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 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.ClassGroup.Basic | {
"line": 454,
"column": 6
} | {
"line": 455,
"column": 59
} | {
"line": 456,
"column": 6
} | [
{
"pp": "case h\nR : Type u_1\nK : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Algebra R K\ninst✝⁶ : IsFractionRing R K\ninst✝⁵ : IsDomain R\nS : Type u_3\nL : Type u_4\ninst✝⁴ : CommRing S\ninst✝³ : IsDomain S\ninst✝² : Field L\ninst✝¹ : Algebra S L\ninst✝ : IsFractionRing S L\nf : R ≃+* S\nI : (... | [
"case h\nR : Type u_1\nK : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Algebra R K\ninst✝⁶ : IsFractionRing R K\ninst✝⁵ : IsDomain R\nS : Type u_3\nL : Type u_4\ninst✝⁴ : CommRing S\ninst✝³ : IsDomain S\ninst✝² : Field L\ninst✝¹ : Algebra S L\ninst✝ : IsFractionRing S L\nf : R ≃+* S\nI : (FractionalId... | rw [← FractionalIdeal.ringEquivOfRingEquiv_spanSingleton,
← FractionalIdeal.ringEquivOfRingEquiv_symm_eq, hu] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Nat.Prime.Int | {
"line": 27,
"column": 6
} | {
"line": 28,
"column": 96
} | {
"line": 28,
"column": 96
} | [
{
"pp": "p : ℕ\nhp : Prime p\na b : ℤ\nh : ↑p ∣ a * b\n⊢ ↑p ∣ a ∨ ↑p ∣ b",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"Int.dvd_natAbs",
"Int.natAbs_mul",
"MulZeroClass.toMul",
"congrArg",
"semigroupDvd",
... | [] | rw [← Int.dvd_natAbs, Int.natCast_dvd_natCast, Int.natAbs_mul, hp.dvd_mul] at h
rwa [← Int.dvd_natAbs, Int.natCast_dvd_natCast, ← Int.dvd_natAbs, Int.natCast_dvd_natCast] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Prime.Int | {
"line": 27,
"column": 6
} | {
"line": 28,
"column": 96
} | {
"line": 28,
"column": 96
} | [
{
"pp": "p : ℕ\nhp : Prime p\na b : ℤ\nh : ↑p ∣ a * b\n⊢ ↑p ∣ a ∨ ↑p ∣ b",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"Int.dvd_natAbs",
"Int.natAbs_mul",
"MulZeroClass.toMul",
"congrArg",
"semigroupDvd",
... | [] | rw [← Int.dvd_natAbs, Int.natCast_dvd_natCast, Int.natAbs_mul, hp.dvd_mul] at h
rwa [← Int.dvd_natAbs, Int.natCast_dvd_natCast, ← Int.dvd_natAbs, Int.natCast_dvd_natCast] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 47
} | {
"line": 68,
"column": 2
} | [
{
"pp": "n : ℕ\na : ZMod (n + 1)\nh✝ : ¬0 ≤ a.valMinAbs\nhe : a.valMinAbs * 2 = ↑(n + 1)\nh : ¬0 ≤ ↑(n + 1)\n⊢ False",
"ppTerm": "?m.112",
"assigned": true,
"usedConstants": [
"Int.instIsStrictOrderedRing",
"Nat.cast_nonneg",
"SemilatticeInf.toPartialOrder",
"instOfNatNat",
... | [] | exacts [h (Nat.cast_nonneg _), zero_lt_two] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 280,
"column": 24
} | {
"line": 280,
"column": 37
} | {
"line": 280,
"column": 38
} | [
{
"pp": "case intro\nR : Type uR\ninst✝⁵ : CommRing R\nι : Type u_1\ninst✝⁴ : Finite ι\nM : ι → Type u_2\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : ∀ (i : ι), Flat R (M i)\ninst✝ : ∀ (i : ι), Module.Finite R (M i)\np : PrimeSpectrum R\nval✝ : Fintype ι\nf : ((i : ι) → M ... | [
"case intro\nR : Type uR\ninst✝⁵ : CommRing R\nι : Type u_1\ninst✝⁴ : Finite ι\nM : ι → Type u_2\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : ∀ (i : ι), Flat R (M i)\ninst✝ : ∀ (i : ι), Module.Finite R (M i)\np : PrimeSpectrum R\nval✝ : Fintype ι\nf : ((i : ι) → M i) →ₗ[R] (i ... | e.finrank_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 350,
"column": 44
} | {
"line": 350,
"column": 57
} | {
"line": 350,
"column": 58
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : Flat R M\ninst✝⁴ : Module.Finite R M\nN : Type u_1\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Module.Finite R N\ninst✝ : Flat R N\np : PrimeSpectrum R\ne : Localization.AtPrime p.asIdeal ⊗... | [
"R : Type uR\nM : Type uM\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : Flat R M\ninst✝⁴ : Module.Finite R M\nN : Type u_1\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Module.Finite R N\ninst✝ : Flat R N\np : PrimeSpectrum R\ne : Localization.AtPrime p.asIdeal ⊗[R] (M ⊗[R] ... | e.finrank_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 711,
"column": 42
} | {
"line": 711,
"column": 66
} | {
"line": 711,
"column": 67
} | [
{
"pp": "case inr.inr\nn p k : ℕ\ninst✝ : Fact (Nat.Prime p)\nhn : n = p ^ k\nhn0 : n ≠ 0\nthis : NeZero n\na : ℕ\nha : (orderOf ↑a).Coprime n\nh : ¬IsUnit ↑(a - 1)\nha0 : 1 ≤ a\n⊢ a ^ p ^ k ≡ 1 [MOD p ^ k]",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"ZMod.isUnit_iff_coprime",... | [
"case inr.inr\nn p k : ℕ\ninst✝ : Fact (Nat.Prime p)\nhn : n = p ^ k\nhn0 : n ≠ 0\nthis : NeZero n\na : ℕ\nha : (orderOf ↑a).Coprime n\nh : ¬(a - 1).Coprime n\nha0 : 1 ≤ a\n⊢ a ^ p ^ k ≡ 1 [MOD p ^ k]"
] | ZMod.isUnit_iff_coprime, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 54
} | {
"line": 46,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : DecidableEq α\ns : Finset α\ni : α → ℕ\np : α\nhps : p ∉ s\nis_prime : ∀ q ∈ insert p s, Prime q\nis_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'\nhp : Prime p\nd : α\nhdprod : d ∣ ∏ p' ∈ s, p' ... | [
"α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : DecidableEq α\ns : Finset α\ni : α → ℕ\np : α\nhps : p ∉ s\nis_prime : ∀ q ∈ insert p s, Prime q\nis_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'\nhp : Prime p\nd : α\nhdprod : d ∣ ∏ p' ∈ s, p' ^ i p'\nhd :... | obtain ⟨q, q_mem, rfl⟩ := Multiset.mem_map.mp q_mem' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative | {
"line": 125,
"column": 13
} | {
"line": 125,
"column": 22
} | {
"line": 125,
"column": 23
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_3\ninst✝ : CommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, IsRel... | [
"case pos\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_3\ninst✝ : CommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, IsRelPrime x y → ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 141,
"column": 2
} | {
"line": 145,
"column": 45
} | {
"line": 146,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsSeparable F E\nhα : IsIntegral F α\nhβ : IsIntegral F β\nf : F[X] := minpoly F α\ng : F[X] := minpoly F β\nιFE : F →+* E := algebraMap F E\nιEE' : E →+* (Polynomial.map ... | [
"F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsSeparable F E\nhα : IsIntegral F α\nhβ : IsIntegral F β\nf : F[X] := minpoly F α\ng : F[X] := minpoly F β\nιFE : F →+* E := algebraMap F E\nιEE' : E →+* (Polynomial.map ιFE g).Split... | have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map_algebraMap, ←
Algebra.smul_def, add_sub_cancel_right, minpoly.aeval]
· rw [eval_map_algebraMap, minpoly.aeval] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 83,
"column": 2
} | {
"line": 91,
"column": 30
} | {
"line": 93,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\n⊢ p.ramificationIdx P = n",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.to... | [] | let Q : ℕ → Prop := fun m => ∀ k : ℕ, map f p ≤ P ^ k → k ≤ m
have : Q n := by
intro k hk
refine le_of_not_gt fun hnk => ?_
exact hgt (hk.trans (Ideal.pow_le_pow_right hnk))
rw [ramificationIdx_eq_find ⟨n, this⟩]
refine le_antisymm (Nat.find_min' _ this) (le_of_not_gt fun h : Nat.find _ < n => ?_)
o... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 83,
"column": 2
} | {
"line": 91,
"column": 30
} | {
"line": 93,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\n⊢ p.ramificationIdx P = n",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.to... | [] | let Q : ℕ → Prop := fun m => ∀ k : ℕ, map f p ≤ P ^ k → k ≤ m
have : Q n := by
intro k hk
refine le_of_not_gt fun hnk => ?_
exact hgt (hk.trans (Ideal.pow_le_pow_right hnk))
rw [ramificationIdx_eq_find ⟨n, this⟩]
refine le_antisymm (Nat.find_min' _ this) (le_of_not_gt fun h : Nat.find _ < n => ?_)
o... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 98,
"column": 4
} | {
"line": 100,
"column": 55
} | {
"line": 101,
"column": 4
} | [
{
"pp": "case succ\nR : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nn : ℕ\nhgt : ¬map f p ≤ P ^ (n + 1)\n⊢ p.ramificationIdx P ≤ n",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Semiring.toModul... | [
"case succ\nR : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nn : ℕ\nhgt : ¬map f p ≤ P ^ (n + 1)\nthis : ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ n\n⊢ p.ramificationIdx P ≤ n"
] | have : ∀ k, map f p ≤ P ^ k → k ≤ n := by
refine fun k hk => le_of_not_gt fun hnk => ?_
exact hgt (hk.trans (Ideal.pow_le_pow_right hnk)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 33
} | {
"line": 188,
"column": 4
} | [
{
"pp": "case h.a\nF : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Finite (IntermediateField F E)\nf : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯\nx y : F\nhneq : ¬x = y\nheq : F⟮α + x • β⟯ = F⟮α + y • β⟯\nαxβ_in_K : α + x • β ∈ ... | [
"case h.a\nF : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Finite (IntermediateField F E)\nf : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯\nx y : F\nhneq : ¬x = y\nheq : F⟮α + x • β⟯ = F⟮α + y • β⟯\nαxβ_in_K : α + x • β ∈ F⟮α + x • β⟯... | simp only [← heq] at αyβ_in_K | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 304,
"column": 71
} | {
"line": 304,
"column": 80
} | {
"line": 305,
"column": 6
} | [
{
"pp": "case neg\nR : Type u\ninst✝⁶ : CommRing R\nS : Type v\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsDedekindDomain S\ninst✝² : IsDedekindDomain R\ninst✝¹ : FaithfulSMul R S\nv : Ideal R\nw : Ideal S\np : Ideal R\nhv : Irreducible v\nhp : Prime p\nhw : Irreducible w\nhw_bot : w ≠ ⊥\ninst✝ : w.L... | [
"case neg\nR : Type u\ninst✝⁶ : CommRing R\nS : Type v\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsDedekindDomain S\ninst✝² : IsDedekindDomain R\ninst✝¹ : FaithfulSMul R S\nv : Ideal R\nw : Ideal S\np : Ideal R\nhv : Irreducible v\nhp : Prime p\nhw : Irreducible w\nhw_bot : w ≠ ⊥\ninst✝ : w.LiesOver v\nh... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 323,
"column": 86
} | {
"line": 323,
"column": 94
} | {
"line": 324,
"column": 6
} | [
{
"pp": "case h₃\nR : Type u\ninst✝⁶ : CommRing R\nS : Type v\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsDedekindDomain S\ninst✝² : IsDedekindDomain R\ninst✝¹ : FaithfulSMul R S\nv : Ideal R\nw : Ideal S\nhv : Irreducible v\nhw : Irreducible w\nhw_bot : w ≠ ⊥\ninst✝ : w.LiesOver v\nI p : Ideal R\nhI... | [
"case h₃\nR : Type u\ninst✝⁶ : CommRing R\nS : Type v\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsDedekindDomain S\ninst✝² : IsDedekindDomain R\ninst✝¹ : FaithfulSMul R S\nv : Ideal R\nw : Ideal S\nhv : Irreducible v\nhw : Irreducible w\nhw_bot : w ≠ ⊥\ninst✝ : w.LiesOver v\nI p : Ideal R\nhI : I ≠ 0\nhp... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 336,
"column": 4
} | {
"line": 336,
"column": 41
} | {
"line": 337,
"column": 2
} | [
{
"pp": "case mp\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\nI : Ideal S\nhI : absNorm I = 0\nx : S\nhx : x ∈ I\n⊢ span {x} ≤ I",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
... | [] | rwa [Ideal.span_singleton_le_iff_mem] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 426,
"column": 6
} | {
"line": 432,
"column": 19
} | {
"line": 433,
"column": 4
} | [
{
"pp": "case inr.refine_1\nS : Type u_1\ninst✝⁴ : CommRing S\ninst✝³ : IsDedekindDomain S\ninst✝² : Free ℤ S\ninst✝¹ : Module.Finite ℤ S\ninst✝ : CharZero S\nn : ℕ\nhn : n > 0\nf : Ideal S → Ideal (S ⧸ span {↑n}) := fun I ↦ map (Quotient.mk (span {↑n})) I\n⊢ (f '' {I | absNorm I = n}).Finite",
"ppTerm": "?... | [] | suffices Finite (S ⧸ @Ideal.span S _ {↑n}) by
let g := ((↑) : Ideal (S ⧸ @Ideal.span S _ {↑n}) → Set (S ⧸ @Ideal.span S _ {↑n}))
refine Set.Finite.of_finite_image (f := g) ?_ SetLike.coe_injective.injOn
exact Set.Finite.subset Set.finite_univ (Set.subset_univ _)
rw [← absNorm_ne_zero_iff, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 426,
"column": 6
} | {
"line": 432,
"column": 19
} | {
"line": 433,
"column": 4
} | [
{
"pp": "case inr.refine_1\nS : Type u_1\ninst✝⁴ : CommRing S\ninst✝³ : IsDedekindDomain S\ninst✝² : Free ℤ S\ninst✝¹ : Module.Finite ℤ S\ninst✝ : CharZero S\nn : ℕ\nhn : n > 0\nf : Ideal S → Ideal (S ⧸ span {↑n}) := fun I ↦ map (Quotient.mk (span {↑n})) I\n⊢ (f '' {I | absNorm I = n}).Finite",
"ppTerm": "?... | [] | suffices Finite (S ⧸ @Ideal.span S _ {↑n}) by
let g := ((↑) : Ideal (S ⧸ @Ideal.span S _ {↑n}) → Set (S ⧸ @Ideal.span S _ {↑n}))
refine Set.Finite.of_finite_image (f := g) ?_ SetLike.coe_injective.injOn
exact Set.Finite.subset Set.finite_univ (Set.subset_univ _)
rw [← absNorm_ne_zero_iff, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Prod.TProd | {
"line": 88,
"column": 75
} | {
"line": 88,
"column": 100
} | {
"line": 90,
"column": 0
} | [
{
"pp": "ι : Type u\nα : ι → Type v\ni j : ι\nl : List ι\ninst✝ : DecidableEq ι\nhj : j ∈ i :: l\nhji : j ≠ i\nv : TProd α (i :: l)\n⊢ v.elim hj = TProd.elim v.2 ⋯",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"of_eq_false",
"eq_false",
"congrArg",
"Membership.mem... | [] | by simp [TProd.elim, hji] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 215,
"column": 2
} | {
"line": 217,
"column": 25
} | {
"line": 219,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np q : α → Prop\nhs : MeasurableSet {x | p x}\nht : MeasurableSet {x | q x}\n⊢ MeasurableSet {x | p x → q x}",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"Compl.compl",
"set... | [] | have h_eq : {x | p x → q x} = {x | p x}ᶜ ∪ {x | q x} := by grind
rw [h_eq]
exact hs.compl.union ht | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 215,
"column": 2
} | {
"line": 217,
"column": 25
} | {
"line": 219,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np q : α → Prop\nhs : MeasurableSet {x | p x}\nht : MeasurableSet {x | q x}\n⊢ MeasurableSet {x | p x → q x}",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"Compl.compl",
"set... | [] | have h_eq : {x | p x → q x} = {x | p x}ᶜ ∪ {x | q x} := by grind
rw [h_eq]
exact hs.compl.union ht | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.Basic | {
"line": 369,
"column": 2
} | {
"line": 369,
"column": 51
} | {
"line": 370,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nC : Set (Set α)\nD : Set (Set β)\ns : ℕ → Set α\nh1s : ∀ (n : ℕ), s n ∈ C\nh2s : ⋃ n, s n = univ\nt : ℕ → Set β\nh1t : ∀ (n : ℕ), t n ∈ D\nh2t : ⋃ n, t n = univ\n⊢ ⋃ n, (fun n ↦ s (Nat.unpair n).1 ×ˢ t (Nat.unpair n).2) n = univ",
"ppTerm": "?m.73",
"assigned": true,... | [] | rw [iUnion_unpair_prod, h2s, h2t, univ_prod_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 52,
"column": 8
} | {
"line": 52,
"column": 30
} | {
"line": 52,
"column": 31
} | [
{
"pp": "case top\nα : Type u_6\ninst✝ : MeasurableSpace α\nf : α → ℕ∞\nh : ∀ (n : ℕ), MeasurableSet (f ⁻¹' {↑n})\n⊢ MeasurableSet (f ⁻¹' {⊤})",
"ppTerm": "?top",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"instTopENat",
"congrArg",
"Set.instSinglet... | [
"case top\nα : Type u_6\ninst✝ : MeasurableSpace α\nf : α → ℕ∞\nh : ∀ (n : ℕ), MeasurableSet (f ⁻¹' {↑n})\n⊢ MeasurableSet (f ⁻¹' {none})"
] | ← WithTop.none_eq_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 367,
"column": 4
} | {
"line": 367,
"column": 85
} | {
"line": 368,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI : FractionalIdeal R⁰ K\nhI : I ≠ 0\na : R\nJ : Ideal R\na₁ : R := choose ⋯\nJ₁ : Ideal R := choose ⋯\nh_aJ : ↑(J₁ * Ideal.span {a}... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI : FractionalIdeal R⁰ K\nhI : I ≠ 0\na : R\nJ : Ideal R\na₁ : R := choose ⋯\nJ₁ : Ideal R := choose ⋯\nh_aJ : ↑(J₁ * Ideal.span {a}) = ↑(J * Id... | rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 548,
"column": 2
} | {
"line": 548,
"column": 60
} | {
"line": 549,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nι : Type u_6\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j ↦ EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h :... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nι : Type u_6\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j ↦ EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h : Inhabited ι... | set f : (⋃ i, t i) → β := iUnionLift t g' ht' _ Subset.rfl | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 379,
"column": 4
} | {
"line": 379,
"column": 85
} | {
"line": 379,
"column": 85
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R⁰ K\nhI : I ≠ 0\nhI' : I' ≠ 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a ≠ 0\nh... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R⁰ K\nhI : I ≠ 0\nhI' : I' ≠ 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a ≠ 0\nhaJ : I = spa... | rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 383,
"column": 4
} | {
"line": 383,
"column": 85
} | {
"line": 383,
"column": 85
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R⁰ K\nhI : I ≠ 0\nhI' : I' ≠ 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a ≠ 0\nh... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R⁰ K\nhI : I ≠ 0\nhI' : I' ≠ 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a ≠ 0\nhaJ : I = spa... | rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 767,
"column": 11
} | {
"line": 767,
"column": 33
} | {
"line": 768,
"column": 2
} | [
{
"pp": "case nil\nδ : Type u_4\nX : δ → Type u_6\ninst✝ : (i : δ) → MeasurableSpace (X i)\n⊢ Measurable (TProd.mk [])",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"TProd.instMeasurableSpace",
"MeasurableSpace.pi",
"List.TProd",
"measurable_const",
"PUnit.un... | [] | exact measurable_const | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 767,
"column": 11
} | {
"line": 767,
"column": 33
} | {
"line": 768,
"column": 2
} | [
{
"pp": "case nil\nδ : Type u_4\nX : δ → Type u_6\ninst✝ : (i : δ) → MeasurableSpace (X i)\n⊢ Measurable (TProd.mk [])",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"TProd.instMeasurableSpace",
"MeasurableSpace.pi",
"List.TProd",
"measurable_const",
"PUnit.un... | [] | exact measurable_const | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 767,
"column": 11
} | {
"line": 767,
"column": 33
} | {
"line": 768,
"column": 2
} | [
{
"pp": "case nil\nδ : Type u_4\nX : δ → Type u_6\ninst✝ : (i : δ) → MeasurableSpace (X i)\n⊢ Measurable (TProd.mk [])",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"TProd.instMeasurableSpace",
"MeasurableSpace.pi",
"List.TProd",
"measurable_const",
"PUnit.un... | [] | exact measurable_const | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 887,
"column": 30
} | {
"line": 887,
"column": 67
} | {
"line": 887,
"column": 67
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\np q : α → Prop\nhp : Measurable p\nhq : Measurable q\n⊢ MeasurableSet {a | p a ↔ q a}",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.MeasurableSpace.Constructions.0.Measurable.iff._simp_1_1",
"Eq... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\np q : α → Prop\nhp : Measurable p\nhq : Measurable q\n⊢ MeasurableSet {a | (p a → q a) ∧ (q a → p a)}"
] | simp_rw [iff_iff_implies_and_implies] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated | {
"line": 176,
"column": 2
} | {
"line": 178,
"column": 73
} | {
"line": 180,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : MeasurableSpace α\nι : Type u_5\ninst✝² : Preorder ι\ninst✝¹ : IsDirectedOrder ι\ninst✝ : atTop.IsCountablyGenerated\ns : ι → Set α\nhsm : Monotone s\nhs : ∀ (i : ι), MeasurableSet (s i)\n⊢ MeasurableSet (⋃ i, s i)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants... | [] | cases isEmpty_or_nonempty ι with
| inl _ => simp
| inr _ => exact .iUnion_of_monotone_of_frequently hsm <| .of_forall hs | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated | {
"line": 176,
"column": 2
} | {
"line": 178,
"column": 73
} | {
"line": 180,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : MeasurableSpace α\nι : Type u_5\ninst✝² : Preorder ι\ninst✝¹ : IsDirectedOrder ι\ninst✝ : atTop.IsCountablyGenerated\ns : ι → Set α\nhsm : Monotone s\nhs : ∀ (i : ι), MeasurableSet (s i)\n⊢ MeasurableSet (⋃ i, s i)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants... | [] | cases isEmpty_or_nonempty ι with
| inl _ => simp
| inr _ => exact .iUnion_of_monotone_of_frequently hsm <| .of_forall hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated | {
"line": 176,
"column": 2
} | {
"line": 178,
"column": 73
} | {
"line": 180,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : MeasurableSpace α\nι : Type u_5\ninst✝² : Preorder ι\ninst✝¹ : IsDirectedOrder ι\ninst✝ : atTop.IsCountablyGenerated\ns : ι → Set α\nhsm : Monotone s\nhs : ∀ (i : ι), MeasurableSet (s i)\n⊢ MeasurableSet (⋃ i, s i)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants... | [] | cases isEmpty_or_nonempty ι with
| inl _ => simp
| inr _ => exact .iUnion_of_monotone_of_frequently hsm <| .of_forall hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.NNReal.Defs | {
"line": 970,
"column": 2
} | {
"line": 970,
"column": 62
} | {
"line": 971,
"column": 2
} | [
{
"pp": "Γ₀ : Type u_1\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nh : Nontrivial Γ₀ˣ\nf : Γ₀ →*₀ ℝ≥0\nhf : StrictMono ⇑f\nr : ℝ≥0\nhr : 0 < r\n⊢ ∃ d, f ↑d < r",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Units.val",
"GroupWithZero.toMonoidWithZero",
... | [
"Γ₀ : Type u_1\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nh : Nontrivial Γ₀ˣ\nf : Γ₀ →*₀ ℝ≥0\nhf : StrictMono ⇑f\nr : ℝ≥0\nhr : 0 < r\ng : Γ₀ˣ\nhg1 : g ≠ 1\n⊢ ∃ d, f ↑d < r"
] | obtain ⟨g, hg1⟩ := (nontrivial_iff_exists_ne (1 : Γ₀ˣ)).mp h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 625,
"column": 2
} | {
"line": 671,
"column": 84
} | {
"line": 673,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI J : Ideal R\nhIJ : I ≤ J\nhI : I ≠ 0\n⊢ ∃ a, I ⊔ Ideal.span {a} = J",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Iff.mpr",
"Ideal.dvd_iff_le",
"Distrib.l... | [] | obtain ⟨I, rfl⟩ := Ideal.dvd_iff_le.mpr hIJ
simp only [ne_eq, mul_eq_zero, not_or] at hI
obtain ⟨hJ, hI⟩ := hI
suffices ∃ a, ∃ K, J * K = Ideal.span {a} ∧ I + K = ⊤ by
obtain ⟨a, K, e, e'⟩ := this
exact ⟨a, by rw [← e, ← Ideal.add_eq_sup, ← mul_add, e', Ideal.mul_top]⟩
let s := (I.finite_factors hI).toF... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 625,
"column": 2
} | {
"line": 671,
"column": 84
} | {
"line": 673,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI J : Ideal R\nhIJ : I ≤ J\nhI : I ≠ 0\n⊢ ∃ a, I ⊔ Ideal.span {a} = J",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Iff.mpr",
"Ideal.dvd_iff_le",
"Distrib.l... | [] | obtain ⟨I, rfl⟩ := Ideal.dvd_iff_le.mpr hIJ
simp only [ne_eq, mul_eq_zero, not_or] at hI
obtain ⟨hJ, hI⟩ := hI
suffices ∃ a, ∃ K, J * K = Ideal.span {a} ∧ I + K = ⊤ by
obtain ⟨a, K, e, e'⟩ := this
exact ⟨a, by rw [← e, ← Ideal.add_eq_sup, ← mul_add, e', Ideal.mul_top]⟩
let s := (I.finite_factors hI).toF... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Operations | {
"line": 547,
"column": 68
} | {
"line": 547,
"column": 97
} | {
"line": 549,
"column": 0
} | [
{
"pp": "a : ℝ≥0∞\ns : Set ℝ≥0∞\n⊢ sInf s + a = ⨅ b ∈ s, b + a",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"iInf",
"congrArg",
"CompletelyDistribLattice.toCompleteLattice",
"Membership.mem",
"ENNReal.iInf_add",
"Conditionally... | [] | simp [sInf_eq_iInf, iInf_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.ENNReal.Operations | {
"line": 547,
"column": 68
} | {
"line": 547,
"column": 97
} | {
"line": 549,
"column": 0
} | [
{
"pp": "a : ℝ≥0∞\ns : Set ℝ≥0∞\n⊢ sInf s + a = ⨅ b ∈ s, b + a",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"iInf",
"congrArg",
"CompletelyDistribLattice.toCompleteLattice",
"Membership.mem",
"ENNReal.iInf_add",
"Conditionally... | [] | simp [sInf_eq_iInf, iInf_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Operations | {
"line": 547,
"column": 68
} | {
"line": 547,
"column": 97
} | {
"line": 549,
"column": 0
} | [
{
"pp": "a : ℝ≥0∞\ns : Set ℝ≥0∞\n⊢ sInf s + a = ⨅ b ∈ s, b + a",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"iInf",
"congrArg",
"CompletelyDistribLattice.toCompleteLattice",
"Membership.mem",
"ENNReal.iInf_add",
"Conditionally... | [] | simp [sInf_eq_iInf, iInf_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Operations | {
"line": 562,
"column": 67
} | {
"line": 562,
"column": 92
} | {
"line": 564,
"column": 0
} | [
{
"pp": "ι : Type u_2\ninst✝¹ : Preorder ι\ninst✝ : IsCodirectedOrder ι\nf g : ι → ℝ≥0∞\nhf : Monotone f\nhg : Monotone g\ni j _k : ι\nx✝ : _k ≤ i ∧ _k ≤ j\nhi : _k ≤ i\nhj : _k ≤ j\n⊢ f _k + g _k ≤ f i + g j",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"E... | [] | by gcongr <;> apply_rules | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Operations | {
"line": 610,
"column": 4
} | {
"line": 610,
"column": 73
} | {
"line": 612,
"column": 0
} | [
{
"pp": "case neg\nι : Sort u_1\nf : ι → ℝ≥0\nh : ¬BddAbove (range f)\n⊢ (⨆ i, ↑(f i)).toNNReal = ⨆ i, f i",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"ENNReal.toNNReal_top",
"NNReal.iSup_of_not_bddAbove",
"ENNReal.ofNNReal",
"co... | [] | rw [NNReal.iSup_of_not_bddAbove h, iSup_coe_eq_top.2 h, toNNReal_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ENNReal.Operations | {
"line": 610,
"column": 4
} | {
"line": 610,
"column": 73
} | {
"line": 612,
"column": 0
} | [
{
"pp": "case neg\nι : Sort u_1\nf : ι → ℝ≥0\nh : ¬BddAbove (range f)\n⊢ (⨆ i, ↑(f i)).toNNReal = ⨆ i, f i",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"ENNReal.toNNReal_top",
"NNReal.iSup_of_not_bddAbove",
"ENNReal.ofNNReal",
"co... | [] | rw [NNReal.iSup_of_not_bddAbove h, iSup_coe_eq_top.2 h, toNNReal_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Operations | {
"line": 610,
"column": 4
} | {
"line": 610,
"column": 73
} | {
"line": 612,
"column": 0
} | [
{
"pp": "case neg\nι : Sort u_1\nf : ι → ℝ≥0\nh : ¬BddAbove (range f)\n⊢ (⨆ i, ↑(f i)).toNNReal = ⨆ i, f i",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"ENNReal.toNNReal_top",
"NNReal.iSup_of_not_bddAbove",
"ENNReal.ofNNReal",
"co... | [] | rw [NNReal.iSup_of_not_bddAbove h, iSup_coe_eq_top.2 h, toNNReal_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Set.OrdConnectedComponent | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 48
} | {
"line": 181,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Set α\nx a : α\nhas : a ∈ s\nha : [[a, x]] ⊆ tᶜ ∧ [[a, x]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ\nb : α\nhbt : b ∈ t\nhb : [[b, x]] ⊆ sᶜ ∧ [[b, x]] ⊆ (t.ordSeparatingSet s).ordConnectedSectionᶜ\nH :\n ∀ {α : Type u_1} [inst : LinearOrder α] ... | [] | exact H b hbt hb a has ha (le_of_not_ge hab) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Interval.Set.OrdConnectedComponent | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 48
} | {
"line": 181,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Set α\nx a : α\nhas : a ∈ s\nha : [[a, x]] ⊆ tᶜ ∧ [[a, x]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ\nb : α\nhbt : b ∈ t\nhb : [[b, x]] ⊆ sᶜ ∧ [[b, x]] ⊆ (t.ordSeparatingSet s).ordConnectedSectionᶜ\nH :\n ∀ {α : Type u_1} [inst : LinearOrder α] ... | [] | exact H b hbt hb a has ha (le_of_not_ge hab) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.OrdConnectedComponent | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 48
} | {
"line": 181,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Set α\nx a : α\nhas : a ∈ s\nha : [[a, x]] ⊆ tᶜ ∧ [[a, x]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ\nb : α\nhbt : b ∈ t\nhb : [[b, x]] ⊆ sᶜ ∧ [[b, x]] ⊆ (t.ordSeparatingSet s).ordConnectedSectionᶜ\nH :\n ∀ {α : Type u_1} [inst : LinearOrder α] ... | [] | exact H b hbt hb a has ha (le_of_not_ge hab) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Operations | {
"line": 688,
"column": 67
} | {
"line": 688,
"column": 92
} | {
"line": 690,
"column": 0
} | [
{
"pp": "ι : Type u_3\ninst✝¹ : Preorder ι\ninst✝ : IsDirectedOrder ι\nf g : ι → ℝ≥0∞\nhf : Monotone f\nhg : Monotone g\ni j _k : ι\nx✝ : i ≤ _k ∧ j ≤ _k\nhi : i ≤ _k\nhj : j ≤ _k\n⊢ f i + g j ≤ f _k + g _k",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"ENN... | [] | by gcongr <;> apply_rules | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Set.OrdConnectedComponent | {
"line": 199,
"column": 2
} | {
"line": 201,
"column": 46
} | {
"line": 202,
"column": 2
} | [
{
"pp": "case inr.inr\nα✝ : Type u_1\ninst✝¹ : LinearOrder α✝\ns✝ t✝ : Set α✝\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Set α\na : α\nhas : a ∈ s\nb : α\nhbt : b ∈ t\nhab : a ≤ b\nhsub : [[a, b]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ\nx : { x // x ∈ s.ordSeparatingSet t }\nha : [[a, ↑x]] ⊆ tᶜ\nhb : [[... | [
"case inr.inr\nα✝ : Type u_1\ninst✝¹ : LinearOrder α✝\ns✝ t✝ : Set α✝\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Set α\na : α\nhas : a ∈ s\nb : α\nhbt : b ∈ t\nhab : a ≤ b\nhsub : [[a, b]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ\nx : { x // x ∈ s.ordSeparatingSet t }\nha : [[a, ↑x]] ⊆ tᶜ\nhb : [[b, ↑x]] ⊆ sᶜ... | have sol1 := fun (hya : y < a) =>
(disjoint_left (t := ordSeparatingSet s t)).1 disjoint_left_ordSeparatingSet has
(hy <| Icc_subset_uIcc' ⟨hya.le, hax⟩) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.EReal.Basic | {
"line": 717,
"column": 4
} | {
"line": 717,
"column": 53
} | {
"line": 718,
"column": 2
} | [
{
"pp": "x : EReal\nhx : x ≤ 0\n⊢ ENNReal.ofReal x.toReal = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"EReal.toReal_nonpos",
"ENNReal.ofReal_of_nonpos",
"EReal.toReal"
],
"usedFVars": [
"x",
"hx"
],
"usedGoals": []
}
] | [] | exact ENNReal.ofReal_of_nonpos (toReal_nonpos hx) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.EReal.Basic | {
"line": 717,
"column": 4
} | {
"line": 717,
"column": 53
} | {
"line": 718,
"column": 2
} | [
{
"pp": "x : EReal\nhx : x ≤ 0\n⊢ ENNReal.ofReal x.toReal = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"EReal.toReal_nonpos",
"ENNReal.ofReal_of_nonpos",
"EReal.toReal"
],
"usedFVars": [
"x",
"hx"
],
"usedGoals": []
}
] | [] | exact ENNReal.ofReal_of_nonpos (toReal_nonpos hx) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Basic | {
"line": 717,
"column": 4
} | {
"line": 717,
"column": 53
} | {
"line": 718,
"column": 2
} | [
{
"pp": "x : EReal\nhx : x ≤ 0\n⊢ ENNReal.ofReal x.toReal = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"EReal.toReal_nonpos",
"ENNReal.ofReal_of_nonpos",
"EReal.toReal"
],
"usedFVars": [
"x",
"hx"
],
"usedGoals": []
}
] | [] | exact ENNReal.ofReal_of_nonpos (toReal_nonpos hx) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Inv | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 26
} | {
"line": 271,
"column": 2
} | [
{
"pp": "a b c : ℝ≥0∞\nh : 0 < b → b < a → c ≠ 0\n⊢ (a - b) / c = a / c - b / c",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"HSub.hSub",
"id",
... | [
"a b c : ℝ≥0∞\nh : 0 < b → b < a → c ≠ 0\n⊢ (a - b) * c⁻¹ = a * c⁻¹ - b * c⁻¹"
] | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Data.ENNReal.Inv | {
"line": 361,
"column": 27
} | {
"line": 361,
"column": 39
} | {
"line": 361,
"column": 39
} | [
{
"pp": "a b : ℝ≥0∞\nh₀ : a ≠ 0\nh₁ : a ≠ ∞\n⊢ a = 0 → b = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"implies_congr",
"True",
"ENNReal",
"of_eq_true",
"Zero.toOfNat0",
"Eq.refl",
"instIsEmptyFalse",
"E... | [] | by simp [h₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Inv | {
"line": 519,
"column": 6
} | {
"line": 519,
"column": 26
} | {
"line": 519,
"column": 27
} | [
{
"pp": "a b c d : ℝ≥0∞\nha : a ≠ 0\nha' : a ≠ ∞\nhb : b ≠ 0\nhb' : b ≠ ∞\n⊢ a * (c / b) = d ↔ b * d = a * c",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"id",
"HDiv.hDiv"... | [
"a b c d : ℝ≥0∞\nha : a ≠ 0\nha' : a ≠ ∞\nhb : b ≠ 0\nhb' : b ≠ ∞\n⊢ a * (c / b) = d ↔ d = a * c / b"
] | ← eq_div_iff hb hb', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.LiminfLimsup | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 25
} | {
"line": 412,
"column": 4
} | [
{
"pp": "case inr.refine_1\nι : Type u_1\nα : Type u_7\nβ : Type u_8\ninst✝³ : ConditionallyCompleteLattice α\ninst✝² : CompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nu : ι → α → β\nc : β\nh_all : ∀ (i : ι), Tendsto (u i) atTop (𝓝 c)\nh_limsup : Tendsto (fun r ↦ limsup (fun i ↦ u ... | [
"case inr.refine_1\nι : Type u_1\nα : Type u_7\nβ : Type u_8\ninst✝³ : ConditionallyCompleteLattice α\ninst✝² : CompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nu : ι → α → β\nc : β\nh_all : ∀ (i : ι), Tendsto (u i) atTop (𝓝 c)\nh_limsup : Tendsto (fun r ↦ limsup (fun i ↦ u i r) cofinit... | filter_upwards with r | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Topology.Order.LiminfLimsup | {
"line": 516,
"column": 2
} | {
"line": 544,
"column": 36
} | {
"line": 546,
"column": 0
} | [
{
"pp": "case a\nR : Type u_4\nS : Type u_5\ninst✝⁶ : ConditionallyCompleteLinearOrder R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : OrderTopology R\ninst✝³ : ConditionallyCompleteLinearOrder S\ninst✝² : TopologicalSpace S\ninst✝¹ : OrderTopology S\nF : Filter R\ninst✝ : F.NeBot\nf : R → S\nf_decr : Antitone f\nf_co... | [] | · by_cases! h' : ∃ c, c < F.limsSup ∧ Set.Ioo c F.limsSup = ∅
· rcases h' with ⟨c, c_lt, hc⟩
have B : ∃ᶠ n in F, F.limsSup ≤ n := by
apply (frequently_lt_of_lt_limsSup cobdd c_lt).mono
intro x hx
by_contra!
have : (Set.Ioo c F.limsSup).Nonempty := ⟨x, ⟨hx, this⟩⟩
simp o... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.EMetricSpace.Diam | {
"line": 80,
"column": 34
} | {
"line": 80,
"column": 77
} | {
"line": 82,
"column": 0
} | [
{
"pp": "X : Type u_2\ns : Set X\nx : X\ninst✝ : PseudoEMetricSpace X\nd : ℝ≥0∞\n⊢ ediam (insert x s) ≤ d ↔ max (⨆ y ∈ s, edist x y) (ediam s) ≤ d",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"PseudoEMetricSpace.edist_comm",
"Pseudo... | [] | simp +contextual [ediam_le_iff, edist_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.EMetricSpace.Diam | {
"line": 80,
"column": 34
} | {
"line": 80,
"column": 77
} | {
"line": 82,
"column": 0
} | [
{
"pp": "X : Type u_2\ns : Set X\nx : X\ninst✝ : PseudoEMetricSpace X\nd : ℝ≥0∞\n⊢ ediam (insert x s) ≤ d ↔ max (⨆ y ∈ s, edist x y) (ediam s) ≤ d",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"PseudoEMetricSpace.edist_comm",
"Pseudo... | [] | simp +contextual [ediam_le_iff, edist_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.Diam | {
"line": 80,
"column": 34
} | {
"line": 80,
"column": 77
} | {
"line": 82,
"column": 0
} | [
{
"pp": "X : Type u_2\ns : Set X\nx : X\ninst✝ : PseudoEMetricSpace X\nd : ℝ≥0∞\n⊢ ediam (insert x s) ≤ d ↔ max (⨆ y ∈ s, edist x y) (ediam s) ≤ d",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"PseudoEMetricSpace.edist_comm",
"Pseudo... | [] | simp +contextual [ediam_le_iff, edist_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 28
} | {
"line": 160,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommMonoid M\ninst✝² : TopologicalSpace M\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Countable β\ninst✝ : CompleteLattice α\nm : α → M\nm0 : m ⊥ = 1\nR : M → M → Prop\nm_iSup : ∀ (s : ℕ → α), R (m (⨆ i, s i)) (∏' (i : ℕ), m (s i))\ns : β → α\n⊢ R (m (⨆ b, s b)) (∏' (b : β), m (s b))",... | [
"case intro\nM : Type u_1\ninst✝³ : CommMonoid M\ninst✝² : TopologicalSpace M\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Countable β\ninst✝ : CompleteLattice α\nm : α → M\nm0 : m ⊥ = 1\nR : M → M → Prop\nm_iSup : ∀ (s : ℕ → α), R (m (⨆ i, s i)) (∏' (i : ℕ), m (s i))\ns : β → α\nval✝ : Encodable β\n⊢ R (m (⨆ b, s b)) (∏'... | cases nonempty_encodable β | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 44
} | {
"line": 185,
"column": 0
} | [
{
"pp": "case neg\nι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝⁴ : CommMonoid α\ninst✝³ : Preorder α\ninst✝² : IsOrderedMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\nh : ∀ (i : ι), f i ≤ 1\nhf : ¬Multipliable f L\n⊢ ∏'[L] (i : ι), f i ≤ 1",
"ppTerm": "?neg✝",
... | [] | rw [tprod_eq_one_of_not_multipliable hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 44
} | {
"line": 185,
"column": 0
} | [
{
"pp": "case neg\nι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝⁴ : CommMonoid α\ninst✝³ : Preorder α\ninst✝² : IsOrderedMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\nh : ∀ (i : ι), f i ≤ 1\nhf : ¬Multipliable f L\n⊢ ∏'[L] (i : ι), f i ≤ 1",
"ppTerm": "?neg✝",
... | [] | rw [tprod_eq_one_of_not_multipliable hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 44
} | {
"line": 185,
"column": 0
} | [
{
"pp": "case neg\nι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝⁴ : CommMonoid α\ninst✝³ : Preorder α\ninst✝² : IsOrderedMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\nh : ∀ (i : ι), f i ≤ 1\nhf : ¬Multipliable f L\n⊢ ∏'[L] (i : ι), f i ≤ 1",
"ppTerm": "?neg✝",
... | [] | rw [tprod_eq_one_of_not_multipliable hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 92,
"column": 2
} | {
"line": 93,
"column": 24
} | {
"line": 95,
"column": 0
} | [
{
"pp": "case h₂\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\nhab : a ≠ b\n⊢ Icc a b ⊆ closure[inst✝³] (Ioc a b)",
"ppTerm": "?h₂",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"congrAr... | [] | · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)
rw [closure_Ioo hab] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.EReal.Operations | {
"line": 418,
"column": 4
} | {
"line": 419,
"column": 44
} | {
"line": 420,
"column": 4
} | [
{
"pp": "case neg\nx y : ℝ\nhy : 0 ≤ ↑y\nhxy : ¬x ≤ y\n⊢ (↑x - ↑y).toENNReal = (↑x).toENNReal - (↑y).toENNReal",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"EReal.toENNReal_of_ne_top",
"LinearOrder.toD... | [
"case neg\nx y : ℝ\nhy : 0 ≤ ↑y\nhxy : ¬x ≤ y\n⊢ ENNReal.ofReal x - ENNReal.ofReal y = (↑x).toENNReal - (↑y).toENNReal"
] | rw [toENNReal_of_ne_top (ne_of_beq_false rfl).symm, ← coe_sub, toReal_coe,
ofReal_sub x (EReal.coe_nonneg.mp hy)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.EReal.Operations | {
"line": 441,
"column": 2
} | {
"line": 451,
"column": 66
} | {
"line": 453,
"column": 0
} | [
{
"pp": "a b c : EReal\nhb : b ≠ ⊥ ∨ c ≠ ⊥\nht : b ≠ ⊤ ∨ c ≠ ⊤\n⊢ a ≤ c - b ↔ a + b ≤ c",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"_private.Mathlib.Data.EReal.Operations.0.EReal.le_sub_iff_add_le._simp_1_3",
"EReal.sub_top",
"False",... | [] | induction b with
| bot =>
simp only [ne_eq, not_true_eq_false, false_or] at hb
simp only [sub_bot hb, le_top, add_bot, bot_le]
| coe b =>
rw [← (addLECancellable_coe b).add_le_add_iff_right, sub_add_cancel]
| top =>
simp only [ne_eq, not_true_eq_false, false_or, sub_top, le_bot_iff] at ht ⊢
re... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Data.EReal.Operations | {
"line": 441,
"column": 2
} | {
"line": 451,
"column": 66
} | {
"line": 453,
"column": 0
} | [
{
"pp": "a b c : EReal\nhb : b ≠ ⊥ ∨ c ≠ ⊥\nht : b ≠ ⊤ ∨ c ≠ ⊤\n⊢ a ≤ c - b ↔ a + b ≤ c",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"_private.Mathlib.Data.EReal.Operations.0.EReal.le_sub_iff_add_le._simp_1_3",
"EReal.sub_top",
"False",... | [] | induction b with
| bot =>
simp only [ne_eq, not_true_eq_false, false_or] at hb
simp only [sub_bot hb, le_top, add_bot, bot_le]
| coe b =>
rw [← (addLECancellable_coe b).add_le_add_iff_right, sub_add_cancel]
| top =>
simp only [ne_eq, not_true_eq_false, false_or, sub_top, le_bot_iff] at ht ⊢
re... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Operations | {
"line": 441,
"column": 2
} | {
"line": 451,
"column": 66
} | {
"line": 453,
"column": 0
} | [
{
"pp": "a b c : EReal\nhb : b ≠ ⊥ ∨ c ≠ ⊥\nht : b ≠ ⊤ ∨ c ≠ ⊤\n⊢ a ≤ c - b ↔ a + b ≤ c",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"_private.Mathlib.Data.EReal.Operations.0.EReal.le_sub_iff_add_le._simp_1_3",
"EReal.sub_top",
"False",... | [] | induction b with
| bot =>
simp only [ne_eq, not_true_eq_false, false_or] at hb
simp only [sub_bot hb, le_top, add_bot, bot_le]
| coe b =>
rw [← (addLECancellable_coe b).add_le_add_iff_right, sub_add_cancel]
| top =>
simp only [ne_eq, not_true_eq_false, false_or, sub_top, le_bot_iff] at ht ⊢
re... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Operations | {
"line": 717,
"column": 20
} | {
"line": 717,
"column": 30
} | {
"line": 717,
"column": 31
} | [
{
"pp": "a b : EReal\n⊢ 0 ≤ -(a * b) ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
"Preorder.toLE",
"id",
"instZer... | [
"a b : EReal\n⊢ 0 ≤ a * -b ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b"
] | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 49
} | {
"line": 162,
"column": 2
} | [
{
"pp": "α : Type u_3\nm' : PseudoMetricSpace α\ntoDist✝ : Dist α\ndist_self✝ : ∀ (x : α), dist x x = 0\ndist_comm✝ : ∀ (x y : α), dist x y = dist y x\ndist_triangle✝ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z\nedist✝ : α → α → ℝ≥0∞\nhed : ∀ (x y : α), edist✝ x y = ENNReal.ofReal (dist x y)\ntoUniformSpace... | [
"α : Type u_3\ntoDist✝¹ : Dist α\ndist_self✝¹ : ∀ (x : α), dist x x = 0\ndist_comm✝¹ : ∀ (x y : α), dist x y = dist y x\ndist_triangle✝¹ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z\nedist✝¹ : α → α → ℝ≥0∞\nhed : ∀ (x y : α), edist✝¹ x y = ENNReal.ofReal (dist x y)\ntoUniformSpace✝¹ : UniformSpace α\nhU : 𝓤 α =... | obtain ⟨_, _, _, _, hed', _, hU', _, hB'⟩ := m' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 26
} | {
"line": 163,
"column": 2
} | [
{
"pp": "α : Type u_3\ntoDist✝¹ : Dist α\ndist_self✝¹ : ∀ (x : α), dist x x = 0\ndist_comm✝¹ : ∀ (x y : α), dist x y = dist y x\ndist_triangle✝¹ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z\nedist✝¹ : α → α → ℝ≥0∞\nhed : ∀ (x y : α), edist✝¹ x y = ENNReal.ofReal (dist x y)\ntoUniformSpace✝¹ : UniformSpace α\... | [
"α : Type u_3\ntoDist✝ : Dist α\ndist_self✝¹ : ∀ (x : α), dist x x = 0\ndist_comm✝¹ : ∀ (x y : α), dist x y = dist y x\ndist_triangle✝¹ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z\nedist✝¹ : α → α → ℝ≥0∞\nhed : ∀ (x y : α), edist✝¹ x y = ENNReal.ofReal (dist x y)\ntoUniformSpace✝¹ : UniformSpace α\nhU : 𝓤 α = ... | obtain rfl : d = d' := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Order.IntermediateValue | {
"line": 314,
"column": 2
} | {
"line": 324,
"column": 89
} | {
"line": 326,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : OrderTopology α\na b : α\ns : Set α\nhs : IsClosed[inst✝²] (s ∩ Icc a b)\nha : a ∈ s\nhab : a ≤ b\nhgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty\n⊢ b ∈ s",
"ppTerm": "?m.30",
"assigned": true,
"u... | [] | let S := s ∩ Icc a b
replace ha : a ∈ S := ⟨ha, left_mem_Icc.2 hab⟩
have Sbd : BddAbove S := ⟨b, fun z hz => hz.2.2⟩
let c := sSup (s ∩ Icc a b)
have c_mem : c ∈ S := hs.csSup_mem ⟨_, ha⟩ Sbd
have c_le : c ≤ b := csSup_le ⟨_, ha⟩ fun x hx => hx.2.2
rcases eq_or_lt_of_le c_le with hc | hc
· exact hc ▸ c_me... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.IntermediateValue | {
"line": 314,
"column": 2
} | {
"line": 324,
"column": 89
} | {
"line": 326,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : OrderTopology α\na b : α\ns : Set α\nhs : IsClosed[inst✝²] (s ∩ Icc a b)\nha : a ∈ s\nhab : a ≤ b\nhgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty\n⊢ b ∈ s",
"ppTerm": "?m.30",
"assigned": true,
"u... | [] | let S := s ∩ Icc a b
replace ha : a ∈ S := ⟨ha, left_mem_Icc.2 hab⟩
have Sbd : BddAbove S := ⟨b, fun z hz => hz.2.2⟩
let c := sSup (s ∩ Icc a b)
have c_mem : c ∈ S := hs.csSup_mem ⟨_, ha⟩ Sbd
have c_le : c ≤ b := csSup_le ⟨_, ha⟩ fun x hx => hx.2.2
rcases eq_or_lt_of_le c_le with hc | hc
· exact hc ▸ c_me... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Pseudo.Constructions | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 57
} | {
"line": 133,
"column": 2
} | [
{
"pp": "a b : ℝ≥0\nthis : ↑a ≤ ↑b + dist a b\n⊢ a ≤ b + nndist a b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"Real.instLE",
"Real",
"NNReal.coe_add",
"coe_nndist",
"congrArg",
"PartialOrder.toPreorder",
... | [] | rwa [← NNReal.coe_le_coe, NNReal.coe_add, coe_nndist] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Topology.MetricSpace.Pseudo.Constructions | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 57
} | {
"line": 133,
"column": 2
} | [
{
"pp": "a b : ℝ≥0\nthis : ↑a ≤ ↑b + dist a b\n⊢ a ≤ b + nndist a b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"Real.instLE",
"Real",
"NNReal.coe_add",
"coe_nndist",
"congrArg",
"PartialOrder.toPreorder",
... | [] | rwa [← NNReal.coe_le_coe, NNReal.coe_add, coe_nndist] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Pseudo.Constructions | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 57
} | {
"line": 133,
"column": 2
} | [
{
"pp": "a b : ℝ≥0\nthis : ↑a ≤ ↑b + dist a b\n⊢ a ≤ b + nndist a b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"Real.instLE",
"Real",
"NNReal.coe_add",
"coe_nndist",
"congrArg",
"PartialOrder.toPreorder",
... | [] | rwa [← NNReal.coe_le_coe, NNReal.coe_add, coe_nndist] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 535,
"column": 2
} | {
"line": 535,
"column": 66
} | {
"line": 537,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx : α\nε : ℝ\n⊢ closedBall x ε = ⋂ δ, ⋂ (_ : δ > ε), ball x δ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"congrArg",
"Set.iInter",
... | [] | ext y; rw [mem_closedBall, ← forall_gt_iff_le, mem_iInter₂]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 535,
"column": 2
} | {
"line": 535,
"column": 66
} | {
"line": 537,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx : α\nε : ℝ\n⊢ closedBall x ε = ⋂ δ, ⋂ (_ : δ > ε), ball x δ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"congrArg",
"Set.iInter",
... | [] | ext y; rw [mem_closedBall, ← forall_gt_iff_le, mem_iInter₂]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.IntermediateValue | {
"line": 425,
"column": 2
} | {
"line": 425,
"column": 88
} | {
"line": 426,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b x y : α\ns t : Set α\nhxy : x ≤ y\nhs : IsClosed[inst✝³] s\nht : IsClosed[inst✝³] t\nhab : Icc a b ⊆ s ∪ t\nhx : x ∈ Icc a b ∩ s\nhy : y ∈ Icc a b ∩ t\nxyab : Ic... | [
"α : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b x y : α\ns t : Set α\nhxy : x ≤ y\nhs : IsClosed[inst✝³] s\nht : IsClosed[inst✝³] t\nhab : Icc a b ⊆ s ∪ t\nhx : x ∈ Icc a b ∩ s\nhy : y ∈ Icc a b ∩ t\nxyab : Icc x y ⊆ Icc ... | have : Ioc z y ⊆ s ∪ t := fun w hw => hab (xyab ⟨le_trans hz.1 (le_of_lt hw.1), hw.2⟩) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.MetricSpace.Pseudo.Pi | {
"line": 60,
"column": 84
} | {
"line": 65,
"column": 35
} | {
"line": 67,
"column": 0
} | [
{
"pp": "β : Type u_2\nX : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (X b)\nf g : (b : β) → X b\nr : ℝ≥0\nhr : 0 < r\n⊢ nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ (b : β), nndist (f b) (g b) ≤ r",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.m... | [] | by
rw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, not_forall, and_comm]
simp_rw [not_lt, and_congr_left_iff, le_antisymm_iff]
intro h
refine exists_congr fun b => ?_
apply (and_iff_right <| h _).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1243,
"column": 4
} | {
"line": 1243,
"column": 67
} | {
"line": 1245,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\ne : β → α\na : α\n⊢ (∀ (i : ℕ), True → ∃ y ∈ range e, y ∈ ball a (1 / (↑i + 1))) ↔ ∀ (n : ℕ), ∃ k, dist a (e k) < 1 / (↑n + 1)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"congrArg",
... | [] | simp only [mem_ball, dist_comm, exists_range_iff, forall_const] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1243,
"column": 4
} | {
"line": 1243,
"column": 67
} | {
"line": 1245,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\ne : β → α\na : α\n⊢ (∀ (i : ℕ), True → ∃ y ∈ range e, y ∈ ball a (1 / (↑i + 1))) ↔ ∀ (n : ℕ), ∃ k, dist a (e k) < 1 / (↑n + 1)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"congrArg",
... | [] | simp only [mem_ball, dist_comm, exists_range_iff, forall_const] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1243,
"column": 4
} | {
"line": 1243,
"column": 67
} | {
"line": 1245,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\ne : β → α\na : α\n⊢ (∀ (i : ℕ), True → ∃ y ∈ range e, y ∈ ball a (1 / (↑i + 1))) ↔ ∀ (n : ℕ), ∃ k, dist a (e k) < 1 / (↑n + 1)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"congrArg",
... | [] | simp only [mem_ball, dist_comm, exists_range_iff, forall_const] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Defs | {
"line": 139,
"column": 2
} | {
"line": 143,
"column": 32
} | {
"line": 145,
"column": 0
} | [
{
"pp": "γ : Type w\ninst✝ : MetricSpace γ\nx : γ\nr : ℝ\nhr : r ≤ 0\n⊢ (closedBall x r).Subsingleton",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
... | [] | rcases hr.lt_or_eq with (hr | rfl)
· rw [closedBall_eq_empty.2 hr]
exact subsingleton_empty
· rw [closedBall_zero]
exact subsingleton_singleton | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Defs | {
"line": 139,
"column": 2
} | {
"line": 143,
"column": 32
} | {
"line": 145,
"column": 0
} | [
{
"pp": "γ : Type w\ninst✝ : MetricSpace γ\nx : γ\nr : ℝ\nhr : r ≤ 0\n⊢ (closedBall x r).Subsingleton",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
... | [] | rcases hr.lt_or_eq with (hr | rfl)
· rw [closedBall_eq_empty.2 hr]
exact subsingleton_empty
· rw [closedBall_zero]
exact subsingleton_singleton | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Basic | {
"line": 63,
"column": 65
} | {
"line": 63,
"column": 79
} | {
"line": 65,
"column": 0
} | [
{
"pp": "γ : Type w\ninst✝² : MetricSpace γ\nα : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : DiscreteTopology α\nε : ℝ\nhε : 0 < ε\nf : α → γ\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"ppTerm": "?m.28",
"assigned": true,
"usedConstan... | [] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.MetricSpace.Basic | {
"line": 63,
"column": 65
} | {
"line": 63,
"column": 79
} | {
"line": 65,
"column": 0
} | [
{
"pp": "γ : Type w\ninst✝² : MetricSpace γ\nα : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : DiscreteTopology α\nε : ℝ\nhε : 0 < ε\nf : α → γ\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"ppTerm": "?m.28",
"assigned": true,
"usedConstan... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Basic | {
"line": 63,
"column": 65
} | {
"line": 63,
"column": 79
} | {
"line": 65,
"column": 0
} | [
{
"pp": "γ : Type w\ninst✝² : MetricSpace γ\nα : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : DiscreteTopology α\nε : ℝ\nhε : 0 < ε\nf : α → γ\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"ppTerm": "?m.28",
"assigned": true,
"usedConstan... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Basic | {
"line": 70,
"column": 66
} | {
"line": 70,
"column": 80
} | {
"line": 72,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nβ : Type u_2\nε : ℝ\nhε : 0 < ε\nf : β → α\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder... | [] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.MetricSpace.Basic | {
"line": 70,
"column": 66
} | {
"line": 70,
"column": 80
} | {
"line": 72,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nβ : Type u_2\nε : ℝ\nhε : 0 < ε\nf : β → α\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Basic | {
"line": 70,
"column": 66
} | {
"line": 70,
"column": 80
} | {
"line": 72,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nβ : Type u_2\nε : ℝ\nhε : 0 < ε\nf : β → α\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Basic | {
"line": 226,
"column": 54
} | {
"line": 226,
"column": 71
} | {
"line": 229,
"column": 0
} | [
{
"pp": "X : Type u_2\nm : PseudoEMetricSpace X\nd : X → X → ℝ≥0∞\nhd : d = edist\n⊢ m.replaceEDist d hd = m",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"PseudoEMetricSpace.ext",
"EDist.ext",
"PseudoEMetricSpace.toEDist",
"PseudoEMetricSpace.replaceEDist"
],
... | [] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Basic | {
"line": 226,
"column": 54
} | {
"line": 226,
"column": 71
} | {
"line": 229,
"column": 0
} | [
{
"pp": "X : Type u_2\nm : PseudoEMetricSpace X\nd : X → X → ℝ≥0∞\nhd : d = edist\n⊢ m.replaceEDist d hd = m",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"PseudoEMetricSpace.ext",
"EDist.ext",
"PseudoEMetricSpace.toEDist",
"PseudoEMetricSpace.replaceEDist"
],
... | [] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Basic | {
"line": 252,
"column": 52
} | {
"line": 252,
"column": 69
} | {
"line": 255,
"column": 0
} | [
{
"pp": "X : Type u_2\nm : PseudoMetricSpace X\nd : X → X → ℝ\nhd : d = dist\n⊢ m.replaceDist d hd = m",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"PseudoMetricSpace.ext",
"Dist.ext",
"PseudoMetricSpace.replaceDist",
"PseudoMetricSpace.toDist"
],
"usedFVa... | [] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Basic | {
"line": 252,
"column": 52
} | {
"line": 252,
"column": 69
} | {
"line": 255,
"column": 0
} | [
{
"pp": "X : Type u_2\nm : PseudoMetricSpace X\nd : X → X → ℝ\nhd : d = dist\n⊢ m.replaceDist d hd = m",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"PseudoMetricSpace.ext",
"Dist.ext",
"PseudoMetricSpace.replaceDist",
"PseudoMetricSpace.toDist"
],
"usedFVa... | [] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Basic | {
"line": 282,
"column": 54
} | {
"line": 282,
"column": 71
} | {
"line": 285,
"column": 0
} | [
{
"pp": "X : Type u_2\nm : EMetricSpace X\nd : X → X → ℝ≥0∞\nhd : d = edist\n⊢ m.replaceEDist d hd = m",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"EMetricSpace.toPseudoEMetricSpace",
"EMetricSpace.ext",
"EDist.ext",
"PseudoEMetricSpace.toEDist",
"EMetricSp... | [] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Basic | {
"line": 282,
"column": 54
} | {
"line": 282,
"column": 71
} | {
"line": 285,
"column": 0
} | [
{
"pp": "X : Type u_2\nm : EMetricSpace X\nd : X → X → ℝ≥0∞\nhd : d = edist\n⊢ m.replaceEDist d hd = m",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"EMetricSpace.toPseudoEMetricSpace",
"EMetricSpace.ext",
"EDist.ext",
"PseudoEMetricSpace.toEDist",
"EMetricSp... | [] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Basic | {
"line": 305,
"column": 52
} | {
"line": 305,
"column": 69
} | {
"line": 308,
"column": 0
} | [
{
"pp": "X : Type u_2\nm : MetricSpace X\nd : X → X → ℝ\nhd : d = dist\n⊢ m.replaceDist d hd = m",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"MetricSpace.ext",
"MetricSpace.replaceDist",
"Dist.ext",
"MetricSpace.toPseudoMetricSpace",
"PseudoMetricSpace.toDi... | [] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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