module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 162,
"column": 42
} | {
"line": 162,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\neif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a\nuif :\n ∀ (f g : Multiset α),\n (∀ x ∈ f, Irreducible x) → (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g\np : α\nthis : Decidable... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 203,
"column": 36
} | {
"line": 203,
"column": 44
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na p : α\nha0 : a ≠ 0\nhp : Irreducible p\nx✝ : p ∣ a\nb : α\nhb : a = p * b\nhb0 : b = 0\n⊢ False",
"usedConstants": [
"False",
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
"False.elim",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 203,
"column": 36
} | {
"line": 203,
"column": 44
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na p : α\nha0 : a ≠ 0\nhp : Irreducible p\nx✝ : p ∣ a\nb : α\nhb : a = p * b\nhb0 : b = 0\n⊢ False",
"usedConstants": [
"False",
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
"False.elim",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 203,
"column": 36
} | {
"line": 203,
"column": 44
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na p : α\nha0 : a ≠ 0\nhp : Irreducible p\nx✝ : p ∣ a\nb : α\nhb : a = p * b\nhb0 : b = 0\n⊢ False",
"usedConstants": [
"False",
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
"False.elim",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Roots | {
"line": 95,
"column": 51
} | {
"line": 95,
"column": 59
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\na : R\nhp0 : 0 < p.degree\nh : p = 0\n⊢ False",
"usedConstants": [
"WithBot.instPreorder",
"False",
"Nat.instMulZeroClass",
"WithBot",
"Preorder.toLT",
"congrArg",
"CommSemiring.toSemiring",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.Roots | {
"line": 95,
"column": 51
} | {
"line": 95,
"column": 59
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\na : R\nhp0 : 0 < p.degree\nh : p = 0\n⊢ False",
"usedConstants": [
"WithBot.instPreorder",
"False",
"Nat.instMulZeroClass",
"WithBot",
"Preorder.toLT",
"congrArg",
"CommSemiring.toSemiring",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Roots | {
"line": 95,
"column": 51
} | {
"line": 95,
"column": 59
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\na : R\nhp0 : 0 < p.degree\nh : p = 0\n⊢ False",
"usedConstants": [
"WithBot.instPreorder",
"False",
"Nat.instMulZeroClass",
"WithBot",
"Preorder.toLT",
"congrArg",
"CommSemiring.toSemiring",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Roots | {
"line": 325,
"column": 84
} | {
"line": 325,
"column": 93
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nhn : 0 < n\na x : R\n⊢ eval x (X ^ n) - a = 0 ↔ x ^ n = a",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.eval_pow",
... | eval_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Roots | {
"line": 644,
"column": 2
} | {
"line": 644,
"column": 10
} | [
{
"pp": "case h\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nh : 0 < n\na x : R\n⊢ x ∈ ↑(nthRootsFinset n a) ↔ x ∈ {r | r ^ n = a}",
"usedConstants": [
"SetLike.mem_coe._simp_1",
"congrArg",
"CommSemiring.toSemiring",
"Finset",
"Membership.mem",
"Monoid.to... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Polynomial.Content | {
"line": 72,
"column": 36
} | {
"line": 72,
"column": 44
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np : R[X]\nhp : Irreducible p\nhp' : p.natDegree ≠ 0\nq : R[X]\nH : IsUnit q\nhq : p = C 0 * q\n⊢ False",
"usedConstants": [
"Polynomial.C",
"RingHom.instRingHomClass",
"False",
"Semigroup.toMul",
"HMul.hM... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Polynomial.Content | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 12
} | [
{
"pp": "case mp.a\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : ∀ x ∈ p.support, p.coeff x = 0\nn : ℕ\n⊢ p.coeff n = coeff 0 n",
"usedConstants": [
"False",
"eq_false",
"Classical.not_not._simp_1",
"congrArg",
"CommSemiring.toSemiring",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 81,
"column": 70
} | {
"line": 81,
"column": 79
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np : R[X]\nt : R\nm : ℕ := rootMultiplicity t p\nhm : m = rootMultiplicity t p\nb : ℕ\nhb : b ∈ range m.succ\nhb0 : b ≠ 0\n⊢ m.choose b •\n (m.descFactorial (m - b) • eval t ((X - C t) ^ (m - (m - b))) * eval t ((⇑derivative)^[b] (p /ₘ (X - C t) ^ m))) =\n 0",
... | eval_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 12
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : UniqueFactorizationMonoid α\ninst✝¹ : DecidableEq (Associates α)\ninst✝ : (p : Associates α) → Decidable (Irreducible p)\na b p : Associates α\nhb : b ≠ 0\nhp : Irreducible p\nh : a.factors ≤ b.factors\nha : a = 0\n⊢ p.count a.factors ≤ p.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 12
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : UniqueFactorizationMonoid α\ninst✝¹ : DecidableEq (Associates α)\ninst✝ : (p : Associates α) → Decidable (Irreducible p)\na b p : Associates α\nhb : b ≠ 0\nhp : Irreducible p\nh : a.factors ≤ b.factors\nha : a = 0\n⊢ p.count a.factors ≤ p.... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 12
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : UniqueFactorizationMonoid α\ninst✝¹ : DecidableEq (Associates α)\ninst✝ : (p : Associates α) → Decidable (Irreducible p)\na b p : Associates α\nhb : b ≠ 0\nhp : Irreducible p\nh : a.factors ≤ b.factors\nha : a = 0\n⊢ p.count a.factors ≤ p.... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 367,
"column": 55
} | {
"line": 374,
"column": 41
} | [
{
"pp": "R : Type u\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : p.degree < q.degree\n⊢ p % q = p",
"usedConstants": [
"Polynomial.modByMonic.eq_1",
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
"instDecidableNot",
"False",
"WithBot",
"Preorder.toLT",
... | by
classical
have : ¬degree (q * C (leadingCoeff q)⁻¹) ≤ degree p :=
not_le_of_gt <| by rwa [degree_mul_leadingCoeff_inv q hq0]
rw [mod_def, modByMonic, dif_pos (monic_mul_leadingCoeff_inv hq0)]
unfold divModByMonicAux
dsimp
simp only [this, false_and, if_false] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 418,
"column": 26
} | {
"line": 418,
"column": 45
} | [
{
"pp": "case neg\nR : Type u\nk : Type y\ninst✝¹ : Field R\np q : R[X]\ninst✝ : Field k\nf : R →+* k\nhq0 : ¬q = 0\n⊢ map f (p %ₘ (q * C q.leadingCoeff⁻¹)) = map f p %ₘ (map f q * C (map f q).leadingCoeff⁻¹)",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"HMul.hMul",
"DivisionCommMo... | leadingCoeff_map f, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 437,
"column": 2
} | {
"line": 439,
"column": 33
} | [
{
"pp": "R : Type u\ninst✝ : Field R\np q : R[X]\nhq : q ≠ 0\n⊢ (p % q).degree < q.degree",
"usedConstants": [
"_private.Mathlib.Algebra.Polynomial.FieldDivision.0.Polynomial.degree_mod_lt._simp_1_1",
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
"GroupWithZero.toMonoidWith... | rw [Polynomial.mod_def]
refine (Polynomial.degree_modByMonic_lt p ?_).trans_eq (by simp)
simp [Polynomial.Monic.def, hq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 437,
"column": 2
} | {
"line": 439,
"column": 33
} | [
{
"pp": "R : Type u\ninst✝ : Field R\np q : R[X]\nhq : q ≠ 0\n⊢ (p % q).degree < q.degree",
"usedConstants": [
"_private.Mathlib.Algebra.Polynomial.FieldDivision.0.Polynomial.degree_mod_lt._simp_1_1",
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
"GroupWithZero.toMonoidWith... | rw [Polynomial.mod_def]
refine (Polynomial.degree_modByMonic_lt p ?_).trans_eq (by simp)
simp [Polynomial.Monic.def, hq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 519,
"column": 7
} | {
"line": 519,
"column": 29
} | [
{
"pp": "case a\nR : Type u\nS : Type v\ninst✝³ : Field R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nι : Type u_1\nf : ι → R[X]\ns : Finset ι\nh : s.prod f ≠ 0\n⊢ ∏ i ∈ s, map (algebraMap R S) (f i) ≠ 0",
"usedConstants": [
"Eq.mpr",
"Algebra.algebraMap",
"congrArg",
... | ← Polynomial.map_prod, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 701,
"column": 2
} | {
"line": 701,
"column": 10
} | [
{
"pp": "case neg\nR : Type u\ninst✝¹ : Field R\ninst✝ : DecidableEq R\na : R[X]\nha : ¬a = 0\n⊢ a * C 1 = a",
"usedConstants": [
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"RingHom.instRingHomClass",
"MulOne.toOne",
"Polynomial.instOne",
"Semigroup.toMul",
"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 701,
"column": 2
} | {
"line": 701,
"column": 10
} | [
{
"pp": "case neg\nR : Type u\ninst✝¹ : Field R\ninst✝ : DecidableEq R\na : R[X]\nha : ¬a = 0\n⊢ a.leadingCoeff ≠ 0",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"False",
"eq_false",
"congrArg",
"CommSemiring.toSemiring",
"DivisionSemiring.toGroupWithZero",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 701,
"column": 2
} | {
"line": 701,
"column": 10
} | [
{
"pp": "case neg.h0\nR : Type u\ninst✝¹ : Field R\ninst✝ : DecidableEq R\na : R[X]\nha : ¬a = 0\n⊢ a.leadingCoeff ≠ 0",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"False",
"eq_false",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.leadingCoeff",
"Fiel... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 701,
"column": 2
} | {
"line": 701,
"column": 10
} | [
{
"pp": "case neg\nR : Type u\ninst✝¹ : Field R\ninst✝ : DecidableEq R\na : R[X]\nha : ¬a = 0\n⊢ a ≠ 0",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"CommSemiring.toCommMonoidWithZero",
"CommMonoidWithZero.toMonoidWithZero",
"Field.toSemifield",
"Polynomial"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Polynomial.UniqueFactorization | {
"line": 39,
"column": 4
} | {
"line": 66,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : WfDvdMonoid R\nf : R[X]\n⊢ WellFounded DvdNotUnit",
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
"WithBot.zeroLEOneClass",
"Polynomial.le... | classical
refine
RelHomClass.wellFounded
(⟨fun p : R[X] =>
((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), ?_⟩ :
DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
(wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wf)
rintro a b ⟨ane0, ⟨c, ⟨not_... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.Polynomial.UniqueFactorization | {
"line": 39,
"column": 4
} | {
"line": 66,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : WfDvdMonoid R\nf : R[X]\n⊢ WellFounded DvdNotUnit",
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
"WithBot.zeroLEOneClass",
"Polynomial.le... | classical
refine
RelHomClass.wellFounded
(⟨fun p : R[X] =>
((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), ?_⟩ :
DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
(wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wf)
rintro a b ⟨ane0, ⟨c, ⟨not_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.UniqueFactorization | {
"line": 39,
"column": 4
} | {
"line": 66,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : WfDvdMonoid R\nf : R[X]\n⊢ WellFounded DvdNotUnit",
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
"WithBot.zeroLEOneClass",
"Polynomial.le... | classical
refine
RelHomClass.wellFounded
(⟨fun p : R[X] =>
((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), ?_⟩ :
DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
(wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wf)
rintro a b ⟨ane0, ⟨c, ⟨not_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.UniqueFactorization | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 43
} | [
{
"pp": "σ : Type v\nD : Type u\ninst✝¹ : CommRing D\ninst✝ : UniqueFactorizationMonoid D\nf : D[X]\nhf : f ≠ 0\nG : Type u := { g // g.Monic ∧ g ∣ f }\ny : Associates D[X] := Associates.mk f\nhy : y ≠ 0\n⊢ Fintype G",
"usedConstants": [
"Dvd.dvd",
"CommSemiring.toSemiring",
"semigroupDvd"... | let H := { x : Associates D[X] // x ∣ y } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 45
} | [
{
"pp": "R : Type u\nS : Type u_1\nA : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Ring A\ninst✝⁶ : Algebra R A\nB : Type u_2\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra S B\nFRS : Type u_3\nFAB : Type u_4\ninst✝³ : FunLike FRS R S\ninst✝² : RingHomClass FRS R S\ninst✝¹ : FunLike FAB A B\ninst✝ : RingHomC... | rw [map_zero, map_aeval_eq_aeval_map h, h2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 677,
"column": 2
} | {
"line": 678,
"column": 51
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : ↥A\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nthis : (aeval ↑x) p = 0\n⊢ (↑x)⁻¹ = (-p.coeff 0)⁻¹ • ↑((aeval x) p.divX)",
"usedConstants": [
"Subalgebra.instS... | rw [inv_eq_of_root_of_coeff_zero_ne_zero this coeff_zero_ne, div_eq_inv_mul, Algebra.smul_def,
aeval_coe, map_inv₀, map_neg, inv_neg, neg_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.DirectedInverseSystem | {
"line": 435,
"column": 4
} | {
"line": 435,
"column": 17
} | [
{
"pp": "case equiv.h.refine_1\nι : Type u_6\nF : ι → Type u_7\nX : ι → Type u_8\ninst✝² : LinearOrder ι\nf : ⦃i j : ι⦄ → i ≤ j → F j → F i\ninst✝¹ : SuccOrder ι\nequivSucc : ⦃i : ι⦄ → ¬IsMax i → F i⁺ ≃ F i × X i\ns : Set ι\ninst✝ : WellFoundedLT ι\nhs : IsLowerSet s\ne₁ : (i : ↑s) → F ↑i ≃ piLT X ↑i\nnat₁ : Is... | ext x ⟨j, hj⟩ | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Order.DirectedInverseSystem | {
"line": 511,
"column": 40
} | {
"line": 513,
"column": 72
} | [
{
"pp": "ι : Type u_6\nF : ι → Type u_7\nX : ι → Type u_8\ninst✝³ : LinearOrder ι\nf : ⦃i j : ι⦄ → i ≤ j → F j → F i\ninst✝² : WellFoundedLT ι\ninst✝¹ : SuccOrder ι\ninst✝ : InverseSystem f\nequivSucc : (i : ι) → ¬IsMax i → { e // ∀ (x : F i⁺), (e x).1 = f ⋯ x }\nequivLim : (i : ι) → IsSuccPrelimit i → { e // ∀... | by
refine (DFunLike.congr_fun ?_ _).trans ((globalEquivAux equivSucc equivLim j).nat le_rfl h h x)
exact pEquivOn_apply_eq ((isLowerSet_Iic _).inter <| isLowerSet_Iic _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.Basic | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 38
} | [
{
"pp": "case a\nσ : Type u\nR : Type v\ninst✝ : CommSemiring R\ns t : Set (σ →₀ ℕ)\n⊢ ∀ x ∈ (fun x ↦ (monomial x) 1) '' s, ∀ y ∈ (fun x ↦ (monomial x) 1) '' t, x * y ∈ ↑(restrictSupport R (s + t))",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
"SetLike.mem_co... | simp +contextual [Set.add_mem_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Finiteness.Small | {
"line": 87,
"column": 19
} | {
"line": 89,
"column": 20
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : Small.{u, u_1} R\nA : Subalgebra R S\nfgS : A.FG\n⊢ Small.{u, u_2} ↥A",
"usedConstants": [
"Subalgebra.instSetLike",
"Subalgebra.FG",
"CommSemiring.toSemiring",
"HEq.r... | by
obtain ⟨s, hs, rfl⟩ := fgS
exact small_adjoin | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Flat.Basic | {
"line": 174,
"column": 2
} | {
"line": 178,
"column": 40
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nf : Flat R M\ni : N →ₗ[R] M\nr : M →ₗ[R] N\nh : r ∘ₗ i = LinearMap.id\n⊢ Flat R N",
"usedConstants": [
"LinearMap.id",
"Eq.mpr",
... | rw [iff_rTensor_injectiveₛ] at *
refine fun P _ _ Q ↦ .of_comp (f := lTensor P i) ?_
rw [← coe_comp, lTensor_comp_rTensor, ← rTensor_comp_lTensor, coe_comp]
refine (f Q).comp (Function.RightInverse.injective (g := lTensor Q r) fun x ↦ ?_)
simp [← comp_apply, ← lTensor_comp, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Flat.Basic | {
"line": 174,
"column": 2
} | {
"line": 178,
"column": 40
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nf : Flat R M\ni : N →ₗ[R] M\nr : M →ₗ[R] N\nh : r ∘ₗ i = LinearMap.id\n⊢ Flat R N",
"usedConstants": [
"LinearMap.id",
"Eq.mpr",
... | rw [iff_rTensor_injectiveₛ] at *
refine fun P _ _ Q ↦ .of_comp (f := lTensor P i) ?_
rw [← coe_comp, lTensor_comp_rTensor, ← rTensor_comp_lTensor, coe_comp]
refine (f Q).comp (Function.RightInverse.injective (g := lTensor Q r) fun x ↦ ?_)
simp [← comp_apply, ← lTensor_comp, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Flat.Basic | {
"line": 209,
"column": 2
} | {
"line": 211,
"column": 45
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nι : Type v\nM : ι → Type w\ninst✝¹ : (i : ι) → AddCommMonoid (M i)\ninst✝ : (i : ι) → Module R (M i)\n⊢ Flat R (⨁ (i : ι), M i) ↔ ∀ (i : ι), Flat R (M i)",
"usedConstants": [
"Eq.mpr",
"Submodule",
"_private.Mathlib.RingTheory.Flat.Basic.0.Modu... | simp_rw [iff_rTensor_injectiveₛ, ← EquivLike.comp_injective _ (directSumRight R R _ _),
← LinearEquiv.coe_coe, ← coe_comp, directSumRight_comp_rTensor, coe_comp, LinearEquiv.coe_coe,
EquivLike.injective_comp, lmap_injective] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Colimit.DirectLimit | {
"line": 518,
"column": 4
} | {
"line": 518,
"column": 50
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝⁹ : Preorder ι\nG : ι → Type u_3\nH : ι → Type u_4\nC : Type u_5\nT : ⦃i j : ι⦄ → i ≤ j → Type u_6\nf : (x x_1 : ι) → (h : x ≤ x_1) → T h\ninst✝⁸ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)\ninst✝⁷ : (i : ι) → FunLike (H i) (G i) C\ninst✝⁶ : DirectedSystem G f... | simp_rw [add_def, smul_def, smul_add, add_def] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Colimit.DirectLimit | {
"line": 518,
"column": 4
} | {
"line": 518,
"column": 50
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝⁹ : Preorder ι\nG : ι → Type u_3\nH : ι → Type u_4\nC : Type u_5\nT : ⦃i j : ι⦄ → i ≤ j → Type u_6\nf : (x x_1 : ι) → (h : x ≤ x_1) → T h\ninst✝⁸ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)\ninst✝⁷ : (i : ι) → FunLike (H i) (G i) C\ninst✝⁶ : DirectedSystem G f... | simp_rw [add_def, smul_def, smul_add, add_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Colimit.DirectLimit | {
"line": 518,
"column": 4
} | {
"line": 518,
"column": 50
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝⁹ : Preorder ι\nG : ι → Type u_3\nH : ι → Type u_4\nC : Type u_5\nT : ⦃i j : ι⦄ → i ≤ j → Type u_6\nf : (x x_1 : ι) → (h : x ≤ x_1) → T h\ninst✝⁸ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)\ninst✝⁷ : (i : ι) → FunLike (H i) (G i) C\ninst✝⁶ : DirectedSystem G f... | simp_rw [add_def, smul_def, smul_add, add_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Flat.Basic | {
"line": 647,
"column": 2
} | {
"line": 647,
"column": 64
} | [
{
"pp": "R : Type u_1\nC : Type u_2\nA : Type u_3\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring C\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Algebra R C\ninst✝ : Module.Flat R C\nh : ∀ (B : Subalgebra R A), B.FG → IsReduced (C ⊗[R] ↥B)\nh_contra : ¬IsReduced (C ⊗[R] A)\n⊢ False",
"usedConstants"... | obtain ⟨x, hx⟩ := exists_isNilpotent_of_not_isReduced h_contra | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Colimit.DirectLimit | {
"line": 912,
"column": 20
} | {
"line": 912,
"column": 35
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝¹¹ : Preorder ι\nG : ι → Type u_3\nH : ι → Type u_4\nC : Type u_5\nT : ⦃i j : ι⦄ → i ≤ j → Type u_6\nf : (x x_1 : ι) → (h : x ≤ x_1) → T h\ninst✝¹⁰ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)\ninst✝⁹ : (i : ι) → FunLike (H i) (G i) C\ninst✝⁸ : DirectedSystem G... | apply lift_smul | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Colimit.DirectLimit | {
"line": 912,
"column": 20
} | {
"line": 912,
"column": 35
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝¹¹ : Preorder ι\nG : ι → Type u_3\nH : ι → Type u_4\nC : Type u_5\nT : ⦃i j : ι⦄ → i ≤ j → Type u_6\nf : (x x_1 : ι) → (h : x ≤ x_1) → T h\ninst✝¹⁰ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)\ninst✝⁹ : (i : ι) → FunLike (H i) (G i) C\ninst✝⁸ : DirectedSystem G... | apply lift_smul | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Colimit.DirectLimit | {
"line": 912,
"column": 20
} | {
"line": 912,
"column": 35
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝¹¹ : Preorder ι\nG : ι → Type u_3\nH : ι → Type u_4\nC : Type u_5\nT : ⦃i j : ι⦄ → i ≤ j → Type u_6\nf : (x x_1 : ι) → (h : x ≤ x_1) → T h\ninst✝¹⁰ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)\ninst✝⁹ : (i : ι) → FunLike (H i) (G i) C\ninst✝⁸ : DirectedSystem G... | apply lift_smul | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Expect | {
"line": 267,
"column": 57
} | {
"line": 268,
"column": 91
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ 𝔼 i ∈ s ×ˢ t, f i.1 i.2 = 𝔼 i ∈ s, 𝔼 j ∈ t, f i j",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"NNRat.instInv",... | by
simp only [expect, card_product, sum_product', smul_sum, mul_inv, mul_smul, Nat.cast_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.Expect | {
"line": 358,
"column": 31
} | {
"line": 358,
"column": 48
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\ninst✝⁴ : Semifield M\ninst✝³ : CharZero M\ninst✝² : Fintype ι\ninst✝¹ : Nonempty ι\ninst✝ : DecidableEq ι\nf : ι → M\ni : ι\n⊢ (↑(Fintype.card ι))⁻¹ • (↑↑(Fintype.card ι) * f i) = f i",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | ← NNRat.smul_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Group.Finset.Indicator | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 77
} | [
{
"pp": "case refine_2\nι : Type u_1\nκ : Type u_2\nβ : Type u_4\ninst✝¹ : CommMonoid β\ns : Finset ι\nf : ι → κ → β\nt : ι → Set κ\ng : ι → κ\ninst✝ : DecidablePred fun i ↦ g i ∈ t i\n⊢ ∏ x ∈ s with g x ∉ t x, (t x).mulIndicator (f x) (g x) = 1",
"usedConstants": [
"instDecidableNot",
"MulOne.t... | · exact prod_eq_one fun x hx ↦ mulIndicator_of_notMem (mem_filter.1 hx).2 _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 92,
"column": 57
} | {
"line": 94,
"column": 31
} | [
{
"pp": "α : Type u_1\nn : ℕ\nf : α → ℕ\ninst✝ : DecidableEq α\ns : Finset α\na : α\nhf : ∀ x ∈ s, x ≠ a → f x ≡ 1 [MOD n]\n⊢ ∏ x ∈ s, f x ≡ if a ∈ s then f a else 1 [MOD n]",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"ZMod.commRing",
"congrArg",
"... | by
simp only [← ZMod.natCast_eq_natCast_iff, cast_one, cast_prod, apply_ite Nat.cast] at *
exact Finset.prod_eq_ite _ hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 187,
"column": 2
} | {
"line": 189,
"column": 31
} | [
{
"pp": "α : Type u_1\nn : ℤ\nf : α → ℤ\ninst✝ : DecidableEq α\ns : Finset α\na : α\nhf : ∀ x ∈ s, x ≠ a → f x ≡ 1 [ZMOD n]\n⊢ ∏ x ∈ s, f x ≡ if a ∈ s then f a else 1 [ZMOD n]",
"usedConstants": [
"Int.instCommMonoid",
"Int.cast",
"Eq.mpr",
"ZMod.commRing",
"congrArg",
"F... | simp only [← modEq_natAbs (n := n), ← ZMod.intCast_eq_intCast_iff, cast_one, cast_prod,
apply_ite Int.cast] at *
exact Finset.prod_eq_ite _ hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 187,
"column": 2
} | {
"line": 189,
"column": 31
} | [
{
"pp": "α : Type u_1\nn : ℤ\nf : α → ℤ\ninst✝ : DecidableEq α\ns : Finset α\na : α\nhf : ∀ x ∈ s, x ≠ a → f x ≡ 1 [ZMOD n]\n⊢ ∏ x ∈ s, f x ≡ if a ∈ s then f a else 1 [ZMOD n]",
"usedConstants": [
"Int.instCommMonoid",
"Int.cast",
"Eq.mpr",
"ZMod.commRing",
"congrArg",
"F... | simp only [← modEq_natAbs (n := n), ← ZMod.intCast_eq_intCast_iff, cast_one, cast_prod,
apply_ite Int.cast] at *
exact Finset.prod_eq_ite _ hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Module | {
"line": 38,
"column": 38
} | {
"line": 38,
"column": 41
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhmn : m < n\nh₁ : ∑ i ∈ Ico (m + 1) n, f i • ∑ i ∈ range i, g i = ∑ i ∈ Ico m (n - 1), f (i + 1) • ∑ i ∈ range (i + 1), g i\nh₂ :\n ∑ i ∈ Ico (m + 1) n, f i • ∑ i ∈ range (i + 1), g... | h₂, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 29,
"column": 20
} | {
"line": 29,
"column": 39
} | [
{
"pp": "ι : Type u_1\ns : Finset ι\nf : ι → ℕ\n⊢ (∑ x ∈ s with ¬Even (f x), f x) % 2 = 0 ↔ Even #({x ∈ s | Odd (f x)})",
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"Finset.sum_nat_mod",
"congrArg",
"Odd",
"id",
"Nat.instMod",
"instHMod",
"instOfNa... | Finset.sum_nat_mod, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 12
} | [
{
"pp": "case refine_2\nι : Type u_1\nM : Type u_2\nf : ι → M\ns : Finset M\nhb : ∀ b ∈ s, {a | f a = b}.Finite\nt : Finset M := ⋯.toFinset\nht : (f ⁻¹' ↑t).Finite\nm : M\nhm : m ∈ t\na✝ : ι\nh : a✝ ∈ {a | f a = m}\n⊢ a✝ ∈ ↑ht.toFinset",
"usedConstants": [
"SetLike.mem_coe._simp_1",
"congrArg",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Finset.Sym | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 12
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : DecidableEq α\na : α\ns : Finset α\nha : a ∈ s\n⊢ ∀ x ∈ s, s(a, x) ∈ s.sym2",
"usedConstants": [
"Sym2.mem_iff._simp_1",
"Sym2.mk",
"congrArg",
"and_self",
"Finset",
"forall_eq_or_imp._simp_1",
"Membership.mem",
"And",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Finset.Sym | {
"line": 74,
"column": 2
} | {
"line": 76,
"column": 47
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq β\nf : α → β\ns : Finset α\n⊢ (image f s).sym2 = image (Sym2.map f) s.sym2",
"usedConstants": [
"Eq.mpr",
"Sym2.map",
"Multiset.map",
"congrArg",
"Multiset.dedup",
"Multiset",
"id",
"Sym2.instDecidableEq... | apply val_injective
dsimp [Finset.sym2]
rw [← Multiset.dedup_sym2, Multiset.sym2_map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Sym | {
"line": 74,
"column": 2
} | {
"line": 76,
"column": 47
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq β\nf : α → β\ns : Finset α\n⊢ (image f s).sym2 = image (Sym2.map f) s.sym2",
"usedConstants": [
"Eq.mpr",
"Sym2.map",
"Multiset.map",
"congrArg",
"Multiset.dedup",
"Multiset",
"id",
"Sym2.instDecidableEq... | apply val_injective
dsimp [Finset.sym2]
rw [← Multiset.dedup_sym2, Multiset.sym2_map] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Sym | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nk : Sym α m\ns : Finset α\nhk : k ∈ s.sym m\n⊢ ∑ i ∈ s, count i ↑k = m",
"usedConstants": [
"Multiset.sum_count_eq_card",
"congrArg",
"Finset",
"Membership.mem",
"Multiset.count",
"Multiset",
"Eq.mp",
"id",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.BigOperators.Sym | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nk : Sym α m\ns : Finset α\nhk : k ∈ s.sym m\n⊢ ∑ i ∈ s, count i ↑k = m",
"usedConstants": [
"Multiset.sum_count_eq_card",
"congrArg",
"Finset",
"Membership.mem",
"Multiset.count",
"Multiset",
"Eq.mp",
"id",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Sym | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nk : Sym α m\ns : Finset α\nhk : k ∈ s.sym m\n⊢ ∑ i ∈ s, count i ↑k = m",
"usedConstants": [
"Multiset.sum_count_eq_card",
"congrArg",
"Finset",
"Membership.mem",
"Multiset.count",
"Multiset",
"Eq.mp",
"id",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Sym | {
"line": 31,
"column": 53
} | {
"line": 32,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nk : Sym α m\ns : Finset α\nhk : k ∈ s.sym m\n⊢ ∑ i ∈ s, count i ↑k = m",
"usedConstants": [
"Multiset.sum_count_eq_card",
"congrArg",
"Finset",
"Membership.mem",
"Multiset.count",
"Multiset",
"Eq.mp",
"id",
... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Sym.Sym2 | {
"line": 321,
"column": 10
} | {
"line": 321,
"column": 17
} | [
{
"pp": "case inr\nα : Type u_1\na b : α\n⊢ Sym2.Mem a s(b, a)",
"usedConstants": [
"Eq.mpr",
"Sym2.mk",
"congrArg",
"id",
"Sym2.eq_swap",
"Eq",
"Sym2",
"Sym2.Mem"
]
}
] | eq_swap | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IsTensorProduct | {
"line": 232,
"column": 21
} | {
"line": 232,
"column": 29
} | [
{
"pp": "case add\nR✝ : Type u_1\ninst✝⁴² : CommSemiring R✝\nM₁✝ : Type u_2\nM₂✝ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝⁴¹ : AddCommMonoid M₁✝\ninst✝⁴⁰ : AddCommMonoid M₂✝\ninst✝³⁹ : AddCommMonoid M\ninst✝³⁸ : AddCommMonoid M'\ninst✝³⁷ : Module R✝ M₁✝\ninst✝³⁶ : Module R✝ M₂✝\ninst✝³⁵ : Module R✝ M\ninst... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.IsTensorProduct | {
"line": 232,
"column": 21
} | {
"line": 232,
"column": 29
} | [
{
"pp": "case add\nR✝ : Type u_1\ninst✝⁴² : CommSemiring R✝\nM₁✝ : Type u_2\nM₂✝ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝⁴¹ : AddCommMonoid M₁✝\ninst✝⁴⁰ : AddCommMonoid M₂✝\ninst✝³⁹ : AddCommMonoid M\ninst✝³⁸ : AddCommMonoid M'\ninst✝³⁷ : Module R✝ M₁✝\ninst✝³⁶ : Module R✝ M₂✝\ninst✝³⁵ : Module R✝ M\ninst... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.IsTensorProduct | {
"line": 232,
"column": 21
} | {
"line": 232,
"column": 29
} | [
{
"pp": "case add\nR✝ : Type u_1\ninst✝⁴² : CommSemiring R✝\nM₁✝ : Type u_2\nM₂✝ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝⁴¹ : AddCommMonoid M₁✝\ninst✝⁴⁰ : AddCommMonoid M₂✝\ninst✝³⁹ : AddCommMonoid M\ninst✝³⁸ : AddCommMonoid M'\ninst✝³⁷ : Module R✝ M₁✝\ninst✝³⁶ : Module R✝ M₂✝\ninst✝³⁵ : Module R✝ M\ninst... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.IsTensorProduct | {
"line": 241,
"column": 6
} | {
"line": 243,
"column": 64
} | [
{
"pp": "case tmul.tmul\nR✝ : Type u_1\ninst✝⁴² : CommSemiring R✝\nM₁✝ : Type u_2\nM₂✝ : Type u_3\nM : Type u_4\nM' : Type u_5\ninst✝⁴¹ : AddCommMonoid M₁✝\ninst✝⁴⁰ : AddCommMonoid M₂✝\ninst✝³⁹ : AddCommMonoid M\ninst✝³⁸ : AddCommMonoid M'\ninst✝³⁷ : Module R✝ M₁✝\ninst✝³⁶ : Module R✝ M₂✝\ninst✝³⁵ : Module R✝ M... | rw [smul_tmul', this, ← f.restrictScalars₁₂_apply_apply R S,
← f.restrictScalars₁₂_apply_apply R S, IsTensorProduct.assocAux_tmul,
IsTensorProduct.assocAux_tmul, TensorProduct.smul_tmul'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.IsTensorProduct | {
"line": 352,
"column": 2
} | {
"line": 352,
"column": 74
} | [
{
"pp": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChang... | have hF : ∀ (s : S) (m : M), h.lift g (s • f m) = s • g m := h.lift_eq _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.ConcreteCategory.Basic | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 12
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type w\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝ : ConcreteCategory C FC\nX Y : C\nf g : X ⟶ Y\nh : (fun f ↦ ⇑(hom f)) f = (fun f ↦ ⇑(hom f)) g\n⊢ ofHom (hom f) = ofHom (hom g)",
"usedConstants": [
"Cate... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.IsTensorProduct | {
"line": 738,
"column": 4
} | {
"line": 738,
"column": 12
} | [
{
"pp": "case H.refine_4\nR : Type u_1\nS : Type v₃\ninst✝¹² : CommSemiring R\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Algebra R S\nR' : Type u_6\nS' : Type u_7\ninst✝⁹ : CommSemiring R'\ninst✝⁸ : CommSemiring S'\ninst✝⁷ : Algebra R R'\ninst✝⁶ : Algebra S S'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra R S'\ninst✝³ : ... | intro s₁ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Combinatorics.Quiver.Symmetric | {
"line": 144,
"column": 29
} | {
"line": 149,
"column": 7
} | [
{
"pp": "V : Type u_2\ninst✝ : Quiver V\nh : HasInvolutiveReverse V\na b : V\np : Path a b\n⊢ p.reverse.reverse = p",
"usedConstants": [
"Eq.mpr",
"Quiver.Hom",
"Quiver.Path.nil",
"congrArg",
"Quiver.Path.rec",
"Quiver.reverse_reverse",
"id",
"Quiver.Path.reve... | by
induction p with
| nil => simp
| cons _ _ h =>
rw [Path.reverse, Path.reverse_comp, h, Path.reverse_toPath, Quiver.reverse_reverse]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.EqToHom | {
"line": 302,
"column": 37
} | {
"line": 302,
"column": 78
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nF G : C ⥤ D\nX Y : C\nf : X ⟶ Y\nH : D ⥤ E\nhobj : ∀ (X : C), F.obj X = G.obj X\nhmap : ∀ {X Y : C} (f : X ⟶ Y), F.map f ≍ G.map f\n⊢ (F ⋙ H).map f ≍ (G ⋙ H).map f",
"used... | by rw [Functor.hext hobj fun _ _ => hmap] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Equivalence | {
"line": 266,
"column": 72
} | {
"line": 267,
"column": 91
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\ne : C ≌ D\nX : C\n⊢ e.counit.app (e.functor.obj X) = e.functor.map (e.unitInv.app X)",
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.Equivalence.unitIso",
"CategoryTheory.CategorySt... | by
simpa using Iso.hom_comp_eq_id (e.functor.mapIso (e.unitIso.app X)) (f := e.counit.app _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Comma.Arrow | {
"line": 152,
"column": 27
} | {
"line": 152,
"column": 35
} | [
{
"pp": "T : Type u\ninst✝ : Category.{v, u} T\nf g : Arrow T\nh₁ : f.left = g.left\nh₂ : f.right = g.right\nh₃ : f.hom = eqToHom h₁ ≫ g.hom ≫ eqToHom ⋯\n⊢ ∃ hX hY, f.hom = eqToHom hX ≫ g.hom ≫ eqToHom ⋯",
"usedConstants": [
"CategoryTheory.Comma.right",
"CategoryTheory.CategoryStruct.toQuiver",... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Comma.Arrow | {
"line": 152,
"column": 27
} | {
"line": 152,
"column": 35
} | [
{
"pp": "T : Type u\ninst✝ : Category.{v, u} T\nf g : Arrow T\nh₁ : f.left = g.left\nh₂ : f.right = g.right\nh₃ : f.hom = eqToHom h₁ ≫ g.hom ≫ eqToHom ⋯\n⊢ ∃ hX hY, f.hom = eqToHom hX ≫ g.hom ≫ eqToHom ⋯",
"usedConstants": [
"CategoryTheory.Comma.right",
"CategoryTheory.CategoryStruct.toQuiver",... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Comma.Arrow | {
"line": 152,
"column": 27
} | {
"line": 152,
"column": 35
} | [
{
"pp": "T : Type u\ninst✝ : Category.{v, u} T\nf g : Arrow T\nh₁ : f.left = g.left\nh₂ : f.right = g.right\nh₃ : f.hom = eqToHom h₁ ≫ g.hom ≫ eqToHom ⋯\n⊢ ∃ hX hY, f.hom = eqToHom hX ≫ g.hom ≫ eqToHom ⋯",
"usedConstants": [
"CategoryTheory.Comma.right",
"CategoryTheory.CategoryStruct.toQuiver",... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Types.Basic | {
"line": 348,
"column": 56
} | {
"line": 348,
"column": 64
} | [
{
"pp": "X : Type u\nx y : X\n⊢ x = y → homOfElement x = homOfElement y",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.homOfElement",
"PUnit",
"True",
"eq_self",
"CategoryTheory.types",
"of_eq_tru... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Types.Basic | {
"line": 348,
"column": 56
} | {
"line": 348,
"column": 64
} | [
{
"pp": "X : Type u\nx y : X\n⊢ x = y → homOfElement x = homOfElement y",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.homOfElement",
"PUnit",
"True",
"eq_self",
"CategoryTheory.types",
"of_eq_tru... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Types.Basic | {
"line": 348,
"column": 56
} | {
"line": 348,
"column": 64
} | [
{
"pp": "X : Type u\nx y : X\n⊢ x = y → homOfElement x = homOfElement y",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.homOfElement",
"PUnit",
"True",
"eq_self",
"CategoryTheory.types",
"of_eq_tru... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adjunction.Basic | {
"line": 191,
"column": 2
} | {
"line": 195,
"column": 82
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj adj' : F ⊣ G\nh : adj.unit = adj'.unit\n⊢ adj = adj'",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"Equiv.instEquivLike",
"CategoryTheory.CategoryStruct.... | suffices h' : adj.counit = adj'.counit by cases adj; cases adj'; aesop
ext X
apply (adj.homEquiv _ _).injective
rw [Adjunction.homEquiv_unit, Adjunction.homEquiv_unit,
Adjunction.right_triangle_components, h, Adjunction.right_triangle_components] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Adjunction.Basic | {
"line": 191,
"column": 2
} | {
"line": 195,
"column": 82
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj adj' : F ⊣ G\nh : adj.unit = adj'.unit\n⊢ adj = adj'",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"Equiv.instEquivLike",
"CategoryTheory.CategoryStruct.... | suffices h' : adj.counit = adj'.counit by cases adj; cases adj'; aesop
ext X
apply (adj.homEquiv _ _).injective
rw [Adjunction.homEquiv_unit, Adjunction.homEquiv_unit,
Adjunction.right_triangle_components, h, Adjunction.right_triangle_components] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Category.Preorder | {
"line": 273,
"column": 8
} | {
"line": 273,
"column": 33
} | [
{
"pp": "case mpr\nX : Type u\ninst✝ : PartialOrder X\na : X\nf : a ⟶ a\n⊢ IsIso f",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsIso",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"PartialOrder.toPreorder",
"CategoryTheory.CategoryStruct.id... | Subsingleton.elim f (𝟙 _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Bicategory.Basic | {
"line": 416,
"column": 2
} | {
"line": 419,
"column": 38
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c : B\nf : a ⟶ b\ng : b ⟶ c\n⊢ f ◁ (ρ_ g).hom = (α_ f g (𝟙 c)).inv ≫ (ρ_ (f ≫ g)).hom",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"CategoryTheory.Bicategory.associator_inv_naturality_middle",
"Quiver.Hom",
... | rw [← whiskerRight_iff, comp_whiskerRight, ← cancel_epi (α_ _ _ _).inv, ←
cancel_epi (f ◁ (α_ _ _ _).inv), pentagon_inv_assoc, triangle_assoc_comp_right, ←
associator_inv_naturality_middle, ← whiskerLeft_comp_assoc, triangle_assoc_comp_right,
associator_inv_naturality_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Bicategory.Basic | {
"line": 416,
"column": 2
} | {
"line": 419,
"column": 38
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c : B\nf : a ⟶ b\ng : b ⟶ c\n⊢ f ◁ (ρ_ g).hom = (α_ f g (𝟙 c)).inv ≫ (ρ_ (f ≫ g)).hom",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"CategoryTheory.Bicategory.associator_inv_naturality_middle",
"Quiver.Hom",
... | rw [← whiskerRight_iff, comp_whiskerRight, ← cancel_epi (α_ _ _ _).inv, ←
cancel_epi (f ◁ (α_ _ _ _).inv), pentagon_inv_assoc, triangle_assoc_comp_right, ←
associator_inv_naturality_middle, ← whiskerLeft_comp_assoc, triangle_assoc_comp_right,
associator_inv_naturality_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Bicategory.Basic | {
"line": 416,
"column": 2
} | {
"line": 419,
"column": 38
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c : B\nf : a ⟶ b\ng : b ⟶ c\n⊢ f ◁ (ρ_ g).hom = (α_ f g (𝟙 c)).inv ≫ (ρ_ (f ≫ g)).hom",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"CategoryTheory.Bicategory.associator_inv_naturality_middle",
"Quiver.Hom",
... | rw [← whiskerRight_iff, comp_whiskerRight, ← cancel_epi (α_ _ _ _).inv, ←
cancel_epi (f ◁ (α_ _ _ _).inv), pentagon_inv_assoc, triangle_assoc_comp_right, ←
associator_inv_naturality_middle, ← whiskerLeft_comp_assoc, triangle_assoc_comp_right,
associator_inv_naturality_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.IsLimit | {
"line": 907,
"column": 48
} | {
"line": 907,
"column": 60
} | [
{
"pp": "J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF : J ⥤ C\nt : Cocone F\nh : IsColimit t\nW : C\np : { p // ∀ {j j' : J} (f : j ⟶ j'), F.map f ≫ p j' = p j }\nj j' : J\nf : j ⟶ j'\n⊢ F.map f ≫ ↑p j' = ↑p j ≫ 𝟙 W",
"use... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Cones | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 33
} | [
{
"pp": "J : Type u₁\ninst✝¹ : Category.{v₁, u₁} J\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF : J ⥤ C\nc d : Cone F\nf : c ≅ d\n⊢ f.hom.hom ≫ f.inv.hom = 𝟙 c.pt",
"usedConstants": [
"CategoryTheory.Limits.Cone",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | simp [← Cone.category_comp_hom] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Cones | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 33
} | [
{
"pp": "J : Type u₁\ninst✝¹ : Category.{v₁, u₁} J\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF : J ⥤ C\nc d : Cone F\nf : c ≅ d\n⊢ f.hom.hom ≫ f.inv.hom = 𝟙 c.pt",
"usedConstants": [
"CategoryTheory.Limits.Cone",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | simp [← Cone.category_comp_hom] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Cones | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 33
} | [
{
"pp": "J : Type u₁\ninst✝¹ : Category.{v₁, u₁} J\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF : J ⥤ C\nc d : Cone F\nf : c ≅ d\n⊢ f.hom.hom ≫ f.inv.hom = 𝟙 c.pt",
"usedConstants": [
"CategoryTheory.Limits.Cone",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | simp [← Cone.category_comp_hom] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Cones | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 33
} | [
{
"pp": "J : Type u₁\ninst✝¹ : Category.{v₁, u₁} J\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF : J ⥤ C\nc d : Cone F\nf : c ≅ d\n⊢ f.inv.hom ≫ f.hom.hom = 𝟙 d.pt",
"usedConstants": [
"CategoryTheory.Limits.Cone",
"CategoryTheory.Iso.inv_hom_id",
"CategoryTheory.CategoryStruct.toQuiver",
... | simp [← Cone.category_comp_hom] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Cones | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 33
} | [
{
"pp": "J : Type u₁\ninst✝¹ : Category.{v₁, u₁} J\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF : J ⥤ C\nc d : Cone F\nf : c ≅ d\n⊢ f.inv.hom ≫ f.hom.hom = 𝟙 d.pt",
"usedConstants": [
"CategoryTheory.Limits.Cone",
"CategoryTheory.Iso.inv_hom_id",
"CategoryTheory.CategoryStruct.toQuiver",
... | simp [← Cone.category_comp_hom] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Cones | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 33
} | [
{
"pp": "J : Type u₁\ninst✝¹ : Category.{v₁, u₁} J\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF : J ⥤ C\nc d : Cone F\nf : c ≅ d\n⊢ f.inv.hom ≫ f.hom.hom = 𝟙 d.pt",
"usedConstants": [
"CategoryTheory.Limits.Cone",
"CategoryTheory.Iso.inv_hom_id",
"CategoryTheory.CategoryStruct.toQuiver",
... | simp [← Cone.category_comp_hom] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.CommSq | {
"line": 207,
"column": 32
} | {
"line": 207,
"column": 43
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nA B X Y : C\nf : A ⟶ X\ni : A ⟶ B\np : X ⟶ Y\ng : B ⟶ Y\nsq : CommSq f i p g\nl : sq.LiftStruct\n⊢ (l.l ≫ p).op = g.op",
"usedConstants": [
"Eq.mpr",
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congr... | l.fac_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.CommSq | {
"line": 207,
"column": 17
} | {
"line": 207,
"column": 44
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nA B X Y : C\nf : A ⟶ X\ni : A ⟶ B\np : X ⟶ Y\ng : B ⟶ Y\nsq : CommSq f i p g\nl : sq.LiftStruct\n⊢ p.op ≫ l.l.op = g.op",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.op_comp",
"Opposite",
"Quiver.opposite",
"CategoryTheory.... | rw [← op_comp, l.fac_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.CommSq | {
"line": 207,
"column": 17
} | {
"line": 207,
"column": 44
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nA B X Y : C\nf : A ⟶ X\ni : A ⟶ B\np : X ⟶ Y\ng : B ⟶ Y\nsq : CommSq f i p g\nl : sq.LiftStruct\n⊢ p.op ≫ l.l.op = g.op",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.op_comp",
"Opposite",
"Quiver.opposite",
"CategoryTheory.... | rw [← op_comp, l.fac_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.CommSq | {
"line": 207,
"column": 17
} | {
"line": 207,
"column": 44
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nA B X Y : C\nf : A ⟶ X\ni : A ⟶ B\np : X ⟶ Y\ng : B ⟶ Y\nsq : CommSq f i p g\nl : sq.LiftStruct\n⊢ p.op ≫ l.l.op = g.op",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.op_comp",
"Opposite",
"Quiver.opposite",
"CategoryTheory.... | rw [← op_comp, l.fac_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.CommSq | {
"line": 216,
"column": 34
} | {
"line": 216,
"column": 45
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nA✝ B✝ X✝ Y✝ : C\nf✝ : A✝ ⟶ X✝\ni✝ : A✝ ⟶ B✝\np✝ : X✝ ⟶ Y✝\ng✝ : B✝ ⟶ Y✝\nA B X Y : Cᵒᵖ\nf : A ⟶ X\ni : A ⟶ B\np : X ⟶ Y\ng : B ⟶ Y\nsq : CommSq f i p g\nl : sq.LiftStruct\n⊢ (l.l ≫ p).unop = g.unop",
"usedConstants": [
"Eq.mpr",
"Opposite",
... | l.fac_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.LiftingProperties.Adjunction | {
"line": 56,
"column": 66
} | {
"line": 56,
"column": 77
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nG : C ⥤ D\nF : D ⥤ C\nA B : C\nX Y : D\ni : A ⟶ B\np : X ⟶ Y\nu : G.obj A ⟶ X\nv : G.obj B ⟶ Y\nsq : CommSq u (G.map i) p v\nadj : G ⊣ F\nl : sq.LiftStruct\n⊢ (adj.homEquiv B Y) (l.l ≫ p) = (adj.homEquiv B Y) v",... | l.fac_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.LiftingProperties.Adjunction | {
"line": 63,
"column": 57
} | {
"line": 63,
"column": 68
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nG : C ⥤ D\nF : D ⥤ C\nA B : C\nX Y : D\ni : A ⟶ B\np : X ⟶ Y\nu : G.obj A ⟶ X\nv : G.obj B ⟶ Y\nsq : CommSq u (G.map i) p v\nadj : G ⊣ F\nl : ⋯.LiftStruct\n⊢ (adj.homEquiv B Y).symm (l.l ≫ F.map p) = v",
"use... | l.fac_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.LiftingProperties.Adjunction | {
"line": 101,
"column": 64
} | {
"line": 101,
"column": 75
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nG : C ⥤ D\nF : D ⥤ C\nA B : C\nX Y : D\ni : A ⟶ B\np : X ⟶ Y\nu : A ⟶ F.obj X\nv : B ⟶ F.obj Y\nsq : CommSq u i (F.map p) v\nadj : G ⊣ F\nl : sq.LiftStruct\n⊢ (adj.homEquiv B Y).symm (l.l ≫ F.map p) = (adj.homEqu... | l.fac_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.LiftingProperties.Adjunction | {
"line": 108,
"column": 45
} | {
"line": 108,
"column": 56
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nG : C ⥤ D\nF : D ⥤ C\nA B : C\nX Y : D\ni : A ⟶ B\np : X ⟶ Y\nu : A ⟶ F.obj X\nv : B ⟶ F.obj Y\nsq : CommSq u i (F.map p) v\nadj : G ⊣ F\nl : ⋯.LiftStruct\n⊢ (adj.homEquiv B Y) (l.l ≫ p) = v",
"usedConstants"... | l.fac_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.PUnit | {
"line": 71,
"column": 21
} | {
"line": 74,
"column": 19
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nh : C ≌ Discrete PUnit.{w + 1}\nx y : C\n⊢ x ⟶ y",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"HEq.refl",
"CategoryTheory.Functor.comp",
"CategoryTheory.Functor.id",
"Eq.casesO... | by
have hx : x ⟶ h.inverse.obj ⟨⟨⟩⟩ := by convert! h.unit.app x
have hy : h.inverse.obj ⟨⟨⟩⟩ ⟶ y := by convert! h.unitInv.app y
exact hx ≫ hy | [anonymous] | Lean.Parser.Term.byTactic |
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