module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 214,
"column": 2
} | {
"line": 215,
"column": 54
} | {
"line": 216,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ\nn d : ℤ\nhqz : q ≠ 0\nqdf : q = n /. d\nhd : d ≠ 0\nc : ℤ\nhc1 : n = c * q.num\nhc2 : d = c * ↑q.den\n⊢ ↑(multiplicity (↑p) q.num) - ↑(multiplicity p q.den) =\n ↑(multiplicity (↑p) (c * q.num)) - ↑(multiplicity (↑p) (c * ↑q.den))",
"ppTerm": "?m.57",
"a... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ\nn d : ℤ\nhqz : q ≠ 0\nqdf : q = n /. d\nhd : d ≠ 0\nc : ℤ\nhc1 : n = c * q.num\nhc2 : d = c * ↑q.den\n⊢ ↑(multiplicity (↑p) q.num) - ↑(multiplicity p q.den) =\n ↑(multiplicity (↑p) c + multiplicity (↑p) q.num) - ↑(multiplicity (↑p) c + multiplicity ↑p ↑q.den)",
"p : ℕ\nh... | rw [multiplicity_mul (Nat.prime_iff_prime_int.1 hp.1),
multiplicity_mul (Nat.prime_iff_prime_int.1 hp.1)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 134,
"column": 19
} | {
"line": 134,
"column": 30
} | {
"line": 134,
"column": 31
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nG : C\ninst✝¹ : Abelian C\nhG : IsSeparator G\nX Y : C\np : X ⟶ Y\ninst✝ : Mono p\nf : (coyoneda.obj (Opposite.op G)).obj Y\nhf : ∀ (x : (coyoneda.obj (Opposite.op G)).obj X), ¬(ConcreteCategory.hom ((yoneda.map p).app (Opposite.op G))) x = f\nh : ∀ (T : C), Func... | [
"C : Type u\ninst✝² : Category.{v, u} C\nG : C\ninst✝¹ : Abelian C\nhG : IsSeparator G\nX Y : C\np : X ⟶ Y\ninst✝ : Mono p\nf : (coyoneda.obj (Opposite.op G)).obj Y\nhf : ∀ (x : (coyoneda.obj (Opposite.op G)).obj X), ¬(ConcreteCategory.hom ((yoneda.map p).app (Opposite.op G))) x = f\nh : ∀ (T : C), Function.Bijecti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 43
} | {
"line": 219,
"column": 44
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ\nn d : ℤ\nhqz : q ≠ 0\nqdf : q = n /. d\nhd : d ≠ 0\nc : ℤ\nhc1 : n = c * q.num\nhc2 : d = c * ↑q.den\n⊢ FiniteMultiplicity (↑p) (c * ↑q.den)",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"IsDomain.to_noZ... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ\nn d : ℤ\nhqz : q ≠ 0\nqdf : q = n /. d\nhd : d ≠ 0\nc : ℤ\nhc1 : n = c * q.num\nhc2 : d = c * ↑q.den\n⊢ ¬c = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 144,
"column": 8
} | {
"line": 144,
"column": 56
} | {
"line": 144,
"column": 57
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nG : C\ninst✝¹ : Abelian C\nhG : IsSeparator G\nX A✝ : C\nf : A✝ ⟶ X\ninst✝ : Mono f\nhA : Subobject.mk f ≠ ⊤\n⊢ ¬IsIso f",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsIso",
"congrArg",
"P... | [
"C : Type u\ninst✝² : Category.{v, u} C\nG : C\ninst✝¹ : Abelian C\nhG : IsSeparator G\nX A✝ : C\nf : A✝ ⟶ X\ninst✝ : Mono f\nhA : Subobject.mk f ≠ ⊤\n⊢ ¬Subobject.mk f = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 48
} | {
"line": 220,
"column": 49
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ\nn d : ℤ\nhqz : q ≠ 0\nqdf : q = n /. d\nhd : d ≠ 0\nc : ℤ\nhc1 : n = c * q.num\nhc2 : d = c * ↑q.den\n⊢ FiniteMultiplicity (↑p) (c * q.num)",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"False",
... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ\nn d : ℤ\nhqz : q ≠ 0\nqdf : q = n /. d\nhd : d ≠ 0\nc : ℤ\nhc1 : n = c * q.num\nhc2 : d = c * ↑q.den\n⊢ ¬c = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 226,
"column": 8
} | {
"line": 226,
"column": 25
} | {
"line": 226,
"column": 26
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nq r : ℚ\nhq : q ≠ 0\nhr : r ≠ 0\n⊢ q * r = q.num * r.num /. (↑q.den * ↑r.den)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Rat.instMul",
"Rat.num",
"Rat.mul_eq_mkRat",
"HMul.hMul",
"congrArg",
... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nq r : ℚ\nhq : q ≠ 0\nhr : r ≠ 0\n⊢ mkRat (q.num * r.num) (q.den * r.den) = q.num * r.num /. (↑q.den * ↑r.den)"
] | Rat.mul_eq_mkRat, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 204,
"column": 16
} | {
"line": 204,
"column": 62
} | {
"line": 204,
"column": 63
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nG : C\ninst✝⁵ : Abelian C\nhG : IsSeparator G\nX : C\ninst✝⁴ : IsGrothendieckAbelian.{w, v, u} C\nA₀ : Subobject X\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nhJ : HasCardinalLT (Subobject X) (Cardinal.... | [
"C : Type u\ninst✝⁶ : Category.{v, u} C\nG : C\ninst✝⁵ : Abelian C\nhG : IsSeparator G\nX : C\ninst✝⁴ : IsGrothendieckAbelian.{w, v, u} C\nA₀ : Subobject X\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nhJ : HasCardinalLT (Subobject X) (Cardinal.mk J)\nh : F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 326,
"column": 37
} | {
"line": 326,
"column": 48
} | {
"line": 326,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nG : C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{w, v, u} C\nκ : Cardinal.{w}\nhκ' : κ.IsRegular\nhκ : HasCardinalLT (Subobject G) κ\nthis : Fact κ.IsRegular\n⊢ Nonempty κ.ord.ToType",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
... | [
"C : Type u\ninst✝² : Category.{v, u} C\nG : C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{w, v, u} C\nκ : Cardinal.{w}\nhκ' : κ.IsRegular\nhκ : HasCardinalLT (Subobject G) κ\nthis : Fact κ.IsRegular\n⊢ ¬κ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 374,
"column": 2
} | {
"line": 375,
"column": 44
} | {
"line": 375,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nG : C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{w, v, u} C\nX : C\nfac : (monomorphisms C).MapFactorizationData (monomorphisms C).rlp 0 := monoMapFactorizationDataRlp 0\n⊢ Injective (monoMapFactorizationDataRlp 0).Z",
"ppTerm": "?m.39",
"assigne... | [
"C : Type u\ninst✝² : Category.{v, u} C\nG : C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{w, v, u} C\nX : C\nfac : (monomorphisms C).MapFactorizationData (monomorphisms C).rlp 0 := monoMapFactorizationDataRlp 0\n⊢ (monomorphisms C).rlp 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 287,
"column": 24
} | {
"line": 287,
"column": 40
} | {
"line": 287,
"column": 41
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nq r : ℚ\nhqr : q + r ≠ 0\nh : padicValRat p q ≤ padicValRat p r\nhq : q = 0\n⊢ padicValRat p q ≤ padicValRat p (q + r)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"padicValRat.zero",
"congrArg",... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nq r : ℚ\nhqr : q + r ≠ 0\nh : padicValRat p q ≤ padicValRat p r\nhq : q = 0\n⊢ 0 ≤ padicValRat p r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNorm | {
"line": 164,
"column": 43
} | {
"line": 164,
"column": 60
} | {
"line": 164,
"column": 61
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nq r : ℚ\nh : padicValRat p q ≤ padicValRat p r\nhnqp : padicNorm p q ≥ 0\nhnrp : padicNorm p r ≥ 0\nhq : ¬q = 0\nhr : ¬r = 0\nhqr : q + r = 0\n⊢ padicNorm p (q + r) ≤ padicNorm p q",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Rat.instOfNat... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nq r : ℚ\nh : padicValRat p q ≤ padicValRat p r\nhnqp : padicNorm p q ≥ 0\nhnrp : padicNorm p r ≥ 0\nhq : ¬q = 0\nhr : ¬r = 0\nhqr : q + r = 0\n⊢ 0 ≤ padicNorm p q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 346,
"column": 2
} | {
"line": 355,
"column": 45
} | {
"line": 357,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nF : ℕ → ℚ\nhF : ∀ i < n, 0 < padicValRat p (F i)\nhn0 : ∑ i ∈ Finset.range n, F i ≠ 0\n⊢ 0 < padicValRat p (∑ i ∈ Finset.range n, F i)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Rat.addCommMonoid",
"Rat.instOfNat",
"Eq... | [] | induction n with
| zero => exact False.elim (hn0 rfl)
| succ d hd =>
rw [Finset.sum_range_succ] at hn0 ⊢
by_cases h : ∑ x ∈ Finset.range d, F x = 0
· rw [h, zero_add]
exact hF d (lt_add_one _)
· refine lt_of_lt_of_le ?_ (min_le_padicValRat_add hn0)
refine lt_min (hd (fun i hi => ?_) h) (... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 346,
"column": 2
} | {
"line": 355,
"column": 45
} | {
"line": 357,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nF : ℕ → ℚ\nhF : ∀ i < n, 0 < padicValRat p (F i)\nhn0 : ∑ i ∈ Finset.range n, F i ≠ 0\n⊢ 0 < padicValRat p (∑ i ∈ Finset.range n, F i)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Rat.addCommMonoid",
"Rat.instOfNat",
"Eq... | [] | induction n with
| zero => exact False.elim (hn0 rfl)
| succ d hd =>
rw [Finset.sum_range_succ] at hn0 ⊢
by_cases h : ∑ x ∈ Finset.range d, F x = 0
· rw [h, zero_add]
exact hF d (lt_add_one _)
· refine lt_of_lt_of_le ?_ (min_le_padicValRat_add hn0)
refine lt_min (hd (fun i hi => ?_) h) (... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 346,
"column": 2
} | {
"line": 355,
"column": 45
} | {
"line": 357,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nF : ℕ → ℚ\nhF : ∀ i < n, 0 < padicValRat p (F i)\nhn0 : ∑ i ∈ Finset.range n, F i ≠ 0\n⊢ 0 < padicValRat p (∑ i ∈ Finset.range n, F i)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Rat.addCommMonoid",
"Rat.instOfNat",
"Eq... | [] | induction n with
| zero => exact False.elim (hn0 rfl)
| succ d hd =>
rw [Finset.sum_range_succ] at hn0 ⊢
by_cases h : ∑ x ∈ Finset.range d, F x = 0
· rw [h, zero_add]
exact hF d (lt_add_one _)
· refine lt_of_lt_of_le ?_ (min_le_padicValRat_add hn0)
refine lt_min (hd (fun i hi => ?_) h) (... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 49
} | {
"line": 398,
"column": 50
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\na n : ℕ\n⊢ padicValNat p (a ^ n) = n * padicValNat p a",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"Rat",
"... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\na n : ℕ\n⊢ padicValRat p (↑a ^ n) = ↑n * padicValRat p ↑a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 66
} | {
"line": 146,
"column": 4
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ IsOpen {y | ‖y‖ ≤ 1}",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ IsOpen {y | ‖y‖ ≤ 1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 249,
"column": 32
} | {
"line": 249,
"column": 60
} | {
"line": 249,
"column": 61
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℤ_[p]\nf : CauSeq ℤ_[p] norm\nx✝ : ℝ\nhε : x✝ > 0\n⊢ ∃ i, ∀ j ≥ i, (fun a ↦ ‖a‖) ((fun n ↦ ↑(↑f n)) j - (fun n ↦ ↑(↑f n)) i) < x✝",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Re... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℤ_[p]\nf : CauSeq ℤ_[p] norm\nx✝ : ℝ\nhε : x✝ > 0\n⊢ ∃ i, ∀ (j : ℕ), i ≤ j → ↑(padicNormE (↑(↑f j) - ↑(↑f i))) < x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 331,
"column": 8
} | {
"line": 331,
"column": 37
} | {
"line": 331,
"column": 38
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℤ_[p]\nhxy : x + y ≠ 0\n⊢ min ↑x.valuation ↑y.valuation ≤ ↑(x + y).valuation",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instLE",
"Real",... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℤ_[p]\nhxy : x + y ≠ 0\n⊢ (↑x).valuation ≤ (↑x + ↑y).valuation ∨ (↑y).valuation ≤ (↑x + ↑y).valuation"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 386,
"column": 2
} | {
"line": 386,
"column": 13
} | {
"line": 386,
"column": 14
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nz : ℤ_[p]\n⊢ ¬IsUnit z ↔ ‖z‖ < 1",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nz : ℤ_[p]\n⊢ ¬IsUnit z ↔ ‖z‖ < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 464,
"column": 4
} | {
"line": 464,
"column": 51
} | {
"line": 464,
"column": 52
} | [
{
"pp": "case mpr\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nhx : x ≠ 0\nn : ℕ\n⊢ n ≤ x.valuation → ↑p ^ n ∣ ↑(unitCoeff hx) * ↑p ^ x.valuation",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"PadicInt",
"Mo... | [
"case mpr\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nhx : x ≠ 0\nn : ℕ\n⊢ n ≤ x.valuation → ↑p ^ n ∣ ↑p ^ x.valuation"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 18
} | {
"line": 219,
"column": 4
} | [
{
"pp": "case mp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nh : (if hf : f ≈ 0 then 0 else padicNorm p (↑f (stationaryPoint hf))) = 0\nhf : ¬f ≈ 0\n⊢ False",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"padicNorm.instIsAbsoluteValueRat",
"Rat.instOfNat",
"congrAr... | [
"case mp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint hf)) = 0\n⊢ False"
] | split_ifs at h | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 239,
"column": 20
} | {
"line": 239,
"column": 35
} | {
"line": 239,
"column": 36
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : ∀ (k : ℕ), padicNorm p (↑f k) = padicNorm p (↑g k)\nhf : f ≈ 0\nε : ℚ\nhε : ε > 0\ni : ℕ\nhi : ∀ j ≥ i, padicNorm p (↑(f - 0) j) < ε\nj : ℕ\nhj : j ≥ i\n⊢ padicNorm p (↑(g - 0) j) < ε",
"ppTerm": "?m.25",
"assigned": true,
"usedConsta... | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : ∀ (k : ℕ), padicNorm p (↑f k) = padicNorm p (↑g k)\nhf : f ≈ 0\nε : ℚ\nhε : ε > 0\ni : ℕ\nhi : ∀ j ≥ i, padicNorm p (↑(f - 0) j) < ε\nj : ℕ\nhj : j ≥ i\n⊢ padicNorm p (↑g j) < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 248,
"column": 42
} | {
"line": 248,
"column": 57
} | {
"line": 248,
"column": 58
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : ¬f ≈ 0\nheq : f.norm = padicNorm p (↑f (stationaryPoint hf))\nh : ↑f (stationaryPoint hf) = 0\n⊢ f.norm = 0",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : ¬f ≈ 0\nheq : f.norm = padicNorm p (↑f (stationaryPoint hf))\nh : ↑f (stationaryPoint hf) = 0\n⊢ f.norm = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 255,
"column": 51
} | {
"line": 255,
"column": 62
} | {
"line": 255,
"column": 63
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nq : ℚ\nhq : q ≠ 0\nh : (const (padicNorm p) q - 0).LimZero\n⊢ (const (padicNorm p) q).LimZero",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nq : ℚ\nhq : q ≠ 0\nh : (const (padicNorm p) q - 0).LimZero\n⊢ (const (padicNorm p) q).LimZero"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.EpiMono | {
"line": 46,
"column": 4
} | {
"line": 46,
"column": 38
} | {
"line": 46,
"column": 39
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : C\nf : X ⟶ Y\nc : PullbackCone f f\nhc : IsLimit c\nhf : Mono f\n⊢ c.fst = c.snd",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Cat... | [
"case mp\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : C\nf : X ⟶ Y\nc : PullbackCone f f\nhc : IsLimit c\nhf : Mono f\n⊢ c.fst ≫ f = c.snd ≫ f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 313,
"column": 2
} | {
"line": 313,
"column": 17
} | {
"line": 313,
"column": 18
} | [
{
"pp": "case h\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : ¬f ≈ 0\nH : ↑f (stationaryPoint hf) = 0\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : n ≥ stationaryPoint hf\n⊢ padicNorm p (↑f (stationaryPoint hf)) < ε",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq... | [
"case h\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : ¬f ≈ 0\nH : ↑f (stationaryPoint hf) = 0\nε : ℚ\nhε : ε > 0\nn : ℕ\nhn : n ≥ stationaryPoint hf\n⊢ 0 < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Functor.ReflectsIso.Jointly | {
"line": 63,
"column": 58
} | {
"line": 63,
"column": 69
} | {
"line": 63,
"column": 70
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{u_4, u_1} C\nI : Type u_2\nD : I → Type u_3\ninst✝² : (i : I) → Category.{u_5, u_3} (D i)\nF : (i : I) → C ⥤ D i\ninst✝¹ : HasEqualizers C\ninst✝ : ∀ (i : I), PreservesLimitsOfShape WalkingParallelPair (F i)\nhF : ∀ ⦃X Y : C⦄ (f : X ⟶ Y) [Mono f], (∀ (i : I), IsIso ((F ... | [
"C : Type u_1\ninst✝³ : Category.{u_4, u_1} C\nI : Type u_2\nD : I → Type u_3\ninst✝² : (i : I) → Category.{u_5, u_3} (D i)\nF : (i : I) → C ⥤ D i\ninst✝¹ : HasEqualizers C\ninst✝ : ∀ (i : I), PreservesLimitsOfShape WalkingParallelPair (F i)\nhF : ∀ ⦃X Y : C⦄ (f : X ⟶ Y) [Mono f], (∀ (i : I), IsIso ((F i).map f)) →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Functor.ReflectsIso.Jointly | {
"line": 62,
"column": 8
} | {
"line": 64,
"column": 71
} | {
"line": 64,
"column": 71
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{u_4, u_1} C\nI : Type u_2\nD : I → Type u_3\ninst✝² : (i : I) → Category.{u_5, u_3} (D i)\nF : (i : I) → C ⥤ D i\ninst✝¹ : HasEqualizers C\ninst✝ : ∀ (i : I), PreservesLimitsOfShape WalkingParallelPair (F i)\nhF : ∀ ⦃X Y : C⦄ (f : X ⟶ Y) [Mono f], (∀ (i : I), IsIso ((F ... | [] | let hc := isLimitForkMapOfIsLimit (F i) _ (equalizerIsEqualizer f g)
obtain ⟨l, hl⟩ := Fork.IsLimit.lift' hc (𝟙 _) (by simpa using hfg i)
exact ⟨l, Fork.IsLimit.hom_ext hc (by cat_disch), by cat_disch⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Functor.ReflectsIso.Jointly | {
"line": 62,
"column": 8
} | {
"line": 64,
"column": 71
} | {
"line": 64,
"column": 71
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{u_4, u_1} C\nI : Type u_2\nD : I → Type u_3\ninst✝² : (i : I) → Category.{u_5, u_3} (D i)\nF : (i : I) → C ⥤ D i\ninst✝¹ : HasEqualizers C\ninst✝ : ∀ (i : I), PreservesLimitsOfShape WalkingParallelPair (F i)\nhF : ∀ ⦃X Y : C⦄ (f : X ⟶ Y) [Mono f], (∀ (i : I), IsIso ((F ... | [] | let hc := isLimitForkMapOfIsLimit (F i) _ (equalizerIsEqualizer f g)
obtain ⟨l, hl⟩ := Fork.IsLimit.lift' hc (𝟙 _) (by simpa using hfg i)
exact ⟨l, Fork.IsLimit.hom_ext hc (by cat_disch), by cat_disch⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.EpiMono | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 37
} | {
"line": 88,
"column": 38
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : C\nf : X ⟶ Y\nc : PushoutCocone f f\nhc : IsColimit c\nhf : Epi f\n⊢ c.inl = c.inr",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.WalkingSpan",
"CategoryTheory.CategoryStr... | [
"case mp\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : C\nf : X ⟶ Y\nc : PushoutCocone f f\nhc : IsColimit c\nhf : Epi f\n⊢ f ≫ c.inl = f ≫ c.inr"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 391,
"column": 2
} | {
"line": 391,
"column": 18
} | {
"line": 391,
"column": 19
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\na : PadicSeq p\nha : ¬a ≈ 0\nk : ℚ\nhk : a.norm = padicNorm p k\nhk' : k ≠ 0\n⊢ ∃ z, a.norm = ↑p ^ (-z)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\na : PadicSeq p\nha : ¬a ≈ 0\nk : ℚ\nhk : a.norm = padicNorm p k\nhk' : k ≠ 0\n⊢ ∃ z, padicNorm p k = (↑p ^ z)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 171,
"column": 2
} | {
"line": 180,
"column": 39
} | {
"line": 182,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝³ : LocallySmall.{w, v, u} C\ninst✝² : HasSheafify J (Type w)\ninst✝¹ : J.WEqualsLocallyBijective (Type w)\nhP : P.IsConservativeFamilyOfPoints\ninst✝ : ObjectProperty.Small.{w, max u w, max (max u v) (... | [] | refine ⟨fun hf Φ x ↦ ?_, fun hf ↦ ?_⟩
· obtain ⟨Z, _, ⟨_, p, _, ⟨i⟩, rfl⟩, z, rfl⟩ := Φ.obj.jointly_surjective _ hf x
exact ⟨i, Φ.obj.fiber.map p z, by simp⟩
· let ι' : Type _ := Σ (Φ : P.FullSubcategory), Φ.obj.fiber.obj X
choose i y hy using fun (j : ι') ↦ hf j.1 j.2
refine J.superset_covering (S := S... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 171,
"column": 2
} | {
"line": 180,
"column": 39
} | {
"line": 182,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝³ : LocallySmall.{w, v, u} C\ninst✝² : HasSheafify J (Type w)\ninst✝¹ : J.WEqualsLocallyBijective (Type w)\nhP : P.IsConservativeFamilyOfPoints\ninst✝ : ObjectProperty.Small.{w, max u w, max (max u v) (... | [] | refine ⟨fun hf Φ x ↦ ?_, fun hf ↦ ?_⟩
· obtain ⟨Z, _, ⟨_, p, _, ⟨i⟩, rfl⟩, z, rfl⟩ := Φ.obj.jointly_surjective _ hf x
exact ⟨i, Φ.obj.fiber.map p z, by simp⟩
· let ι' : Type _ := Σ (Φ : P.FullSubcategory), Φ.obj.fiber.obj X
choose i y hy using fun (j : ι') ↦ hf j.1 j.2
refine J.superset_covering (S := S... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 454,
"column": 38
} | {
"line": 454,
"column": 87
} | {
"line": 454,
"column": 88
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : (f - 0).LimZero\n⊢ LimZero (f + g - g)",
"ppTerm": "?m.124",
"assigned": true,
"usedConstants": [
"padicNorm.instIsAbsoluteValueRat",
"CauSeq.addGroup",
"Eq.mpr",
"congrArg",
"Rat",
... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : (f - 0).LimZero\n⊢ LimZero f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 463,
"column": 40
} | {
"line": 463,
"column": 88
} | {
"line": 463,
"column": 89
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : (g - 0).LimZero\n⊢ LimZero (f + g - f)",
"ppTerm": "?m.227",
"assigned": true,
"usedConstants": [
"padicNorm.instIsAbsoluteValueRat",
"Eq.mpr",
"congrArg",
"add_sub_cancel_left",
... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : (g - 0).LimZero\n⊢ LimZero g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 498,
"column": 2
} | {
"line": 498,
"column": 29
} | {
"line": 498,
"column": 30
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : f + g ≈ 0\nthis✝¹ : (f + g - 0).LimZero\nthis✝ : f ≈ -g\nthis : f.norm = (-g).norm\n⊢ f.norm = g.norm",
"ppTerm": "?m.73",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : f + g ≈ 0\nthis✝¹ : (f + g - 0).LimZero\nthis✝ : f ≈ -g\nthis : f.norm = (-g).norm\n⊢ f.norm = g.norm"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 211,
"column": 4
} | {
"line": 211,
"column": 15
} | {
"line": 211,
"column": 16
} | [
{
"pp": "case inr\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nhP :\n ∀ ⦃X : C⦄ (S : Sieve X),\n (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fi... | [
"case inr\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nhP :\n ∀ ⦃X : C⦄ (S : Sieve X),\n (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 226,
"column": 4
} | {
"line": 226,
"column": 15
} | {
"line": 226,
"column": 16
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nhP :\n ∀ ⦃X : C⦄ (S : Sieve X),\n (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.o... | [
"case refine_1\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nhP :\n ∀ ⦃X : C⦄ (S : Sieve X),\n (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Representable | {
"line": 205,
"column": 18
} | {
"line": 205,
"column": 40
} | {
"line": 205,
"column": 41
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nY : D\nb✝ : C\nf' : F.obj b✝ ⟶ Y\nhf' : F.relativelyRepresentable f'\na✝ : C\ng : F.obj a✝ ⟶ Y\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nc : C\na b : c ⟶ hf'.pullback g\nh₁ : a ≫ hf'.fst' g = b ≫ hf'.fst' g\nh₂... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nY : D\nb✝ : C\nf' : F.obj b✝ ⟶ Y\nhf' : F.relativelyRepresentable f'\na✝ : C\ng : F.obj a✝ ⟶ Y\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nc : C\na b : c ⟶ hf'.pullback g\nh₁ : a ≫ hf'.fst' g = b ≫ hf'.fst' g\nh₂ : a ≫ hf'.s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Representable | {
"line": 291,
"column": 9
} | {
"line": 291,
"column": 20
} | {
"line": 291,
"column": 21
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF✝ : C ⥤ D\nF G H : D\nf : F ⟶ G\ng : G ⟶ H\nhf : F✝.relativelyRepresentable f\nhg : F✝.relativelyRepresentable g\nX : C\nh : F✝.obj X ⟶ H\n⊢ IsPullback (hf.fst (hg.fst h)) (F✝.map (hf.snd (hg.fst h) ≫ hg.snd h)) (f ≫ ... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF✝ : C ⥤ D\nF G H : D\nf : F ⟶ G\ng : G ⟶ H\nhf : F✝.relativelyRepresentable f\nhg : F✝.relativelyRepresentable g\nX : C\nh : F✝.obj X ⟶ H\n⊢ IsPullback (hf.fst (hg.fst h)) (F✝.map (hf.snd (hg.fst h)) ≫ F✝.map (hg.snd h)) (f ≫ g) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Representable | {
"line": 296,
"column": 70
} | {
"line": 296,
"column": 81
} | {
"line": 296,
"column": 82
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Y' X' : D\nf : X ⟶ X'\ng : Y ⟶ X'\nf' : Y' ⟶ Y\ng' : Y' ⟶ X\nP₁ : IsPullback f' g' g f\nhg : F.relativelyRepresentable g\na : C\nh : F.obj a ⟶ X\n⊢ hg.fst (h ≫ f) ≫ g = (F.map (hg.snd (h ≫ f)) ≫ h) ≫ f",... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Y' X' : D\nf : X ⟶ X'\ng : Y ⟶ X'\nf' : Y' ⟶ Y\ng' : Y' ⟶ X\nP₁ : IsPullback f' g' g f\nhg : F.relativelyRepresentable g\na : C\nh : F.obj a ⟶ X\n⊢ hg.fst (h ≫ f) ≫ g = F.map (hg.snd (h ≫ f)) ≫ h ≫ f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Sites.Representability | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 69
} | {
"line": 167,
"column": 70
} | [
{
"pp": "F : Sheaf zariskiTopology (Type u)\nι : Type u\nX : ι → Scheme\nf : (i : ι) → yoneda.obj (X i) ⟶ F.obj\nhf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)\ni✝ j✝ k : ι\nU : Scheme\nα β : U ⟶ (glueData hf).glued\nmem :\n Sieve.ofArrows (glueData hf).openCover.X (glueData hf).openCover.f ∈\n (grothendiec... | [
"F : Sheaf zariskiTopology (Type u)\nι : Type u\nX : ι → Scheme\nf : (i : ι) → yoneda.obj (X i) ⟶ F.obj\nhf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)\ni✝ j✝ k : ι\nU : Scheme\nα β : U ⟶ (glueData hf).glued\nmem :\n Sieve.ofArrows (glueData hf).openCover.X (glueData hf).openCover.f ∈\n (grothendieckTopology Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Representable | {
"line": 404,
"column": 42
} | {
"line": 404,
"column": 53
} | {
"line": 404,
"column": 54
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nF✝ : C ⥤ D\nX✝ Y : D\nP : MorphismProperty C\ninst✝² : F✝.Faithful\ninst✝¹ : F✝.Full\ninst✝ : P.IsStableUnderComposition\nF G H : D\nf : F ⟶ G\ng : G ⟶ H\nhf : relative F✝ P f\nhg : relative F✝ P g\nZ X : C\np : F✝.ob... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nF✝ : C ⥤ D\nX✝ Y : D\nP : MorphismProperty C\ninst✝² : F✝.Faithful\ninst✝¹ : F✝.Full\ninst✝ : P.IsStableUnderComposition\nF G H : D\nf : F ⟶ G\ng : G ⟶ H\nhf : relative F✝ P f\nhg : relative F✝ P g\nZ X : C\np : F✝.obj Z ⟶ H\nfst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Representable | {
"line": 418,
"column": 30
} | {
"line": 418,
"column": 41
} | {
"line": 418,
"column": 42
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\nF : C ⥤ D\nX✝ Y✝ : D\nP : MorphismProperty C\ninst✝³ : F.Faithful\ninst✝² : F.Full\ninst✝¹ : P.IsMultiplicative\ninst✝ : P.RespectsIso\nX : D\nY : C\ng : F.obj Y ⟶ X\n⊢ IsPullback g (F.map (𝟙 Y)) (𝟙 X) g",
"ppTe... | [
"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\nF : C ⥤ D\nX✝ Y✝ : D\nP : MorphismProperty C\ninst✝³ : F.Faithful\ninst✝² : F.Full\ninst✝¹ : P.IsMultiplicative\ninst✝ : P.RespectsIso\nX : D\nY : C\ng : F.obj Y ⟶ X\n⊢ IsPullback g (𝟙 (F.obj Y)) (𝟙 X) g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Representable | {
"line": 435,
"column": 4
} | {
"line": 435,
"column": 15
} | {
"line": 435,
"column": 16
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nF G : Cᵒᵖ ⥤ Type v₁\nf : F ⟶ G\nhf : (monomorphisms C).presheaf f\nX : C\na b : yoneda.obj X ⟶ F\nh : a ≫ f = b ≫ f\nthis : ⋯.lift a (𝟙 X) ⋯ = ⋯.lift b (𝟙 X) ⋯\n⊢ a = b",
"ppTerm": "?m.132",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nF G : Cᵒᵖ ⥤ Type v₁\nf : F ⟶ G\nhf : (monomorphisms C).presheaf f\nX : C\na b : yoneda.obj X ⟶ F\nh : a ≫ f = b ≫ f\nthis : ⋯.lift a (𝟙 X) ⋯ = ⋯.lift b (𝟙 X) ⋯\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 782,
"column": 18
} | {
"line": 782,
"column": 36
} | {
"line": 782,
"column": 37
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx y z : ℚ_[p]\n⊢ dist x z ≤ max (dist x y) (dist y z)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Lattice.toSemilatticeSup",
"padicNormE",
"congrArg",
"Real.instRatCas... | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx y z : ℚ_[p]\n⊢ padicNormE (x - z) ≤ padicNormE (x - y) ∨ padicNormE (x - z) ≤ padicNormE (y - z)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 817,
"column": 44
} | {
"line": 817,
"column": 55
} | {
"line": 817,
"column": 56
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nq : ℚ_[p]\nε : ℝ\nhε : 0 < ε\nε' : ℚ\nhε'l : 0 < ↑ε'\nhε'r : ↑ε' < ε\n⊢ 0 < ε'",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nq : ℚ_[p]\nε : ℝ\nhε : 0 < ε\nε' : ℚ\nhε'l : 0 < ↑ε'\nhε'r : ↑ε' < ε\n⊢ 0 < ε'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 818,
"column": 19
} | {
"line": 818,
"column": 42
} | {
"line": 818,
"column": 43
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nq : ℚ_[p]\nε : ℝ\nhε : 0 < ε\nε' : ℚ\nhε'l : 0 < ↑ε'\nhε'r : ↑ε' < ε\nr : ℚ\nhr : padicNormE (q - ↑r) < ε'\n⊢ ‖q - ↑r‖ < ↑ε'",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"Preorder.toLT",
... | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nq : ℚ_[p]\nε : ℝ\nhε : 0 < ε\nε' : ℚ\nhε'l : 0 < ↑ε'\nhε'r : ↑ε' < ε\nr : ℚ\nhr : padicNormE (q - ↑r) < ε'\n⊢ padicNormE (q - ↑r) < ε'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.DoldKan.Projections | {
"line": 71,
"column": 2
} | {
"line": 72,
"column": 6
} | {
"line": 74,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nq : ℕ\n⊢ P q + Q q = 𝟙 K[X]",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainComplex",
"HomologicalComplex.instCategory",
"Nat.instOne",
"Cate... | [] | rw [Q]
abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.DoldKan.Projections | {
"line": 71,
"column": 2
} | {
"line": 72,
"column": 6
} | {
"line": 74,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nq : ℕ\n⊢ P q + Q q = 𝟙 K[X]",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainComplex",
"HomologicalComplex.instCategory",
"Nat.instOne",
"Cate... | [] | rw [Q]
abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.DoldKan.Faces | {
"line": 62,
"column": 49
} | {
"line": 62,
"column": 79
} | {
"line": 62,
"column": 80
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X _⦋n + 1⦌\nv : HigherFacesVanish (q + 1) φ\nj : Fin (n + 1)\nhj : n + 1 ≤ ↑j + q\n⊢ n + 1 ≤ ↑j + (q + 1)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X _⦋n + 1⦌\nv : HigherFacesVanish (q + 1) φ\nj : Fin (n + 1)\nhj : n + 1 ≤ ↑j + q\n⊢ n + 1 ≤ ↑j + q + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 938,
"column": 30
} | {
"line": 938,
"column": 41
} | {
"line": 938,
"column": 42
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ\n⊢ ‖↑x‖ ≤ 1",
"ppTerm": "?m.105",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ\n⊢ ‖↑x‖ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 946,
"column": 2
} | {
"line": 946,
"column": 39
} | {
"line": 946,
"column": 40
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ ‖↑n‖ < 1 ↔ p ∣ n",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ ‖↑n‖ < 1 ↔ p ∣ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 959,
"column": 2
} | {
"line": 959,
"column": 30
} | {
"line": 959,
"column": 31
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ ‖↑n‖ = 1 ↔ p.Coprime n",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Nat.Coprime",
"Real",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"AddMonoidWithOne.to... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ ‖↑n‖ = 1 ↔ n.Coprime p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 984,
"column": 2
} | {
"line": 984,
"column": 35
} | {
"line": 984,
"column": 36
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nz1 z2 : ℚ_[p]\nh : ‖z1 - z2‖ < ‖z1‖\n⊢ ‖z2 - z1‖ < ‖z1‖",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nz1 z2 : ℚ_[p]\nh : ‖z1 - z2‖ < ‖z1‖\n⊢ ‖z2 - z1‖ < ‖z1‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 989,
"column": 2
} | {
"line": 990,
"column": 70
} | {
"line": 992,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ ‖↑(p - 1)‖ = 1",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"Eq.mpr",
"Nat.Coprime",
"Real",
"Nat.Prime",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"HSub.hSub... | [] | rw [norm_natCast_eq_one_iff]
exact (coprime_self_sub_right hp.out.one_le).mpr p.coprime_one_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 989,
"column": 2
} | {
"line": 990,
"column": 70
} | {
"line": 992,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ ‖↑(p - 1)‖ = 1",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"Eq.mpr",
"Nat.Coprime",
"Real",
"Nat.Prime",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"HSub.hSub... | [] | rw [norm_natCast_eq_one_iff]
exact (coprime_self_sub_right hp.out.one_le).mpr p.coprime_one_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1030,
"column": 2
} | {
"line": 1030,
"column": 30
} | {
"line": 1031,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nu : ℕ → ℚ_[p]\nhu : CauchySeq u\nc : CauSeq ℚ_[p] norm := ⟨u, ⋯⟩\n⊢ ∃ a, Tendsto u atTop (nhds a)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Norm.norm",
"NormedCommRing.toSeminormedCommRing",
"R... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nu : ℕ → ℚ_[p]\nhu : CauchySeq u\nc : CauSeq ℚ_[p] norm := ⟨u, ⋯⟩\ns : Set ℚ_[p]\nh : s ∈ nhds c.lim\n⊢ s ∈ Filter.map u atTop"
] | refine ⟨c.lim, fun s h ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicTopology.DoldKan.PInfty | {
"line": 54,
"column": 4
} | {
"line": 55,
"column": 47
} | {
"line": 55,
"column": 48
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ (P (n + 1)).f (n + 1) ≫ K[X].d (n + 1) n = K[X].d (n + 1) n ≫ (P n).f n",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"instHSMul",
"... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ (P (n + 1)).f (n + 1) ≫ ∑ i, (-1) ^ ↑i • X.δ i = (∑ i, (-1) ^ ↑i • X.δ i) ≫ (P (n + 1)).f n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1062,
"column": 4
} | {
"line": 1062,
"column": 15
} | {
"line": 1062,
"column": 16
} | [
{
"pp": "case h\np : ℕ\nhp : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : f - 0 ≈ 0\n⊢ f ≈ const (padicNorm p) 0",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"padicNorm.instIsAbsoluteValueRat",
"Rat",
"Rat.linearOrder",
"id",
"Rat.instDivisionRing",
... | [
"case h\np : ℕ\nhp : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : f - 0 ≈ 0\n⊢ f ≈ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1075,
"column": 45
} | {
"line": 1075,
"column": 63
} | {
"line": 1075,
"column": 63
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\n⊢ padicValRat p ↑n = ↑(padicValInt p n)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"congrArg",
"padicValInt",
"Rat",
"Rat.instIntCast",
"id",
"Int",
"padicValRat... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\n⊢ ↑(padicValInt p n) = ↑(padicValInt p n)"
] | padicValRat.of_int | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1097,
"column": 4
} | {
"line": 1097,
"column": 35
} | {
"line": 1097,
"column": 36
} | [
{
"pp": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : x + y ≠ 0\nhx : x = 0\n⊢ min x.valuation y.valuation ≤ (x + y).valuation",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instAddPadic",
"instZeroPadic",
"congrArg",
"AddMonoid.toA... | [
"case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : x + y ≠ 0\nhx : x = 0\n⊢ min (valuation 0) y.valuation ≤ y.valuation"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1099,
"column": 4
} | {
"line": 1099,
"column": 35
} | {
"line": 1099,
"column": 36
} | [
{
"pp": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : x + y ≠ 0\nhx : ¬x = 0\nhy : y = 0\n⊢ min x.valuation y.valuation ≤ (x + y).valuation",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instAddPadic",
"instZeroPadic",
"congrArg",
"... | [
"case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : x + y ≠ 0\nhx : ¬x = 0\nhy : y = 0\n⊢ min x.valuation (valuation 0) ≤ x.valuation"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.DoldKan.Decomposition | {
"line": 70,
"column": 10
} | {
"line": 70,
"column": 17
} | {
"line": 70,
"column": 18
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn q : ℕ\nhq : (Q q).f (n + 1) = ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev\nhqn : q ≤ n\na : ℕ\nha : q + a = n\n⊢ (Q (q + 1)).f (n + 1) = ∑ i with ↑i < q + 1, (P ↑i).f (n + 1) ≫ X.... | [
"case neg\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn q : ℕ\nhq : (Q q).f (n + 1) = ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev\nhqn : q ≤ n\na : ℕ\nha : q + a = n\n⊢ (Q q - P q ≫ Hσ q).f (n + 1) = ∑ i with ↑i < q + 1, (P ↑i).f (n + 1) ≫ X.δ i.r... | Q_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1189,
"column": 4
} | {
"line": 1189,
"column": 15
} | {
"line": 1189,
"column": 16
} | [
{
"pp": "case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nx✝¹ x✝ : ℚ_[p]\nh✝² : ¬x✝¹ + x✝ = 0\nh✝¹ : ¬x✝¹ = 0\nh✝ : ¬x✝ = 0\n⊢ x✝¹.valuation ≤ (x✝¹ + x✝).valuation ∨ x✝.valuation ≤ (x✝¹ + x✝).valuation",
"ppTerm": "?neg✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nx✝¹ x✝ : ℚ_[p]\nh✝² : ¬x✝¹ + x✝ = 0\nh✝¹ : ¬x✝¹ = 0\nh✝ : ¬x✝ = 0\n⊢ x✝¹.valuation ≤ (x✝¹ + x✝).valuation ∨ x✝.valuation ≤ (x✝¹ + x✝).valuation"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.DoldKan.Decomposition | {
"line": 79,
"column": 6
} | {
"line": 81,
"column": 11
} | {
"line": 83,
"column": 0
} | [
{
"pp": "case neg.e_a.e_s\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn q : ℕ\nhq : (Q q).f (n + 1) = ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev\nhqn : q ≤ n\na : ℕ\nha : q + a = n\nq' : Fin (n + 1) := ⟨q, ⋯⟩\n⊢ {i | ↑i < q + 1}.erase q' = {i... | [] | · ext ⟨i, hi⟩
simp_rw [Finset.mem_erase, Finset.mem_filter_univ, q', ne_eq, Fin.mk.injEq]
lia | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplicialObject.Split | {
"line": 97,
"column": 8
} | {
"line": 97,
"column": 39
} | {
"line": 97,
"column": 40
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nΔ₁ : SimplexCategory\nα₁ : { α // Epi α }\nΔ₂ : SimplexCategory\nα₂ : { α // Epi α }\nh₁ : Δ₁.len = Δ₂.len ∧ ⇑(Hom.toOrderHom (e ⟨op Δ₁, α₁⟩)) ≍ ⇑(Hom.toOrderHom (e ⟨op Δ₂, ... | [
"C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nΔ₁ : SimplexCategory\nα₁ : { α // Epi α }\nΔ₂ : SimplexCategory\nα₂ : { α // Epi α }\nh₁ : Δ₁.len = Δ₂.len ∧ ⇑(Hom.toOrderHom (e ⟨op Δ₁, α₁⟩)) ≍ ⇑(Hom.toOrderHom (e ⟨op Δ₂, α₂⟩))\n⊢ Δ₁.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.DoldKan.FunctorGamma | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 49
} | {
"line": 109,
"column": 50
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK : ChainComplex C ℕ\nΔ : SimplexCategory\ni : Δ ⟶ Δ\ninst✝ : Mono i\nhi : Isδ₀ i\n⊢ False",
"ppTerm": "?m.48",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK : ChainComplex C ℕ\nΔ : SimplexCategory\ni : Δ ⟶ Δ\ninst✝ : Mono i\nhi : Isδ₀ i\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.DoldKan.Faces | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 69
} | {
"line": 187,
"column": 70
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nq a m : ℕ\nφ : Y ⟶ X _⦋m + 1 + 1⦌\nv : HigherFacesVanish q φ\nj : Fin (m + 1 + 1)\nhj₁ : m + 1 + 1 ≤ ↑j + (q + 1)\nhqn : q ≤ m + 1\nha : q + a = m + 1\nhj₂ : ¬a = ↑j\nhaj : a < ↑j\nham : a ≤ m\... | [
"case neg\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nq a m : ℕ\nφ : Y ⟶ X _⦋m + 1 + 1⦌\nv : HigherFacesVanish q φ\nj : Fin (m + 1 + 1)\nhj₁ : m + 1 + 1 ≤ ↑j + (q + 1)\nhqn : q ≤ m + 1\nha : q + a = m + 1\nhj₂ : ¬a = ↑j\nhaj : a < ↑j\nham : a ≤ m\n⊢ a < ↑j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.DoldKan.HomotopyEquivalence | {
"line": 67,
"column": 6
} | {
"line": 68,
"column": 43
} | {
"line": 68,
"column": 44
} | [
{
"pp": "case zero\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\n⊢ PInfty.f 0 =\n (((dNext 0) fun i j ↦ (homotopyPToId X (j + 1)).hom i j) + (prevD 0) fun i j ↦ (homotopyPToId X (j + 1)).hom i j) +\n (𝟙 K[X]).f 0",
"ppTerm": "?zero",
"assigned": t... | [
"case zero\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\n⊢ 𝟙 (K[X].X 0) = (homotopyPToId X (0 + 1 + 1)).hom 0 (0 + 1) ≫ K[X].d (0 + 1) 0 + (𝟙 K[X]).f 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.DoldKan.HomotopyEquivalence | {
"line": 69,
"column": 6
} | {
"line": 73,
"column": 13
} | {
"line": 73,
"column": 14
} | [
{
"pp": "case succ\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ PInfty.f (n + 1) =\n (((dNext (n + 1)) fun i j ↦ (homotopyPToId X (j + 1)).hom i j) +\n (prevD (n + 1)) fun i j ↦ (homotopyPToId X (j + 1)).hom i j) +\n (𝟙 K[X]).f (n + 1)",
... | [
"case succ\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ (P (n + 1 + 1)).f (n + 1) =\n K[X].d (n + 1) n ≫ (homotopyPToId X (n + 1 + 1)).hom n (n + 1) +\n (homotopyPToId X (n.succ + 1)).hom n.succ (n.succ + 1) ≫ K[X].d (n + 1 + 1) (n + 1) +\n 𝟙... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.DoldKan.Degeneracies | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 37
} | {
"line": 188,
"column": 4
} | [
{
"pp": "case succ\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nT : C\nn : ℕ\nf : X _⦋n + 1⦌ ⟶ T\n⊢ DegeneraciesVanish f ↔ QInfty.f (n + 1) ≫ f = 0",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Nat.instOne",
"HomologicalComple... | [
"case succ.refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nT : C\nn : ℕ\nf : X _⦋n + 1⦌ ⟶ T\nhf : DegeneraciesVanish f\n⊢ QInfty.f (n + 1) ≫ f = 0",
"case succ.refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject... | refine ⟨fun hf ↦ ?_, fun hf ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicTopology.DoldKan.Normalized | {
"line": 117,
"column": 4
} | {
"line": 119,
"column": 45
} | {
"line": 121,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Category.{v_1, u_1} A\ninst✝ : Abelian A\nX✝ X : SimplicialObject A\n⊢ inclusionOfMooreComplexMap X ≫ PInftyToNormalizedMooreComplex X = 𝟙 ((normalizedMooreComplex A).obj X)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPre... | [] | simp only [← cancel_mono (inclusionOfMooreComplexMap X), assoc, id_comp,
PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap,
inclusionOfMooreComplexMap_comp_PInfty] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.DoldKan.Normalized | {
"line": 117,
"column": 4
} | {
"line": 119,
"column": 45
} | {
"line": 121,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Category.{v_1, u_1} A\ninst✝ : Abelian A\nX✝ X : SimplicialObject A\n⊢ inclusionOfMooreComplexMap X ≫ PInftyToNormalizedMooreComplex X = 𝟙 ((normalizedMooreComplex A).obj X)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPre... | [] | simp only [← cancel_mono (inclusionOfMooreComplexMap X), assoc, id_comp,
PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap,
inclusionOfMooreComplexMap_comp_PInfty] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.DoldKan.Normalized | {
"line": 117,
"column": 4
} | {
"line": 119,
"column": 45
} | {
"line": 121,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Category.{v_1, u_1} A\ninst✝ : Abelian A\nX✝ X : SimplicialObject A\n⊢ inclusionOfMooreComplexMap X ≫ PInftyToNormalizedMooreComplex X = 𝟙 ((normalizedMooreComplex A).obj X)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPre... | [] | simp only [← cancel_mono (inclusionOfMooreComplexMap X), assoc, id_comp,
PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap,
inclusionOfMooreComplexMap_comp_PInfty] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Idempotents.HomologicalComplex | {
"line": 74,
"column": 6
} | {
"line": 74,
"column": 50
} | {
"line": 74,
"column": 51
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nP : Karoubi (HomologicalComplex C c)\nn : ι\n⊢ P.p.f n ≫ P.p.f n = P.p.f n",
"ppTerm": "?m.71",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nP : Karoubi (HomologicalComplex C c)\nn : ι\n⊢ P.p.f n ≫ P.p.f n = P.p.f n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.HomologicalComplex | {
"line": 105,
"column": 8
} | {
"line": 105,
"column": 33
} | {
"line": 105,
"column": 34
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nK : HomologicalComplex (Karoubi C) c\ni j k : ι\nx✝¹ : c.Rel i j\nx✝ : c.Rel j k\n⊢ (K.d i j).f ≫ (K.d j k).f = 0",
"ppTerm": "?m.76",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nK : HomologicalComplex (Karoubi C) c\ni j k : ι\nx✝¹ : c.Rel i j\nx✝ : c.Rel j k\n⊢ (K.d i j).f ≫ (K.d j k).f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.DoldKan.FunctorGamma | {
"line": 338,
"column": 42
} | {
"line": 338,
"column": 69
} | {
"line": 338,
"column": 70
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nn : ℕ\nj : Fin (n + 1)\na✝ : n + 1 ≤ ↑j + (n + 1)\neq :\n ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫\n (Γ₀.obj K).map (SimplexCategory.δ... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nn : ℕ\nj : Fin (n + 1)\na✝ : n + 1 ≤ ↑j + (n + 1)\neq :\n ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫\n (Γ₀.obj K).map (SimplexCategory.δ j.succ).op ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.KaroubiKaroubi | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 40
} | {
"line": 35,
"column": 41
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : Karoubi (Karoubi C)\n⊢ P.p.f ≫ P.p.f = P.p.f",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : Karoubi (Karoubi C)\n⊢ P.p.f ≫ P.p.f = P.p.f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.KaroubiKaroubi | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 40
} | {
"line": 39,
"column": 41
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP Q : Karoubi (Karoubi C)\nf : P ⟶ Q\n⊢ P.p.f ≫ f.f.f = f.f.f ≫ Q.p.f",
"ppTerm": "?m.46",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP Q : Karoubi (Karoubi C)\nf : P ⟶ Q\n⊢ P.p.f ≫ f.f.f = f.f.f ≫ Q.p.f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.DoldKan.GammaCompN | {
"line": 64,
"column": 10
} | {
"line": 64,
"column": 37
} | {
"line": 64,
"column": 38
} | [
{
"pp": "case h₂\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nn : ℕ\ni : Fin (n + 2)\nhi : i ≠ 0\n⊢ ¬Isδ₀ (SimplexCategory.δ i)",
"ppTerm": "?h₂",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathl... | [
"case h₂\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nn : ℕ\ni : Fin (n + 2)\nhi : i ≠ 0\n⊢ ¬i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.DoldKan.NCompGamma | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 73
} | {
"line": 56,
"column": 2
} | [
{
"pp": "case mk.zero\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nm : ℕ\nh₁ : ⦋m⦌.len ≠ m + 1\nj : Fin (m + 2)\nhi : Mono (SimplexCategory.δ j)\nh₂ : ¬j = 0\nh₃ : 1 ≤ ↑j\n⊢ PInfty.f (m + 1) ≫ X.map (SimplexCategory.δ j).op = 0",
"ppTerm": "?mk.zero",
"as... | [] | exact (HigherFacesVanish.of_P (m + 1) m).comp_δ_eq_zero j h₂ (by lia) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Quiver.ReflQuiver | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 13
} | {
"line": 78,
"column": 14
} | [
{
"pp": "V : Type u\ninst✝¹ : ReflQuiver V\nW : Type u₂\ninst✝ : ReflQuiver W\nF_obj : V → W\nmap✝¹ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nmap_id✝¹ : ∀ (X : V), { obj := F_obj, map := map✝¹ }.map (𝟙rq X) = 𝟙rq ({ obj := F_obj, map := map✝¹ }.obj X)\nmap✝ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nmap_... | [
"V : Type u\ninst✝¹ : ReflQuiver V\nW : Type u₂\ninst✝ : ReflQuiver W\nF_obj : V → W\nmap✝¹ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nmap_id✝¹ : ∀ (X : V), { obj := F_obj, map := map✝¹ }.map (𝟙rq X) = 𝟙rq ({ obj := F_obj, map := map✝¹ }.obj X)\nmap✝ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nmap_id✝ : ∀ (X :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 13
} | {
"line": 234,
"column": 14
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : X.Splitting\ninst✝ : Preadditive C\n⊢ s.toKaroubiNondegComplexIsoN₁.hom.f ≫ PInfty = s.toKaroubiNondegComplexIsoN₁.hom.f",
"ppTerm": "?m.41",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : X.Splitting\ninst✝ : Preadditive C\n⊢ s.toKaroubiNondegComplexIsoN₁.hom.f ≫ PInfty = s.toKaroubiNondegComplexIsoN₁.hom.f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.DoldKan.NCompGamma | {
"line": 231,
"column": 2
} | {
"line": 239,
"column": 36
} | {
"line": 240,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\n⊢ IsIso Γ₂N₂.natTrans",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"ChainComplex",
"HomologicalComplex.instCategory",... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nthis : ∀ (P : Karoubi (SimplicialObject C)), IsIso (Γ₂N₂.natTrans.app P)\n⊢ IsIso Γ₂N₂.natTrans"
] | have : ∀ P : Karoubi (SimplicialObject C), IsIso (Γ₂N₂.natTrans.app P) := by
intro P
have : IsIso (N₂.map (Γ₂N₂.natTrans.app P)) := by
have h := identity_N₂_objectwise P
dsimp only [Functor.id_obj, Functor.comp_obj] at h
rw [hom_comp_eq_id] at h
rw [h]
infer_instance
exact isIs... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.SingleObj | {
"line": 251,
"column": 42
} | {
"line": 251,
"column": 61
} | {
"line": 251,
"column": 62
} | [
{
"pp": "X✝ Y✝ : MonCat\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : toCat.map a₁✝ = toCat.map a₂✝\n⊢ Hom.hom a₁✝ = Hom.hom a₂✝",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X✝ Y✝ : MonCat\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : toCat.map a₁✝ = toCat.map a₂✝\n⊢ Hom.hom a₁✝ = Hom.hom a₂✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps | {
"line": 88,
"column": 4
} | {
"line": 89,
"column": 11
} | {
"line": 89,
"column": 12
} | [
{
"pp": "case η\nX✝ : Type u_1\nY✝ : Type u_2\ninst✝³ : TopologicalSpace X✝\ninst✝² : TopologicalSpace Y✝\nf g : C(X✝, Y✝)\nX : Type u_3\nY : Type u_4\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nhequiv : X ≃ₕ Y\n⊢ 𝟭 (FundamentalGroupoid X) ≅ map hequiv.toFun ⋙ map hequiv.invFun",
"ppTerm": "?... | [
"case η\nX✝ : Type u_1\nY✝ : Type u_2\ninst✝³ : TopologicalSpace X✝\ninst✝² : TopologicalSpace Y✝\nf g : C(X✝, Y✝)\nX : Type u_3\nY : Type u_4\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nhequiv : X ≃ₕ Y\n⊢ 𝟭 (FundamentalGroupoid X) ≅ map hequiv.toFun ⋙ map hequiv.invFun"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps | {
"line": 90,
"column": 4
} | {
"line": 91,
"column": 11
} | {
"line": 91,
"column": 12
} | [
{
"pp": "case ε\nX✝ : Type u_1\nY✝ : Type u_2\ninst✝³ : TopologicalSpace X✝\ninst✝² : TopologicalSpace Y✝\nf g : C(X✝, Y✝)\nX : Type u_3\nY : Type u_4\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nhequiv : X ≃ₕ Y\n⊢ map hequiv.invFun ⋙ map hequiv.toFun ≅ 𝟭 (FundamentalGroupoid Y)",
"ppTerm": "?... | [
"case ε\nX✝ : Type u_1\nY✝ : Type u_2\ninst✝³ : TopologicalSpace X✝\ninst✝² : TopologicalSpace Y✝\nf g : C(X✝, Y✝)\nX : Type u_3\nY : Type u_4\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nhequiv : X ≃ₕ Y\n⊢ map hequiv.invFun ⋙ map hequiv.toFun ≅ 𝟭 (FundamentalGroupoid Y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyConnected | {
"line": 126,
"column": 6
} | {
"line": 126,
"column": 17
} | {
"line": 126,
"column": 18
} | [
{
"pp": "Y : Type u_2\ninst✝ : TopologicalSpace Y\nhpc : PathConnectedSpace Y\nx y : Y\np₁ p₂ : Path x y\nhloops : ∀ (x : Y) (γ : Path x x), ⟦γ⟧ = ⟦Path.refl x⟧\n⊢ Path.Homotopic.Quotient.trans ⟦p₁⟧ (Path.Homotopic.Quotient.symm ⟦p₂⟧) = Path.Homotopic.Quotient.refl x",
"ppTerm": "?m.114",
"assigned": tr... | [
"Y : Type u_2\ninst✝ : TopologicalSpace Y\nhpc : PathConnectedSpace Y\nx y : Y\np₁ p₂ : Path x y\nhloops : ∀ (x : Y) (γ : Path x x), ⟦γ⟧ = ⟦Path.refl x⟧\n⊢ (mk p₁).trans (mk p₂).symm = Path.Homotopic.Quotient.refl x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyConnected | {
"line": 111,
"column": 89
} | {
"line": 129,
"column": 26
} | {
"line": 131,
"column": 0
} | [
{
"pp": "Y : Type u_2\ninst✝ : TopologicalSpace Y\n⊢ SimplyConnectedSpace Y ↔ PathConnectedSpace Y ∧ ∀ (x : Y) (γ : Path x x), γ.Homotopic (Path.refl x)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"_private.Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyConnected.0.simply_co... | [] | by
rw [simply_connected_iff_paths_homotopic']
constructor
· -- Forward: all paths homotopic implies all loops null-homotopic
intro ⟨hpc, hall⟩
exact ⟨hpc, fun x γ => hall γ (Path.refl x)⟩
· -- Backward: all loops null-homotopic implies all paths homotopic
intro ⟨hpc, hloops⟩
refine ⟨hpc, fun {x ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyConnected | {
"line": 173,
"column": 25
} | {
"line": 173,
"column": 36
} | {
"line": 173,
"column": 37
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : ∀ (x : ↑s) (γ : Path x x), γ.Homotopic (Path.refl x)\nx : X\np : Path x x\nhp : ∀ (t : ↑unitInterval), p t ∈ s\n⊢ x ∈ s",
"ppTerm": "?m.59",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : ∀ (x : ↑s) (γ : Path x x), γ.Homotopic (Path.refl x)\nx : X\np : Path x x\nhp : ∀ (t : ↑unitInterval), p t ∈ s\n⊢ x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.Path | {
"line": 420,
"column": 2
} | {
"line": 420,
"column": 51
} | {
"line": 421,
"column": 2
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nx₀ x₁ x₃ : X\np₁ : Path x₀ x₁\np₂ : Path x₀ x₃\nhp : ∀ (t : ↑I), p₁ t = p₂ t\n⊢ ⟦p₁⟧ ≍ ⟦p₂⟧",
"ppTerm": "?m.129",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Real",
"cong... | [
"X : Type u\ninst✝ : TopologicalSpace X\nx₀ x₁ : X\np₁ p₂ : Path x₀ x₁\nhp : ∀ (t : ↑I), p₁ t = p₂ t\n⊢ ⟦p₁⟧ ≍ ⟦p₂⟧"
] | obtain rfl : x₁ = x₃ := by convert! hp 1 <;> simp | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.OfAdjunction | {
"line": 53,
"column": 6
} | {
"line": 53,
"column": 89
} | {
"line": 53,
"column": 89
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\ninst✝² : Category.{v_1, u_1} C₁\ninst✝¹ : Category.{v_2, u_2} C₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nW : MorphismProperty C₁\ninst✝ : W.IsMultiplicative\nhW : W.IsInvertedBy G\nhW' : W.functorCategory C₁ adj.unit\nX Y T : C₁\ns : T ⟶ X\nw✝ : W s\nf : T ⟶ Y\nthis✝ : IsI... | [] | rw [reassoc_of% this, Functor.map_inv, IsIso.hom_inv_id_assoc, adj.unit_naturality] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.ModelCategory.CofibrantObjectHomotopy | {
"line": 198,
"column": 36
} | {
"line": 198,
"column": 47
} | {
"line": 198,
"column": 48
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : ModelCategory C\nX : C\n⊢ LeftHomotopyRel (resolutionMap (𝟙 X) ≫ pResolutionObj X) (𝟙 (mk (resolutionObj X)).obj ≫ pResolutionObj X)",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : ModelCategory C\nX : C\n⊢ LeftHomotopyRel (pResolutionObj X) (pResolutionObj X)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.CofibrantObjectHomotopy | {
"line": 201,
"column": 36
} | {
"line": 201,
"column": 47
} | {
"line": 201,
"column": 48
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : ModelCategory C\nX₁ X₂ X₃ : C\nf : X₁ ⟶ X₂\ng : X₂ ⟶ X₃\n⊢ LeftHomotopyRel (resolutionMap (f ≫ g) ≫ pResolutionObj X₃)\n (((homMk (resolutionMap f)).hom ≫ (homMk (resolutionMap g)).hom) ≫ pResolutionObj X₃)",
"ppTerm": "?m.114",
"assigned... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : ModelCategory C\nX₁ X₂ X₃ : C\nf : X₁ ⟶ X₂\ng : X₂ ⟶ X₃\n⊢ LeftHomotopyRel (pResolutionObj X₁ ≫ f ≫ g) (pResolutionObj X₁ ≫ f ≫ g)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.CofibrantObjectHomotopy | {
"line": 211,
"column": 4
} | {
"line": 213,
"column": 38
} | {
"line": 213,
"column": 39
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : ModelCategory C\nx✝² x✝¹ : C\nx✝ : x✝² ⟶ x✝¹\nh : weakEquivalences C x✝\n⊢ (weakEquivalences (HoCat C)).inverseImage resolution x✝",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.has... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : ModelCategory C\nx✝² x✝¹ : C\nx✝ : x✝² ⟶ x✝¹\nh : weakEquivalences C x✝\n⊢ WeakEquivalence x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.OfAdjunction | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 13
} | {
"line": 92,
"column": 14
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\ninst✝³ : Category.{v_1, u_1} C₁\ninst✝² : Category.{v_2, u_2} C₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nadj : F ⊣ G\nW : MorphismProperty C₁\nhW : W.IsInvertedBy G\nhW' : W.functorCategory C₁ adj.counit\n⊢ G.IsLocalization W",
"ppTerm": "?m.37"... | [
"C₁ : Type u_1\nC₂ : Type u_2\ninst✝³ : Category.{v_1, u_1} C₁\ninst✝² : Category.{v_2, u_2} C₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nadj : F ⊣ G\nW : MorphismProperty C₁\nhW : W.IsInvertedBy G\nhW' : W.functorCategory C₁ adj.counit\n⊢ G.IsLocalization W"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic | {
"line": 52,
"column": 4
} | {
"line": 55,
"column": 33
} | {
"line": 56,
"column": 2
} | [
{
"pp": "case pos\nx : ↑I × ↑I\nh✝ : ↑x.2 ≤ 1 / 2\n⊢ ↑x.1 * 2 * ↑x.2 ∈ I",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"mul_nonneg",
"Real.partialOrder",
"Semigroup.toMul",
"Real",
"IsOrderedRing.toPosMulMono",... | [] | constructor
· apply mul_nonneg <;> grind
· rw [mul_assoc]
apply mul_le_one₀ <;> grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic | {
"line": 52,
"column": 4
} | {
"line": 55,
"column": 33
} | {
"line": 56,
"column": 2
} | [
{
"pp": "case pos\nx : ↑I × ↑I\nh✝ : ↑x.2 ≤ 1 / 2\n⊢ ↑x.1 * 2 * ↑x.2 ∈ I",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"mul_nonneg",
"Real.partialOrder",
"Semigroup.toMul",
"Real",
"IsOrderedRing.toPosMulMono",... | [] | constructor
· apply mul_nonneg <;> grind
· rw [mul_assoc]
apply mul_le_one₀ <;> grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.ModelCategory.DerivabilityStructureCofibrant | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 40
} | {
"line": 38,
"column": 41
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : ModelCategory C\nX : C\nR : (localizerMorphism C).LeftResolution X\n⊢ WeakEquivalence R.w",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.hasFiniteCoproducts_of_hasFiniteColimits",
... | [
"C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : ModelCategory C\nX : C\nR : (localizerMorphism C).LeftResolution X\n⊢ weakEquivalences C R.w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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