_insert]
exact (hf i).inter h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.ElementarySubstructures | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 19
} | {
"line": 213,
"column": 2
} | [
{
"pp": "L : Language\nM : Type u_1\ninst✝ : L.Structure M\nS : L.ElementarySubstructure M\nD : Set M\nx : M\nhx : x ∈ D\nφ : L[[↑↑S]].Formula (Fin 1)\nhφ : ∀ (x : Fin 1 → M), x 0 ∈ D ↔ φ.Realize x\n⊢ φ.Realize ![x]",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Set.Definable₁._pro... | [] | simp [← hφ, hx] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Skolem | {
"line": 145,
"column": 18
} | {
"line": 145,
"column": 33
} | {
"line": 145,
"column": 34
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\ns : Set M\nκ : Cardinal.{w'}\nh3 : lift.{w', max u v} L.card ≤ lift.{max u v, w'} κ\nh4 : lift.{w, w'} κ ≤ lift.{w', w} #M\ns' : Set (ULift.{w', w} M)\nh2 : lift.{w', w} #↑s ≤ #↑s'\nh1 : ℵ₀ ≤ #↑s'\nhs' : #↑s' = lift.{w, w'} κ\nthis : Nonempty M\nh : lift... | [
"L : Language\nM : Type w\ninst✝ : L.Structure M\ns : Set M\nκ : Cardinal.{w'}\nh3 : lift.{w', max u v} L.card ≤ lift.{max u v, w'} κ\nh4 : lift.{w, w'} κ ≤ lift.{w', w} #M\ns' : Set (ULift.{w', w} M)\nh2 : lift.{w', w} #↑s ≤ #↑s'\nh1 : ℵ₀ ≤ #↑s'\nhs' : #↑s' = lift.{w, w'} κ\nthis : Nonempty M\nh : lift.{w', w} #↑(... | aleph0_le_lift, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Skolem | {
"line": 149,
"column": 45
} | {
"line": 149,
"column": 60
} | {
"line": 149,
"column": 61
} | [
{
"pp": "case refine_2\nL : Language\nM : Type w\ninst✝ : L.Structure M\ns : Set M\nκ : Cardinal.{w'}\nh3 : lift.{w', max u v} L.card ≤ lift.{max u v, w'} κ\nh4 : lift.{w, w'} κ ≤ lift.{w', w} #M\ns' : Set (ULift.{w', w} M)\nh2 : lift.{w', w} #↑s ≤ #↑s'\nh1 : ℵ₀ ≤ #↑s'\nhs' : #↑s' = lift.{w, w'} κ\nthis : Nonem... | [
"case refine_2\nL : Language\nM : Type w\ninst✝ : L.Structure M\ns : Set M\nκ : Cardinal.{w'}\nh3 : lift.{w', max u v} L.card ≤ lift.{max u v, w'} κ\nh4 : lift.{w, w'} κ ≤ lift.{w', w} #M\ns' : Set (ULift.{w', w} M)\nh2 : lift.{w', w} #↑s ≤ #↑s'\nh1 : ℵ₀ ≤ #↑s'\nhs' : #↑s' = lift.{w, w'} κ\nthis : Nonempty M\nh : l... | aleph0_le_lift, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Skolem | {
"line": 156,
"column": 8
} | {
"line": 156,
"column": 23
} | {
"line": 156,
"column": 24
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\ns : Set M\nκ : Cardinal.{w'}\nh3 : lift.{w', max u v} L.card ≤ lift.{max u v, w'} κ\nh4 : lift.{w, w'} κ ≤ lift.{w', w} #M\ns' : Set (ULift.{w', w} M)\nh2 : lift.{w', w} #↑s ≤ #↑s'\nh1 : ℵ₀ ≤ #↑s'\nhs' : #↑s' = lift.{w, w'} κ\nthis : Nonempty M\nh : lift... | [
"L : Language\nM : Type w\ninst✝ : L.Structure M\ns : Set M\nκ : Cardinal.{w'}\nh3 : lift.{w', max u v} L.card ≤ lift.{max u v, w'} κ\nh4 : lift.{w, w'} κ ≤ lift.{w', w} #M\ns' : Set (ULift.{w', w} M)\nh2 : lift.{w', w} #↑s ≤ #↑s'\nh1 : ℵ₀ ≤ #↑s'\nhs' : #↑s' = lift.{w, w'} κ\nthis : Nonempty M\nh : lift.{w', w} #↑(... | aleph0_le_lift, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.OreLocalization.Cardinality | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 24
} | {
"line": 81,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type v\ninst✝ : MulAction R X\nh✝¹ : Infinite X\nh✝ : Infinite ↥S\n⊢ #(X × ↥S) = max (lift.{v, u} #↥S) (lift.{u, v} #X)",
"ppTerm": "?m.171",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatt... | [
"R : Type u\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type v\ninst✝ : MulAction R X\nh✝¹ : Infinite X\nh✝ : Infinite ↥S\n⊢ lift.{v, u} #↥S * lift.{u, v} #X = max (lift.{v, u} #↥S) (lift.{u, v} #X)"
] | rw [mk_prod, mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Cardinal.Divisibility | {
"line": 131,
"column": 2
} | {
"line": 134,
"column": 21
} | {
"line": 136,
"column": 0
} | [
{
"pp": "a : Cardinal.{u_1}\n⊢ Prime a ↔ ℵ₀ ≤ a ∨ ∃ p, a = ↑p ∧ Nat.Prime p",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"not_le",
"Iff.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"Nat.Prime",
"Preorder.toLT",
"eq_false",
"Car... | [] | rcases le_or_gt ℵ₀ a with h | h
· simp [h]
lift a to ℕ using id h
simp [not_le.mpr h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Divisibility | {
"line": 131,
"column": 2
} | {
"line": 134,
"column": 21
} | {
"line": 136,
"column": 0
} | [
{
"pp": "a : Cardinal.{u_1}\n⊢ Prime a ↔ ℵ₀ ≤ a ∨ ∃ p, a = ↑p ∧ Nat.Prime p",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"not_le",
"Iff.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"Nat.Prime",
"Preorder.toLT",
"eq_false",
"Car... | [] | rcases le_or_gt ℵ₀ a with h | h
· simp [h]
lift a to ℕ using id h
simp [not_le.mpr h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Differential.Basic | {
"line": 44,
"column": 93
} | {
"line": 46,
"column": 38
} | {
"line": 48,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Field R\ninst✝ : Differential R\na b : R\nha : a ≠ 0\nhb : b ≠ 0\n⊢ logDeriv (a / b) = logDeriv a - logDeriv b",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Derivation",
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Mathlib.Tactic.FieldSimp.NF.... | [] | by
unfold logDeriv
simp [field, Derivation.leibniz_div] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 140,
"column": 6
} | {
"line": 140,
"column": 30
} | {
"line": 141,
"column": 6
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝⁹ : Field F\ninst✝⁸ : Field K\ninst✝⁷ : Differential F\ninst✝⁶ : Differential K\ninst✝⁵ : Algebra F K\ninst✝⁴ : DifferentialAlgebra F K\ninst✝³ : CharZero F\ninst✝² : FiniteDimensional F K\ninst✝¹ : IsGalois F K\na : F\nι : Type\ninst✝ : Fintype ι\nc : ι → F\nhc : ∀ (x ... | [
"F : Type u_1\nK : Type u_2\ninst✝⁹ : Field F\ninst✝⁸ : Field K\ninst✝⁷ : Differential F\ninst✝⁶ : Differential K\ninst✝⁵ : Algebra F K\ninst✝⁴ : DifferentialAlgebra F K\ninst✝³ : CharZero F\ninst✝² : FiniteDimensional F K\ninst✝¹ : IsGalois F K\na : F\nι : Type\ninst✝ : Fintype ι\nc : ι → F\nhc : ∀ (x : ι), (c x)′... | simp only [u₁, map_prod] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 144,
"column": 13
} | {
"line": 144,
"column": 17
} | {
"line": 144,
"column": 18
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝⁹ : Field F\ninst✝⁸ : Field K\ninst✝⁷ : Differential F\ninst✝⁶ : Differential K\ninst✝⁵ : Algebra F K\ninst✝⁴ : DifferentialAlgebra F K\ninst✝³ : CharZero F\ninst✝² : FiniteDimensional F K\ninst✝¹ : IsGalois F K\na : F\nι : Type\ninst✝ : Fintype ι\nc : ι → F\nhc : ∀ (x ... | [
"F : Type u_1\nK : Type u_2\ninst✝⁹ : Field F\ninst✝⁸ : Field K\ninst✝⁷ : Differential F\ninst✝⁶ : Differential K\ninst✝⁵ : Algebra F K\ninst✝⁴ : DifferentialAlgebra F K\ninst✝³ : CharZero F\ninst✝² : FiniteDimensional F K\ninst✝¹ : IsGalois F K\na : F\nι : Type\ninst✝ : Fintype ι\nc : ι → F\nhc : ∀ (x : ι), (c x)′... | ffb, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 156,
"column": 8
} | {
"line": 156,
"column": 12
} | {
"line": 156,
"column": 13
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝⁹ : Field F\ninst✝⁸ : Field K\ninst✝⁷ : Differential F\ninst✝⁶ : Differential K\ninst✝⁵ : Algebra F K\ninst✝⁴ : DifferentialAlgebra F K\ninst✝³ : CharZero F\ninst✝² : FiniteDimensional F K\ninst✝¹ : IsGalois F K\na : F\nι : Type\ninst✝ : Fintype ι\nc : ι → F\nhc : ∀ (x ... | [
"F : Type u_1\nK : Type u_2\ninst✝⁹ : Field F\ninst✝⁸ : Field K\ninst✝⁷ : Differential F\ninst✝⁶ : Differential K\ninst✝⁵ : Algebra F K\ninst✝⁴ : DifferentialAlgebra F K\ninst✝³ : CharZero F\ninst✝² : FiniteDimensional F K\ninst✝¹ : IsGalois F K\na : F\nι : Type\ninst✝ : Fintype ι\nc : ι → F\nhc : ∀ (x : ι), (c x)′... | ffb, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Differential.Basic | {
"line": 101,
"column": 6
} | {
"line": 119,
"column": 34
} | {
"line": 119,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Field R\ninst✝⁵ : Differential R\na b : R\nF : Type u_2\ninst✝⁴ : Field F\ninst✝³ : Differential F\ninst✝² : CharZero F\np : F[X]\ninst✝¹ : Fact (Irreducible p)\ninst✝ : Fact p.Monic\n⊢ ∀ (x : F[X]),\n (AdjoinRoot.mk p).toIntAlgHom x = 0 →\n (AdjoinRoot.mk p).toIntAlgHom\... | [] | rintro x hx
simp_all only [RingHom.toIntAlgHom_apply, AdjoinRoot.mk_eq_zero]
obtain ⟨q, rfl⟩ := hx
simp only [Derivation.leibniz, smul_eq_mul]
apply dvd_add (dvd_mul_right ..)
apply dvd_mul_of_dvd_right
rw [← AdjoinRoot.mk_eq_zero]
unfold implicitDeriv
simp only [AdjoinRo... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Differential.Basic | {
"line": 101,
"column": 6
} | {
"line": 119,
"column": 34
} | {
"line": 119,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Field R\ninst✝⁵ : Differential R\na b : R\nF : Type u_2\ninst✝⁴ : Field F\ninst✝³ : Differential F\ninst✝² : CharZero F\np : F[X]\ninst✝¹ : Fact (Irreducible p)\ninst✝ : Fact p.Monic\n⊢ ∀ (x : F[X]),\n (AdjoinRoot.mk p).toIntAlgHom x = 0 →\n (AdjoinRoot.mk p).toIntAlgHom\... | [] | rintro x hx
simp_all only [RingHom.toIntAlgHom_apply, AdjoinRoot.mk_eq_zero]
obtain ⟨q, rfl⟩ := hx
simp only [Derivation.leibniz, smul_eq_mul]
apply dvd_add (dvd_mul_right ..)
apply dvd_mul_of_dvd_right
rw [← AdjoinRoot.mk_eq_zero]
unfold implicitDeriv
simp only [AdjoinRo... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 40,
"column": 4
} | {
"line": 40,
"column": 16
} | {
"line": 40,
"column": 16
} | [
{
"pp": "case mul_X\nσ : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : MvPolynomial σ (ZMod p)\n⊢ ∀ (p_1 : MvPolynomial σ (ZMod p)) (n : σ),\n (frobenius (MvPolynomial σ (ZMod p)) p) p_1 = (expand p) p_1 →\n (frobenius (MvPolynomial σ (ZMod p)) p) p_1 * (frobenius (MvPolynomial σ (ZMod p)) p) (X n) =... | [
"case mul_X\nσ : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf p✝ : MvPolynomial σ (ZMod p)\nn✝ : σ\nhf : (frobenius (MvPolynomial σ (ZMod p)) p) p✝ = (expand p) p✝\n⊢ (frobenius (MvPolynomial σ (ZMod p)) p) p✝ * (frobenius (MvPolynomial σ (ZMod p)) p) (X n✝) = (expand p) p✝ * X n✝ ^ p"
] | intro _ _ hf | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.FieldTheory.Finite.Valuation | {
"line": 26,
"column": 2
} | {
"line": 26,
"column": 42
} | {
"line": 27,
"column": 2
} | [
{
"pp": "Fq : Type u_1\nA : Type u_2\nΓ : Type u_3\ninst✝⁴ : Field Fq\ninst✝³ : Finite Fq\ninst✝² : Ring A\ninst✝¹ : Algebra Fq A\ninst✝ : LinearOrderedCommMonoidWithZero Γ\nv : Valuation A Γ\na : Fq\nha : a ≠ 0\n⊢ v ((algebraMap Fq A) a) = 1",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [... | [
"Fq : Type u_1\nA : Type u_2\nΓ : Type u_3\ninst✝⁴ : Field Fq\ninst✝³ : Finite Fq\ninst✝² : Ring A\ninst✝¹ : Algebra Fq A\ninst✝ : LinearOrderedCommMonoidWithZero Γ\nv : Valuation A Γ\na : Fq\nha : a ≠ 0\nthis : Fintype Fq\n⊢ v ((algebraMap Fq A) a) = 1"
] | have : Fintype Fq := Fintype.ofFinite Fq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.PerfectClosure | {
"line": 123,
"column": 32
} | {
"line": 123,
"column": 52
} | {
"line": 123,
"column": 53
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx1 x2 y : ℕ × K\nH : R K p x1 x2\nn : ℕ\nx : K\n⊢ R K p ((n, x).1 + y.1, (⇑(frobenius K p))^[y.1] (n, x).2 * (⇑(frobenius K p))^[(n, x).1] y.2)\n ((n + 1, (frobenius K p) x).1 + y.1,\n (⇑(frobenius K p))^[y.... | [
"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx1 x2 y : ℕ × K\nH : R K p x1 x2\nn : ℕ\nx : K\n⊢ R K p ((n, x).1 + y.1, (⇑(frobenius K p))^[y.1] (n, x).2 * (⇑(frobenius K p))^[(n, x).1] y.2)\n ((n + 1, (frobenius K p) x).1 + y.1,\n (frobenius K p) ((⇑(frobenius K p)... | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PerfectClosure | {
"line": 123,
"column": 53
} | {
"line": 123,
"column": 73
} | {
"line": 123,
"column": 74
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx1 x2 y : ℕ × K\nH : R K p x1 x2\nn : ℕ\nx : K\n⊢ R K p ((n, x).1 + y.1, (⇑(frobenius K p))^[y.1] (n, x).2 * (⇑(frobenius K p))^[(n, x).1] y.2)\n ((n + 1, (frobenius K p) x).1 + y.1,\n (frobenius K p) ((⇑(fr... | [
"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx1 x2 y : ℕ × K\nH : R K p x1 x2\nn : ℕ\nx : K\n⊢ R K p ((n, x).1 + y.1, (⇑(frobenius K p))^[y.1] (n, x).2 * (⇑(frobenius K p))^[(n, x).1] y.2)\n ((n + 1, (frobenius K p) x).1 + y.1,\n (frobenius K p) ((⇑(frobenius K p)... | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PerfectClosure | {
"line": 134,
"column": 32
} | {
"line": 134,
"column": 52
} | {
"line": 134,
"column": 53
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y1 y2 : ℕ × K\nH : R K p y1 y2\nn : ℕ\ny : K\n⊢ R K p (x.1 + (n, y).1, (⇑(frobenius K p))^[(n, y).1] x.2 * (⇑(frobenius K p))^[x.1] (n, y).2)\n (x.1 + (n + 1, (frobenius K p) y).1,\n (⇑(frobenius K p))^[(n... | [
"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y1 y2 : ℕ × K\nH : R K p y1 y2\nn : ℕ\ny : K\n⊢ R K p (x.1 + (n, y).1, (⇑(frobenius K p))^[(n, y).1] x.2 * (⇑(frobenius K p))^[x.1] (n, y).2)\n (x.1 + (n + 1, (frobenius K p) y).1, (frobenius K p) ((⇑(frobenius K p))^[n] x... | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PerfectClosure | {
"line": 134,
"column": 53
} | {
"line": 134,
"column": 73
} | {
"line": 134,
"column": 74
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y1 y2 : ℕ × K\nH : R K p y1 y2\nn : ℕ\ny : K\n⊢ R K p (x.1 + (n, y).1, (⇑(frobenius K p))^[(n, y).1] x.2 * (⇑(frobenius K p))^[x.1] (n, y).2)\n (x.1 + (n + 1, (frobenius K p) y).1, (frobenius K p) ((⇑(frobenius... | [
"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y1 y2 : ℕ × K\nH : R K p y1 y2\nn : ℕ\ny : K\n⊢ R K p (x.1 + (n, y).1, (⇑(frobenius K p))^[(n, y).1] x.2 * (⇑(frobenius K p))^[x.1] (n, y).2)\n (x.1 + (n + 1, (frobenius K p) y).1,\n (frobenius K p) ((⇑(frobenius K p)... | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PerfectClosure | {
"line": 191,
"column": 32
} | {
"line": 191,
"column": 52
} | {
"line": 191,
"column": 53
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx1 x2 y : ℕ × K\nH : R K p x1 x2\nn : ℕ\nx : K\n⊢ R K p ((n, x).1 + y.1, (⇑(frobenius K p))^[y.1] (n, x).2 + (⇑(frobenius K p))^[(n, x).1] y.2)\n ((n + 1, (frobenius K p) x).1 + y.1,\n (⇑(frobenius K p))^[y.... | [
"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx1 x2 y : ℕ × K\nH : R K p x1 x2\nn : ℕ\nx : K\n⊢ R K p ((n, x).1 + y.1, (⇑(frobenius K p))^[y.1] (n, x).2 + (⇑(frobenius K p))^[(n, x).1] y.2)\n ((n + 1, (frobenius K p) x).1 + y.1,\n (frobenius K p) ((⇑(frobenius K p)... | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PerfectClosure | {
"line": 191,
"column": 53
} | {
"line": 191,
"column": 73
} | {
"line": 191,
"column": 74
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx1 x2 y : ℕ × K\nH : R K p x1 x2\nn : ℕ\nx : K\n⊢ R K p ((n, x).1 + y.1, (⇑(frobenius K p))^[y.1] (n, x).2 + (⇑(frobenius K p))^[(n, x).1] y.2)\n ((n + 1, (frobenius K p) x).1 + y.1,\n (frobenius K p) ((⇑(fr... | [
"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx1 x2 y : ℕ × K\nH : R K p x1 x2\nn : ℕ\nx : K\n⊢ R K p ((n, x).1 + y.1, (⇑(frobenius K p))^[y.1] (n, x).2 + (⇑(frobenius K p))^[(n, x).1] y.2)\n ((n + 1, (frobenius K p) x).1 + y.1,\n (frobenius K p) ((⇑(frobenius K p)... | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PerfectClosure | {
"line": 202,
"column": 32
} | {
"line": 202,
"column": 52
} | {
"line": 202,
"column": 53
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y1 y2 : ℕ × K\nH : R K p y1 y2\nn : ℕ\ny : K\n⊢ R K p (x.1 + (n, y).1, (⇑(frobenius K p))^[(n, y).1] x.2 + (⇑(frobenius K p))^[x.1] (n, y).2)\n (x.1 + (n + 1, (frobenius K p) y).1,\n (⇑(frobenius K p))^[(n... | [
"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y1 y2 : ℕ × K\nH : R K p y1 y2\nn : ℕ\ny : K\n⊢ R K p (x.1 + (n, y).1, (⇑(frobenius K p))^[(n, y).1] x.2 + (⇑(frobenius K p))^[x.1] (n, y).2)\n (x.1 + (n + 1, (frobenius K p) y).1, (frobenius K p) ((⇑(frobenius K p))^[n] x... | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PerfectClosure | {
"line": 202,
"column": 53
} | {
"line": 202,
"column": 73
} | {
"line": 202,
"column": 74
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y1 y2 : ℕ × K\nH : R K p y1 y2\nn : ℕ\ny : K\n⊢ R K p (x.1 + (n, y).1, (⇑(frobenius K p))^[(n, y).1] x.2 + (⇑(frobenius K p))^[x.1] (n, y).2)\n (x.1 + (n + 1, (frobenius K p) y).1, (frobenius K p) ((⇑(frobenius... | [
"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y1 y2 : ℕ × K\nH : R K p y1 y2\nn : ℕ\ny : K\n⊢ R K p (x.1 + (n, y).1, (⇑(frobenius K p))^[(n, y).1] x.2 + (⇑(frobenius K p))^[x.1] (n, y).2)\n (x.1 + (n + 1, (frobenius K p) y).1,\n (frobenius K p) ((⇑(frobenius K p)... | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Galois.Profinite | {
"line": 137,
"column": 6
} | {
"line": 140,
"column": 88
} | {
"line": 140,
"column": 89
} | [
{
"pp": "k✝ : Type u_1\nK✝ : Type u_2\ninst✝⁵ : Field k✝\ninst✝⁴ : Field K✝\ninst✝³ : Algebra k✝ K✝\nk : Type u_3\nK : Type u_4\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\nσ : Gal(K/k)\nL₁ L₂ : (FiniteGaloisIntermediateField k K)ᵒᵖ\nπ : L₁ ⟶ L₂\n⊢ (Hom.hom ((asProfiniteGaloisGroupFunctor k K).map ... | [] | algebraize [Subsemiring.inclusion π.1.le]
have : IsScalarTower k L₂.unop L₁.unop := IsScalarTower.of_algebraMap_eq (congrFun rfl)
have : IsScalarTower L₂.unop L₁.unop K := IsScalarTower.of_algebraMap_eq (congrFun rfl)
apply (IsScalarTower.AlgEquiv.restrictNormalHom_comp_apply L₂.unop L₁.unop σ).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Galois.Profinite | {
"line": 137,
"column": 6
} | {
"line": 140,
"column": 88
} | {
"line": 140,
"column": 89
} | [
{
"pp": "k✝ : Type u_1\nK✝ : Type u_2\ninst✝⁵ : Field k✝\ninst✝⁴ : Field K✝\ninst✝³ : Algebra k✝ K✝\nk : Type u_3\nK : Type u_4\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\nσ : Gal(K/k)\nL₁ L₂ : (FiniteGaloisIntermediateField k K)ᵒᵖ\nπ : L₁ ⟶ L₂\n⊢ (Hom.hom ((asProfiniteGaloisGroupFunctor k K).map ... | [] | algebraize [Subsemiring.inclusion π.1.le]
have : IsScalarTower k L₂.unop L₁.unop := IsScalarTower.of_algebraMap_eq (congrFun rfl)
have : IsScalarTower L₂.unop L₁.unop K := IsScalarTower.of_algebraMap_eq (congrFun rfl)
apply (IsScalarTower.AlgEquiv.restrictNormalHom_comp_apply L₂.unop L₁.unop σ).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Galois.Profinite | {
"line": 264,
"column": 4
} | {
"line": 268,
"column": 87
} | {
"line": 269,
"column": 2
} | [
{
"pp": "k✝ : Type u_1\nK✝ : Type u_2\ninst✝⁶ : Field k✝\ninst✝⁵ : Field K✝\ninst✝⁴ : Algebra k✝ K✝\nk : Type u_3\nK : Type u_4\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\ng : ↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop\nx y : K\n⊢ toAlgEquivAux g (x + y) ... | [] | have hx : x ∈ (adjoin k {x, y}).1 := subset_adjoin _ _ (Set.mem_insert x {y})
have hy : y ∈ (adjoin k {x, y}).1 := subset_adjoin _ _ (Set.mem_insert_of_mem x rfl)
simp only [toAlgEquivAux_eq_liftNormal g x (adjoin k {x, y}) hx,
toAlgEquivAux_eq_liftNormal g y (adjoin k {x, y}) hy,
toAlgEquivAux_eq_l... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Galois.Profinite | {
"line": 264,
"column": 4
} | {
"line": 268,
"column": 87
} | {
"line": 269,
"column": 2
} | [
{
"pp": "k✝ : Type u_1\nK✝ : Type u_2\ninst✝⁶ : Field k✝\ninst✝⁵ : Field K✝\ninst✝⁴ : Algebra k✝ K✝\nk : Type u_3\nK : Type u_4\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\ng : ↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop\nx y : K\n⊢ toAlgEquivAux g (x + y) ... | [] | have hx : x ∈ (adjoin k {x, y}).1 := subset_adjoin _ _ (Set.mem_insert x {y})
have hy : y ∈ (adjoin k {x, y}).1 := subset_adjoin _ _ (Set.mem_insert_of_mem x rfl)
simp only [toAlgEquivAux_eq_liftNormal g x (adjoin k {x, y}) hx,
toAlgEquivAux_eq_liftNormal g y (adjoin k {x, y}) hy,
toAlgEquivAux_eq_l... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.CardinalEmb | {
"line": 177,
"column": 66
} | {
"line": 177,
"column": 84
} | {
"line": 177,
"column": 84
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nrank_inf : Fact (ℵ₀ ≤ Module.rank F E)\ninst✝ : Algebra.IsAlgebraic F E\ni : (Module.rank F E).ord.ToType\n⊢ adjoin F (⇑b ∘ φ '' Iio i ∪ {(⇑b ∘ φ) i}) =\n IntermediateField.restrictScalars F (↥(adjoin F (⇑b ∘ φ '' Iio ... | [
"F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nrank_inf : Fact (ℵ₀ ≤ Module.rank F E)\ninst✝ : Algebra.IsAlgebraic F E\ni : (Module.rank F E).ord.ToType\n⊢ adjoin F (⇑b ∘ φ '' Iio i ∪ {(⇑b ∘ φ) i}) = adjoin F (⇑b ∘ φ '' Iio i ∪ {b (φ i)})"
] | adjoin_adjoin_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.IsRealClosed.Basic | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 38
} | {
"line": 100,
"column": 0
} | [
{
"pp": "case h.inr\nR : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nn : ℕ\nih : ∀ m < n, ∀ {x : R}, IsSquare x → m ≠ 0 → ∃ r, x = r ^ m\nx : R\nhx : IsSquare x\nhn : n ≠ 0\nodd : Odd n\n⊢ ∃ r, x = r ^ n",
"ppTerm": "?h.inr",
"assigned": true,
"usedConstants": [
"IsRealClosed.exists_eq_p... | [] | · exact exists_eq_pow_of_odd x odd | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.Isaacs | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 84
} | {
"line": 55,
"column": 2
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nalg : Algebra.IsAlgebraic F E\nh : ∀ (x : E), ∃ y, (aeval y) (minpoly F x) = 0\nS : Finset E\np : K[X] := ∏ x ∈ S, Polynomial.map (algebraMap F K) (minpoly F x)\nK'... | [
"F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nalg : Algebra.IsAlgebraic F E\nh : ∀ (x : E), ∃ y, (aeval y) (minpoly F x) = 0\nS : Finset E\np : K[X] := ∏ x ∈ S, Polynomial.map (algebraMap F K) (minpoly F x)\nK' : Type u_3 ... | have := finiteDimensional_adjoin (S := (S : Set E)) fun _ _ ↦ (alg.isIntegral).1 _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.KummerExtension | {
"line": 125,
"column": 48
} | {
"line": 127,
"column": 82
} | {
"line": 128,
"column": 4
} | [
{
"pp": "p n : ℕ\nhp : Nat.Prime p\nIH :\n ∀ {K : Type u} [inst : Field K],\n Odd n → ∀ {a : K}, (∀ (p : ℕ), Nat.Prime p → p ∣ n → ∀ (b : K), b ^ p ≠ a) → Irreducible (X ^ n - C a)\nK : Type u\ninst✝² : Field K\nhn : Odd (p * n)\na : K\nha : ∀ (p_1 : ℕ), Nat.Prime p_1 → p_1 ∣ p * n → ∀ (b : K), b ^ p_1 ≠ a\... | [] | by
simpa only [degree_zero, degree_X_pow_sub_C hp.pos,
WithBot.natCast_ne_bot] using congr_arg degree (hx.symm.trans (dif_neg h)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.KummerExtension | {
"line": 131,
"column": 4
} | {
"line": 132,
"column": 56
} | {
"line": 133,
"column": 4
} | [
{
"pp": "case prime_mul\np n : ℕ\nhp : Nat.Prime p\nIH :\n ∀ {K : Type u} [inst : Field K],\n Odd n → ∀ {a : K}, (∀ (p : ℕ), Nat.Prime p → p ∣ n → ∀ (b : K), b ^ p ≠ a) → Irreducible (X ^ n - C a)\nK : Type u\ninst✝² : Field K\nhn : Odd (p * n)\na : K\nha : ∀ (p_1 : ℕ), Nat.Prime p_1 → p_1 ∣ p * n → ∀ (b : ... | [
"case prime_mul\np n : ℕ\nhp : Nat.Prime p\nIH :\n ∀ {K : Type u} [inst : Field K],\n Odd n → ∀ {a : K}, (∀ (p : ℕ), Nat.Prime p → p ∣ n → ∀ (b : K), b ^ p ≠ a) → Irreducible (X ^ n - C a)\nK : Type u\ninst✝² : Field K\nhn : Odd (p * n)\na : K\nha : ∀ (p_1 : ℕ), Nat.Prime p_1 → p_1 ∣ p * n → ∀ (b : K), b ^ p_1 ... | rw [← map_pow, hb, ← adjoin.powerBasis_gen this,
Algebra.PowerBasis.norm_gen_eq_coeff_zero_minpoly] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.KummerExtension | {
"line": 509,
"column": 4
} | {
"line": 509,
"column": 75
} | {
"line": 510,
"column": 4
} | [
{
"pp": "case refine_2\nK : Type u\ninst✝⁵ : Field K\nL : Type u_1\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : FiniteDimensional K L\nhK : (primitiveRoots (finrank K L) K).Nonempty\ninst✝¹ : IsGalois K L\ninst✝ : IsCyclic Gal(L/K)\nζ : K\nhζ : IsPrimitiveRoot ζ (finrank K L)\nσ : Gal(L/K)\nhσ : Function.S... | [
"case refine_2\nK : Type u\ninst✝⁵ : Field K\nL : Type u_1\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : FiniteDimensional K L\nhK : (primitiveRoots (finrank K L) K).Nonempty\ninst✝¹ : IsGalois K L\ninst✝ : IsCyclic Gal(L/K)\nζ : K\nhζ : IsPrimitiveRoot ζ (finrank K L)\nσ : Gal(L/K)\nhσ : Function.Surjective fu... | rw [pow_mul, ← IsGalois.card_aut_eq_finrank, pow_card_eq_one', one_pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Minpoly.ConjRootClass | {
"line": 193,
"column": 2
} | {
"line": 199,
"column": 24
} | {
"line": 201,
"column": 0
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : Algebra.IsSeparable K L\ninst✝¹ : Normal K L\nc : ConjRootClass K L\ninst✝ : Fintype ↑c.carrier\n⊢ map (algebraMap K L) c.minpoly = ∏ x ∈ c.carrier.toFinset, (X - C x)",
"ppTerm": "?m.65",
"assigned":... | [] | classical
simp_rw [← rootSet_minpoly_eq_carrier, Finset.prod_eq_multiset_prod, rootSet_def,
Finset.toFinset_coe, Multiset.toFinset_val]
rw [Multiset.dedup_eq_self.mpr (nodup_roots c.separable_minpoly.map),
prod_multiset_X_sub_C_of_monic_of_roots_card_eq (c.monic_minpoly.map _)]
rw [← splits_iff_card_roots... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.FieldTheory.Minpoly.ConjRootClass | {
"line": 193,
"column": 2
} | {
"line": 199,
"column": 24
} | {
"line": 201,
"column": 0
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : Algebra.IsSeparable K L\ninst✝¹ : Normal K L\nc : ConjRootClass K L\ninst✝ : Fintype ↑c.carrier\n⊢ map (algebraMap K L) c.minpoly = ∏ x ∈ c.carrier.toFinset, (X - C x)",
"ppTerm": "?m.65",
"assigned":... | [] | classical
simp_rw [← rootSet_minpoly_eq_carrier, Finset.prod_eq_multiset_prod, rootSet_def,
Finset.toFinset_coe, Multiset.toFinset_val]
rw [Multiset.dedup_eq_self.mpr (nodup_roots c.separable_minpoly.map),
prod_multiset_X_sub_C_of_monic_of_roots_card_eq (c.monic_minpoly.map _)]
rw [← splits_iff_card_roots... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Minpoly.ConjRootClass | {
"line": 193,
"column": 2
} | {
"line": 199,
"column": 24
} | {
"line": 201,
"column": 0
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : Algebra.IsSeparable K L\ninst✝¹ : Normal K L\nc : ConjRootClass K L\ninst✝ : Fintype ↑c.carrier\n⊢ map (algebraMap K L) c.minpoly = ∏ x ∈ c.carrier.toFinset, (X - C x)",
"ppTerm": "?m.65",
"assigned":... | [] | classical
simp_rw [← rootSet_minpoly_eq_carrier, Finset.prod_eq_multiset_prod, rootSet_def,
Finset.toFinset_coe, Multiset.toFinset_val]
rw [Multiset.dedup_eq_self.mpr (nodup_roots c.separable_minpoly.map),
prod_multiset_X_sub_C_of_monic_of_roots_card_eq (c.monic_minpoly.map _)]
rw [← splits_iff_card_roots... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.NormalizedTrace | {
"line": 74,
"column": 55
} | {
"line": 74,
"column": 69
} | {
"line": 74,
"column": 70
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝⁴ : Field F\ninst✝³ : Field K\ninst✝² : Algebra F K\ninst✝¹ : CharZero F\ninst✝ : FiniteDimensional F K\na : K\nh : ↑(Module.finrank (↥F⟮a⟯) K) ≠ 0\n⊢ normalizedTraceAux F K a = (↑(Module.finrank F ↥F⟮a⟯))⁻¹ * (trace F ↥F⟮a⟯) (AdjoinSimple.gen F a)",
"ppTerm": "?m.1... | [
"F : Type u_1\nK : Type u_2\ninst✝⁴ : Field F\ninst✝³ : Field K\ninst✝² : Algebra F K\ninst✝¹ : CharZero F\ninst✝ : FiniteDimensional F K\na : K\nh : ↑(Module.finrank (↥F⟮a⟯) K) ≠ 0\n⊢ normalizedTraceAux F K a = (↑(Module.finrank F ↥F⟮a⟯))⁻¹ • (trace F ↥F⟮a⟯) (AdjoinSimple.gen F a)"
] | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.RatFunc.Degree | {
"line": 92,
"column": 8
} | {
"line": 92,
"column": 11
} | {
"line": 92,
"column": 12
} | [
{
"pp": "case pos\nK : Type u\ninst✝ : Field K\nx : K⟮X⟯\nhx : x = 0\n⊢ (-x).intDegree = x.intDegree",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"RatFunc.instNeg",
"id",
"Int",
"Field.toCommRing",
"RatFunc",
"Zero.to... | [
"case pos\nK : Type u\ninst✝ : Field K\nx : K⟮X⟯\nhx : x = 0\n⊢ (-0).intDegree = intDegree 0"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LinearDisjoint | {
"line": 761,
"column": 4
} | {
"line": 763,
"column": 20
} | {
"line": 764,
"column": 2
} | [
{
"pp": "case pos\nR : Type u\nS : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nA B : Subalgebra R S\ninst✝³ : Free R ↥A\ninst✝² : Free R ↥B\ninst✝¹ : Free ↥A ↥(Algebra.adjoin ↥A ↑B)\ninst✝ : Free ↥B ↥(Algebra.adjoin ↥B ↑A)\nH : (finrank R ↥A).Coprime (finrank R ↥B)\na✝ : Nontrivial R... | [] | rw [h2, Nat.coprime_zero_right] at H
rw [eq_bot_of_finrank_one H]
exact bot_left _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LinearDisjoint | {
"line": 761,
"column": 4
} | {
"line": 763,
"column": 20
} | {
"line": 764,
"column": 2
} | [
{
"pp": "case pos\nR : Type u\nS : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nA B : Subalgebra R S\ninst✝³ : Free R ↥A\ninst✝² : Free R ↥B\ninst✝¹ : Free ↥A ↥(Algebra.adjoin ↥A ↑B)\ninst✝ : Free ↥B ↥(Algebra.adjoin ↥B ↑A)\nH : (finrank R ↥A).Coprime (finrank R ↥B)\na✝ : Nontrivial R... | [] | rw [h2, Nat.coprime_zero_right] at H
rw [eq_bot_of_finrank_one H]
exact bot_left _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.PurelyInseparable.Tower | {
"line": 256,
"column": 58
} | {
"line": 259,
"column": 27
} | {
"line": 261,
"column": 0
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁸ : Field F\ninst✝⁷ : Field E\ninst✝⁶ : Algebra F E\nK : Type w\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\ninst✝³ : Algebra E K\ninst✝² : IsScalarTower F E K\nS : IntermediateField F K\ninst✝¹ : Algebra.IsAlgebraic F ↥S\ninst✝ : IsPurelyInseparable F E\n⊢ sepDegree E ↥(adjoin... | [] | by
have : Algebra.IsAlgebraic F (adjoin F (S : Set K)) := by rwa [adjoin_self]
have := sepDegree_adjoin_eq_of_isAlgebraic_of_isPurelyInseparable (F := F) E (S : Set K)
rwa [adjoin_self] at this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Relrank | {
"line": 97,
"column": 2
} | {
"line": 99,
"column": 37
} | {
"line": 101,
"column": 0
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B : Subfield E\n⊢ A.relrank B = 1 ↔ B ≤ A",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Subfield.subtype",
"Eq.mpr",
"Subfield.toDivisionRing",
"Lattice.toSemilatticeSup",
"Subfield.relrank",
"Subfield.toAlgebra... | [] | rw [relrank, IntermediateField.rank_eq_one_iff, ← IntermediateField.toSubfield_inj,
extendScalars_toSubfield, IntermediateField.bot_toSubfield, algebraMap_ofSubfield,
fieldRange_subtype, right_eq_inf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Relrank | {
"line": 97,
"column": 2
} | {
"line": 99,
"column": 37
} | {
"line": 101,
"column": 0
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B : Subfield E\n⊢ A.relrank B = 1 ↔ B ≤ A",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Subfield.subtype",
"Eq.mpr",
"Subfield.toDivisionRing",
"Lattice.toSemilatticeSup",
"Subfield.relrank",
"Subfield.toAlgebra... | [] | rw [relrank, IntermediateField.rank_eq_one_iff, ← IntermediateField.toSubfield_inj,
extendScalars_toSubfield, IntermediateField.bot_toSubfield, algebraMap_ofSubfield,
fieldRange_subtype, right_eq_inf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Relrank | {
"line": 97,
"column": 2
} | {
"line": 99,
"column": 37
} | {
"line": 101,
"column": 0
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B : Subfield E\n⊢ A.relrank B = 1 ↔ B ≤ A",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Subfield.subtype",
"Eq.mpr",
"Subfield.toDivisionRing",
"Lattice.toSemilatticeSup",
"Subfield.relrank",
"Subfield.toAlgebra... | [] | rw [relrank, IntermediateField.rank_eq_one_iff, ← IntermediateField.toSubfield_inj,
extendScalars_toSubfield, IntermediateField.bot_toSubfield, algebraMap_ofSubfield,
fieldRange_subtype, right_eq_inf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.RatFunc.Valuation | {
"line": 64,
"column": 8
} | {
"line": 64,
"column": 11
} | {
"line": 64,
"column": 12
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : DecidableEq F⟮X⟯\nx y : F⟮X⟯\nhx : x = 0\n⊢ (if x * y = 0 then 0 else exp (x * y).intDegree) =\n (if x = 0 then 0 else exp x.intDegree) * if y = 0 then 0 else exp y.intDegree",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
... | [
"case pos\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : DecidableEq F⟮X⟯\nx y : F⟮X⟯\nhx : x = 0\n⊢ (if 0 * y = 0 then 0 else exp (0 * y).intDegree) =\n (if 0 = 0 then 0 else exp (intDegree 0)) * if y = 0 then 0 else exp y.intDegree"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Convex.Cone.DualFinite | {
"line": 84,
"column": 13
} | {
"line": 84,
"column": 24
} | {
"line": 84,
"column": 24
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ns : Finset M\n⊢ dual p ↑s = dual p ↑(Submodule.span { c // 0 ≤ c } ... | [
"case h\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ns : Finset M\n⊢ dual p ↑(hull R ↑s) = dual p ↑(Submodule.span { c // 0 ≤ c } ↑s... | ← dual_hull | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.RatFunc.IntermediateField | {
"line": 61,
"column": 12
} | {
"line": 61,
"column": 29
} | {
"line": 61,
"column": 29
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nf : K⟮X⟯\n⊢ (algebraMap K[X] K⟮X⟯) f.num - f * (algebraMap K[X] K⟮X⟯) f.denom = 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"RatFunc.denom",
"HMul.hMul",
"Algebra.algebraMap",
"AddGroupW... | [
"K : Type u_1\ninst✝ : Field K\nf : K⟮X⟯\n⊢ (algebraMap K[X] K⟮X⟯) f.num -\n (algebraMap K[X] K⟮X⟯) f.num / (algebraMap K[X] K⟮X⟯) f.denom * (algebraMap K[X] K⟮X⟯) f.denom =\n 0"
] | ← num_div_denom f | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Convex.Hull | {
"line": 111,
"column": 46
} | {
"line": 111,
"column": 72
} | {
"line": 111,
"column": 72
} | [
{
"pp": "R : Type u_1\nX : Type u_2\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R X\ns : Set X\nx : X\nhs : IsConvexSet R s\nhx : x ∉ (convexHull R) (s \\ {x})\ny : X\nhy : y ∈ (convexHull R) (s \\ {x})\n⊢ y ∉ {x}",
"ppTerm": "?m.88",
"assigned": tr... | [] | by rintro rfl; exact hx hy | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Diffeology.Basic | {
"line": 387,
"column": 6
} | {
"line": 387,
"column": 38
} | {
"line": 388,
"column": 6
} | [
{
"pp": "X : Type u_1\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace ℝ X\ninst✝ : FiniteDimensional ℝ X\nn✝ : ℕ\nu✝ : Set (EuclideanSpace ℝ (Fin n✝))\nx✝¹ : IsOpen u✝\nx✝ : EuclideanSpace ℝ (Fin n✝) → X\nh : ∀ x ∈ u✝, ∃ v, ∃ (_ : IsOpen v), x ∈ v ∧ ContDiffOn ℝ ∞ x✝ v\nx : EuclideanSpace ℝ (Fin n✝)\nhxu :... | [
"X : Type u_1\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace ℝ X\ninst✝ : FiniteDimensional ℝ X\nn✝ : ℕ\nu✝ : Set (EuclideanSpace ℝ (Fin n✝))\nx✝¹ : IsOpen u✝\nx✝ : EuclideanSpace ℝ (Fin n✝) → X\nh : ∀ x ∈ u✝, ∃ v, ∃ (_ : IsOpen v), x ∈ v ∧ ContDiffOn ℝ ∞ x✝ v\nx : EuclideanSpace ℝ (Fin n✝)\nhxu : x ∈ u✝\nv :... | let ⟨v, hv, hxv, hv'⟩ := h x hxu | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | {
"line": 199,
"column": 33
} | {
"line": 199,
"column": 56
} | {
"line": 199,
"column": 56
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nhx : x ≠ 0\nhy : y ≠ 0\nθ : Real.Angle\n⊢ ↑(↑(-θ).toCircle).arg + ↑((o.kahler x) y).arg = ↑((o.kahler x) y).arg - θ",
"ppTerm": "?m.54",
"assigned":... | [
"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nhx : x ≠ 0\nhy : y ≠ 0\nθ : Real.Angle\n⊢ -θ + ↑((o.kahler x) y).arg = ↑((o.kahler x) y).arg - θ",
"case hx\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : Inn... | Real.Angle.arg_toCircle | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | {
"line": 209,
"column": 33
} | {
"line": 209,
"column": 56
} | {
"line": 209,
"column": 56
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nhx : x ≠ 0\nhy : y ≠ 0\nθ : Real.Angle\n⊢ ↑(↑θ.toCircle).arg + ↑((o.kahler x) y).arg = ↑((o.kahler x) y).arg + θ",
"ppTerm": "?m.54",
"assigned": tr... | [
"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nhx : x ≠ 0\nhy : y ≠ 0\nθ : Real.Angle\n⊢ θ + ↑((o.kahler x) y).arg = ↑((o.kahler x) y).arg + θ",
"case hx\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : Inne... | Real.Angle.arg_toCircle | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Projection | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 40
} | {
"line": 69,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np x : P\nhx : x ∈ s\n⊢ s.direc... | [
"𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np x : P\nhx : x ∈ s\n⊢ ↑(Classical.arbitra... | Submodule.starProjection_eq_self_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Projection | {
"line": 67,
"column": 2
} | {
"line": 70,
"column": 51
} | {
"line": 72,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np x : P\nhx : x ∈ s\n⊢ ↑((orth... | [] | rw [orthogonalProjection_apply, coe_vadd, vadd_eq_vadd_iff_sub_eq_vsub, ← Submodule.coe_sub,
← map_sub, vsub_sub_vsub_cancel_left, Submodule.coe_orthogonalProjectionOnto_apply,
Submodule.starProjection_eq_self_iff]
exact s.vsub_mem_direction (SetLike.coe_mem _) hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Projection | {
"line": 67,
"column": 2
} | {
"line": 70,
"column": 51
} | {
"line": 72,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np x : P\nhx : x ∈ s\n⊢ ↑((orth... | [] | rw [orthogonalProjection_apply, coe_vadd, vadd_eq_vadd_iff_sub_eq_vsub, ← Submodule.coe_sub,
← map_sub, vsub_sub_vsub_cancel_left, Submodule.coe_orthogonalProjectionOnto_apply,
Submodule.starProjection_eq_self_iff]
exact s.vsub_mem_direction (SetLike.coe_mem _) hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 263,
"column": 27
} | {
"line": 263,
"column": 78
} | {
"line": 263,
"column": 78
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : dist p₃ p₁ = dist p₃ p₂\n⊢ ∠ p₃ (midpoint ℝ p₂ p₁) p₂ = π / 2",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : dist p₃ p₁ = dist p₃ p₂\n⊢ π / 2 = π / 2"
] | angle_left_midpoint_eq_pi_div_two_of_dist_eq h.symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 21
} | {
"line": 262,
"column": 2
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ p₅ p₆ : P\nh₂ : p₂ ∉ line[ℝ, p₁, p₃]\nh₄ : p₄ ∈ line[ℝ, p₁, p₃]\nh₆ : p₆ ∈ line[ℝ, p₁,... | [
"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ p₅ p₆ : P\nh₂ : p₂ ∉ line[ℝ, p₁, p₃]\nh₄ : p₄ ∈ line[ℝ, p₁, p₃]\nh₆ : p₆ ∈ line[ℝ, p₁, p₃]\nh₁₂₄₅ ... | rw [oangle, oangle] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Projection | {
"line": 636,
"column": 2
} | {
"line": 636,
"column": 48
} | {
"line": 638,
"column": 0
} | [
{
"pp": "case h\n𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace 𝕜 V\nV₂ : Type u_4\nP₂ : Type u_5\ninst✝⁵ : NormedAddCommGroup V₂\ninst✝⁴ : InnerProductSpace 𝕜 V₂\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ninst✝¹ : MetricSpa... | [] | simp [AffineSubspace.map_span, Set.range_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 21
} | {
"line": 320,
"column": 2
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nhn : p₂ ≠ p₃\nh : ‖p₁ -ᵥ p₂‖ = ‖p₁ -ᵥ p₃‖\n⊢ ∡ p₃ p₁ p₂ = ↑π - 2 • ∡ p₁ p₂ p₃",
... | [
"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nhn : p₂ ≠ p₃\nh : ‖p₁ -ᵥ p₂‖ = ‖p₁ -ᵥ p₃‖\n⊢ o.oangle (p₃ -ᵥ p₁) (p₂ -ᵥ p₁) = ↑π - 2 • o.oangle ... | rw [oangle, oangle] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 384,
"column": 34
} | {
"line": 384,
"column": 63
} | {
"line": 385,
"column": 4
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\n⊢ SameRay ℝ x y ∨ o.oangle x y = ↑π ↔ ¬LinearIndependent ℝ ![x, y]",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\n⊢ SameRay ℝ x y ∨ x ≠ 0 ∧ y ≠ 0 ∧ SameRay ℝ x (-y) ↔ ¬LinearIndependent ℝ ![x, y]"
] | oangle_eq_pi_iff_sameRay_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 100
} | {
"line": 172,
"column": 2
} | [
{
"pp": "case neg.h.inl\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫ = 0\nh0 : ¬x = 0\n⊢ 0 < ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖",
"ppTerm": "?neg.h.inl✝",
"assigned": true,
"usedConstants": [
"mul_self_nonneg",
"Iff.mpr",
"AddGroup.toSubtracti... | [] | exact Left.add_pos_of_pos_of_nonneg (mul_self_pos.2 (norm_ne_zero_iff.2 h0)) (mul_self_nonneg _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 100
} | {
"line": 172,
"column": 2
} | [
{
"pp": "case neg.h.inl\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫ = 0\nh0 : ¬x = 0\n⊢ 0 < ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖",
"ppTerm": "?neg.h.inl✝",
"assigned": true,
"usedConstants": [
"mul_self_nonneg",
"Iff.mpr",
"AddGroup.toSubtracti... | [] | exact Left.add_pos_of_pos_of_nonneg (mul_self_pos.2 (norm_ne_zero_iff.2 h0)) (mul_self_nonneg _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 100
} | {
"line": 172,
"column": 2
} | [
{
"pp": "case neg.h.inl\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫ = 0\nh0 : ¬x = 0\n⊢ 0 < ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖",
"ppTerm": "?neg.h.inl✝",
"assigned": true,
"usedConstants": [
"mul_self_nonneg",
"Iff.mpr",
"AddGroup.toSubtracti... | [] | exact Left.add_pos_of_pos_of_nonneg (mul_self_pos.2 (norm_ne_zero_iff.2 h0)) (mul_self_nonneg _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 209,
"column": 91
} | {
"line": 215,
"column": 59
} | {
"line": 217,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nh : o.oangle x y = ↑(π / 2)\n⊢ ‖y‖ / (o.oangle x (x + y)).sin = ‖x + y‖",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Norm.norm"... | [] | by
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe,
InnerProductGeometry.norm_div_sin_angle_add_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inr (o.right_ne... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 253,
"column": 50
} | {
"line": 255,
"column": 55
} | {
"line": 257,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫ = 0\n⊢ Real.cos (angle x (x - y)) = ‖x‖ / ‖x - y‖",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"InnerProductGeometry.cos_angle_add_of_inner_eq_zero",
"Norm.norm",
"... | [] | by
rw [← neg_eq_zero, ← inner_neg_right] at h
rw [sub_eq_add_neg, cos_angle_add_of_inner_eq_zero h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 449,
"column": 2
} | {
"line": 450,
"column": 71
} | {
"line": 451,
"column": 2
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nhx : x ≠ 0\nhy : y ≠ 0\nhz : z ≠ 0\n⊢ ↑(↑‖y‖ ^ 2 * (o.kahler x) z).arg = ↑((o.kahler x) z).arg",
"ppTerm": "?m.61",
"assigned": true,
"usedCon... | [
"case hx\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nhx : x ≠ 0\nhy : y ≠ 0\nhz : z ≠ 0\n⊢ (o.kahler x) y ≠ 0",
"case hy\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : F... | · congr 1
exact mod_cast Complex.arg_real_mul _ (by positivity : 0 < ‖y‖ ^ 2) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 540,
"column": 2
} | {
"line": 540,
"column": 82
} | {
"line": 541,
"column": 2
} | [
{
"pp": "case neg\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nhp₁p₂ : ¬p₁ = p₂\nhp₃p₂ : ¬p₃ = p₂\n⊢ ∡ p₁ p₂ p₃ = 0 ↔ Wbtw ℝ p₂ p₁ p₃ ∨ W... | [
"case neg\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nhp₁p₂ : ¬p₁ = p₂\nhp₃p₂ : ¬p₃ = p₂\n⊢ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ ... | rw [oangle_eq_zero_iff_angle_eq_zero hp₁p₂ hp₃p₂, angle_eq_zero_iff_ne_and_wbtw] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 657,
"column": 15
} | {
"line": 657,
"column": 29
} | {
"line": 657,
"column": 30
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p : P\nh : p₁ ≠ p₂\nhd : ⟪p -ᵥ p₁, p -ᵥ p₁⟫ = ⟪p -ᵥ p₂, p -ᵥ p₂⟫\n⊢ ⟪p₂ -ᵥ p₁, p -ᵥ midpoint... | [
"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p : P\nh : p₁ ≠ p₂\nhd : ⟪p -ᵥ p₁, p -ᵥ p₁⟫ = ⟪p -ᵥ p₂, p -ᵥ p₂⟫\n⊢ ⟪p₂ -ᵥ p₁, ⅟2 • (p -ᵥ p₁) + ⅟2 • (p ... | vsub_midpoint, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 658,
"column": 30
} | {
"line": 658,
"column": 55
} | {
"line": 658,
"column": 55
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p : P\nh : p₁ ≠ p₂\nhd : ⟪p -ᵥ p₁, p -ᵥ p₁⟫ = ⟪p -ᵥ p₂, p -ᵥ p₂⟫\n⊢ ⅟2 * ⟪p -ᵥ p₂, p -ᵥ p₂⟫ ... | [
"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p : P\nh : p₁ ≠ p₂\nhd : ⟪p -ᵥ p₁, p -ᵥ p₁⟫ = ⟪p -ᵥ p₂, p -ᵥ p₂⟫\n⊢ ⅟2 * ⟪p -ᵥ p₂, p -ᵥ p₂⟫ + ⅟2 * ⟪p -ᵥ... | real_inner_comm (p -ᵥ p₁) | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 452,
"column": 2
} | {
"line": 452,
"column": 47
} | {
"line": 453,
"column": 2
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nh : x ≠ 0\nr : ℝ\n⊢ o.oangle x (x + r • (o.rotation ↑(π / 2)) x) = ↑(Real.arctan r)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"... | [
"case inl\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nh : x ≠ 0\nr : ℝ\nhr : r < 0\n⊢ o.oangle x (x + r • (o.rotation ↑(π / 2)) x) = ↑(Real.arctan r)",
"case inr.inl\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\nins... | rcases lt_trichotomy r 0 with (hr | rfl | hr) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 664,
"column": 4
} | {
"line": 664,
"column": 22
} | {
"line": 665,
"column": 2
} | [
{
"pp": "case refine_1\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p : P\nh : p₁ ≠ p₂\nhd : dist p₁ p = dist p₂ p\nr : ℝ\nhr : p = r • (o.rotati... | [] | exact ⟨r, hr.symm⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 461,
"column": 36
} | {
"line": 461,
"column": 50
} | {
"line": 461,
"column": 51
} | [
{
"pp": "case inl\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nh : x ≠ 0\nr : ℝ\nhr : r < 0\nha : (-o).oangle x (r • (o.rotation ↑(π / 2)) x) = ↑(π / 2)\n⊢ -↑(Real.arctan (‖r‖ * ‖x‖ / ‖x‖)) = ↑(Real.arctan r)",
... | [
"case inl\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nh : x ≠ 0\nr : ℝ\nhr : r < 0\nha : (-o).oangle x (r • (o.rotation ↑(π / 2)) x) = ↑(π / 2)\n⊢ -↑(Real.arctan (‖r‖ * (‖x‖ / ‖x‖))) = ↑(Real.arctan r)"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
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