module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.GroupTheory.OrderOfElement | {
"line": 1206,
"column": 15
} | {
"line": 1206,
"column": 33
} | {
"line": 1206,
"column": 34
} | [
{
"pp": "G : Type u_6\ninst✝ : Group G\na : G\nn : ℕ\n⊢ a ^ n = a ^ (n % Nat.card G)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"pow_mod_orderOf",
"id",
"Nat.instMod",
"Nat.card",
"instHMod",
"DivInvMonoid.toMonoid"... | [
"G : Type u_6\ninst✝ : Group G\na : G\nn : ℕ\n⊢ a ^ (n % orderOf a) = a ^ (n % Nat.card G)"
] | ← pow_mod_orderOf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Tactic.ComputeDegree | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 39
} | {
"line": 132,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nm n : ℕ\np : R[X]\nh_pow : p.natDegree ≤ n\nh_exp : m * n ≤ m * n\n⊢ (p ^ m).coeff (m * n) = p.coeff n ^ m",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Polynomial.coeff_pow_of_natDegree_le"
],
"usedFVars": [
"R",
... | [] | exact coeff_pow_of_natDegree_le ‹_› | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.Inductions | {
"line": 169,
"column": 33
} | {
"line": 169,
"column": 44
} | {
"line": 169,
"column": 44
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nP : R[X] → Prop\np : R[X]\nh0 : 0 < p.degree\nhC : ∀ {a : R}, a ≠ 0 → P (C a * X)\nhX : ∀ {p : R[X]}, 0 < p.degree → P p → P (p * X)\nhadd : ∀ {p : R[X]} {a : R}, 0 < p.degree → P p → P (p + C a)\nh : 0 < degree 0\n⊢ P 0",
"ppTerm": "?m.88",
"assigned": true,
... | [
"R : Type u\ninst✝ : Semiring R\nP : R[X] → Prop\np : R[X]\nh0 : 0 < p.degree\nhC : ∀ {a : R}, a ≠ 0 → P (C a * X)\nhX : ∀ {p : R[X]}, 0 < p.degree → P p → P (p * X)\nhadd : ∀ {p : R[X]} {a : R}, 0 < p.degree → P p → P (p + C a)\nh : 0 < ⊥\n⊢ P 0"
] | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.RingDivision | {
"line": 96,
"column": 12
} | {
"line": 96,
"column": 21
} | {
"line": 96,
"column": 22
} | [
{
"pp": "case pos\nR : Type u\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nhp : ¬p = 0\nhq : q = 0\n⊢ (p * 0).trailingDegree = p.trailingDegree + trailingDegree 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
... | [
"case pos\nR : Type u\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nhp : ¬p = 0\nhq : q = 0\n⊢ trailingDegree 0 = p.trailingDegree + trailingDegree 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Derivative | {
"line": 67,
"column": 25
} | {
"line": 67,
"column": 34
} | {
"line": 67,
"column": 35
} | [
{
"pp": "case h₀.zero\nR : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\na✝¹ : 0 ∈ p.support\na✝ : 0 ≠ n + 1\n⊢ (p.coeff 0 * 0 * if n = 0 - 1 then 1 else 0) = 0",
"ppTerm": "?h₀.zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hM... | [
"case h₀.zero\nR : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\na✝¹ : 0 ∈ p.support\na✝ : 0 ≠ n + 1\n⊢ (0 * if n = 0 - 1 then 1 else 0) = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Expand | {
"line": 67,
"column": 24
} | {
"line": 67,
"column": 54
} | {
"line": 67,
"column": 54
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\np q : ℕ\nf✝ f g : R[X]\nihf : (expand R p) ((expand R q) f) = (expand R (p * q)) f\nihg : (expand R p) ((expand R q) g) = (expand R (p * q)) g\n⊢ (expand R p) ((expand R q) (f + g)) = (expand R (p * q)) (f + g)",
"ppTerm": "?m.26",
"assigned": true,
"used... | [] | by simp_rw [map_add, ihf, ihg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Expand | {
"line": 100,
"column": 10
} | {
"line": 100,
"column": 16
} | {
"line": 100,
"column": 17
} | [
{
"pp": "case pos.h₀.hnc\nR : Type u\ninst✝ : CommSemiring R\np : ℕ\nhp : 0 < p\nf : R[X]\nn : ℕ\nh : p ∣ n\nb : ℕ\na✝ : b ∈ f.support\nhb2 : b ≠ n / p\nhb3 : p * b = n\n⊢ b = n / p",
"ppTerm": "?pos.h₀.hnc✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
... | [
"case pos.h₀.hnc\nR : Type u\ninst✝ : CommSemiring R\np : ℕ\nhp : 0 < p\nf : R[X]\nn : ℕ\nh : p ∣ n\nb : ℕ\na✝ : b ∈ f.support\nhb2 : b ≠ n / p\nhb3 : p * b = n\n⊢ b = p * b / p"
] | ← hb3, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Expand | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 23
} | {
"line": 104,
"column": 2
} | [
{
"pp": "case pos.h₁\nR : Type u\ninst✝ : CommSemiring R\np : ℕ\nhp : 0 < p\nf : R[X]\nn : ℕ\nh : p ∣ n\nhn : n / p ∉ f.support\n⊢ (if p * (n / p) = n then 0 else 0) = 0",
"ppTerm": "?pos.h₁✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"congrArg"... | [] | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Polynomial.Expand | {
"line": 136,
"column": 34
} | {
"line": 136,
"column": 43
} | {
"line": 136,
"column": 44
} | [
{
"pp": "case inl\nR : Type u\ninst✝ : CommSemiring R\np : ℕ\nf : R[X]\nhp : p = 0\n⊢ (eval₂ C 1 f).natDegree = f.natDegree * 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"MulOne.toOne",
"Nat.instMulZeroClass",
"HMul.hMul",
... | [
"case inl\nR : Type u\ninst✝ : CommSemiring R\np : ℕ\nf : R[X]\nhp : p = 0\n⊢ (eval₂ C 1 f).natDegree = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 181,
"column": 51
} | {
"line": 182,
"column": 38
} | {
"line": 184,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np : R[X]\n⊢ 0 %ₘ p = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Algebra.Polynomial.Div.0.Polynomial.zero_modByMonic._proof_1_4"
],
"usedFVars": [
"R",
"inst✝",
"p"
],
"usedGoals": []
}
] | [] | by
grind [modByMonic, divModByMonicAux] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Div | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 49
} | {
"line": 226,
"column": 2
} | [
{
"pp": "case pos.inr\nR : Type u\ninst✝ : Ring R\np q : R[X]\na✝ : Nontrivial R\nhq : q.Monic\nh✝ : q.degree ≤ p.degree\n⊢ (p %ₘ q).degree ≤ p.degree",
"ppTerm": "?pos.inr✝",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"WithBot",
"Polynomial.degree_modByMonic_le",... | [] | · exact (degree_modByMonic_le p hq).trans ‹_› | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.Div | {
"line": 251,
"column": 59
} | {
"line": 251,
"column": 67
} | {
"line": 251,
"column": 68
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np q : R[X]\nthis : DecidableEq R := Classical.decEq R\nhq : q.Monic\nh : q.degree ≤ p.degree ∧ p ≠ 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\nih :\n ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonic... | [
"R : Type u\ninst✝ : Ring R\np q : R[X]\nthis : DecidableEq R := Classical.decEq R\nhq : q.Monic\nh : q.degree ≤ p.degree ∧ p ≠ 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\nih :\n ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux hq).2 =\... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Expand | {
"line": 221,
"column": 36
} | {
"line": 221,
"column": 52
} | {
"line": 221,
"column": 52
} | [
{
"pp": "case h\nR : Type u\ninst✝ : CommSemiring R\np : ℕ\nhp : p ≠ 0\nf g : R[X]\nn : ℕ\n⊢ ∀ ⦃x : ℕ × ℕ⦄, (∃ a ∈ antidiagonal n, (a.1 * p, a.2 * p) = x) → x ∈ antidiagonal (n * p)",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"AddMonoid.toAddSemigr... | [
"case h\nR : Type u\ninst✝ : CommSemiring R\np : ℕ\nhp : p ≠ 0\nf g : R[X]\nn : ℕ\n⊢ ∀ ⦃x : ℕ × ℕ⦄, (∃ a, a.1 + a.2 = n ∧ (a.1 * p, a.2 * p) = x) → x.1 + x.2 = n * p"
] | mem_antidiagonal | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 255,
"column": 56
} | {
"line": 255,
"column": 65
} | {
"line": 255,
"column": 66
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np q : R[X]\nthis : DecidableEq R := Classical.decEq R\nhq : q.Monic\nh : ¬(q.degree ≤ p.degree ∧ p ≠ 0)\n⊢ (0, p).2 = p - q * (0, p).1",
"ppTerm": "?m.203",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
... | [
"R : Type u\ninst✝ : Ring R\np q : R[X]\nthis : DecidableEq R := Classical.decEq R\nhq : q.Monic\nh : ¬(q.degree ≤ p.degree ∧ p ≠ 0)\n⊢ (0, p).2 = p - 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Expand | {
"line": 222,
"column": 22
} | {
"line": 222,
"column": 38
} | {
"line": 222,
"column": 38
} | [
{
"pp": "case hf\nR : Type u\ninst✝ : CommSemiring R\np : ℕ\nhp : p ≠ 0\nf g : R[X]\nn : ℕ\n⊢ ∀ x ∈ antidiagonal (n * p),\n (¬∃ a ∈ antidiagonal n, (a.1 * p, a.2 * p) = x) → f.coeff x.1 * ((expand R p) g).coeff x.2 = 0",
"ppTerm": "?hf",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case hf\nR : Type u\ninst✝ : CommSemiring R\np : ℕ\nhp : p ≠ 0\nf g : R[X]\nn : ℕ\n⊢ ∀ (x : ℕ × ℕ),\n x.1 + x.2 = n * p → (¬∃ a, a.1 + a.2 = n ∧ (a.1 * p, a.2 * p) = x) → f.coeff x.1 * ((expand R p) g).coeff x.2 = 0"
] | mem_antidiagonal | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 257,
"column": 76
} | {
"line": 257,
"column": 85
} | {
"line": 257,
"column": 86
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np q : R[X]\nthis : DecidableEq R := Classical.decEq R\nhq : ¬q.Monic\n⊢ p = p - q * 0",
"ppTerm": "?m.221",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"HSub.hSub",
"id",
... | [
"R : Type u\ninst✝ : Ring R\np q : R[X]\nthis : DecidableEq R := Classical.decEq R\nhq : ¬q.Monic\n⊢ p = p - 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 274,
"column": 12
} | {
"line": 274,
"column": 21
} | {
"line": 274,
"column": 22
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\np q : R[X]\ninst✝ : Nontrivial R\nhq : q.Monic\nh : p /ₘ q = 0\nthis : p %ₘ q + q * 0 = p\n⊢ p.degree < q.degree",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"Eq.mp",
"Poly... | [
"R : Type u\ninst✝¹ : Ring R\np q : R[X]\ninst✝ : Nontrivial R\nhq : q.Monic\nh : p /ₘ q = 0\nthis : p %ₘ q + 0 = p\n⊢ p.degree < q.degree"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Expand | {
"line": 259,
"column": 20
} | {
"line": 259,
"column": 32
} | {
"line": 259,
"column": 32
} | [
{
"pp": "case zero\nR : Type u\ninst✝³ : CommSemiring R\ninst✝² : NoZeroDivisors R\nf : R[X]\nhf : derivative f = 0\ninst✝¹ : CharZero R\ninst✝ : ExpChar R 1\n⊢ contract 1 f = f",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring... | [
"case zero\nR : Type u\ninst✝³ : CommSemiring R\ninst✝² : NoZeroDivisors R\nf : R[X]\nhf : derivative f = 0\ninst✝¹ : CharZero R\ninst✝ : ExpChar R 1\n⊢ f = f"
] | contract_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Eval.SMul | {
"line": 76,
"column": 70
} | {
"line": 76,
"column": 81
} | {
"line": 76,
"column": 82
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : Semiring R\ninst✝¹ : SMulZeroClass S R\ninst✝ : IsScalarTower S R R\ns : S\np q : R[X]\n⊢ (s • 1) • eval₂ C q p = s • eval₂ C q p",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCom... | [
"R : Type u\nS : Type v\ninst✝² : Semiring R\ninst✝¹ : SMulZeroClass S R\ninst✝ : IsScalarTower S R R\ns : S\np q : R[X]\n⊢ s • 1 • eval₂ C q p = s • eval₂ C q p"
] | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Expand | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 69
} | {
"line": 262,
"column": 0
} | [
{
"pp": "case prime\nR : Type u\ninst✝² : CommSemiring R\np : ℕ\ninst✝¹ : ExpChar R p\ninst✝ : NoZeroDivisors R\nf : R[X]\nhf : derivative f = 0\nhprime : Nat.Prime p\nhchar : CharP R p\n⊢ (expand R p) (contract p f) = f",
"ppTerm": "?prime",
"assigned": true,
"usedConstants": [
"Polynomial.ex... | [] | · haveI := Fact.mk hchar; exact expand_contract p hf hprime.ne_zero | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.Div | {
"line": 409,
"column": 50
} | {
"line": 409,
"column": 59
} | {
"line": 409,
"column": 60
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np q : R[X]\nhq : q.Monic\nh✝ : q ∣ p\na✝ : Nontrivial R\nr : R[X]\nhr : p = q * r\nhpq0 : ¬p %ₘ q = 0\nhmod : p %ₘ q = q * (r - p /ₘ q)\nthis : (q * (r - p /ₘ q)).degree < q.degree\nh : (r - p /ₘ q).leadingCoeff = 0\n⊢ (q * 0).leadingCoeff = 0",
"ppTerm": "?m.134",
"... | [
"R : Type u\ninst✝ : Ring R\np q : R[X]\nhq : q.Monic\nh✝ : q ∣ p\na✝ : Nontrivial R\nr : R[X]\nhr : p = q * r\nhpq0 : ¬p %ₘ q = 0\nhmod : p %ₘ q = q * (r - p /ₘ q)\nthis : (q * (r - p /ₘ q)).degree < q.degree\nh : (r - p /ₘ q).leadingCoeff = 0\n⊢ leadingCoeff 0 = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 452,
"column": 6
} | {
"line": 452,
"column": 17
} | {
"line": 452,
"column": 17
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np q : R[X]\nhmo : q.Monic\na✝ : Nontrivial R\n⊢ degree 0 < q.degree",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"Preorder.toLT",
"congrArg",
"id",
"Bot.bot",
... | [
"R : Type u\ninst✝ : Ring R\np q : R[X]\nhmo : q.Monic\na✝ : Nontrivial R\n⊢ ⊥ < q.degree"
] | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 677,
"column": 35
} | {
"line": 677,
"column": 43
} | {
"line": 677,
"column": 44
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np₁ p₂ q : R[X]\nhq : q.Monic\na✝ : Nontrivial R\nf : R[X]\nsub_eq : p₁ - p₂ = q * f\n⊢ p₂ %ₘ q + q * (p₂ /ₘ q + f) = q * f + p₂",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Semigroup.toMul"... | [
"R : Type u\ninst✝ : CommRing R\np₁ p₂ q : R[X]\nhq : q.Monic\na✝ : Nontrivial R\nf : R[X]\nsub_eq : p₁ - p₂ = q * f\n⊢ p₂ %ₘ q + (q * (p₂ /ₘ q) + q * f) = q * f + p₂"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 686,
"column": 16
} | {
"line": 686,
"column": 24
} | {
"line": 686,
"column": 25
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nq p₁ p₂ : R[X]\nhq : q.Monic\nhR : Nontrivial R\n⊢ p₁ %ₘ q + p₂ %ₘ q + q * (p₁ /ₘ q + p₂ /ₘ q) = p₁ + p₂",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Co... | [
"R : Type u\ninst✝ : CommRing R\nq p₁ p₂ : R[X]\nhq : q.Monic\nhR : Nontrivial R\n⊢ p₁ %ₘ q + p₂ %ₘ q + (q * (p₁ /ₘ q) + q * (p₂ /ₘ q)) = p₁ + p₂"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 680,
"column": 2
} | {
"line": 690,
"column": 45
} | {
"line": 692,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nq p₁ p₂ : R[X]\n⊢ (p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Distrib.leftDistribClass",
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"Preorder.toLT",
"L... | [] | by_cases hq : q.Monic
· rcases subsingleton_or_nontrivial R with hR | hR
· simp only [eq_iff_true_of_subsingleton]
· exact
(div_modByMonic_unique (p₁ /ₘ q + p₂ /ₘ q) _ hq
⟨by
rw [mul_add, add_left_comm, add_assoc, modByMonic_add_div, ← add_assoc,
add_comm (q * _), mod... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Div | {
"line": 680,
"column": 2
} | {
"line": 690,
"column": 45
} | {
"line": 692,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nq p₁ p₂ : R[X]\n⊢ (p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Distrib.leftDistribClass",
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"Preorder.toLT",
"L... | [] | by_cases hq : q.Monic
· rcases subsingleton_or_nontrivial R with hR | hR
· simp only [eq_iff_true_of_subsingleton]
· exact
(div_modByMonic_unique (p₁ /ₘ q + p₂ /ₘ q) _ hq
⟨by
rw [mul_add, add_left_comm, add_assoc, modByMonic_add_div, ← add_assoc,
add_comm (q * _), mod... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Div | {
"line": 789,
"column": 6
} | {
"line": 789,
"column": 56
} | {
"line": 790,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\np : R[X]\ninst✝ : IsDomain R\nhi : Irreducible p\nx : R\nhx : p.IsRoot x\ng : R[X]\nhg : p = (X - C x) * g\nthis : IsUnit (X - C x) ∨ IsUnit g\nh : IsUnit (X - C x)\n⊢ p.degree = 1",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Polynomial.C... | [
"R : Type u\ninst✝¹ : CommRing R\np : R[X]\ninst✝ : IsDomain R\nhi : Irreducible p\nx : R\nhx : p.IsRoot x\ng : R[X]\nhg : p = (X - C x) * g\nthis : IsUnit (X - C x) ∨ IsUnit g\nh : IsUnit (X - C x)\nh₁ : (X - C x).degree = 1\n⊢ p.degree = 1"
] | have h₁ : degree (X - C x) = 1 := degree_X_sub_C x | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 170,
"column": 10
} | {
"line": 170,
"column": 97
} | {
"line": 171,
"column": 10
} | [
{
"pp": "case a\nα : 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 : D... | [
"case a\nα : 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 : DecidableEq α... | · exact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm ▸ hpi) (hfx.1 _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.Roots | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 94
} | {
"line": 84,
"column": 0
} | [
{
"pp": "case neg\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nhp0 : ¬p = 0\n⊢ p.roots.card ≤ p.natDegree",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"WithBot.some",
"WithBot",
"Polynomial.roots",
"Polynomial.degr... | [] | exact WithBot.coe_le_coe.1 (le_trans (card_roots hp0) (le_of_eq <| degree_eq_natDegree hp0)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 348,
"column": 10
} | {
"line": 349,
"column": 38
} | {
"line": 350,
"column": 8
} | [
{
"pp": "case neg.calc_1\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\npf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a\na b : α\nane0 : a ≠ 0\nc : α\nhc : ¬IsUnit c\nb_eq : b = a * c\nh : ¬b = 0\ncne0 : c ≠ 0\ncon : Classical.choose ⋯ = 0\n⊢ c ~ᵤ 1",
"ppTerm": "?neg.... | [] | convert! (Classical.choose_spec (pf c cne0)).2.symm
rw [con, Multiset.prod_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 348,
"column": 10
} | {
"line": 349,
"column": 38
} | {
"line": 350,
"column": 8
} | [
{
"pp": "case neg.calc_1\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\npf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a\na b : α\nane0 : a ≠ 0\nc : α\nhc : ¬IsUnit c\nb_eq : b = a * c\nh : ¬b = 0\ncne0 : c ≠ 0\ncon : Classical.choose ⋯ = 0\n⊢ c ~ᵤ 1",
"ppTerm": "?neg.... | [] | convert! (Classical.choose_spec (pf c cne0)).2.symm
rw [con, Multiset.prod_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 379,
"column": 12
} | {
"line": 379,
"column": 21
} | {
"line": 379,
"column": 22
} | [
{
"pp": "R : Type u\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : p / q = 0\nthis : q * 0 + p % q = p\n⊢ p.degree < q.degree",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp",
... | [
"R : Type u\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : p / q = 0\nthis : 0 + p % q = p\n⊢ p.degree < q.degree"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet | {
"line": 270,
"column": 21
} | {
"line": 270,
"column": 81
} | {
"line": 272,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\ns t : Associates α\nd : Associates α\neq : t = s * d\n⊢ s.factors ≤ t.factors",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"Associate... | [] | by rw [eq, factors_mul]; exact le_add_of_nonneg_right bot_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.UniqueFactorization | {
"line": 65,
"column": 8
} | {
"line": 65,
"column": 61
} | {
"line": 67,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : WfDvdMonoid R\nf a : R[X]\nane0 : a ≠ 0\nc : R[X]\nnot_unit_c : ¬IsUnit c\nhac : ¬a * c = 0\ncne0 : c ≠ 0\nhdeg : ¬c.natDegree = 0\n⊢ a.natDegree < a.natDegree + c.natDegree",
"ppTerm": "?neg✝",
"assigned": true... | [] | exact lt_add_of_pos_right _ (Nat.pos_of_ne_zero hdeg) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 41
} | {
"line": 43,
"column": 0
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nh : Subsingleton R\n⊢ ¬IsAlgebraic R a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"is_transcendental_of_subsingleton"
],
"usedFVars": [
"R",
"A",
"inst✝²"... | [] | apply is_transcendental_of_subsingleton | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet | {
"line": 591,
"column": 37
} | {
"line": 591,
"column": 46
} | {
"line": 591,
"column": 47
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : UniqueFactorizationMonoid α\ninst✝¹ : DecidableEq (Associates α)\ninst✝ : (p : Associates α) → Decidable (Irreducible p)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh✝ : a ∣ p ^ n\na✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\neq_zer... | [
"case neg\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : UniqueFactorizationMonoid α\ninst✝¹ : DecidableEq (Associates α)\ninst✝ : (p : Associates α) → Decidable (Irreducible p)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh✝ : a ∣ p ^ n\na✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\neq_zero_of_ne : ∀ ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 665,
"column": 4
} | {
"line": 665,
"column": 36
} | {
"line": 666,
"column": 2
} | [
{
"pp": "case e'_3\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\n⊢ -((aeval x) p.divX / (algebraMap K L) (p.coeff 0)) = (aeval x) p.divX / ((aeval x) p - (algebraMap K L) (p.coeff 0))",
"ppTer... | [] | rw [aeval_eq, zero_sub, div_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 665,
"column": 4
} | {
"line": 665,
"column": 36
} | {
"line": 666,
"column": 2
} | [
{
"pp": "case e'_3\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\n⊢ -((aeval x) p.divX / (algebraMap K L) (p.coeff 0)) = (aeval x) p.divX / ((aeval x) p - (algebraMap K L) (p.coeff 0))",
"ppTer... | [] | rw [aeval_eq, zero_sub, div_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 665,
"column": 4
} | {
"line": 665,
"column": 36
} | {
"line": 666,
"column": 2
} | [
{
"pp": "case e'_3\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\n⊢ -((aeval x) p.divX / (algebraMap K L) (p.coeff 0)) = (aeval x) p.divX / ((aeval x) p - (algebraMap K L) (p.coeff 0))",
"ppTer... | [] | rw [aeval_eq, zero_sub, div_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 691,
"column": 4
} | {
"line": 691,
"column": 35
} | {
"line": 692,
"column": 4
} | [
{
"pp": "case refine_3\nK : 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]\n⊢ ∀ (p : K[X]), p ≠ 0 → (p ≠ 0 → (aeval x) p = 0 → (↑x)⁻¹ ∈ A) → p * X ≠ 0 → (aeval x) (p * X) = 0 → (↑x)⁻¹ ∈ A",
"ppTerm": "?refine_3",
"assigned": true,
... | [
"case refine_3\nK : 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✝ p : K[X]\nhp : p ≠ 0\nih : p ≠ 0 → (aeval x) p = 0 → (↑x)⁻¹ ∈ A\n_ne_zero : p * X ≠ 0\naeval_eq : (aeval x) (p * X) = 0\n⊢ (↑x)⁻¹ ∈ A"
] | intro p hp ih _ne_zero aeval_eq | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Order.DirectedInverseSystem | {
"line": 311,
"column": 54
} | {
"line": 313,
"column": 5
} | {
"line": 315,
"column": 0
} | [
{
"pp": "ι : Type u_6\ninst✝ : LinearOrder ι\nX : ι → Type u_7\ni : ι\nhi : IsSuccPrelimit i\nf : ↑(limit piLTProj i)\nk l : ↑(Iio i)\nhl : ↑l < ↑k\n⊢ (piLTLim hi).symm f l = ↑f k ⟨↑l, hl⟩",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"E... | [] | by
conv_rhs => rw [← (piLTLim hi).right_inv f]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.Basic | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 60
} | {
"line": 87,
"column": 2
} | [
{
"pp": "σ : Type u\nR : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : R →+* S\n⊢ Finsupp.mapRange ⇑f ⋯ p = (map f) p",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"RingHom.instRi... | [
"σ : Type u\nR : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np : MvPolynomial σ R\nf : R →+* S\n⊢ ∑ x ∈ p.support, Finsupp.mapRange ⇑f ⋯ ((monomial x) (coeff x p)) =\n ∑ x ∈ p.support, (map f) ((monomial x) (coeff x p))"
] | rw [p.as_sum, Finsupp.mapRange_finsetSum, map_sum (map f)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Finiteness.Small | {
"line": 30,
"column": 2
} | {
"line": 30,
"column": 29
} | {
"line": 31,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nP Q : Submodule R M\nsmallP : Small.{u, u_2} ↥P\nsmallQ : Small.{u, u_2} ↥Q\n⊢ Small.{u, u_2} ↥(P ⊔ Q)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
... | [
"R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nP Q : Submodule R M\nsmallP : Small.{u, u_2} ↥P\nsmallQ : Small.{u, u_2} ↥Q\n⊢ Small.{u, u_2} ↥(P.subtype.coprod Q.subtype).range"
] | rw [Submodule.sup_eq_range] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.Basic | {
"line": 178,
"column": 57
} | {
"line": 180,
"column": 5
} | {
"line": 182,
"column": 0
} | [
{
"pp": "σ : Type u\nR : Type v\ninst✝ : CommSemiring R\nm : ℕ\np : MvPolynomial σ R\n⊢ p ∈ restrictTotalDegree σ R m ↔ p.totalDegree ≤ m",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Nat.instMulZeroClass",
"Nat.instLattice",
"Lattice... | [] | by
rw [totalDegree, Finset.sup_le_iff]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 126,
"column": 4
} | {
"line": 128,
"column": 32
} | {
"line": 130,
"column": 0
} | [
{
"pp": "case add\nR : Type u_1\ninst✝⁶ : CommSemiring R\nN : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : AddCommMonoid P\ninst✝³ : AddCommMonoid Q\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\ng : N →ₗ[R] P\nhg : Function.Surjective ⇑g\nx y : Q ⊗[R] P\nhx : ∃ a, (l... | [] | obtain ⟨x, rfl⟩ := hx
obtain ⟨y, rfl⟩ := hy
exact ⟨x + y, map_add _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 126,
"column": 4
} | {
"line": 128,
"column": 32
} | {
"line": 130,
"column": 0
} | [
{
"pp": "case add\nR : Type u_1\ninst✝⁶ : CommSemiring R\nN : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : AddCommMonoid P\ninst✝³ : AddCommMonoid Q\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\ng : N →ₗ[R] P\nhg : Function.Surjective ⇑g\nx y : Q ⊗[R] P\nhx : ∃ a, (l... | [] | obtain ⟨x, rfl⟩ := hx
obtain ⟨y, rfl⟩ := hy
exact ⟨x + y, map_add _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 157,
"column": 4
} | {
"line": 159,
"column": 32
} | {
"line": 161,
"column": 0
} | [
{
"pp": "case add\nR : Type u_1\ninst✝⁶ : CommSemiring R\nN : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : AddCommMonoid P\ninst✝³ : AddCommMonoid Q\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\ng : N →ₗ[R] P\nhg : Function.Surjective ⇑g\nx y : P ⊗[R] Q\nhx : ∃ a, (r... | [] | obtain ⟨x, rfl⟩ := hx
obtain ⟨y, rfl⟩ := hy
exact ⟨x + y, map_add _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 157,
"column": 4
} | {
"line": 159,
"column": 32
} | {
"line": 161,
"column": 0
} | [
{
"pp": "case add\nR : Type u_1\ninst✝⁶ : CommSemiring R\nN : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : AddCommMonoid P\ninst✝³ : AddCommMonoid Q\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\ng : N →ₗ[R] P\nhg : Function.Surjective ⇑g\nx y : P ⊗[R] Q\nhx : ∃ a, (r... | [] | obtain ⟨x, rfl⟩ := hx
obtain ⟨y, rfl⟩ := hy
exact ⟨x + y, map_add _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Flat.Basic | {
"line": 160,
"column": 11
} | {
"line": 160,
"column": 34
} | {
"line": 160,
"column": 35
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ Flat R M ↔\n ∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] (N : Submodule R P), Injective ⇑(lTensor M N.subtype)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
... | [
"R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ (∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] (N : Submodule R P),\n Injective ⇑(rTensor M N.subtype)) ↔\n ∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] (N : Submodule R P), Inj... | iff_rTensor_injectiveₛ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 475,
"column": 8
} | {
"line": 476,
"column": 51
} | {
"line": 477,
"column": 6
} | [
{
"pp": "case a.refine_4.zero\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal A\nx✝ : A ⊗[R] B\nhx : x✝ ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I)))\nx : ↥(Subm... | [] | use 0
simp only [map_zero, smul_eq_mul, zero_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 475,
"column": 8
} | {
"line": 476,
"column": 51
} | {
"line": 477,
"column": 6
} | [
{
"pp": "case a.refine_4.zero\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal A\nx✝ : A ⊗[R] B\nhx : x✝ ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I)))\nx : ↥(Subm... | [] | use 0
simp only [map_zero, smul_eq_mul, zero_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Flat.Basic | {
"line": 176,
"column": 18
} | {
"line": 176,
"column": 39
} | {
"line": 176,
"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 : ∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] (N : Submodule R P), Injective ⇑(rTensor M N.subtype)\ni : N →ₗ[R] M\nr : M →ₗ[R]... | [
"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 : ∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] (N : Submodule R P), Injective ⇑(rTensor M N.subtype)\ni : N →ₗ[R] M\nr : M →ₗ[R] N\nh : r ∘ₗ... | lTensor_comp_rTensor, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 209,
"column": 11
} | {
"line": 209,
"column": 34
} | {
"line": 209,
"column": 35
} | [
{
"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)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"... | [
"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⊢ (∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] (N : Submodule R P),\n Injective ⇑(rTensor (⨁ (i : ι), M i) N.subtype)) ↔\n ∀ (i : ι) ⦃P : Type u⦄ [ins... | iff_rTensor_injectiveₛ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 419,
"column": 2
} | {
"line": 422,
"column": 11
} | {
"line": 424,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_3\nB : Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\ninst✝ : Module.Flat R B\nha : Function.Injective ⇑(algebraMap R A)\n⊢ Function.Injective ⇑includeRight",
"ppTerm": "?m.39",
"assigned": true,
... | [] | convert!
Module.Flat.rTensor_preserves_injective_linearMap (M := B) (Algebra.linearMap R A) ha |>.comp
(_root_.TensorProduct.lid R B).symm.injective
ext; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Flat.Basic | {
"line": 419,
"column": 2
} | {
"line": 422,
"column": 11
} | {
"line": 424,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_3\nB : Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\ninst✝ : Module.Flat R B\nha : Function.Injective ⇑(algebraMap R A)\n⊢ Function.Injective ⇑includeRight",
"ppTerm": "?m.39",
"assigned": true,
... | [] | convert!
Module.Flat.rTensor_preserves_injective_linearMap (M := B) (Algebra.linearMap R A) ha |>.comp
(_root_.TensorProduct.lid R B).symm.injective
ext; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 539,
"column": 8
} | {
"line": 540,
"column": 51
} | {
"line": 541,
"column": 6
} | [
{
"pp": "case a.refine_4.zero\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal B\nx✝ : A ⊗[R] B\nhx : x✝ ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)))\nx : A ⊗[R... | [] | use 0
simp only [map_zero, smul_eq_mul, zero_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 539,
"column": 8
} | {
"line": 540,
"column": 51
} | {
"line": 541,
"column": 6
} | [
{
"pp": "case a.refine_4.zero\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal B\nx✝ : A ⊗[R] B\nhx : x✝ ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)))\nx : A ⊗[R... | [] | use 0
simp only [map_zero, smul_eq_mul, zero_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Colimit.DirectLimit | {
"line": 391,
"column": 53
} | {
"line": 391,
"column": 67
} | {
"line": 391,
"column": 68
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝⁷ : 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✝⁶ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)\ninst✝⁵ : (i : ι) → FunLike (H i) (G i) C\ninst✝⁴ : DirectedSystem G f... | [
"R : Type u_1\nι : Type u_2\ninst✝⁷ : 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✝⁶ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)\ninst✝⁵ : (i : ι) → FunLike (H i) (G i) C\ninst✝⁴ : DirectedSystem G fun x1 x2 x3 ... | Nat.cast_succ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 636,
"column": 6
} | {
"line": 636,
"column": 25
} | {
"line": 636,
"column": 25
} | [
{
"pp": "R : Type u_4\nS : Type u_5\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\nA : Type u_6\nB : Type u_7\nC : Type u_8\nD : Type u_9\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Ring B\ninst✝⁹ : Ring C\ninst✝⁸ : Ring D\ninst✝⁷ : Algebra R A\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra R C\ninst✝⁴ : Algebra... | [
"R : Type u_4\nS : Type u_5\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\nA : Type u_6\nB : Type u_7\nC : Type u_8\nD : Type u_9\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Ring B\ninst✝⁹ : Ring C\ninst✝⁸ : Ring D\ninst✝⁷ : Algebra R A\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra R C\ninst✝⁴ : Algebra R D\ninst✝³... | ← RingHom.comap_ker | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Expect | {
"line": 287,
"column": 43
} | {
"line": 287,
"column": 83
} | {
"line": 289,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\nN : Type u_5\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module ℚ≥0 M\ninst✝³ : AddCommMonoid N\ninst✝² : Module ℚ≥0 N\nF : Type u_6\ninst✝¹ : FunLike F M N\ninst✝ : LinearMapClass F ℚ≥0 M N\ng : F\nf : ι → M\ns : Finset ι\n⊢ g (𝔼 i ∈ s, f i) = 𝔼 i ∈ s, g (f i)",
"ppTerm": "?m... | [] | by simp only [expect, map_smul, map_sum] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.EvenFunction | {
"line": 143,
"column": 36
} | {
"line": 143,
"column": 41
} | {
"line": 143,
"column": 42
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝² : AddCommGroup β\ninst✝¹ : IsAddTorsionFree β\nf : α → β\ninst✝ : Neg α\nhe : Function.Even f\nho : Function.Odd f\nr : α\n⊢ -f r = f r",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"N... | [
"α : Type u_3\nβ : Type u_4\ninst✝² : AddCommGroup β\ninst✝¹ : IsAddTorsionFree β\nf : α → β\ninst✝ : Neg α\nhe : Function.Even f\nho : Function.Odd f\nr : α\n⊢ f (-r) = f r"
] | ← ho, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Int.Interval | {
"line": 162,
"column": 28
} | {
"line": 162,
"column": 36
} | {
"line": 162,
"column": 37
} | [
{
"pp": "n a : ℤ\nh : 0 ≤ a\nha : 0 < a\ni : ℤ\nhi₀ : 0 ≤ i\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ n % a + a * (n / a + 1) = n + a",
"ppTerm": "?m.303",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Int.instDiv",
"instHDiv"... | [
"n a : ℤ\nh : 0 ≤ a\nha : 0 < a\ni : ℤ\nhi₀ : 0 ≤ i\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ n % a + (a * (n / a) + a * 1) = n + a"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Group.Finset.Powerset | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 55
} | {
"line": 39,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Finset α\na : α\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\nha : a ∉ s\nf : Finset α → β\n⊢ Disjoint s.powerset (image (insert a) s.powerset)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"False... | [] | · aesop (add simp [disjoint_left, insert_subset_iff]) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 57,
"column": 95
} | {
"line": 58,
"column": 36
} | {
"line": 60,
"column": 0
} | [
{
"pp": "n : ℕ\ns : Multiset ℕ\nh : ∀ x ∈ s, x ≡ 1 [MOD n]\n⊢ s.prod ≡ 1 [MOD n]",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Multiset.map",
"congrArg",
"Multiset.prod",
"Multiset",
"Eq.mp",
"instOfNatNat",
"Nat.ModEq.multisetProd_map_one",
... | [] | by
simpa using multisetProd_map_one h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 150,
"column": 97
} | {
"line": 151,
"column": 36
} | {
"line": 153,
"column": 0
} | [
{
"pp": "n : ℤ\ns : Multiset ℤ\nh : ∀ x ∈ s, x ≡ 1 [ZMOD n]\n⊢ s.prod ≡ 1 [ZMOD n]",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Int.instCommMonoid",
"Multiset.map",
"congrArg",
"Multiset.prod",
"Multiset",
"Eq.mp",
"Int",
"instOfNat",
... | [] | by
simpa using multisetProd_map_one h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Sym.Sym2 | {
"line": 282,
"column": 18
} | {
"line": 284,
"column": 40
} | {
"line": 286,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\na b₁✝ b₁ : α\nh : (fun b ↦ s(a, b)) b₁✝ = (fun b ↦ s(a, b)) b₁\n⊢ b₁✝ = b₁",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Sym2.Rel",
"Sym2.eq._simp_1",
"Sym2.mk",
"congrArg",
"Eq.mp",
"Prod.mk",
... | [] | by
simp only [Sym2.eq, Sym2.rel_iff', Prod.mk.injEq, true_and, Prod.swap_prod_mk] at h
obtain rfl | ⟨rfl, rfl⟩ := h <;> rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Sym.Sym2 | {
"line": 412,
"column": 2
} | {
"line": 414,
"column": 36
} | {
"line": 416,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf g : α → β\ns : Sym2 α\nh : ∀ x ∈ s, f x = g x\ny : β\n⊢ (∃ a ∈ s, f a = y) ↔ ∃ a ∈ s, g a = y",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"congrArg",
"Membership.mem",
"Exists",
"And.casesOn",
"And",
"Exists.cas... | [] | constructor <;>
· rintro ⟨w, hw, rfl⟩
exact ⟨w, hw, by simp [hw, h]⟩ | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Sym.Sym2 | {
"line": 559,
"column": 67
} | {
"line": 560,
"column": 34
} | {
"line": 562,
"column": 0
} | [
{
"pp": "α : Type u_1\n⊢ Set.range diag = diagSet",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"Sym2.Rel",
"Sym2.eq._simp_1",
"Sym2.mk",
"congrArg",
"Quot.ind",
"Membership.mem",
"Exists",
"_private.Mathlib.Data.Sym.Sym2.0.... | [] | by
ext ⟨a, b⟩; simp [diag, eq_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Opposite | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 47
} | {
"line": 111,
"column": 2
} | [
{
"pp": "α : Sort u\nX : Type v\ninst✝ : Small.{u, v} X\n⊢ Small.{u, v} Xᵒᵖ",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"Opposite",
"Small.equiv_small",
"Exists",
"Equiv",
"Exists.casesOn",
"Nonempty.casesOn",
"Nonempty",
"Small"
],
... | [
"α : Sort u\nX : Type v\ninst✝ : Small.{u, v} X\nS : Type u\ne : X ≃ S\n⊢ Small.{u, v} Xᵒᵖ"
] | obtain ⟨S, ⟨e⟩⟩ := Small.equiv_small (α := X) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.IsTensorProduct | {
"line": 353,
"column": 2
} | {
"line": 354,
"column": 35
} | {
"line": 356,
"column": 0
} | [
{
"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 _
convert! hF 1 x <;> rw [one_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.IsTensorProduct | {
"line": 353,
"column": 2
} | {
"line": 354,
"column": 35
} | {
"line": 356,
"column": 0
} | [
{
"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 _
convert! hF 1 x <;> rw [one_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.ObjectProperty.Basic | {
"line": 106,
"column": 39
} | {
"line": 106,
"column": 62
} | {
"line": 106,
"column": 62
} | [
{
"pp": "C : Type u\ninst✝ : CategoryStruct.{v, u} C\nι : Type u'\nX : ι → C\nP : ObjectProperty C\nh : ∀ (i : ι), P (X i)\n⊢ ofObj X ≤ P",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CategoryTheory.ObjectProperty.ofObj.casesOn",
"CategoryTheory.ObjectProperty.ofObj",
... | [] | rintro _ ⟨i⟩; exact h i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.ObjectProperty.Basic | {
"line": 106,
"column": 39
} | {
"line": 106,
"column": 62
} | {
"line": 106,
"column": 62
} | [
{
"pp": "C : Type u\ninst✝ : CategoryStruct.{v, u} C\nι : Type u'\nX : ι → C\nP : ObjectProperty C\nh : ∀ (i : ι), P (X i)\n⊢ ofObj X ≤ P",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CategoryTheory.ObjectProperty.ofObj.casesOn",
"CategoryTheory.ObjectProperty.ofObj",
... | [] | rintro _ ⟨i⟩; exact h i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.IsTensorProduct | {
"line": 708,
"column": 62
} | {
"line": 709,
"column": 34
} | {
"line": 711,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type v₃\ninst✝¹³ : CommSemiring R\ninst✝¹² : CommSemiring S\ninst✝¹¹ : Algebra R S\nR' : Type u_6\nS' : Type u_7\ninst✝¹⁰ : CommSemiring R'\ninst✝⁹ : CommSemiring S'\ninst✝⁸ : Algebra R R'\ninst✝⁷ : Algebra S S'\ninst✝⁶ : Algebra R' S'\ninst✝⁵ : Algebra R S'\ninst✝⁴ : IsScalarTower R ... | [] | by
simp [Algebra.pushoutDesc_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.ConcreteCategory.Basic | {
"line": 159,
"column": 2
} | {
"line": 160,
"column": 36
} | {
"line": 162,
"column": 0
} | [
{
"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 Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ ⇑(hom (f ≫ g)) = ⇑(hom g) ∘ ⇑(hom f)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants":... | [] | ext
simp [ConcreteCategory.comp_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.ConcreteCategory.Basic | {
"line": 159,
"column": 2
} | {
"line": 160,
"column": 36
} | {
"line": 162,
"column": 0
} | [
{
"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 Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ ⇑(hom (f ≫ g)) = ⇑(hom g) ∘ ⇑(hom f)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants":... | [] | ext
simp [ConcreteCategory.comp_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Lemmas | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 19
} | {
"line": 38,
"column": 0
} | [
{
"pp": "α : Sort u_1\np q : Prop\ninst✝¹ : Decidable p\ninst✝ : Decidable q\na : p → α\nb : ¬p → q → α\nc : ¬p → ¬q → α\n⊢ (dite p a fun hp ↦ dite q (b hp) (c hp)) = if hq : q then dite p a fun hp ↦ b hp hq else dite p a fun hp ↦ c hp hq",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
... | [] | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Logic.Lemmas | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 19
} | {
"line": 38,
"column": 0
} | [
{
"pp": "α : Sort u_1\np q : Prop\ninst✝¹ : Decidable p\ninst✝ : Decidable q\na : p → α\nb : ¬p → q → α\nc : ¬p → ¬q → α\n⊢ (dite p a fun hp ↦ dite q (b hp) (c hp)) = if hq : q then dite p a fun hp ↦ b hp hq else dite p a fun hp ↦ c hp hq",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
... | [] | split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Lemmas | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 19
} | {
"line": 38,
"column": 0
} | [
{
"pp": "α : Sort u_1\np q : Prop\ninst✝¹ : Decidable p\ninst✝ : Decidable q\na : p → α\nb : ¬p → q → α\nc : ¬p → ¬q → α\n⊢ (dite p a fun hp ↦ dite q (b hp) (c hp)) = if hq : q then dite p a fun hp ↦ b hp hq else dite p a fun hp ↦ c hp hq",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
... | [] | split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Lemmas | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 19
} | {
"line": 43,
"column": 0
} | [
{
"pp": "α : Sort u_1\np q : Prop\ninst✝¹ : Decidable p\ninst✝ : Decidable q\na : p → q → α\nb : p → ¬q → α\nc : ¬p → α\n⊢ dite p (fun hp ↦ dite q (a hp) (b hp)) c = if hq : q then dite p (fun hp ↦ a hp hq) c else dite p (fun hp ↦ b hp hq) c",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
... | [] | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Logic.Lemmas | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 19
} | {
"line": 43,
"column": 0
} | [
{
"pp": "α : Sort u_1\np q : Prop\ninst✝¹ : Decidable p\ninst✝ : Decidable q\na : p → q → α\nb : p → ¬q → α\nc : ¬p → α\n⊢ dite p (fun hp ↦ dite q (a hp) (b hp)) c = if hq : q then dite p (fun hp ↦ a hp hq) c else dite p (fun hp ↦ b hp hq) c",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
... | [] | split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Lemmas | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 19
} | {
"line": 43,
"column": 0
} | [
{
"pp": "α : Sort u_1\np q : Prop\ninst✝¹ : Decidable p\ninst✝ : Decidable q\na : p → q → α\nb : p → ¬q → α\nc : ¬p → α\n⊢ dite p (fun hp ↦ dite q (a hp) (b hp)) c = if hq : q then dite p (fun hp ↦ a hp hq) c else dite p (fun hp ↦ b hp hq) c",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
... | [] | split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Quiver.Path | {
"line": 211,
"column": 30
} | {
"line": 211,
"column": 51
} | {
"line": 213,
"column": 0
} | [
{
"pp": "V : Type u\ninst✝ : Quiver V\na b : V\np : Path a b\nx✝ d : V\nq : Path b d\na✝ : d ⟶ x✝\n⊢ (p.comp (q.cons a✝)).toList = (q.cons a✝).toList ++ p.toList",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"congrArg",
"Quiver.Path.toList",
"Quiver.Path.toList.eq_2",
... | [] | by simp [toList_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Equivalence | {
"line": 509,
"column": 32
} | {
"line": 509,
"column": 76
} | {
"line": 511,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\ne : C ≌ D\nW X X' Y Y' Z : D\nf : W ⟶ X\ng : X ⟶ Y\nh : Y ⟶ Z\nf' : W ⟶ X'\ng' : X' ⟶ Y'\nh' : Y' ⟶ Z\n⊢ f ≫ g ≫ h ≫ e.counitInv.app Z = f' ≫ g' ≫ h' ≫ e.counitInv.app Z ↔ f ≫ g ≫ h = f' ≫ g' ≫ h'",
"ppTerm": "?m.8... | [] | by simp only [← Category.assoc, cancel_mono] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Cones | {
"line": 708,
"column": 14
} | {
"line": 708,
"column": 36
} | {
"line": 708,
"column": 37
} | [
{
"pp": "J : Type u₁\ninst✝⁴ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝³ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝² : Category.{v₃, u₃} C\nD : Type u₄\ninst✝¹ : Category.{v₄, u₄} D\nE : Type u₅\ninst✝ : Category.{v₅, u₅} E\nF : J ⥤ C\nX Y : Cocone F\nf : X ⟶ Y\nj : Jᵒᵖ\n⊢ X.ι.app (unop j) ≫ f.hom = Y.ι.app (unop... | [] | apply CoconeMorphism.w | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Retract | {
"line": 158,
"column": 53
} | {
"line": 158,
"column": 74
} | {
"line": 158,
"column": 74
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ Z✝ W✝ : C\nf✝ : X✝ ⟶ Y✝\ng✝ : Z✝ ⟶ W✝\nh✝ : RetractArrow f✝ g✝\nX Y Z W : Cᵒᵖ\nf : X ⟶ Y\ng : Z ⟶ W\nh : RetractArrow f g\n⊢ (Arrow.Hom.right h.r).unop ≫ (Arrow.mk g.unop).hom = (Arrow.mk f.unop).hom ≫ (Arrow.Hom.le... | [] | by simp [← unop_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Retract | {
"line": 159,
"column": 53
} | {
"line": 159,
"column": 74
} | {
"line": 159,
"column": 74
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ Z✝ W✝ : C\nf✝ : X✝ ⟶ Y✝\ng✝ : Z✝ ⟶ W✝\nh✝ : RetractArrow f✝ g✝\nX Y Z W : Cᵒᵖ\nf : X ⟶ Y\ng : Z ⟶ W\nh : RetractArrow f g\n⊢ (Arrow.Hom.right h.i).unop ≫ (Arrow.mk f.unop).hom = (Arrow.mk g.unop).hom ≫ (Arrow.Hom.le... | [] | by simp [← unop_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.HasLimits | {
"line": 149,
"column": 27
} | {
"line": 149,
"column": 43
} | {
"line": 149,
"column": 43
} | [
{
"pp": "J : Type u₁\ninst✝³ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝² : Category.{v₂, u₂} K\nC : Type u\ninst✝¹ : Category.{v, u} C\nF✝ F : J ⥤ C\ninst✝ : HasLimit F\nj j' : J\nhj : j = j'\n⊢ F.obj j = F.obj j'",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.ndrec",
"Eq.r... | [] | by subst hj; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.IsTerminal | {
"line": 426,
"column": 8
} | {
"line": 427,
"column": 45
} | {
"line": 427,
"column": 46
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : Type u\ninst✝¹ : Category.{v, u} J\nX : J\nhX : IsInitial X\nF : J ⥤ C\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\ni j : J\nf : i ⟶ j\n⊢ F.map f ≫ inv (F.map (hX.to j)) = inv (F.map (hX.to i)) ≫ 𝟙 (F.obj X)",
"ppTerm": "?m.60",
"assigned": ... | [] | simp only [IsIso.eq_inv_comp, IsIso.comp_inv_eq, Category.comp_id, ← F.map_comp,
hX.hom_ext (hX.to i ≫ f) (hX.to j)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.HasLimits | {
"line": 713,
"column": 13
} | {
"line": 713,
"column": 29
} | {
"line": 713,
"column": 29
} | [
{
"pp": "J : Type u₁\ninst✝³ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝² : Category.{v₂, u₂} K\nC : Type u\ninst✝¹ : Category.{v, u} C\nF✝ F : J ⥤ C\ninst✝ : HasColimit F\nj j' : J\nhj : j = j'\n⊢ F.obj j' = F.obj j",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.ndrec",
"Eq... | [] | by subst hj; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.HasLimits | {
"line": 1029,
"column": 82
} | {
"line": 1033,
"column": 41
} | {
"line": 1035,
"column": 0
} | [
{
"pp": "J : Type u₁\ninst✝⁵ : Category.{v₁, u₁} J\nC : Type u\ninst✝⁴ : Category.{v, u} C\nF : J ⥤ C\nD : Type u'\ninst✝³ : Category.{v', u'} D\ninst✝² : HasColimit F\nG : C ⥤ D\ninst✝¹ : HasColimit (F ⋙ G)\nE : Type u''\ninst✝ : Category.{v'', u''} E\nH : D ⥤ E\nh : HasColimit ((F ⋙ G) ⋙ H)\n⊢ post (F ⋙ G) H ... | [] | by
ext j
rw [← assoc, colimit.ι_post, ← H.map_comp, colimit.ι_post]
haveI : HasColimit (F ⋙ G ⋙ H) := h
exact (colimit.ι_post F (G ⋙ H) j).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Cospan | {
"line": 232,
"column": 24
} | {
"line": 232,
"column": 57
} | {
"line": 232,
"column": 57
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : WalkingCospan ⥤ C\nj : WalkingCospan\n⊢ F.obj j = (cospan (F.map inl) (F.map inr)).obj j",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.WalkingCospan.Hom.inl",
"CategoryTheory.Limits.WalkingCospan.right... | [] | rcases j with (⟨⟩ | ⟨⟨⟩⟩) <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Cospan | {
"line": 232,
"column": 24
} | {
"line": 232,
"column": 57
} | {
"line": 232,
"column": 57
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : WalkingCospan ⥤ C\nj : WalkingCospan\n⊢ F.obj j = (cospan (F.map inl) (F.map inr)).obj j",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.WalkingCospan.Hom.inl",
"CategoryTheory.Limits.WalkingCospan.right... | [] | rcases j with (⟨⟩ | ⟨⟨⟩⟩) <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Cospan | {
"line": 232,
"column": 24
} | {
"line": 232,
"column": 57
} | {
"line": 232,
"column": 57
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : WalkingCospan ⥤ C\nj : WalkingCospan\n⊢ F.obj j = (cospan (F.map inl) (F.map inr)).obj j",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.WalkingCospan.Hom.inl",
"CategoryTheory.Limits.WalkingCospan.right... | [] | rcases j with (⟨⟩ | ⟨⟨⟩⟩) <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Cospan | {
"line": 239,
"column": 24
} | {
"line": 239,
"column": 57
} | {
"line": 239,
"column": 57
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : WalkingSpan ⥤ C\nj : WalkingSpan\n⊢ F.obj j = (span (F.map fst) (F.map snd)).obj j",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.WalkingSpan",
"Option.casesOn",
"CategoryTheory.Limits.WalkingSpan... | [] | rcases j with (⟨⟩ | ⟨⟨⟩⟩) <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Cospan | {
"line": 239,
"column": 24
} | {
"line": 239,
"column": 57
} | {
"line": 239,
"column": 57
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : WalkingSpan ⥤ C\nj : WalkingSpan\n⊢ F.obj j = (span (F.map fst) (F.map snd)).obj j",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.WalkingSpan",
"Option.casesOn",
"CategoryTheory.Limits.WalkingSpan... | [] | rcases j with (⟨⟩ | ⟨⟨⟩⟩) <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Cospan | {
"line": 239,
"column": 24
} | {
"line": 239,
"column": 57
} | {
"line": 239,
"column": 57
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : WalkingSpan ⥤ C\nj : WalkingSpan\n⊢ F.obj j = (span (F.map fst) (F.map snd)).obj j",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.WalkingSpan",
"Option.casesOn",
"CategoryTheory.Limits.WalkingSpan... | [] | rcases j with (⟨⟩ | ⟨⟨⟩⟩) <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 190,
"column": 2
} | {
"line": 196,
"column": 28
} | {
"line": 198,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\n⊢ IsZero X ↔ f = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.retraction.congr_simp",
"CategoryTheory.CategoryStruct.toQ... | [] | rw [iff_id_eq_zero]
constructor
· intro h
rw [← Category.id_comp f, h, zero_comp]
· intro h
rw [← IsSplitMono.id f]
simp only [h, zero_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 190,
"column": 2
} | {
"line": 196,
"column": 28
} | {
"line": 198,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\n⊢ IsZero X ↔ f = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.retraction.congr_simp",
"CategoryTheory.CategoryStruct.toQ... | [] | rw [iff_id_eq_zero]
constructor
· intro h
rw [← Category.id_comp f, h, zero_comp]
· intro h
rw [← IsSplitMono.id f]
simp only [h, zero_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.ExactFunctor | {
"line": 190,
"column": 9
} | {
"line": 190,
"column": 65
} | {
"line": 190,
"column": 65
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝¹ : PreservesFiniteLimits F\ninst✝ : PreservesFiniteColimits F\n⊢ exactFunctor C D F",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.P... | [] | simp only [exactFunctor_iff]; constructor <;> assumption | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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