module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 90
} | {
"line": 183,
"column": 2
} | [
{
"pp": "case refine_3\nR : Type u_3\nm : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\ni : ℕ\nhim : i < m\nthis :\n supDegree (⇑toLex) (∑ j ∈ powersetCard (i + 1) univ, (monomial (∑ j ∈ j, fun₀ | j => 1)) 1) =\n supDegree (⇑toLex) ((monomial (∑ j ∈ Iic ⟨i, him⟩, fun₀ | j => 1)) 1) ∧\n leadingCoeff... | [
"case refine_1\nR : Type u_3\nm : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\ni : ℕ\nhim : i < m\n⊢ Iic ⟨i, him⟩ ∈ powersetCard (i + 1) univ",
"case refine_2\nR : Type u_3\nm : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\ni : ℕ\nhim : i < m\n⊢ ∀ j ∈ powersetCard (i + 1) univ,\n j ≠ Iic ⟨i, him⟩ →\n ... | · rwa [← esymm_eq_sum_monomial, ← Finsupp.indicator_eq_sum_single, ← single_eq_monomial,
supDegree_single_ne_zero _ one_ne_zero, leadingCoeff_single toLex.injective] at this | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.LaurentSeries | {
"line": 903,
"column": 37
} | {
"line": 903,
"column": 43
} | {
"line": 903,
"column": 44
} | [
{
"pp": "case h\nK : Type u_2\ninst✝ : Field K\nx : K⸨X⸩\nhx : ¬Valued.v x = 0\n⊢ (Valued.v RatFunc.X ^ (Valued.v x).log)⁻¹ = Valued.v x",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Multiplicative.group",
"Eq.mpr",
"Int.instAddCommMonoid",
"LinearOrderedCommGroupWi... | [
"case h\nK : Type u_2\ninst✝ : Field K\nx : K⸨X⸩\nhx : ¬Valued.v x = 0\n⊢ ((valuation K⟮X⟯ (idealX K)) RatFunc.X ^ (Valued.v x).log)⁻¹ = Valued.v x"
] | v_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 946,
"column": 78
} | {
"line": 946,
"column": 84
} | {
"line": 947,
"column": 8
} | [
{
"pp": "case mpr.refine_2\nK : Type u_2\ninst✝ : Field K\nS : Set (K⟮X⟯ × K⟮X⟯)\nw✝ : Set K⟮X⟯\nhT : w✝ ∈ nhds 0\npre_T : (fun x ↦ x.2 - x.1) ⁻¹' w✝ ⊆ S\nd : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ\nhd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝\nX : Set K⸨X⸩ := {f | Valued.v f < embedding ↑d}\nX_def... | [
"case mpr.refine_2\nK : Type u_2\ninst✝ : Field K\nS : Set (K⟮X⟯ × K⟮X⟯)\nw✝ : Set K⟮X⟯\nhT : w✝ ∈ nhds 0\npre_T : (fun x ↦ x.2 - x.1) ⁻¹' w✝ ⊆ S\nd : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ\nhd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝\nX : Set K⸨X⸩ := {f | Valued.v f < embedding ↑d}\nX_def : X = {f | ... | v_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Restricted | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 43
} | {
"line": 87,
"column": 2
} | [
{
"pp": "M : Type u_3\nS : Type u_4\ninst✝³ : AddMonoid M\ninst✝² : Finset.HasAntidiagonal M\ninst✝¹ : NormedRing S\ninst✝ : IsUltrametricDist S\nf g : M → S\nC : M → ℝ\nhC : ∀ (a b : M), C (a + b) = C a * C b\nhf : Tendsto (fun i ↦ ‖f i‖ * C i) cofinite (𝓝 0)\nhg : Tendsto (fun i ↦ ‖g i‖ * C i) cofinite (𝓝 0... | [
"M : Type u_3\nS : Type u_4\ninst✝³ : AddMonoid M\ninst✝² : Finset.HasAntidiagonal M\ninst✝¹ : NormedRing S\ninst✝ : IsUltrametricDist S\nf g : M → S\nC : M → ℝ\nhC : ∀ (a b : M), C (a + b) = C a * C b\nhf : Tendsto (fun i ↦ ‖f i‖ * C i) cofinite (𝓝 0)\nhg : Tendsto (fun i ↦ ‖g i‖ * C i) cofinite (𝓝 0)\nhC' : 0 ≤... | refine Finset.sup'_mono_fun fun x hx ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Noetherian.OfPrime | {
"line": 57,
"column": 20
} | {
"line": 57,
"column": 34
} | {
"line": 57,
"column": 35
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R\nhsup : (I ⊔ span {a}).FG\nhcolon : (Submodule.colon I ↑(span {a})).FG\nw✝¹ : ℕ\nf : Fin w✝¹ → R\nhf : span (Set.range f) = I ⊔ span {a}\nw✝ : ℕ\ni : Fin w✝ → R\nhi : Submodule.span R (Set.range i) = Submodule.colon I ↑(span {a})\nr p ... | [
"case refine_2\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R\nhsup : (I ⊔ span {a}).FG\nhcolon : (Submodule.colon I ↑(span {a})).FG\nw✝¹ : ℕ\nf : Fin w✝¹ → R\nhf : span (Set.range f) = I ⊔ span {a}\nw✝ : ℕ\ni : Fin w✝ → R\nhi : Submodule.span R (Set.range i) = Submodule.colon I ↑(span {a})\nr p : Fin w✝¹ → ... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.WittPolynomial | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 25
} | {
"line": 227,
"column": 26
} | [
{
"pp": "case ind\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)\n⊢ n ∈ (X n - ∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars ∧\n (X n - ∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars ⊆ range ... | [
"case ind\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)\n⊢ n ∈ (X n).vars ∪ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars ∧\n (X n).vars ∪ (∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p ℚ i ^ p ^ (n - i)).vars ⊆ rang... | vars_sub_of_disjoint, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 160,
"column": 4
} | {
"line": 161,
"column": 63
} | {
"line": 162,
"column": 4
} | [
{
"pp": "case refine_2\np : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℚ\nφ : ℕ → MvPolynomial (idx × ℕ) ℚ\nH : ∀ (n : ℕ), (bind₁ φ) (W_ ℚ n) = (bind₁ fun i ↦ (rename (Prod.mk i)) (W_ ℚ n)) Φ\nn : ℕ\n⊢ φ n = wittStructureRat p Φ n",
"ppTerm": "?refine_2",
"assigned": true,
"use... | [
"case refine_2\np : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℚ\nφ : ℕ → MvPolynomial (idx × ℕ) ℚ\nH : ∀ (n : ℕ), (bind₁ φ) (W_ ℚ n) = (bind₁ fun i ↦ (rename (Prod.mk i)) (W_ ℚ n)) Φ\nn : ℕ\n⊢ (bind₁ φ) ((bind₁ (W_ ℚ)) (xInTermsOfW p ℚ n)) = wittStructureRat p Φ n"
] | rw [show φ n = bind₁ φ (bind₁ (W_ ℚ) (xInTermsOfW p ℚ n)) by
rw [bind₁_wittPolynomial_xInTermsOfW p, bind₁_X_right]] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 9
} | {
"line": 221,
"column": 9
} | [
{
"pp": "p : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m\nkey :\n (bind₁ fun i ↦ (expand p) ((rename (Prod.mk i)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) ... | [
"p : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m\nkey :\n (bind₁ fun i ↦ (expand p) ((rename (Prod.mk i)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =\n (bind... | key | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 245,
"column": 6
} | {
"line": 245,
"column": 9
} | {
"line": 245,
"column": 9
} | [
{
"pp": "case succ\np : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m\nkey :\n (bind₁ fun i ↦ (rename fun i_1 ↦ (i, i_1)) ((expand p) (W_ (ZMod (p ^ (n + 1))... | [
"case succ\np : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m\nkey :\n (bind₁ fun i ↦ (rename fun i_1 ↦ (i, i_1)) ((expand p) (W_ (ZMod (p ^ (n + 1))) n)))\n ... | key | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 92,
"column": 2
} | {
"line": 93,
"column": 56
} | {
"line": 97,
"column": 0
} | [
{
"pp": "p : ℕ\nR : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ IsPoly p fun R _Rcr ↦ verschiebungFun",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"WittVector.aeval_verschiebung_poly'",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"C... | [] | use verschiebungPoly
simp only [aeval_verschiebung_poly', forall₃_true_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 92,
"column": 2
} | {
"line": 93,
"column": 56
} | {
"line": 97,
"column": 0
} | [
{
"pp": "p : ℕ\nR : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ IsPoly p fun R _Rcr ↦ verschiebungFun",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"WittVector.aeval_verschiebung_poly'",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"C... | [] | use verschiebungPoly
simp only [aeval_verschiebung_poly', forall₃_true_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.Identities | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 31
} | {
"line": 69,
"column": 2
} | [
{
"pp": "case zero.hn\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nj : ℕ\nhj : j ≠ 0\n⊢ 0 < j",
"ppTerm": "?zero.hn",
"assigned": true,
"usedConstants": [
"Nat.pos_of_ne_zero"
],
"usedFVars": [
"j",
"hj"
],
"usedGoals": []
... | [] | exact Nat.pos_of_ne_zero hj | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.WittVector.Frobenius | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 32
} | {
"line": 157,
"column": 33
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nh1 : ↑p ^ n * ⅟↑p ^ n = 1\ni : ℕ\nhi : i < n\n⊢ (C (↑p ^ i) *\n ∑ k ∈ range (p ^ (n - i)),\n (C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p i)) ^ (k + 1) *\n (X i ^ p) ^ (p ^ (n - i) - (k + 1)) *\n ... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nh1 : ↑p ^ n * ⅟↑p ^ n = 1\ni : ℕ\nhi : i < n\n⊢ (C (↑p ^ i) *\n ∑ k ∈ range (p ^ (n - i)),\n (C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p i)) ^ (k + 1) *\n (X i ^ p) ^ (p ^ (n - i) - (k + 1)) *\n ↑(... | Nat.succ_eq_add_one (n - i), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.Identities | {
"line": 170,
"column": 21
} | {
"line": 170,
"column": 41
} | {
"line": 170,
"column": 42
} | [
{
"pp": "case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ k : ℕ\nih : ((⇑verschiebung)^[k] x).coeff (k✝ + k) = x.coeff k✝\n⊢ ((⇑verschiebung)^[k + 1] x).coeff (k✝ + (k + 1)) = x.coeff k✝",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Functio... | [
"case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ k : ℕ\nih : ((⇑verschiebung)^[k] x).coeff (k✝ + k) = x.coeff k✝\n⊢ (verschiebung ((⇑verschiebung)^[k] x)).coeff (k✝ + (k + 1)) = x.coeff k✝"
] | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.Identities | {
"line": 177,
"column": 8
} | {
"line": 177,
"column": 28
} | {
"line": 177,
"column": 29
} | [
{
"pp": "case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\ni : ℕ\nih : ∀ (y : 𝕎 R), (⇑verschiebung)^[i] x * y = (⇑verschiebung)^[i] (x * (⇑frobenius)^[i] y)\ny : 𝕎 R\n⊢ (⇑verschiebung)^[i + 1] x * y = (⇑verschiebung)^[i + 1] (x * (⇑frobenius)^[i + 1] y)",
"ppTerm": "?s... | [
"case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\ni : ℕ\nih : ∀ (y : 𝕎 R), (⇑verschiebung)^[i] x * y = (⇑verschiebung)^[i] (x * (⇑frobenius)^[i] y)\ny : 𝕎 R\n⊢ verschiebung ((⇑verschiebung)^[i] x) * y = (⇑verschiebung)^[i + 1] (x * (⇑frobenius)^[i + 1] y)"
] | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.Identities | {
"line": 177,
"column": 63
} | {
"line": 177,
"column": 83
} | {
"line": 178,
"column": 6
} | [
{
"pp": "case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\ni : ℕ\nih : ∀ (y : 𝕎 R), (⇑verschiebung)^[i] x * y = (⇑verschiebung)^[i] (x * (⇑frobenius)^[i] y)\ny : 𝕎 R\n⊢ verschiebung ((⇑verschiebung)^[i] (x * (⇑frobenius)^[i] (frobenius y))) =\n (⇑verschiebung)^[i + 1] (... | [
"case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\ni : ℕ\nih : ∀ (y : 𝕎 R), (⇑verschiebung)^[i] x * y = (⇑verschiebung)^[i] (x * (⇑frobenius)^[i] y)\ny : 𝕎 R\n⊢ verschiebung ((⇑verschiebung)^[i] (x * (⇑frobenius)^[i] (frobenius y))) =\n verschiebung ((⇑verschiebung)^[i] (x ... | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Perfection | {
"line": 730,
"column": 17
} | {
"line": 730,
"column": 20
} | {
"line": 730,
"column": 21
} | [
{
"pp": "K : Type u₁\ninst✝⁴ : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝³ : CommRing O\ninst✝² : Algebra O K\nhv : v.Integers O\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fact ¬IsUnit ↑p\nf : PreTilt O p\nh : ∃ n, (coeff n) f ≠ 0\nk : ℕ\nih :\n (coeff (Nat.find h + k)) f ≠ 0 →\n ModP.preVal K v O p... | [
"K : Type u₁\ninst✝⁴ : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝³ : CommRing O\ninst✝² : Algebra O K\nhv : v.Integers O\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fact ¬IsUnit ↑p\nf : PreTilt O p\nh : ∃ n, (coeff n) f ≠ 0\nk : ℕ\nih :\n (coeff (Nat.find h + k)) f ≠ 0 →\n ModP.preVal K v O p ((coeff (Na... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.Identities | {
"line": 205,
"column": 21
} | {
"line": 205,
"column": 41
} | {
"line": 205,
"column": 42
} | [
{
"pp": "case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nx : 𝕎 R\nk i : ℕ\nih : ((⇑frobenius)^[i] x).coeff k = x.coeff k ^ p ^ i\n⊢ ((⇑frobenius)^[i + 1] x).coeff k = x.coeff k ^ p ^ (i + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
... | [
"case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nx : 𝕎 R\nk i : ℕ\nih : ((⇑frobenius)^[i] x).coeff k = x.coeff k ^ p ^ i\n⊢ (frobenius ((⇑frobenius)^[i] x)).coeff k = x.coeff k ^ p ^ (i + 1)"
] | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.Frobenius | {
"line": 292,
"column": 6
} | {
"line": 292,
"column": 53
} | {
"line": 293,
"column": 4
} | [
{
"pp": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : CharP R p\ninst✝ : PerfectRing R p\nf : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nn : ℕ\n⊢ ((map ↑(_root_.frobeniusEquiv R p).symm) ((map (_root_.frobenius R p)) f)).coeff n... | [] | exact frobeniusEquiv_symm_apply_frobenius R p _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.WittVector.Identities | {
"line": 229,
"column": 8
} | {
"line": 229,
"column": 28
} | {
"line": 229,
"column": 29
} | [
{
"pp": "case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nx : 𝕎 R\nn : ℕ\nih : (⇑verschiebung ∘ ⇑frobenius)^[n] x = x * ↑p ^ n\n⊢ (⇑verschiebung ∘ ⇑frobenius)^[n + 1] x = x * ↑p ^ (n + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
... | [
"case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nx : 𝕎 R\nn : ℕ\nih : (⇑verschiebung ∘ ⇑frobenius)^[n] x = x * ↑p ^ n\n⊢ (⇑verschiebung ∘ ⇑frobenius) ((⇑verschiebung ∘ ⇑frobenius)^[n] x) = x * ↑p ^ (n + 1)"
] | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 187,
"column": 8
} | {
"line": 187,
"column": 26
} | {
"line": 188,
"column": 6
} | [
{
"pp": "case succ.refine_1\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nn : ℕ\nhn : p.natDegree < n\nd : ℕ\nhd :\n (PowerSeries.coeff n) (↑p * ↑(invOneSubPow F d)) =\n eval (↑n)\n (match d with\n | 0 => 0\n | d.succ => ∑ i ∈ p.support, p.coeff i • preHilbertPoly F d i)\nh_l... | [] | exact h_le ⟨s, hs⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 110,
"column": 8
} | {
"line": 110,
"column": 100
} | {
"line": 111,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nh : IsLeftCancelMulZero R[X]\na b c : R\neq : (fun x ↦ a + x) b = (fun x ↦ a + x) c\na✝ : Nontrivial R\ntrinomial : R → R[X] := fun r ↦ a • X ^ 2 + r • X + C a\nr : R\n⊢ (X + C 1) * trinomial r = a • X ^ 3 + (a + r) • X ^ 2 + (a + r) • X + C a",
"ppTerm": "?m.240",... | [
"R : Type u_1\ninst✝ : Semiring R\nh : IsLeftCancelMulZero R[X]\na b c : R\neq : (fun x ↦ a + x) b = (fun x ↦ a + x) c\na✝ : Nontrivial R\ntrinomial : R → R[X] := fun r ↦ a • X ^ 2 + r • X + C a\nr : R\n⊢ C a * X * X ^ 2 + C a * X ^ 2 + (C r * X * X + C r * X) + (C a * X + C a) =\n C a * X ^ 3 + (C a * X ^ 2 + C... | simp only [trinomial, mul_add, add_mul, ← C_mul', C_1, one_mul, ← mul_assoc, X_mul_C, C_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 31
} | {
"line": 63,
"column": 4
} | [
{
"pp": "n : ℕ\n⊢ (⇑derivative)^[n] (∑ m ∈ range (n + 1), n.choose m • (-1) ^ m * X ^ (n + m)) =\n ∑ x ∈ range (n + 1), ↑((n + x)! / x !) * C (↑(n.choose x) * (-1) ^ x) * X ^ x",
"ppTerm": "?m.213",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Polynom... | [
"n : ℕ\n⊢ ∑ b ∈ range (n + 1), (⇑derivative)^[n] (n.choose b • (-1) ^ b * X ^ (n + b)) =\n ∑ x ∈ range (n + 1), ↑((n + x)! / x !) * C (↑(n.choose x) * (-1) ^ x) * X ^ x"
] | rw [iterate_derivative_sum] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
"line": 57,
"column": 68
} | {
"line": 58,
"column": 95
} | {
"line": 60,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : Semiring A\nI : Ideal A\nf g : A⟦X⟧\nn : ℕ\nhg : ∀ i ≤ n, (coeff i) g ∈ I\n⊢ ∀ i ≤ n, (coeff i) (f * g) ∈ I",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Submodule",
"Submodule.mem_top._simp_1",
"Semiring.toModule",
"HMul.hMul",
... | [] | by
simpa using coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal (I := ⊤) (f := f) n (by simp) hg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PowerSeries.Expand | {
"line": 89,
"column": 28
} | {
"line": 89,
"column": 42
} | {
"line": 89,
"column": 43
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : R⟦X⟧\nm : ℕ\n⊢ (MvPowerSeries.coeff (Finsupp.single () (p * m))) ((MvPowerSeries.expand p hp) φ) =\n (MvPowerSeries.coeff (Finsupp.single () m)) φ",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"MvPowerSeries.expan... | [
"R : Type u_2\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : R⟦X⟧\nm : ℕ\n⊢ (MvPowerSeries.coeff (Finsupp.single () (p • m))) ((MvPowerSeries.expand p hp) φ) =\n (MvPowerSeries.coeff (Finsupp.single () m)) φ"
] | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
"line": 96,
"column": 4
} | {
"line": 99,
"column": 21
} | {
"line": 100,
"column": 2
} | [
{
"pp": "R : Type u\nN : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R N\nι : Type u_3\ninst✝² : Preorder ι\nF : ι → Type u_4\ninst✝¹ : (i : ι) → AddCommMonoid (F i)\ninst✝ : (i : ι) → Module R (F i)\nf : ⦃i j : ι⦄ → i ≤ j → F i →ₗ[R] F j\nD : DirectedSystem F fun x x_1 h ↦ ⇑(f ... | [] | rw [← id_apply (R := R) t]
apply DFunLike.congr_fun
ext m n
simp [D.map_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
"line": 96,
"column": 4
} | {
"line": 99,
"column": 21
} | {
"line": 100,
"column": 2
} | [
{
"pp": "R : Type u\nN : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R N\nι : Type u_3\ninst✝² : Preorder ι\nF : ι → Type u_4\ninst✝¹ : (i : ι) → AddCommMonoid (F i)\ninst✝ : (i : ι) → Module R (F i)\nf : ⦃i j : ι⦄ → i ≤ j → F i →ₗ[R] F j\nD : DirectedSystem F fun x x_1 h ↦ ⇑(f ... | [] | rw [← id_apply (R := R) t]
apply DFunLike.congr_fun
ext m n
simp [D.map_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
"line": 149,
"column": 4
} | {
"line": 152,
"column": 21
} | {
"line": 153,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type u_1\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nι : Type u_3\ninst✝² : Preorder ι\nF : ι → Type u_4\ninst✝¹ : (i : ι) → AddCommMonoid (F i)\ninst✝ : (i : ι) → Module R (F i)\nf : ⦃i j : ι⦄ → i ≤ j → F i →ₗ[R] F j\nD : DirectedSystem F fun x x_1 h ↦ ⇑(f ... | [] | rw [← id_apply (R := R) t]
apply DFunLike.congr_fun
ext m n
simp [D.map_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
"line": 149,
"column": 4
} | {
"line": 152,
"column": 21
} | {
"line": 153,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type u_1\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nι : Type u_3\ninst✝² : Preorder ι\nF : ι → Type u_4\ninst✝¹ : (i : ι) → AddCommMonoid (F i)\ninst✝ : (i : ι) → Module R (F i)\nf : ⦃i j : ι⦄ → i ≤ j → F i →ₗ[R] F j\nD : DirectedSystem F fun x x_1 h ↦ ⇑(f ... | [] | rw [← id_apply (R := R) t]
apply DFunLike.congr_fun
ext m n
simp [D.map_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Regular.Free | {
"line": 52,
"column": 77
} | {
"line": 52,
"column": 91
} | {
"line": 53,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : FinitePresentation R M\nx : R\nmem : x ∈ ⊥.jacobson\nreg : IsSMulRegular M x\nfree : Free (R ⧸ Ideal.span {x}) (QuotSMulTop x M)\nthis : Module.Finite (R ⧸ Ideal.span {x}) (QuotSMulTop x M)\nI : Type ... | [
"R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : FinitePresentation R M\nx : R\nmem : x ∈ ⊥.jacobson\nreg : IsSMulRegular M x\nfree : Free (R ⧸ Ideal.span {x}) (QuotSMulTop x M)\nthis : Module.Finite (R ⧸ Ideal.span {x}) (QuotSMulTop x M)\nI : Type u_2 := Free.... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 344,
"column": 8
} | {
"line": 344,
"column": 79
} | {
"line": 344,
"column": 80
} | [
{
"pp": "case inr\nA : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsHausdorff I A\nq : A⟦X⟧\nr : A[X]\nhdeg : r.degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nheq :\n ((X ^ ((map (Ideal.Quotient.mk I)) g).order.toNat *\n mk fun i ↦ (coeff (i +... | [
"case inr\nA : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsHausdorff I A\nq : A⟦X⟧\nr : A[X]\nhdeg : r.degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nheq :\n ((X ^ ((map (Ideal.Quotient.mk I)) g).order.toNat *\n mk fun i ↦ (coeff (i + ((map (Idea... | Polynomial.coeff_eq_zero_of_degree_lt (lt_of_lt_of_le hdeg (by simpa)), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 35
} | {
"line": 366,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsHausdorff I A\nq q' : A⟦X⟧\nr r' : A[X]\nhr : r.degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nhr' : r'.degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nheq : g * q + ↑r = g * q' + ↑r'\n⊢ g... | [
"A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsHausdorff I A\nq q' : A⟦X⟧\nr r' : A[X]\nhr : r.degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nhr' : r'.degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nheq : g * q = g * q' + ↑r' - ↑r\n⊢ g * (q - q') ... | rw [← eq_sub_iff_add_eq] at heq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 415,
"column": 4
} | {
"line": 415,
"column": 60
} | {
"line": 416,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\na✝ : A\nf✝ f'✝ : A⟦X⟧\ninst✝ : IsAdicComplete I A\nf f' : A⟦X⟧\nhf : f ≈ f'\na : A⟦X⟧\nha : a * g = f - f'\n⊢ H.mod f = H.mod f'",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"WithBot... | [
"A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\na✝ : A\nf✝ f'✝ : A⟦X⟧\ninst✝ : IsAdicComplete I A\nf f' : A⟦X⟧\nhf : f ≈ f'\na : A⟦X⟧\nha : a * g = f - f'\nhf1 : (H.mod f).degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nhf2 : f = g * H.div f + ↑(H.mod f)\n⊢ H.mod f ... | obtain ⟨hf1, hf2⟩ := H.isWeierstrassDivisionAt_div_mod f | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.RegularLocalRing.Polynomial | {
"line": 70,
"column": 25
} | {
"line": 70,
"column": 61
} | {
"line": 70,
"column": 62
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsRegularLocalRing R\np : Ideal R[X]\ninst✝ : p.IsPrime\nmax : comap C p = maximalIdeal R\nq : Ideal R[X] := Ideal.map C (maximalIdeal R)\nqle : q ≤ p\nreg : ↑(Submodule.spanFinrank (maximalIdeal R)) = ringKrullDim R\nfg' : (maximalIdeal R).FG\nfg : (Submodul... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsRegularLocalRing R\np : Ideal R[X]\ninst✝ : p.IsPrime\nmax : comap C p = maximalIdeal R\nq : Ideal R[X] := Ideal.map C (maximalIdeal R)\nqle : q ≤ p\nreg : ↑(Submodule.spanFinrank (maximalIdeal R)) = ringKrullDim R\nfg' : (maximalIdeal R).FG\nfg : (Submodule.generators... | maximalIdeal_height_eq_ringKrullDim, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 176,
"column": 45
} | {
"line": 176,
"column": 59
} | {
"line": 176,
"column": 60
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsLocalRing R\ninst✝ : IsNoetherianRing R\nrs : List R\nreg : RingTheory.Sequence.IsRegular R rs\nmem_max : ∀ x ∈ rs, x ∈ maximalIdeal R\n⊢ Submodule.map (↑(Shrink.linearEquiv R R).symm) (Ideal.ofList rs * ⊤) = Ideal.ofList rs • ⊤",
... | [
"R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsLocalRing R\ninst✝ : IsNoetherianRing R\nrs : List R\nreg : RingTheory.Sequence.IsRegular R rs\nmem_max : ∀ x ∈ rs, x ∈ maximalIdeal R\n⊢ Submodule.map (↑(Shrink.linearEquiv R R).symm) (Ideal.ofList rs • ⊤) = Ideal.ofList rs • ⊤"
] | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 81
} | {
"line": 92,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\ninst✝¹ : IsSimpleRing R\ninst✝ : IsArtinianRing R\nn : ℕ\nhn : NeZero n\nS : Submodule R R\nhS : IsSimpleModule R ↥S\ne : R ≃ₗ[R] Fin n → ↥S\n⊢ ∃ n, ∃ (_ : NeZero n), ∃ I, ∃ (_ : IsSimpleModule R ↥I), Nonempty (R ≃+* Matrix (Fin n) (Fin n) (Module.End R ↥I)ᵐᵒᵖ)",
"ppTer... | [
"R : Type u\ninst✝² : Ring R\ninst✝¹ : IsSimpleRing R\ninst✝ : IsArtinianRing R\nn : ℕ\nhn : NeZero n\nS : Submodule R R\nhS : IsSimpleModule R ↥S\ne : R ≃ₗ[R] Fin n → ↥S\n⊢ Rᵐᵒᵖ ≃+* Matrix (Fin n) (Fin n) (Module.End R ↥S)"
] | refine ⟨n, hn, S, hS, ⟨.trans (.opOp R) <| .trans (.op ?_) (.symm .mopMatrix)⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 32
} | {
"line": 39,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\nι : Type u_2\ns : Finset ι\nf : ι → R\nhs : ∀ x ∈ s, f x ∈ I\n⊢ I.ringCon (s.sum f) (∑ _x ∈ ?m.23, 0)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZero... | [] | exact I.ringCon.finsetSum s hs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.WittVector.Compare | {
"line": 53,
"column": 57
} | {
"line": 53,
"column": 79
} | {
"line": 53,
"column": 79
} | [
{
"pp": "case h\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharP R p\ni : ℕ\nhin : i < n\n⊢ ¬(↑p ^ i).coeff i = 0",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WittVector.instNatCast",
"congrArg",
"CommSemiring.toSem... | [
"case h\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharP R p\ni : ℕ\nhin : i < n\n⊢ ¬1 = 0"
] | WittVector.coeff_p_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 55
} | {
"line": 75,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹⁰ : CommRing R\nS : Type u_2\ninst✝⁹ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁸ : CommRing R'\ninst✝⁷ : CommRing S'\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R R'\ninst✝⁴ : Algebra R' S'\ninst✝³ : Algebra S S'\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\ninst✝ : IsSc... | [] | simp [eqmap, Ideal.comap_map_of_surjective' _ surjRS] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Valuation.ValuativeRel.Trivial | {
"line": 52,
"column": 2
} | {
"line": 53,
"column": 71
} | {
"line": 55,
"column": 0
} | [
{
"pp": "R Γ : Type\ninst✝³ : Ring R\ninst✝² : DecidableEq R\ninst✝¹ : IsDomain R\ninst✝ : LinearOrderedCommGroupWithZero Γ\nh : ValuativeRel R\nhv : Valuation.Compatible 1\nx✝¹ x✝ : R\n⊢ 1 x✝¹ ≤ 1 x✝ ↔ if x✝ = 0 then x✝¹ = 0 else True",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
... | [] | split_ifs <;>
simp_all [Valuation.one_apply_of_ne_zero, Valuation.one_apply_le_one] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 102,
"column": 4
} | {
"line": 103,
"column": 33
} | {
"line": 104,
"column": 4
} | [
{
"pp": "case refine_1\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range (n + 1)\n⊢ (Finsupp.single (0, x) (p ^ (n + 1 - x))).support ⊆ univ ×ˢ range (n + 1)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Finset.singleton_subset_iff._simp_1",
"Eq.mpr",
"Finse... | [
"case refine_1\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range (n + 1)\n⊢ ↑p ^ x ≠ 0"
] | · apply Subset.trans Finsupp.support_single_subset
simpa using mem_range.mp hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 102,
"column": 4
} | {
"line": 103,
"column": 33
} | {
"line": 104,
"column": 4
} | [
{
"pp": "case refine_2\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range (n + 1)\n⊢ (Finsupp.single (1, x) (p ^ (n + 1 - x))).support ⊆ univ ×ˢ range (n + 1)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Finset.singleton_subset_iff._simp_1",
"Eq.mpr",
"Finse... | [
"case refine_2\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range (n + 1)\n⊢ ↑p ^ x ≠ 0"
] | · apply Subset.trans Finsupp.support_single_subset
simpa using mem_range.mp hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 211,
"column": 4
} | {
"line": 211,
"column": 33
} | {
"line": 212,
"column": 4
} | [
{
"pp": "case e'_2\np : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\na₁ a₂ : 𝕎 k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\nn : ℕ\n⊢ truncateFun (n + 1) (frobenius (mk p (frobeniusRotationCoeff p ha₁ ha₂))) = fun v ↦\n frobeniusRotationCoeff p ha₁ ... | [
"case e'_2\np : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\na₁ a₂ : 𝕎 k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\nn : ℕ\n⊢ ∀ (i : Fin (n + 1)),\n TruncatedWittVector.coeff i (truncateFun (n + 1) (frobenius (mk p (frobeniusRotationCoeff p ha₁ ha₂)))) ... | apply TruncatedWittVector.ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.SetTheory.Ordinal.Topology | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 35
} | {
"line": 124,
"column": 35
} | [
{
"pp": "case mpr\ns : Set Ordinal.{u}\n⊢ (∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → ⨆ i, f i ∈ s) → IsClosed s",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"closure_subset_iff_isClosed",
"iSup",
"Memb... | [
"case mpr\ns : Set Ordinal.{u}\n⊢ (∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → ⨆ i, f i ∈ s) → closure s ⊆ s"
] | ← closure_subset_iff_isClosed | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.ZFC.Rank | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 80
} | {
"line": 44,
"column": 4
} | [
{
"pp": "α✝¹ : Type u_1\nA✝¹ : α✝¹ → PSet.{u_1}\nα✝ : Type u_1\nA✝ : α✝ → PSet.{u_1}\nαβ : ∀ (a : α✝¹), ∃ b, (A✝¹ a).Equiv (A✝ b)\nβα : ∀ (b : α✝), ∃ a, (A✝¹ a).Equiv (A✝ b)\nx✝ : Ordinal.{u_1}\n⊢ (x✝ ∈ Set.range fun a ↦ succ (A✝¹ a).rank) ↔ x✝ ∈ Set.range fun a ↦ succ (A✝ a).rank",
"ppTerm": "?m.57",
"... | [
"case mp\nα✝¹ : Type u_1\nA✝¹ : α✝¹ → PSet.{u_1}\nα✝ : Type u_1\nA✝ : α✝ → PSet.{u_1}\nαβ : ∀ (a : α✝¹), ∃ b, (A✝¹ a).Equiv (A✝ b)\nβα : ∀ (b : α✝), ∃ a, (A✝¹ a).Equiv (A✝ b)\nx✝ : Ordinal.{u_1}\na : α✝¹\nh : succ (A✝¹ a).rank = x✝\n⊢ ∃ y, succ (A✝ y).rank = x✝",
"case mpr\nα✝¹ : Type u_1\nA✝¹ : α✝¹ → PSet.{u_1}\... | constructor <;> simp only [Set.mem_range, forall_exists_index] <;> intro a h | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 427,
"column": 29
} | {
"line": 446,
"column": 37
} | {
"line": 448,
"column": 0
} | [
{
"pp": "e : ONote\nn : ℕ+\na o : ONote\nh₁ : (e.oadd n a).NF\nh₂ : o.NF\n⊢ (e.oadd n a + o).repr = (e.oadd n a).repr + o.repr",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"PNat.val",
"Iff.mpr",
"Ordering.gt",
"ONote.NF",
"Eq.mpr",
"lt_of_le_of_lt",
... | [] | by
haveI := h₁.snd; have h' := repr_add a o
conv_lhs at h' => simp [HAdd.hAdd, Add.add]
have nf := ONote.add_nf a o
conv at nf => simp [HAdd.hAdd, Add.add]
conv in _ + o => simp [HAdd.hAdd, Add.add]
rcases h : add a o with - | ⟨e', n', a'⟩ <;>
simp only [add, addAux, h'.symm, h, add_assoc,... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 457,
"column": 6
} | {
"line": 464,
"column": 46
} | {
"line": 465,
"column": 4
} | [
{
"pp": "case eq\ne₁ : ONote\nn₁ : ℕ+\na₁ e₂ : ONote\nn₂ : ℕ+\na₂ : ONote\nb : Ordinal.{0}\nh₁ : (e₁.oadd n₁ a₁).NFBelow b\nh₂ : (e₂.oadd n₂ a₂).NF\nh' : (a₁.sub a₂).NFBelow e₁.repr\nthis : (e₁.cmp e₂).Compares e₁ e₂\nh : e₁.cmp e₂ = Ordering.eq\n⊢ (match Ordering.eq with\n | Ordering.lt => 0\n | Orde... | [] | rw [Nat.sub_eq]
simp only [h, Ordering.compares_eq] at this
subst e₂
cases (n₁ : ℕ) - n₂
· by_cases en : n₁ = n₂ <;> simp only [en, ↓reduceIte]
· exact h'.mono (le_of_lt h₁.lt)
· exact NFBelow.zero
· exact NFBelow.oadd h₁.fst h₁.snd h₁.lt | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 457,
"column": 6
} | {
"line": 464,
"column": 46
} | {
"line": 465,
"column": 4
} | [
{
"pp": "case eq\ne₁ : ONote\nn₁ : ℕ+\na₁ e₂ : ONote\nn₂ : ℕ+\na₂ : ONote\nb : Ordinal.{0}\nh₁ : (e₁.oadd n₁ a₁).NFBelow b\nh₂ : (e₂.oadd n₂ a₂).NF\nh' : (a₁.sub a₂).NFBelow e₁.repr\nthis : (e₁.cmp e₂).Compares e₁ e₂\nh : e₁.cmp e₂ = Ordering.eq\n⊢ (match Ordering.eq with\n | Ordering.lt => 0\n | Orde... | [] | rw [Nat.sub_eq]
simp only [h, Ordering.compares_eq] at this
subst e₂
cases (n₁ : ℕ) - n₂
· by_cases en : n₁ = n₂ <;> simp only [en, ↓reduceIte]
· exact h'.mono (le_of_lt h₁.lt)
· exact NFBelow.zero
· exact NFBelow.oadd h₁.fst h₁.snd h₁.lt | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.ZFC.VonNeumann | {
"line": 48,
"column": 59
} | {
"line": 48,
"column": 74
} | {
"line": 48,
"column": 74
} | [
{
"pp": "o : Ordinal.{u}\nx : ZFSet.{u}\n⊢ x ∈ V_ o ↔ ∃ a < o, x ⊆ V_ a",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"ZFSet.vonNeumann.eq_1",
"Ordinal.partialOrder",
"congrArg",
"ZFSet",
"PartialOrder.toPreorder",
... | [
"o : Ordinal.{u}\nx : ZFSet.{u}\n⊢ x ∈ ⋃ (a : ↑(Set.Iio o)), (V_ ↑a).powerset ↔ ∃ a < o, x ⊆ V_ a"
] | rw [vonNeumann] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.ZFC.VonNeumann | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 17
} | {
"line": 52,
"column": 2
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ (V_ o).IsTransitive",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ZFSet.vonNeumann.eq_1",
"Ordinal.partialOrder",
"congrArg",
"ZFSet",
"PartialOrder.toPreorder",
"Membership.mem",
"Set.Elem",
"i... | [
"o : Ordinal.{u_1}\n⊢ (⋃ (a : ↑(Set.Iio o)), (V_ ↑a).powerset).IsTransitive"
] | rw [vonNeumann] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.ZFC.VonNeumann | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 17
} | {
"line": 56,
"column": 17
} | [
{
"pp": "a b : Ordinal.{u}\nh : a < b\n⊢ V_ a ∈ V_ b",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ZFSet.vonNeumann.eq_1",
"Ordinal.partialOrder",
"congrArg",
"ZFSet",
"PartialOrder.toPreorder",
"Membership.mem",
"Set.Elem",
... | [
"a b : Ordinal.{u}\nh : a < b\n⊢ V_ a ∈ ⋃ (a : ↑(Set.Iio b)), (V_ ↑a).powerset"
] | rw [vonNeumann] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.ZFC.Ordinal | {
"line": 292,
"column": 6
} | {
"line": 292,
"column": 13
} | {
"line": 292,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_1}\nx : PSet.{u_1}\n⊢ x ∈ o.toPSet ↔ ∃ a < o, x.Equiv a.toPSet",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Ordinal.ToType.toOrd",
"PSet.instMembership",
"Ordinal.partialOrder",
"congrArg",
"Par... | [
"o : Ordinal.{u_1}\nx : PSet.{u_1}\n⊢ (x ∈ PSet.mk o.ToType fun a ↦ (↑a.toOrd).toPSet) ↔ ∃ a < o, x.Equiv a.toPSet"
] | toPSet, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.ZFC.Ordinal | {
"line": 297,
"column": 6
} | {
"line": 297,
"column": 13
} | {
"line": 297,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ o.toPSet.rank = o",
"ppTerm": "?m.2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.ToType.toOrd",
"Ordinal.partialOrder",
"congrArg",
"PartialOrder.toPreorder",
"Membership.mem",
"id",
"PSet.mk",
"Ordinal.... | [
"o : Ordinal.{u_1}\n⊢ (PSet.mk o.ToType fun a ↦ (↑a.toOrd).toPSet).rank = o"
] | toPSet, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic | {
"line": 190,
"column": 2
} | {
"line": 192,
"column": 61
} | {
"line": 194,
"column": 0
} | [
{
"pp": "m : UnitMonomial\nleft right : Basis\n⊢ (List.replicate (List.length left) 0 ++ m).toFun (left ++ right) = m.toFun right",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"List.replicate",
"Real.instPow",
"Real",
"HMul.hMul",
"Tactic.ComputeAsymptotics.... | [] | induction left with
| nil => rfl
| cons left_hd left_tl ih => simp [List.replicate_succ, ih] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic | {
"line": 190,
"column": 2
} | {
"line": 192,
"column": 61
} | {
"line": 194,
"column": 0
} | [
{
"pp": "m : UnitMonomial\nleft right : Basis\n⊢ (List.replicate (List.length left) 0 ++ m).toFun (left ++ right) = m.toFun right",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"List.replicate",
"Real.instPow",
"Real",
"HMul.hMul",
"Tactic.ComputeAsymptotics.... | [] | induction left with
| nil => rfl
| cons left_hd left_tl ih => simp [List.replicate_succ, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic | {
"line": 190,
"column": 2
} | {
"line": 192,
"column": 61
} | {
"line": 194,
"column": 0
} | [
{
"pp": "m : UnitMonomial\nleft right : Basis\n⊢ (List.replicate (List.length left) 0 ++ m).toFun (left ++ right) = m.toFun right",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"List.replicate",
"Real.instPow",
"Real",
"HMul.hMul",
"Tactic.ComputeAsymptotics.... | [] | induction left with
| nil => rfl
| cons left_hd left_tl ih => simp [List.replicate_succ, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 38
} | {
"line": 139,
"column": 2
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : Basis\ns : Stream'.Seq (ℝ × MultiseriesExpansion basis_tl)\n⊢ s.destruct =\n Option.map\n (fun x ↦\n match x with\n | (exp, coef, tl) => ((exp, coef), tl))\n (destruct s)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
... | [
"basis_hd : ℝ → ℝ\nbasis_tl : Basis\ns : Stream'.Seq (ℝ × MultiseriesExpansion basis_tl)\n⊢ s.destruct =\n Option.map\n ((fun x ↦ ((x.1, x.2.1), x.2.2)) ∘ fun x ↦\n match x with\n | ((exp, coef), tl) => (exp, coef, tl))\n s.destruct"
] | simp only [destruct, Option.map_map] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs | {
"line": 338,
"column": 2
} | {
"line": 341,
"column": 28
} | {
"line": 343,
"column": 0
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nmotive : MultiseriesExpansion (basis_hd :: basis_tl) → Sort u_1\nnil : (f : ℝ → ℝ) → motive (mk Multiseries.nil f)\ncons :\n (exp : ℝ) →\n (coef : MultiseriesExpansion basis_tl) →\n (tl : Multiseries basis_hd basis_tl) → (f : ℝ → ℝ) → motive (mk (Mult... | [] | let ⟨s, f⟩ := ms
cases s with
| nil => apply nil
| cons hd tl => apply cons | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs | {
"line": 338,
"column": 2
} | {
"line": 341,
"column": 28
} | {
"line": 343,
"column": 0
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nmotive : MultiseriesExpansion (basis_hd :: basis_tl) → Sort u_1\nnil : (f : ℝ → ℝ) → motive (mk Multiseries.nil f)\ncons :\n (exp : ℝ) →\n (coef : MultiseriesExpansion basis_tl) →\n (tl : Multiseries basis_hd basis_tl) → (f : ℝ → ℝ) → motive (mk (Mult... | [] | let ⟨s, f⟩ := ms
cases s with
| nil => apply nil
| cons hd tl => apply cons | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs | {
"line": 510,
"column": 2
} | {
"line": 515,
"column": 74
} | {
"line": 517,
"column": 0
} | [
{
"pp": "case seq\nbasis_hd : ℝ → ℝ\nbasis_tl : Basis\nexp : ℝ\ncoef : MultiseriesExpansion basis_tl\ntl : Multiseries basis_hd basis_tl\nh_coef : coef.Sorted\nh_comp : tl.leadingExp < ↑exp\nh_tl_coef : ∀ x ∈ (mk tl 0).seq, x.2.Sorted\nh_tl_tl : Seq.Pairwise (fun x1 x2 ↦ x1 > x2) (mk tl 0).seq\n⊢ (Multiseries.c... | [] | constructor
· simp at h_tl_coef ⊢
grind
· cases tl
· exact Seq.Pairwise_cons_nil
· exact h_tl_tl.cons_cons_of_trans (by simpa [lt_iff_lt] using h_comp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs | {
"line": 510,
"column": 2
} | {
"line": 515,
"column": 74
} | {
"line": 517,
"column": 0
} | [
{
"pp": "case seq\nbasis_hd : ℝ → ℝ\nbasis_tl : Basis\nexp : ℝ\ncoef : MultiseriesExpansion basis_tl\ntl : Multiseries basis_hd basis_tl\nh_coef : coef.Sorted\nh_comp : tl.leadingExp < ↑exp\nh_tl_coef : ∀ x ∈ (mk tl 0).seq, x.2.Sorted\nh_tl_tl : Seq.Pairwise (fun x1 x2 ↦ x1 > x2) (mk tl 0).seq\n⊢ (Multiseries.c... | [] | constructor
· simp at h_tl_coef ⊢
grind
· cases tl
· exact Seq.Pairwise_cons_nil
· exact h_tl_tl.cons_cons_of_trans (by simpa [lt_iff_lt] using h_comp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 1133,
"column": 2
} | {
"line": 1133,
"column": 18
} | {
"line": 1135,
"column": 0
} | [
{
"pp": "o : ONote\n⊢ o.fastGrowing =\n match (motive := (x : Option ONote ⊕ (ℕ → ONote)) → o.FundamentalSequenceProp x → ℕ → ℕ) o.fundamentalSequence,\n ⋯ with\n | Sum.inl none, x => Nat.succ\n | Sum.inl (some a), x => fun i ↦ a.fastGrowing^[i] i\n | Sum.inr f, x => fun i ↦ (f i).fastGrowing i... | [] | rw [fastGrowing] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 1246,
"column": 58
} | {
"line": 1246,
"column": 72
} | {
"line": 1247,
"column": 4
} | [
{
"pp": "case lt\na : ONote\nha : a.NF\nb : ONote\nhb : b.NF\nh : a.cmp b = Ordering.lt\nthis : Ordering.lt.Compares a b\n⊢ Ordering.lt.Compares ⟨a, ha⟩ ⟨b, hb⟩",
"ppTerm": "?lt",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"this"
],
"usedGoals": []
}
] | [] | try exact this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 1246,
"column": 58
} | {
"line": 1246,
"column": 72
} | {
"line": 1247,
"column": 4
} | [
{
"pp": "case eq\na : ONote\nha : a.NF\nb : ONote\nhb : b.NF\nh : a.cmp b = Ordering.eq\nthis : Ordering.eq.Compares a b\n⊢ Ordering.eq.Compares ⟨a, ha⟩ ⟨b, hb⟩",
"ppTerm": "?eq",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case eq\na : ONote\nha : a.NF\nb : ONote\nhb : b.NF\nh : a.cmp b = Ordering.eq\nthis : Ordering.eq.Compares a b\n⊢ Ordering.eq.Compares ⟨a, ha⟩ ⟨b, hb⟩"
] | try exact this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 1246,
"column": 58
} | {
"line": 1246,
"column": 72
} | {
"line": 1247,
"column": 4
} | [
{
"pp": "case gt\na : ONote\nha : a.NF\nb : ONote\nhb : b.NF\nh : a.cmp b = Ordering.gt\nthis : Ordering.gt.Compares a b\n⊢ Ordering.gt.Compares ⟨a, ha⟩ ⟨b, hb⟩",
"ppTerm": "?gt",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"this"
],
"usedGoals": []
}
] | [] | try exact this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.Tactic.NormNum.Irrational | {
"line": 49,
"column": 6
} | {
"line": 49,
"column": 9
} | {
"line": 49,
"column": 10
} | [
{
"pp": "q : ℚ\na b : ℕ\nh : ∀ (p : ℚ), q ^ a ≠ p ^ b\nhb : 0 < b\nhq : 0 ≤ q\nx : ℚ\nhx : ↑x = ↑q ^ (↑a / ↑b)\n⊢ ↑q ^ a = ↑x ^ b",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"DivisionRing.toRatCast",
"con... | [
"q : ℚ\na b : ℕ\nh : ∀ (p : ℚ), q ^ a ≠ p ^ b\nhb : 0 < b\nhq : 0 ≤ q\nx : ℚ\nhx : ↑x = ↑q ^ (↑a / ↑b)\n⊢ ↑q ^ a = (↑q ^ (↑a / ↑b)) ^ b"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Category.ProfiniteGrp.Completion | {
"line": 90,
"column": 33
} | {
"line": 90,
"column": 58
} | {
"line": 91,
"column": 4
} | [
{
"pp": "G : GrpCat\n⊢ Function.Injective ⇑(ConcreteCategory.hom (eta G)) ↔ Group.ResiduallyFinite ↑G",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"GrpCat.instConcreteCategoryMonoidHomCarrier",
"Eq.mpr",
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"GrpC... | [
"G : GrpCat\n⊢ (∀ (a : ↑G), (ConcreteCategory.hom (eta G)) a = 1 → a = 1) ↔ Group.ResiduallyFinite ↑G"
] | injective_iff_map_eq_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Group.OpenMapping | {
"line": 57,
"column": 4
} | {
"line": 72,
"column": 40
} | {
"line": 73,
"column": 2
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : Group G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : MulAction G X\ninst✝⁴ : SigmaCompactSpace G\ninst✝³ : BaireSpace X\ninst✝² : T2Space X\ninst✝¹ : ContinuousSMul G X\ninst✝ : IsPretransitive G X\nU : Set G\nhU... | [] | have : Nonempty X := ⟨x⟩
have : Encodable s := Countable.toEncodable s_count
apply nonempty_interior_of_iUnion_of_closed
· rintro ⟨n, ⟨g, hg⟩⟩
apply IsCompact.isClosed
suffices H : IsCompact ((fun (g : G) ↦ g • x) '' (K n ∩ g • V)) by
simpa only [F, smul_singleton] using H
apply Is... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Group.OpenMapping | {
"line": 57,
"column": 4
} | {
"line": 72,
"column": 40
} | {
"line": 73,
"column": 2
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : Group G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : MulAction G X\ninst✝⁴ : SigmaCompactSpace G\ninst✝³ : BaireSpace X\ninst✝² : T2Space X\ninst✝¹ : ContinuousSMul G X\ninst✝ : IsPretransitive G X\nU : Set G\nhU... | [] | have : Nonempty X := ⟨x⟩
have : Encodable s := Countable.toEncodable s_count
apply nonempty_interior_of_iUnion_of_closed
· rintro ⟨n, ⟨g, hg⟩⟩
apply IsCompact.isClosed
suffices H : IsCompact ((fun (g : G) ↦ g • x) '' (K n ∩ g • V)) by
simpa only [F, smul_singleton] using H
apply Is... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Group.CompactOpen | {
"line": 179,
"column": 4
} | {
"line": 179,
"column": 64
} | {
"line": 180,
"column": 4
} | [
{
"pp": "X : Type u_7\nY : Type u_8\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : Group X\ninst✝⁵ : IsTopologicalGroup X\ninst✝⁴ : UniformSpace Y\ninst✝³ : CommGroup Y\ninst✝² : IsUniformGroup Y\ninst✝¹ : T0Space Y\ninst✝ : CompactSpace Y\nU : Set X\nV : Set Y\nhU : IsCompact U\nhV : V ∈ 𝓝 1\nW : Set Y\nhWo : W ∈ 𝓝 ... | [
"X : Type u_7\nY : Type u_8\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : Group X\ninst✝⁵ : IsTopologicalGroup X\ninst✝⁴ : UniformSpace Y\ninst✝³ : CommGroup Y\ninst✝² : IsUniformGroup Y\ninst✝¹ : T0Space Y\ninst✝ : CompactSpace Y\nU : Set X\nV : Set Y\nhU : IsCompact U\nhV : V ∈ 𝓝 1\nW : Set Y\nhWo : W ∈ 𝓝 1\nhWV : W ⊆... | have h2 : T ⊆ S2 := fun f hf ↦ hf.mono_right interior_subset | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.UniformSpace.Ascoli | {
"line": 380,
"column": 2
} | {
"line": 387,
"column": 39
} | {
"line": 389,
"column": 0
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nα : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : UniformSpace α\nF : ι → X → α\n𝔖 : Set (Set X)\n𝔖_compact : ∀ K ∈ 𝔖, IsCompact K\nF_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K\nH : IsClosed[UniformOnFun.topologicalSpace X α 𝔖] (range (⇑(UniformOnFun.ofFun 𝔖) ∘ F))\n⊢ IsC... | [] | rcases isEmpty_or_nonempty α with _ | _
· simp [isClosed_discrete]
-- This follows from the previous lemmas and the characterization of the closure using filters.
simp_rw [isClosed_iff_clusterPt, ← Filter.map_top, ← mapClusterPt_def,
mapClusterPt_iff_ultrafilter, range_comp, Subtype.coe_injective.surjective_c... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Ascoli | {
"line": 380,
"column": 2
} | {
"line": 387,
"column": 39
} | {
"line": 389,
"column": 0
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nα : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : UniformSpace α\nF : ι → X → α\n𝔖 : Set (Set X)\n𝔖_compact : ∀ K ∈ 𝔖, IsCompact K\nF_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K\nH : IsClosed[UniformOnFun.topologicalSpace X α 𝔖] (range (⇑(UniformOnFun.ofFun 𝔖) ∘ F))\n⊢ IsC... | [] | rcases isEmpty_or_nonempty α with _ | _
· simp [isClosed_discrete]
-- This follows from the previous lemmas and the characterization of the closure using filters.
simp_rw [isClosed_iff_clusterPt, ← Filter.map_top, ← mapClusterPt_def,
mapClusterPt_iff_ultrafilter, range_comp, Subtype.coe_injective.surjective_c... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Category.Compactum | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 44
} | {
"line": 222,
"column": 4
} | [
{
"pp": "case h\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F ∈ basic (cl A)\nfsu : Type u_1 := ⋯\nssu : Type u_1 := ⋯\nι : fsu → ssu := ⋯\nC0 : ssu := ⋯\nAA : Set (Ultrafilter X.A) := ⋯\nC1 : ssu := ⋯\nC2 : Set (Set (Ultrafilter X.A)) := ⋯\nQ : Set X.A\nhQ : Q ∈ F\nR : Set X.A\nhR : R ∈ F\n⊢ Q ∩ R ∈... | [
"case h\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F ∈ basic (cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x ↦ ↑x\nC0 : ssu := {Z | ∃ B ∈ F, X.str ⁻¹' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A ∈ G}\nC1 : ssu := insert AA ... | simp only [and_true, Set.preimage_inter] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Convenient.ContinuousMapGeneratedBy | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 32
} | {
"line": 78,
"column": 2
} | [
{
"pp": "ι : Type t\nX : ι → Type u\ninst✝³ : (i : ι) → TopologicalSpace (X i)\nY : Type v\ninst✝² : TopologicalSpace Y\nZ : Type v'\ninst✝¹ : TopologicalSpace Z\ng : Y → Z\nhg : ContinuousGeneratedBy X g\nT : Type u_1\ninst✝ : TopologicalSpace T\nf : T → Y\nhf : ContinuousGeneratedBy X f\n⊢ ContinuousGenerated... | [
"ι : Type t\nX : ι → Type u\ninst✝³ : (i : ι) → TopologicalSpace (X i)\nY : Type v\ninst✝² : TopologicalSpace Y\nZ : Type v'\ninst✝¹ : TopologicalSpace Z\ng : Y → Z\nhg : ContinuousGeneratedBy X g\nT : Type u_1\ninst✝ : TopologicalSpace T\nf : T → Y\nhf : ContinuousGeneratedBy X f\n⊢ Continuous (⇑WithGeneratedByTop... | rw [continuousGeneratedBy_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Convenient.Category | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 36
} | {
"line": 175,
"column": 6
} | [
{
"pp": "ι : Type t\nX : ι → Type u\ninst✝ : (i : ι) → TopologicalSpace (X i)\nY Z : ContinuousGeneratedByCat X\ng : (toTopCat X).obj Y ⟶ (toTopCat X).obj Z\n⊢ ContinuousGeneratedBy X (⇑WithGeneratedByTopology.equiv ∘ ⇑(TopCat.Hom.hom g) ∘ ⇑WithGeneratedByTopology.equiv.symm)",
"ppTerm": "?m.48",
"assig... | [
"ι : Type t\nX : ι → Type u\ninst✝ : (i : ι) → TopologicalSpace (X i)\nY Z : ContinuousGeneratedByCat X\ng : (toTopCat X).obj Y ⟶ (toTopCat X).obj Z\n⊢ Continuous\n (⇑WithGeneratedByTopology.equiv.symm ∘\n (⇑WithGeneratedByTopology.equiv ∘ ⇑(TopCat.Hom.hom g) ∘ ⇑WithGeneratedByTopology.equiv.symm) ∘\n ... | rw [continuousGeneratedBy_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Category.Profinite.Product | {
"line": 128,
"column": 12
} | {
"line": 128,
"column": 15
} | {
"line": 128,
"column": 16
} | [
{
"pp": "ι : Type u\nX : ι → Type\ninst✝² : (i : ι) → TopologicalSpace (X i)\nC : Set ((i : ι) → X i)\nJ✝ K✝ : ι → Prop\ninst✝¹ : ∀ (i : ι), T2Space (X i)\ninst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)\nhC : IsCompact C\nthis : CompactSpace ↑C\na : (fun X ↦ ↑X.toTop) (limitCone (indexFunctor hC)).pt\nhc : ∀... | [
"ι : Type u\nX : ι → Type\ninst✝² : (i : ι) → TopologicalSpace (X i)\nC : Set ((i : ι) → X i)\nJ✝ K✝ : ι → Prop\ninst✝¹ : ∀ (i : ι), T2Space (X i)\ninst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)\nhC : IsCompact C\nthis : CompactSpace ↑C\na : (fun X ↦ ↑X.toTop) (limitCone (indexFunctor hC)).pt\nhc : ∀ (J : Finset... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Category.Profinite.Nobeling.Span | {
"line": 135,
"column": 8
} | {
"line": 135,
"column": 11
} | {
"line": 135,
"column": 12
} | [
{
"pp": "case false\nI : Type u\nC : Set (I → Bool)\ninst✝ : LinearOrder I\ns : Finset I\nx y : ↑(π C fun x ↦ x ∈ s)\nh : y ≠ x\na : I\nha : ↑y a ≠ ↑x a\nhx : ↑x a = false\n⊢ ∃ a ∈ factors C s x, (LocallyConstant.evalMonoidHom y) a = 0",
"ppTerm": "?false",
"assigned": true,
"usedConstants": [
... | [
"case false\nI : Type u\nC : Set (I → Bool)\ninst✝ : LinearOrder I\ns : Finset I\nx y : ↑(π C fun x ↦ x ∈ s)\nh : y ≠ x\na : I\nha : ↑y a ≠ false\nhx : ↑x a = false\n⊢ ∃ a ∈ factors C s x, (LocallyConstant.evalMonoidHom y) a = 0"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Category.Profinite.Nobeling.Basic | {
"line": 183,
"column": 4
} | {
"line": 189,
"column": 33
} | {
"line": 190,
"column": 2
} | [
{
"pp": "case refine_1\nI : Type u\nC : Set (I → Bool)\nJ : I → Prop\ninst✝ : (i : I) → Decidable (J i)\na b : ↑(π C J)\nh : (iso_map C J) a = (iso_map C J) b\n⊢ a = b",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Pi.topologicalSpace",
"congrArg",
"ContinuousMap",
... | [] | ext i
rw [Subtype.ext_iff] at h
by_cases hi : J i
· exact congr_fun h ⟨i, hi⟩
· rcases a with ⟨_, c, hc, rfl⟩
rcases b with ⟨_, d, hd, rfl⟩
simp only [Proj, if_neg hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.Profinite.Nobeling.Basic | {
"line": 183,
"column": 4
} | {
"line": 189,
"column": 33
} | {
"line": 190,
"column": 2
} | [
{
"pp": "case refine_1\nI : Type u\nC : Set (I → Bool)\nJ : I → Prop\ninst✝ : (i : I) → Decidable (J i)\na b : ↑(π C J)\nh : (iso_map C J) a = (iso_map C J) b\n⊢ a = b",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Pi.topologicalSpace",
"congrArg",
"ContinuousMap",
... | [] | ext i
rw [Subtype.ext_iff] at h
by_cases hi : J i
· exact congr_fun h ⟨i, hi⟩
· rcases a with ⟨_, c, hc, rfl⟩
rcases b with ⟨_, d, hd, rfl⟩
simp only [Proj, if_neg hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Category.Profinite.Nobeling.Span | {
"line": 157,
"column": 2
} | {
"line": 168,
"column": 53
} | {
"line": 170,
"column": 0
} | [
{
"pp": "I : Type u\nC : Set (I → Bool)\ninst✝ : LinearOrder I\ns : Finset I\na : I\nas : List I\nha : List.IsChain (fun x1 x2 ↦ x1 > x2) (a :: as)\nc : Products I →₀ ℤ\nhc : ↑c.support ⊆ {m | ↑m ≤ as}\n⊢ (c.sum fun a_1 b ↦ e (π C fun x ↦ x ∈ s) a * b • Products.eval (π C fun x ↦ x ∈ s) a_1) ∈\n Submodule.sp... | [] | apply Submodule.finsuppSum_mem
intro m hm
have hsm := (LinearMap.mulLeft ℤ (e (π C (· ∈ s)) a)).map_smul
dsimp at hsm
rw [hsm]
apply Submodule.smul_mem
apply Submodule.subset_span
have hmas : m.val ≤ as := by
apply hc
simpa only [Finset.mem_coe, Finsupp.mem_support_iff] using hm
refine ⟨⟨a :: m.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.Profinite.Nobeling.Span | {
"line": 157,
"column": 2
} | {
"line": 168,
"column": 53
} | {
"line": 170,
"column": 0
} | [
{
"pp": "I : Type u\nC : Set (I → Bool)\ninst✝ : LinearOrder I\ns : Finset I\na : I\nas : List I\nha : List.IsChain (fun x1 x2 ↦ x1 > x2) (a :: as)\nc : Products I →₀ ℤ\nhc : ↑c.support ⊆ {m | ↑m ≤ as}\n⊢ (c.sum fun a_1 b ↦ e (π C fun x ↦ x ∈ s) a * b • Products.eval (π C fun x ↦ x ∈ s) a_1) ∈\n Submodule.sp... | [] | apply Submodule.finsuppSum_mem
intro m hm
have hsm := (LinearMap.mulLeft ℤ (e (π C (· ∈ s)) a)).map_smul
dsimp at hsm
rw [hsm]
apply Submodule.smul_mem
apply Submodule.subset_span
have hmas : m.val ≤ as := by
apply hc
simpa only [Finset.mem_coe, Finsupp.mem_support_iff] using hm
refine ⟨⟨a :: m.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Category.Compactum | {
"line": 454,
"column": 10
} | {
"line": 454,
"column": 38
} | {
"line": 455,
"column": 10
} | [
{
"pp": "D : CompHaus\nx✝ : Set (Compactum.ofTopologicalSpace ↑D.toTop).A\nh1 : IsOpen x✝\n⊢ IsOpen (id ⁻¹' x✝)",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"CategoryTheory.ofTypeMonad",
"Compactum.ofTopologica... | [
"D : CompHaus\nx✝ : Set (Compactum.ofTopologicalSpace ↑D.toTop).A\nh1 : IsOpen x✝\n⊢ ∀ (F : Ultrafilter ↑D.1), F.lim ∈ id ⁻¹' x✝ → id ⁻¹' x✝ ∈ ↑F"
] | rw [isOpen_iff_ultrafilter'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 194,
"column": 70
} | {
"line": 205,
"column": 23
} | {
"line": 207,
"column": 0
} | [
{
"pp": "I : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\n⊢ ∀ (y : LocallyConstant ↑(π C fun x ↦ ord I x < o) ℤ), (Linear_CC' C hsC ho) ((πs C o) y) = 0",
"ppTerm": "?m.31",
"ass... | [] | by
intro y
ext x
dsimp [Linear_CC', Linear_CC'₀, Linear_CC'₁, LocallyConstant.sub_apply]
simp only [sub_eq_zero]
congr 1
ext i
dsimp [CC'₀, CC'₁, ProjRestrict, Proj]
apply if_ctx_congr Iff.rfl _ (fun _ ↦ rfl)
simp only [SwapTrue, ite_eq_right_iff]
intro h₁ h₂
exact (h₁.ne h₂).elim | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 580,
"column": 6
} | {
"line": 580,
"column": 96
} | {
"line": 581,
"column": 4
} | [
{
"pp": "case hi.inl\nX : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝¹ : RelCWComplex C D\ninst✝ : T2Space X\nA : Set X\nhAC : A ⊆ C\nhDA : IsClosed[t] (A ∩ D)\nn : ℕ\nhn : ∀ m ≤ n, ∀ (j : cell C m), IsClosed[t] (A ∩ closedCell m j)\nj : cell C (n + 1)\nh1 : IsClosed[t] (A ∩ openCell n.succ j)\n⊢ IsClo... | [] | exact (isClosed_inter_cellFrontier_succ_of_le_isClosed_inter_closedCell hn j hDA).union h1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 568,
"column": 81
} | {
"line": 581,
"column": 14
} | {
"line": 583,
"column": 0
} | [
{
"pp": "X : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝¹ : RelCWComplex C D\ninst✝ : T2Space X\nA : Set X\nhAC : A ⊆ C\nhDA : IsClosed[t] (A ∩ D)\nh : ∀ (n : ℕ), 0 < n → ∀ (j : cell C n), IsClosed[t] (A ∩ openCell n j) ∨ IsClosed[t] (A ∩ closedCell n j)\n⊢ IsClosed[t] A",
"ppTerm": "?m.43",
"a... | [] | by
rw [closed C A hAC]
refine ⟨?_, hDA⟩
intro n j
induction n using Nat.case_strong_induction_on with
| hz =>
rw [closedCell_zero_eq_singleton]
exact isClosed_inter_singleton
| hi n hn =>
specialize h n.succ n.zero_lt_succ j
rcases h with h1 | h2
· rw [← cellFrontier_union_openCell_eq_cl... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 625,
"column": 4
} | {
"line": 626,
"column": 75
} | {
"line": 627,
"column": 4
} | [
{
"pp": "case hi\nX : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝ : RelCWComplex C D\nn : ℕ\ni : cell C (n + 1)\nJ : (m : ℕ) → Finset (cell C m)\nhJ : cellFrontier n.succ i ⊆ D ∪ ⋃ m, ⋃ (_ : m < n.succ), ⋃ j ∈ J m, closedCell m j\np : (m : ℕ) → m ≤ n → cell C m → (m : ℕ) → Finset (cell C m)\nhp :\n ∀ ... | [
"case hi\nX : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝ : RelCWComplex C D\nn : ℕ\ni : cell C (n + 1)\nJ : (m : ℕ) → Finset (cell C m)\nhJ : cellFrontier n.succ i ⊆ D ∪ ⋃ m, ⋃ (_ : m < n.succ), ⋃ j ∈ J m, closedCell m j\np : (m : ℕ) → m ≤ n → cell C m → (m : ℕ) → Finset (cell C m)\nhp :\n ∀ (m : ℕ) (a :... | let I m := J m ∪ ((Finset.range n.succ).biUnion
(fun l ↦ (J l).biUnion (fun y ↦ if h : l ≤ n then p l h y m else ∅))) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 46
} | {
"line": 78,
"column": 6
} | [
{
"pp": "case refine_1\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : CompactSpace β\nA : Set (α →ᵇ β)\nclosed : IsClosed A\nH : ∀ (x₀ : α), ∀ ε > 0, ∃ U ∈ nhds x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ↑A), dist (↑i x) (↑i x') < ε\nε : ℝ\nε0 : ε > 0\nε₁... | [
"case refine_2\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : CompactSpace β\nA : Set (α →ᵇ β)\nclosed : IsClosed A\nH : ∀ (x₀ : α), ∀ ε > 0, ∃ U ∈ nhds x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ↑A), dist (↑i x) (↑i x') < ε\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : ... | · exact (hU x').2.2 _ hx' _ (hU x').1 hf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Compactness.CountablyCompact | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 72
} | {
"line": 184,
"column": 2
} | [
{
"pp": "E : Type u_2\ninst✝ : TopologicalSpace E\nA : Set E\nhA : IsSeqCompact A\nx : ℕ → E\nhx : ∀ᶠ (n : ℕ) in atTop, x n ∈ A\n⊢ ∃ a ∈ A, MapClusterPt a atTop x",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Nat.instLattice",
"Lattice.toSemilatticeSup",
"StrictMono",
... | [
"E : Type u_2\ninst✝ : TopologicalSpace E\nA : Set E\nhA : IsSeqCompact A\nx : ℕ → E\nhx : ∀ᶠ (n : ℕ) in atTop, x n ∈ A\na : E\nha : a ∈ A\nφ : ℕ → ℕ\nhφ : StrictMono φ\nhφa : Tendsto (x ∘ φ) atTop (𝓝 a)\n⊢ ∃ a ∈ A, MapClusterPt a atTop x"
] | obtain ⟨a, ha, φ, hφ, hφa⟩ := hA.subseq_of_frequently_in hx.frequently | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.ContinuousMap.SecondCountableSpace | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 39
} | {
"line": 101,
"column": 4
} | [
{
"pp": "case h.refine_2\nX : Type u_1\nY : Type u_2\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace Y\ninst✝² : SecondCountableTopology X\ninst✝¹ : LocallyCompactSpace X\ninst✝ : SecondCountableTopology Y\nthis : ∀ (U : ↑(countableBasis X)), LocallyCompactSpace ↑↑U\nK : (U : ↑(countableBasis X)) → Comp... | [
"case h.refine_2\nX : Type u_1\nY : Type u_2\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace Y\ninst✝² : SecondCountableTopology X\ninst✝¹ : LocallyCompactSpace X\ninst✝ : SecondCountableTopology Y\nthis : ∀ (U : ↑(countableBasis X)), LocallyCompactSpace ↑↑U\nK : (U : ↑(countableBasis X)) → CompactExhaustio... | lift U to countableBasis X using hU | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.Topology.DerivedSet | {
"line": 73,
"column": 11
} | {
"line": 79,
"column": 19
} | {
"line": 81,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nA : Set X\nh : derivedSet A ⊆ A\n⊢ IsClosed[inst✝] A",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"eq_false",
"congrArg",
"Classical.byContradiction",
"Membership.mem",
"Eq.mp... | [] | by
rw [isClosed_iff_clusterPt]
intro a ha
by_contra! nh
have : A = A \ {a} := by simp [nh]
rw [this, ← accPt_principal_iff_clusterPt] at ha
exact nh (h ha) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Germ | {
"line": 98,
"column": 6
} | {
"line": 98,
"column": 60
} | {
"line": 99,
"column": 6
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝ : TopologicalSpace X\nf✝ g : X → Y\nA✝ : Set X\nx✝ : X\nP : (x : X) → (𝓝 x).Germ Y → Prop\nA : Set X\nx : X\nφ : (𝓝 x).Germ Y\nf f' : X → Y\nhff' : f =ᶠ[𝓝 x] f'\nhf : ∀ᶠ (y : X) in 𝓝 x, P y ↑f\n⊢ ∀ᶠ (y : X) in 𝓝 x, P y ↑f'",
"ppTerm": "?m.65",
... | [
"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝ : TopologicalSpace X\nf✝ g : X → Y\nA✝ : Set X\nx✝ : X\nP : (x : X) → (𝓝 x).Germ Y → Prop\nA : Set X\nx : X\nφ : (𝓝 x).Germ Y\nf f' : X → Y\nhff' : f =ᶠ[𝓝 x] f'\nhf : ∀ᶠ (y : X) in 𝓝 x, P y ↑f\n⊢ ∀ (x : X), (P x ↑f ∧ ∀ᶠ (x : X) in 𝓝 x, f x = f' x) → P x ↑f'"
] | apply (hf.and <| Eventually.eventually_nhds hff').mono | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Maps.Proper.UniversallyClosed | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 22
} | {
"line": 106,
"column": 0
} | [
{
"pp": "case mpr\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nH : Continuous f ∧ ∀ (Z : Type u) [inst : TopologicalSpace Z], IsClosedMap (Prod.map f id)\n⊢ Continuous f ∧ IsClosedMap (Prod.map f id)",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants... | [] | exact ⟨H.1, H.2 _⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Maps.Strict.Group | {
"line": 46,
"column": 2
} | {
"line": 47,
"column": 41
} | {
"line": 48,
"column": 2
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝³ : Group G\ninst✝² : Group H\nf : G →* H\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalSpace H\n⊢ IsStrictMap ⇑f ↔ IsEmbedding ⇑(kerLift f)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"QuotientGroup.isQuotientMap_mk",
"Eq.mpr",
... | [
"G : Type u_1\nH : Type u_2\ninst✝³ : Group G\ninst✝² : Group H\nf : G →* H\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalSpace H\n⊢ IsStrictMap ⇑f ↔ IsStrictMap (⇑(kerLift f) ∘ QuotientGroup.mk)"
] | simp_rw [isEmbedding_iff_isStrictMap_injective, kerLift_injective, and_true,
(isQuotientMap_mk _).isStrictMap_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.List | {
"line": 132,
"column": 4
} | {
"line": 134,
"column": 96
} | {
"line": 136,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\na : α\nn : ℕ\na' : α\nl : List α\nthis : 𝓝 a ×ˢ 𝓝 (a' :: l) = Filter.map (fun p ↦ (p.1, p.2.1 :: p.2.2)) (𝓝 a ×ˢ 𝓝 a' ×ˢ 𝓝 l)\n⊢ Tendsto ((fun p ↦ p.2.insertIdx (n + 1) p.1) ∘ fun p ↦ (p.1, p.2.1 :: p.2.2)) (𝓝 a ×ˢ 𝓝 a' ×ˢ 𝓝 l)\n (𝓝 ((a' :: l).inser... | [] | exact
(tendsto_fst.comp tendsto_snd).cons
((@tendsto_insertIdx' _ n l).comp <| tendsto_fst.prodMk <| tendsto_snd.comp tendsto_snd) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.BundledFun | {
"line": 144,
"column": 36
} | {
"line": 145,
"column": 48
} | {
"line": 147,
"column": 0
} | [
{
"pp": "X : Type u_1\nR : Type u_2\ninst✝³ : AddCommMonoid R\ninst✝² : LinearOrder R\ninst✝¹ : AddLeftStrictMono R\ninst✝ : IsOrderedAddMonoid R\nY : Type u_3\nf : Y → PseudoMetric X R\ns : Finset Y\nhs : s.Nonempty\n⊢ ⇑(s.sup f) = ⇑(s.sup' hs fun x ↦ f x)",
"ppTerm": "?m.20",
"assigned": true,
"us... | [] | by
simpa using (Finset.sup'_eq_sup hs (f ·)).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 569,
"column": 2
} | {
"line": 569,
"column": 56
} | {
"line": 570,
"column": 2
} | [
{
"pp": "N : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\nx : X\ninst✝ : DecidableEq N\ni j : N\nf g : ↑(Ω^ N X x)\n⊢ ⟦fromLoop i (Path.trans (toLoop i f) (toLoop i g))⟧ = ⟦fromLoop j (Path.trans (toLoop j f) (toLoop j g))⟧",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"HMu... | [
"N : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\nx : X\ninst✝ : DecidableEq N\ni j : N\nf g : ↑(Ω^ N X x)\nm : (G : Type ?u.36) → Group G → G → G → G := fun G x x1 x2 ↦ x1 * x2\n⊢ ⟦fromLoop i (Path.trans (toLoop i f) (toLoop i g))⟧ = ⟦fromLoop j (Path.trans (toLoop j f) (toLoop j g))⟧"
] | let m := fun (G) (_ : Group G) ↦ ((· * ·) : G → G → G) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 22
} | {
"line": 77,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\n⊢ IsClopen {∅}",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.powerset_empty",
"TopologicalSpace.vietoris",
"Set.powerset",
"Set.instSingletonSet",
"id",
"Set.instE... | [
"α : Type u_1\ninst✝ : TopologicalSpace α\n⊢ IsClopen (𝒫 ∅)"
] | ← powerset_empty | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Closeds | {
"line": 64,
"column": 37
} | {
"line": 65,
"column": 61
} | {
"line": 67,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nU : SetRel α α\ninst✝ : U.IsSymm\n⊢ (hausdorffEntourage U).IsSymm",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetRel",
"congrArg",
"SetRel.inv",
"id",
"inv_hausdorffEntourage",
"SetRe... | [] | by
rw [← inv_eq_self_iff, inv_hausdorffEntourage, inv_eq_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 221,
"column": 56
} | {
"line": 221,
"column": 73
} | {
"line": 221,
"column": 74
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nι : Type u_4\ninst✝ : Finite ι\n⊢ (∀ (U : Set α), IsOpen[inst✝¹] U → IsOpen[Pi.topologicalSpace] {a | range a ⊆ U}) ∧\n ∀ (F : Set α), IsClosed[inst✝¹] F → IsClosed[Pi.topologicalSpace] {a | range a ⊆ F}",
"ppTerm": "?m.10",
"assigned": true,
"u... | [
"α : Type u_1\ninst✝¹ : TopologicalSpace α\nι : Type u_4\ninst✝ : Finite ι\n⊢ (∀ (U : Set α), IsOpen[inst✝¹] U → IsOpen[Pi.topologicalSpace] {a | ∀ (y : ι), a y ∈ U}) ∧\n ∀ (F : Set α), IsClosed[inst✝¹] F → IsClosed[Pi.topologicalSpace] {a | ∀ (y : ι), a y ∈ F}"
] | range_subset_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 87
} | {
"line": 222,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nι : Type u_4\ninst✝ : Finite ι\n⊢ Continuous[Pi.topologicalSpace, TopologicalSpace.vietoris α] range",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continuous",
"Pi.topologicalSpace",
"congrArg",
"... | [
"α : Type u_1\ninst✝¹ : TopologicalSpace α\nι : Type u_4\ninst✝ : Finite ι\n⊢ (∀ (U : Set α), IsOpen[inst✝¹] U → IsOpen[Pi.topologicalSpace] (⋂ i, {x | x i ∈ U})) ∧\n ∀ (F : Set α), IsClosed[inst✝¹] F → IsClosed[Pi.topologicalSpace] (⋂ i, {x | x i ∈ F})"
] | simp_rw [continuous_iff, powerset, preimage_setOf_eq, range_subset_iff, setOf_forall] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
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