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.Algebra.Module.ZLattice.Summable | {
"line": 144,
"column": 8
} | {
"line": 144,
"column": 13
} | {
"line": 145,
"column": 4
} | [
{
"pp": "case e_f\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\nι : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι ℤ ↥L\nn : ℕ\nr : ℝ\ns : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset f... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 147,
"column": 66
} | {
"line": 147,
"column": 81
} | {
"line": 147,
"column": 81
} | [
{
"pp": "case e_a.e_f\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\nι : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι ℤ ↥L\nn : ℕ\nr : ℝ\ns : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFins... | [
"case e_a.e_f\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\nι : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι ℤ ↥L\nn : ℕ\nr : ℝ\ns : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ I... | Nat.cast_sub hd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 19
} | {
"line": 167,
"column": 19
} | [
{
"pp": "α : Type u_1\nf : α → ℕ\na b : α\ninst✝ : DecidableEq α\nh : a ≠ b\n⊢ ((f a).succ + (f b).succ).choose (f a).succ =\n ((f a).succ + if b = a then (f a).succ else f b).choose (f a).succ + (f a + (f b).succ).choose (f a)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"α : Type u_1\nf : α → ℕ\na b : α\ninst✝ : DecidableEq α\nh : a ≠ b\n⊢ ((f a).succ + (f b).succ).choose (f a).succ = ((f a).succ + f b).choose (f a).succ + (f a + (f b).succ).choose (f a)"
] | if_neg h.symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Funext | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 71
} | {
"line": 51,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nih :\n ∀ {p : MvPolynomial (Fin n) R} (s : Fin n → Set R),\n (∀ (i : Fin n), (s i).Infinite) → (∀ x ∈ Set.univ.pi s, (eval x) p = 0) → p = 0\np : MvPolynomial (Fin (n + 1)) R\ns : Fin (n + 1) → Set R\nhs : ∀ (i : Fin (n + 1)),... | [] | exact h _ fun i _ ↦ i.cases (by simpa using hr) (by simpa using hx) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 401,
"column": 26
} | {
"line": 401,
"column": 61
} | {
"line": 402,
"column": 4
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : DecidableEq α\nm : Fin (n + 1)\ns : Sym α (n - ↑m)\nx : α\nhx : x ∉ ↑s\nj : α\nhj : j ∈ Multiset.replicate (↑m) x\n⊢ ¬x ≠ j",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"Membership.mem",
"Multiset",
... | [] | simp [Multiset.mem_replicate.mp hj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 401,
"column": 26
} | {
"line": 401,
"column": 61
} | {
"line": 402,
"column": 4
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : DecidableEq α\nm : Fin (n + 1)\ns : Sym α (n - ↑m)\nx : α\nhx : x ∉ ↑s\nj : α\nhj : j ∈ Multiset.replicate (↑m) x\n⊢ ¬x ≠ j",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"Membership.mem",
"Multiset",
... | [] | simp [Multiset.mem_replicate.mp hj] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 401,
"column": 26
} | {
"line": 401,
"column": 61
} | {
"line": 402,
"column": 4
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : DecidableEq α\nm : Fin (n + 1)\ns : Sym α (n - ↑m)\nx : α\nhx : x ∉ ↑s\nj : α\nhj : j ∈ Multiset.replicate (↑m) x\n⊢ ¬x ≠ j",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"Membership.mem",
"Multiset",
... | [] | simp [Multiset.mem_replicate.mp hj] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 191,
"column": 6
} | {
"line": 191,
"column": 30
} | {
"line": 193,
"column": 0
} | [
{
"pp": "case right\nR : Type u_1\nσ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\np : MvPolynomial σ R\ni : σ\nr : R\nhr : r ≠ 0\nb : R\nhb : b ∣ r\nhp : p = b • (monomial (Finsupp.single i 1)) 1\n⊢ p = (monomial (Finsupp.single i 1)) b",
"ppTerm": "?right",
"assigned": true,
"usedCons... | [] | simp [hp, smul_monomial] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 35
} | {
"line": 201,
"column": 0
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\np : MvPolynomial σ R\ni : σ\na✝ : Nontrivial R\nr : R\n⊢ r ∣ 1 ∧ (p = C r ∨ p = r • X i) ↔ IsUnit r ∧ (p = C r ∨ p = r • 1 • X i)",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroCl... | [] | rw [isUnit_iff_dvd_one, one_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.NonAssoc.LieAdmissible.Defs | {
"line": 141,
"column": 15
} | {
"line": 143,
"column": 8
} | {
"line": 145,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Ring L\n⊢ ∀ (x y z : L),\n associator x y z + associator z x y + associator y z x = associator y x z + associator z y x + associator x z y",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddSemigroup",
... | [] | by
suffices ∀ a b c : L, associator a b c = 0 by simp
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.UniqueFactorizationDomain.Nat | {
"line": 40,
"column": 4
} | {
"line": 40,
"column": 12
} | {
"line": 42,
"column": 0
} | [
{
"pp": "case succ.succ\na n✝ : ℕ\nh : DvdNotUnit (a + 1) (n✝ + 1)\nh1 : a + 1 ∣ n✝ + 1\nh2 : ¬n✝ + 1 ∣ a + 1\ncon : a + 1 = n✝ + 1\n⊢ n✝ + 1 ∣ a + 1",
"ppTerm": "?succ.succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"Nat.instMonoid",
"semig... | [] | rw [con] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 384,
"column": 53
} | {
"line": 402,
"column": 12
} | {
"line": 404,
"column": 0
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf g : MvPolynomial σ R\n⊢ m.toSyn (m.degree (f * g)) ≤ m.toSyn (m.degree f + m.degree g)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Finsupp.instHasAntidiagonal",
"Finsupp.instAddZeroClass",
... | [] | by
classical
rw [degree_le_iff]
intro c
rw [← not_lt, mem_support_iff, not_imp_not]
intro hc
rw [coeff_mul]
apply Finset.sum_eq_zero
rintro ⟨d, e⟩ hde
simp only [Finset.mem_antidiagonal] at hde
dsimp only
by_cases hd : m.degree f ≺[m] d
· rw [m.coeff_eq_zero_of_lt hd, zero_mul]
· suffices m.de... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 545,
"column": 61
} | {
"line": 550,
"column": 87
} | {
"line": 552,
"column": 0
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf : MvPolynomial σ R\nn : ℕ\n⊢ coeff (n • m.degree f) (f ^ n) = m.leadingCoeff f ^ n",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mul... | [] | by
induction n with
| zero => simp
| succ n hrec =>
simp only [add_smul, one_smul, pow_add, pow_one]
rw [m.coeff_mul_of_add_of_degree_le (m.degree_pow_le _) le_rfl, hrec, leadingCoeff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 757,
"column": 51
} | {
"line": 759,
"column": 63
} | {
"line": 761,
"column": 0
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\ns : σ →₀ ℕ\nc : R\n⊢ m.leadingTerm ((monomial s) c) = (monomial s) c",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MonomialOrder.degree_monomial",
"Nat.instMulZeroClass",
"AddMonoidAlgebr... | [] | by
classical
by_cases h : c = 0 <;> simp [leadingTerm, degree_monomial, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 880,
"column": 81
} | {
"line": 881,
"column": 42
} | {
"line": 882,
"column": 2
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf g : MvPolynomial σ R\nthis :\n ∀ (f : MvPolynomial σ ?m.74) (g : MvPolynomial σ (?m.96 f)),\n m.degree g - m.degree f = m.degree f ⊔ m.degree g - m.degree f\n⊢ m.sPolynomial f g =\n (monomial (m.degree f ⊔ m.degree g - m.degr... | [] | by
rw [sPolynomial, this, this, sup_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Factorization.PrimePow | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 33
} | {
"line": 58,
"column": 34
} | [
{
"pp": "n : ℕ\n⊢ (∃ p k, 0 < k ∧ n.factorization = Finsupp.single p k) ↔ n.factorization.support.card = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"_private.Mathlib.Data.Nat.Factorization.PrimePow.0.isPrimePow_iff... | [
"n : ℕ\n⊢ (∃ p k, 0 < k ∧ n.factorization = Finsupp.single p k) ↔ ∃ a b, b ≠ 0 ∧ n.factorization = Finsupp.single a b"
] | Finsupp.card_support_eq_one', | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 9
} | {
"line": 125,
"column": 10
} | [
{
"pp": "k n : ℕ\n⊢ (pow k) n = if k = 0 ∧ n = 0 then 0 else n ^ k",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"Nat.instMonoid",
"instOfNatNat",
"Monoid.toPow",
"Nat.casesAuxOn",
"Ari... | [
"case zero\nn : ℕ\n⊢ (pow 0) n = if 0 = 0 ∧ n = 0 then 0 else n ^ 0",
"case succ\nn n✝ : ℕ\n⊢ (pow (n✝ + 1)) n = if n✝ + 1 = 0 ∧ n = 0 then 0 else n ^ (n✝ + 1)"
] | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 308,
"column": 4
} | {
"line": 308,
"column": 11
} | {
"line": 308,
"column": 12
} | [
{
"pp": "case pos\nm k : ℕ\nhm : m = 0\n⊢ Ω (m ^ k) = k * Ω m",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"HMul.hMul",
"ArithmeticFunction.instFunLikeNat",
"Nat.instMonoid",
"ArithmeticFunction.cardFactors",
"instMulNat",
... | [
"case pos.zero\nm : ℕ\nhm : m = 0\n⊢ Ω (m ^ 0) = 0 * Ω m",
"case pos.succ\nm : ℕ\nhm : m = 0\nn✝ : ℕ\n⊢ Ω (m ^ (n✝ + 1)) = (n✝ + 1) * Ω m"
] | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Algebra.Order.Antidiag.Nat | {
"line": 257,
"column": 2
} | {
"line": 257,
"column": 17
} | {
"line": 258,
"column": 2
} | [
{
"pp": "case refine_2\nn : ℕ\nhn : Squarefree n\na : Fin 3 → ℕ\nha : a ∈ finMulAntidiag 3 n\n⊢ (a 0 * a 1, a 0 * a 2).2 ∣ a 0 * a 1 * a 2",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMo... | [
"case refine_3\nn : ℕ\nhn : Squarefree n\na : Fin 3 → ℕ\nha : a ∈ finMulAntidiag 3 n\n⊢ match (a 0 * a 1, a 0 * a 2) with\n | (x, y) => x.lcm y = a 0 * a 1 * a 2"
] | · use a 1; ring | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.Archimedean.Hom | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 19
} | {
"line": 63,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : Field α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : Archimedean α\nf : α →+*o α\nx : α\n⊢ f x = x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"OrderRingHom.eq_id",
"congrArg",
"PartialOrder.toPreorde... | [] | rw [f.eq_id]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Archimedean.Hom | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 19
} | {
"line": 63,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : Field α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : Archimedean α\nf : α →+*o α\nx : α\n⊢ f x = x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"OrderRingHom.eq_id",
"congrArg",
"PartialOrder.toPreorde... | [] | rw [f.eq_id]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Archimedean.Real.Hom | {
"line": 36,
"column": 31
} | {
"line": 36,
"column": 43
} | {
"line": 36,
"column": 43
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedAddMonoid R\ninst✝³ : Ring S\ninst✝² : LinearOrder S\ninst✝¹ : IsOrderedAddMonoid S\ninst✝ : PosMulMono S\nhR : ∀ (r : R), 0 ≤ r → IsSquare r\nf : R →+* S\ns : R\nh : 0 ≤ s * s\n⊢ 0 ≤ f (s * s)",
"ppTerm": "?m.5... | [
"R : Type u_1\nS : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedAddMonoid R\ninst✝³ : Ring S\ninst✝² : LinearOrder S\ninst✝¹ : IsOrderedAddMonoid S\ninst✝ : PosMulMono S\nhR : ∀ (r : R), 0 ≤ r → IsSquare r\nf : R →+* S\ns : R\nh : 0 ≤ s * s\n⊢ 0 ≤ f s * f s"
] | rw [map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 530,
"column": 8
} | {
"line": 530,
"column": 24
} | {
"line": 530,
"column": 25
} | [
{
"pp": "case h\nR : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : ℕ\ncop : m.Coprime n\nb1 b2 : ℕ\nh : (b1, b2).1 * (b1, b2).2 = m * n ∧ m * n ≠ 0\n⊢ ((((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).1.1 * ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gc... | [
"case h\nR : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : ℕ\ncop : m.Coprime n\nb1 b2 : ℕ\nh : (b1, b2).1 * (b1, b2).2 = m * n ∧ m * n ≠ 0\n⊢ ((((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).1.1 * ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).1.2 = m... | ← cop.gcd_mul _, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 530,
"column": 25
} | {
"line": 530,
"column": 41
} | {
"line": 530,
"column": 42
} | [
{
"pp": "case h\nR : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : ℕ\ncop : m.Coprime n\nb1 b2 : ℕ\nh : (b1, b2).1 * (b1, b2).2 = m * n ∧ m * n ≠ 0\n⊢ ((((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).1.1 * ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gc... | [
"case h\nR : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : ℕ\ncop : m.Coprime n\nb1 b2 : ℕ\nh : (b1, b2).1 * (b1, b2).2 = m * n ∧ m * n ≠ 0\n⊢ ((((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).1.1 * ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).1.2 = m... | ← cop.gcd_mul _, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 583,
"column": 51
} | {
"line": 583,
"column": 64
} | {
"line": 583,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nf : ArithmeticFunction R\nhf : f.IsMultiplicative\nx y : ℕ\nhx : ¬x = 0\nhy : ¬y = 0\nhgcd_ne_zero : x.gcd y ≠ 0\nhlcm_ne_zero : x.lcm y ≠ 0\nhfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1\n⊢ x.lcm y ≠ 0",
"ppTerm": "?m.95",
"assigned": true,
"usedConstants"... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 583,
"column": 51
} | {
"line": 583,
"column": 64
} | {
"line": 583,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nf : ArithmeticFunction R\nhf : f.IsMultiplicative\nx y : ℕ\nhx : ¬x = 0\nhy : ¬y = 0\nhgcd_ne_zero : x.gcd y ≠ 0\nhlcm_ne_zero : x.lcm y ≠ 0\nhfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1\n⊢ x.gcd y ≠ 0",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 583,
"column": 51
} | {
"line": 583,
"column": 64
} | {
"line": 583,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nf : ArithmeticFunction R\nhf : f.IsMultiplicative\nx y : ℕ\nhx : ¬x = 0\nhy : ¬y = 0\nhgcd_ne_zero : x.gcd y ≠ 0\nhlcm_ne_zero : x.lcm y ≠ 0\nhfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1\n⊢ x ≠ 0",
"ppTerm": "?m.137",
"assigned": true,
"usedConstants": [],... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 583,
"column": 51
} | {
"line": 583,
"column": 64
} | {
"line": 583,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nf : ArithmeticFunction R\nhf : f.IsMultiplicative\nx y : ℕ\nhx : ¬x = 0\nhy : ¬y = 0\nhgcd_ne_zero : x.gcd y ≠ 0\nhlcm_ne_zero : x.lcm y ≠ 0\nhfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1\n⊢ y ≠ 0",
"ppTerm": "?m.158",
"assigned": true,
"usedConstants": [],... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Interval.Multiset | {
"line": 30,
"column": 16
} | {
"line": 30,
"column": 20
} | {
"line": 30,
"column": 21
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : AddCommMonoid α\ninst✝³ : PartialOrder α\ninst✝² : IsOrderedCancelAddMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map (fun x ↦ c + x) (Ico a b) = Ico (c + a) (c + b)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.... | [
"α : Type u_1\ninst✝⁴ : AddCommMonoid α\ninst✝³ : PartialOrder α\ninst✝² : IsOrderedCancelAddMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map (fun x ↦ c + x) (Finset.Ico a b).val = Ico (c + a) (c + b)"
] | Ico, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Interval.Multiset | {
"line": 30,
"column": 21
} | {
"line": 30,
"column": 25
} | {
"line": 30,
"column": 26
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : AddCommMonoid α\ninst✝³ : PartialOrder α\ninst✝² : IsOrderedCancelAddMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map (fun x ↦ c + x) (Finset.Ico a b).val = Ico (c + a) (c + b)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [... | [
"α : Type u_1\ninst✝⁴ : AddCommMonoid α\ninst✝³ : PartialOrder α\ninst✝² : IsOrderedCancelAddMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map (fun x ↦ c + x) (Finset.Ico a b).val = (Finset.Ico (c + a) (c + b)).val"
] | Ico, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Interval.Basic | {
"line": 476,
"column": 93
} | {
"line": 484,
"column": 35
} | {
"line": 486,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\ns t : NonemptyInterval α\n⊢ s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"instIsRightCancelMulOfMulRightReflectLE",
"IsLeft... | [] | by
refine ⟨fun h => ?_, ?_⟩
· rw [NonemptyInterval.ext_iff, Prod.ext_iff] at h
have := (mul_le_mul_iff_of_ge s.fst_le_snd t.fst_le_snd).1 (h.2.trans h.1.symm).le
refine ⟨s.fst, t.fst, ?_, ?_, h.1⟩ <;> apply NonemptyInterval.ext <;> dsimp [pure]
· nth_rw 2 [this.1]
· nth_rw 2 [this.2]
· rintro ⟨b, ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 414,
"column": 4
} | {
"line": 414,
"column": 51
} | {
"line": 416,
"column": 0
} | [
{
"pp": "case neg\nΓ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : Zero Γ\nx : R⟦Γ⟧\nh : ¬x = 0\n⊢ 0 ≤ x.orderTop ↔ 0 ≤ x.order",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"Iff.mpr",
"HahnSeries.order",
"WithTo... | [] | simp_all [order_of_ne h, orderTop_of_ne_zero h] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 414,
"column": 4
} | {
"line": 414,
"column": 51
} | {
"line": 416,
"column": 0
} | [
{
"pp": "case neg\nΓ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : Zero Γ\nx : R⟦Γ⟧\nh : ¬x = 0\n⊢ 0 ≤ x.orderTop ↔ 0 ≤ x.order",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"Iff.mpr",
"HahnSeries.order",
"WithTo... | [] | simp_all [order_of_ne h, orderTop_of_ne_zero h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 414,
"column": 4
} | {
"line": 414,
"column": 51
} | {
"line": 416,
"column": 0
} | [
{
"pp": "case neg\nΓ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : Zero Γ\nx : R⟦Γ⟧\nh : ¬x = 0\n⊢ 0 ≤ x.orderTop ↔ 0 ≤ x.order",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"Iff.mpr",
"HahnSeries.order",
"WithTo... | [] | simp_all [order_of_ne h, orderTop_of_ne_zero h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder | {
"line": 50,
"column": 6
} | {
"line": 50,
"column": 21
} | {
"line": 51,
"column": 6
} | [
{
"pp": "case mp\nM : Type u_1\ninst✝⁴ : CancelCommMonoid M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedMonoid M\ninst✝¹ : LocallyFiniteOrder M\ninst✝ : ExistsMulOfLE M\na b c x : M\n⊢ a * c ≤ x ∧ x < b * c → ∃ a_2, (a ≤ a_2 ∧ a_2 < b) ∧ a_2 * c = x",
"ppTerm": "?mp",
"assigned": true,
"usedConstants... | [
"case mp\nM : Type u_1\ninst✝⁴ : CancelCommMonoid M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedMonoid M\ninst✝¹ : LocallyFiniteOrder M\ninst✝ : ExistsMulOfLE M\na b c x : M\nh₁ : a * c ≤ x\nh₂ : x < b * c\n⊢ ∃ a_1, (a ≤ a_1 ∧ a_1 < b) ∧ a_1 * c = x"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder | {
"line": 52,
"column": 43
} | {
"line": 52,
"column": 80
} | {
"line": 52,
"column": 80
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : CancelCommMonoid M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedMonoid M\ninst✝¹ : LocallyFiniteOrder M\ninst✝ : ExistsMulOfLE M\na b c d : M\nh₁ : a * c ≤ a * c * d\nh₂ : a * c * d < b * c\n⊢ a * d < b",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"i... | [] | simpa [mul_right_comm a c d] using h₂ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder | {
"line": 52,
"column": 43
} | {
"line": 52,
"column": 80
} | {
"line": 52,
"column": 80
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : CancelCommMonoid M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedMonoid M\ninst✝¹ : LocallyFiniteOrder M\ninst✝ : ExistsMulOfLE M\na b c d : M\nh₁ : a * c ≤ a * c * d\nh₂ : a * c * d < b * c\n⊢ a * d < b",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"i... | [] | simpa [mul_right_comm a c d] using h₂ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder | {
"line": 52,
"column": 43
} | {
"line": 52,
"column": 80
} | {
"line": 52,
"column": 80
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : CancelCommMonoid M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedMonoid M\ninst✝¹ : LocallyFiniteOrder M\ninst✝ : ExistsMulOfLE M\na b c d : M\nh₁ : a * c ≤ a * c * d\nh₂ : a * c * d < b * c\n⊢ a * d < b",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"i... | [] | simpa [mul_right_comm a c d] using h₂ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 19
} | {
"line": 142,
"column": 4
} | [
{
"pp": "case h.mp\nG : Type u_2\ninst✝⁴ : AddCommGroup G\ninst✝³ : LinearOrder G\ninst✝² : IsOrderedAddMonoid G\ninst✝¹ : LocallyFiniteOrder G\ninst✝ : Nontrivial G\na : G\nha : 0 < a\nb : ↥(Icc 0 a)\nhb : ⟨0, ⋯⟩ ⋖ b\nthis : 0 ≤ ↑b\nx : G\n⊢ 0 ≤ x ∧ x < ↑b → x = 0",
"ppTerm": "?h.mp",
"assigned": true,... | [
"case h.mp\nG : Type u_2\ninst✝⁴ : AddCommGroup G\ninst✝³ : LinearOrder G\ninst✝² : IsOrderedAddMonoid G\ninst✝¹ : LocallyFiniteOrder G\ninst✝ : Nontrivial G\na : G\nha : 0 < a\nb : ↥(Icc 0 a)\nhb : ⟨0, ⋯⟩ ⋖ b\nthis : 0 ≤ ↑b\nx : G\nh₁ : 0 ≤ x\nh₂ : x < ↑b\n⊢ x = 0"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 47
} | {
"line": 161,
"column": 4
} | [
{
"pp": "case inl\nΓ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : LinearOrder R\ninst✝¹ : AddCommGroup R\ninst✝ : IsOrderedAddMonoid R\nx : Lex R⟦Γ⟧\nhlt : x < 0\n⊢ (ofLex |x|).leadingCoeff = |(ofLex x).leadingCoeff|",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.m... | [
"case inl\nΓ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : LinearOrder R\ninst✝¹ : AddCommGroup R\ninst✝ : IsOrderedAddMonoid R\nx : Lex R⟦Γ⟧\nhlt : x < 0\nhlt' : (ofLex x).leadingCoeff < 0\n⊢ (ofLex |x|).leadingCoeff = |(ofLex x).leadingCoeff|"
] | obtain hlt' := leadingCoeff_neg_iff.mpr hlt | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 232,
"column": 2
} | {
"line": 234,
"column": 63
} | {
"line": 235,
"column": 2
} | [
{
"pp": "case inl\nΓ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : LinearOrder R\ninst✝¹ : AddCommGroup R\ninst✝ : IsOrderedAddMonoid R\nx y : Lex R⟦Γ⟧\nhlt : (ofLex x).orderTop < (ofLex y).orderTop\n⊢ ArchimedeanClass.mk x ≤ ArchimedeanClass.mk y ↔\n (ofLex x).orderTop < (ofLex y).orderTop ∨\n ... | [
"case inr.inl\nΓ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : LinearOrder R\ninst✝¹ : AddCommGroup R\ninst✝ : IsOrderedAddMonoid R\nx y : Lex R⟦Γ⟧\nheq : (ofLex x).orderTop = (ofLex y).orderTop\n⊢ ArchimedeanClass.mk x ≤ ArchimedeanClass.mk y ↔\n (ofLex x).orderTop < (ofLex y).orderTop ∨\n (of... | · -- when `x`'s order is less than `y`'s, this reduces to abs_lt_abs_of_orderTop_ofLex
simpa [ArchimedeanClass.mk_le_mk, hlt] using
⟨1, by simpa using (abs_lt_abs_of_orderTop_ofLex hlt).le⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 378,
"column": 34
} | {
"line": 378,
"column": 66
} | {
"line": 379,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_3\nV : Type u_5\ninst✝⁵ : AddCommMonoid V\nΓ : Type u_6\ninst✝⁴ : AddCommMonoid Γ\ninst✝³ : LinearOrder Γ\ninst✝² : IsOrderedCancelAddMonoid Γ\ninst✝¹ : Zero R\ninst✝ : SMulWithZero R V\nx : R⟦Γ⟧\ny : HahnModule Γ R V\nhx : x = 0\n⊢ ((of R).symm (x • y)).coeff (x.order + ((of R).sy... | [] | simp [HahnSeries.coeff_zero, hx] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 378,
"column": 34
} | {
"line": 378,
"column": 66
} | {
"line": 379,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_3\nV : Type u_5\ninst✝⁵ : AddCommMonoid V\nΓ : Type u_6\ninst✝⁴ : AddCommMonoid Γ\ninst✝³ : LinearOrder Γ\ninst✝² : IsOrderedCancelAddMonoid Γ\ninst✝¹ : Zero R\ninst✝ : SMulWithZero R V\nx : R⟦Γ⟧\ny : HahnModule Γ R V\nhx : x = 0\n⊢ ((of R).symm (x • y)).coeff (x.order + ((of R).sy... | [] | simp [HahnSeries.coeff_zero, hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 378,
"column": 34
} | {
"line": 378,
"column": 66
} | {
"line": 379,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_3\nV : Type u_5\ninst✝⁵ : AddCommMonoid V\nΓ : Type u_6\ninst✝⁴ : AddCommMonoid Γ\ninst✝³ : LinearOrder Γ\ninst✝² : IsOrderedCancelAddMonoid Γ\ninst✝¹ : Zero R\ninst✝ : SMulWithZero R V\nx : R⟦Γ⟧\ny : HahnModule Γ R V\nhx : x = 0\n⊢ ((of R).symm (x • y)).coeff (x.order + ((of R).sy... | [] | simp [HahnSeries.coeff_zero, hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.IsNonarchimedean | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 54
} | {
"line": 189,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : Semiring R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\nF : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝² : CommRing α\ninst✝¹ : FunLike F α R\ninst✝ : AddGroupSeminormClass F α R\nf : F\nhf_na : IsNonarchimedean ⇑f\ns : Finset β\nb : β → α\nm : ℕ\n⊢ ∃ u, f (∑ t ∈ powe... | [
"R : Type u_1\ninst✝⁵ : Semiring R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\nF : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝² : CommRing α\ninst✝¹ : FunLike F α R\ninst✝ : AddGroupSeminormClass F α R\nf : F\nhf_na : IsNonarchimedean ⇑f\ns : Finset β\nb : β → α\nm : ℕ\ng : Finset β → α := fun t ↦ ∏ i ... | set g := fun t : Finset β ↦ t.prod fun i : β ↦ - b i | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 381,
"column": 82
} | {
"line": 381,
"column": 96
} | {
"line": 382,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_3\nV : Type u_5\ninst✝⁵ : AddCommMonoid V\nΓ : Type u_6\ninst✝⁴ : AddCommMonoid Γ\ninst✝³ : LinearOrder Γ\ninst✝² : IsOrderedCancelAddMonoid Γ\ninst✝¹ : Zero R\ninst✝ : SMulWithZero R V\nx : R⟦Γ⟧\ny : HahnModule Γ R V\nhx : ¬x = 0\nhy : ¬(of R).symm y = 0\n⊢ ∑ ij ∈ VAddAntidiagonal... | [
"case neg\nR : Type u_3\nV : Type u_5\ninst✝⁵ : AddCommMonoid V\nΓ : Type u_6\ninst✝⁴ : AddCommMonoid Γ\ninst✝³ : LinearOrder Γ\ninst✝² : IsOrderedCancelAddMonoid Γ\ninst✝¹ : Zero R\ninst✝ : SMulWithZero R V\nx : R⟦Γ⟧\ny : HahnModule Γ R V\nhx : ¬x = 0\nhy : ¬(of R).symm y = 0\n⊢ ∑ ij ∈ VAddAntidiagonal (⋯.min ⋯ +ᵥ... | ← vadd_eq_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Ring.IsNonarchimedean | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 22
} | {
"line": 219,
"column": 4
} | [
{
"pp": "case cons\nR : Type u_1\ninst✝⁴ : Semiring R\ninst✝³ : LinearOrder R\nα : Type u_2\nβ : Type u_3\nF : Type u_4\ninst✝² : AddCommGroup α\ninst✝¹ : FunLike F α R\ninst✝ : AddGroupSeminormClass F α R\nf : F\nnonarch : IsNonarchimedean ⇑f\ns✝ : Finset β\nl : β → α\na : β\ns : Finset β\nh✝ : a ∉ s\nhs : s.N... | [] | | cons a s _ hs _ => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 165,
"column": 21
} | {
"line": 165,
"column": 48
} | {
"line": 165,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP : RingPreordering R\ninst✝ : HasMemOrNegMem P\nx a : R\nha : a ∈ P.supportAddSubgroup\nhx : x ∈ P\n⊢ x * a ∈ ↑P",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"NonUnitalCommRing.toNonUnita... | [] | simpa using mul_mem hx ha.1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 165,
"column": 21
} | {
"line": 165,
"column": 48
} | {
"line": 165,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP : RingPreordering R\ninst✝ : HasMemOrNegMem P\nx a : R\nha : a ∈ P.supportAddSubgroup\nhx : x ∈ P\n⊢ x * a ∈ ↑P",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"NonUnitalCommRing.toNonUnita... | [] | simpa using mul_mem hx ha.1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 165,
"column": 21
} | {
"line": 165,
"column": 48
} | {
"line": 165,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP : RingPreordering R\ninst✝ : HasMemOrNegMem P\nx a : R\nha : a ∈ P.supportAddSubgroup\nhx : x ∈ P\n⊢ x * a ∈ ↑P",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"NonUnitalCommRing.toNonUnita... | [] | simpa using mul_mem hx ha.1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 166,
"column": 53
} | {
"line": 166,
"column": 80
} | {
"line": 166,
"column": 80
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP : RingPreordering R\ninst✝ : HasMemOrNegMem P\nx a : R\nha : a ∈ P.supportAddSubgroup\nhx : -x ∈ P\n⊢ x * a ∈ -↑P",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"SetLike.mem_coe._simp_1",
... | [] | simpa using mul_mem hx ha.1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 166,
"column": 53
} | {
"line": 166,
"column": 80
} | {
"line": 166,
"column": 80
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP : RingPreordering R\ninst✝ : HasMemOrNegMem P\nx a : R\nha : a ∈ P.supportAddSubgroup\nhx : -x ∈ P\n⊢ x * a ∈ -↑P",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"SetLike.mem_coe._simp_1",
... | [] | simpa using mul_mem hx ha.1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 166,
"column": 53
} | {
"line": 166,
"column": 80
} | {
"line": 166,
"column": 80
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP : RingPreordering R\ninst✝ : HasMemOrNegMem P\nx a : R\nha : a ∈ P.supportAddSubgroup\nhx : -x ∈ P\n⊢ x * a ∈ -↑P",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"SetLike.mem_coe._simp_1",
... | [] | simpa using mul_mem hx ha.1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 644,
"column": 6
} | {
"line": 645,
"column": 10
} | {
"line": 646,
"column": 2
} | [
{
"pp": "case neg\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\nx : R⟦Γ⟧\nh : ¬NeZero 1\n⊢ 0 ≤ orderTop 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Nontrivial",
"MulOne.toOne",
... | [] | have : Subsingleton R := not_nontrivial_iff_subsingleton.mp fun _ ↦ h NeZero.one
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 644,
"column": 6
} | {
"line": 645,
"column": 10
} | {
"line": 646,
"column": 2
} | [
{
"pp": "case neg\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\nx : R⟦Γ⟧\nh : ¬NeZero 1\n⊢ 0 ≤ orderTop 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Nontrivial",
"MulOne.toOne",
... | [] | have : Subsingleton R := not_nontrivial_iff_subsingleton.mp fun _ ↦ h NeZero.one
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 310,
"column": 59
} | {
"line": 310,
"column": 77
} | {
"line": 310,
"column": 77
} | [
{
"pp": "K : Type u_1\ninst✝¹¹ : DivisionRing K\ninst✝¹⁰ : LinearOrder K\ninst✝⁹ : IsOrderedRing K\ninst✝⁸ : Archimedean K\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : LinearOrder M\ninst✝⁵ : IsOrderedAddMonoid M\ninst✝⁴ : Module K M\ninst✝³ : IsOrderedModule K M\nR : Type u_3\ninst✝² : AddCommGroup R\ninst... | [] | by simpa using! hc | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 1004,
"column": 6
} | {
"line": 1005,
"column": 84
} | {
"line": 1006,
"column": 6
} | [
{
"pp": "case hfg\nΓ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nS : Type u_4\nV : Type u_5\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : IsCancelAdd R\ninst✝ : IsCancelMulZero R\nx : R⟦Γ⟧\nhx : x ≠ 0\ny z : R⟦Γ⟧\nthis✝ : AddC... | [
"case hfg\nΓ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nS : Type u_4\nV : Type u_5\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : IsCancelAdd R\ninst✝ : IsCancelMulZero R\nx : R⟦Γ⟧\nhx : x ≠ 0\ny z : R⟦Γ⟧\nthis✝ : AddCancelCommMon... | simp +contextual only [mem_union, mem_addAntidiagonal, mul_eq_mul_right_iff, Prod.mk.injEq,
ne_eq, ← or_and_right, or_false, and_imp, Prod.forall, mem_support, not_and] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 679,
"column": 14
} | {
"line": 679,
"column": 22
} | {
"line": 680,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹³ : DivisionRing K\ninst✝¹² : LinearOrder K\ninst✝¹¹ : IsOrderedRing K\ninst✝¹⁰ : Archimedean K\nM : Type u_2\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : LinearOrder M\ninst✝⁷ : IsOrderedAddMonoid M\ninst✝⁶ : Module K M\ninst✝⁵ : IsOrderedModule K M\nR : Type u_3\ninst✝⁴ : AddCommGroup R\nin... | [
"K : Type u_1\ninst✝¹³ : DivisionRing K\ninst✝¹² : LinearOrder K\ninst✝¹¹ : IsOrderedRing K\ninst✝¹⁰ : Archimedean K\nM : Type u_2\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : LinearOrder M\ninst✝⁷ : IsOrderedAddMonoid M\ninst✝⁶ : Module K M\ninst✝⁵ : IsOrderedModule K M\nR : Type u_3\ninst✝⁴ : AddCommGroup R\ninst✝³ : Linea... | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 313,
"column": 20
} | {
"line": 313,
"column": 43
} | {
"line": 313,
"column": 43
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx : K\n⊢ (if h : 0 ≤ mk (-x) then (Classical.ofNonempty.comp FiniteResidueField.mk) (FiniteElement.mk (-x) h) else 0) =\n -if h : 0 ≤ mk x then (Classical.ofNonempty.comp FiniteResidueField.mk) (FiniteElement.mk x h) el... | [
"K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx : K\n⊢ (if h : 0 ≤ mk x then (Classical.ofNonempty.comp FiniteResidueField.mk) (FiniteElement.mk (-x) ⋯) else 0) =\n -if h : 0 ≤ mk x then (Classical.ofNonempty.comp FiniteResidueField.mk) (FiniteElement.mk x h) else 0"
] | ArchimedeanClass.mk_neg | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.UpperLower | {
"line": 242,
"column": 70
} | {
"line": 242,
"column": 88
} | {
"line": 243,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : Preorder α\ninst✝ : IsOrderedMonoid α\ns t : Set α\n⊢ ⋃ a ∈ s, a • ↑(lowerClosure t) = ↑(⨆ i ∈ s, lowerClosure (i • t))",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"Iff... | [
"α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : Preorder α\ninst✝ : IsOrderedMonoid α\ns t : Set α\n⊢ ⋃ a ∈ s, a • ↑(lowerClosure t) = ↑(⨆ i ∈ s, i • lowerClosure t)"
] | lowerClosure_smul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 369,
"column": 57
} | {
"line": 371,
"column": 47
} | {
"line": 373,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nq : ℚ\n⊢ stdPart ↑q = ↑q",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"ArchimedeanClass.FiniteResidueField.mk_ratCast",
"Eq.mpr",
"Classical.ofNonempty",
"Real",
"IsDo... | [] | by
rw [stdPart_of_mk_nonneg Classical.ofNonempty (mk_ratCast_nonneg q), FiniteElement.mk_ratCast,
FiniteResidueField.mk_ratCast, map_ratCast] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Derivation.MapCoeffs | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 68
} | {
"line": 104,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Algebra R A\nB : Type u_4\nM' : Type u_5\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra R B\ninst✝⁶ : Algebra A B\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module B M'\ninst✝³ : Module R M'\ninst✝² : Module A M'\ninst✝¹ : IsScalarTower ... | [
"R : Type u_1\nA : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Algebra R A\nB : Type u_4\nM' : Type u_5\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra R B\ninst✝⁶ : Algebra A B\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module B M'\ninst✝³ : Module R M'\ninst✝² : Module A M'\ninst✝¹ : IsScalarTower R A B\ninst✝... | convert! apply_aeval_eq' (d.compAlgebraMap A) d LinearMap.id _ x p | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 240,
"column": 4
} | {
"line": 243,
"column": 81
} | {
"line": 245,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\na b : R\n⊢ (C a * X - C b).natDegree < 2",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Nat.instMulZeroClass",
"Nat.instLattice",
"Trans.trans",
"Lattice.toSemilatticeSup",
"... | [] | calc
_ ≤ max (C a * X).natDegree (C b).natDegree := natDegree_sub_le ..
_ = (C a * X).natDegree := by simp
_ < 2 := natDegree_C_mul_le .. |>.trans natDegree_X_le |>.trans_lt one_lt_two | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Algebra.Polynomial.Eval.Irreducible | {
"line": 45,
"column": 55
} | {
"line": 56,
"column": 11
} | {
"line": 58,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nh_mon : f.Monic\nh_irr : Irreducible (map φ f)\n⊢ Irreducible f",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonU... | [] | by
refine ⟨h_irr.not_isUnit ∘ IsUnit.map (mapRingHom φ), fun a b h => ?_⟩
dsimp [Monic] at h_mon
have q := (leadingCoeff_mul a b).symm
rw [← h, h_mon] at q
refine (h_irr.isUnit_or_isUnit <|
(congr_arg (Polynomial.map φ) h).trans (Polynomial.map_mul φ)).imp ?_ ?_ <;>
apply isUnit_of_isUnit_leadingCoe... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 974,
"column": 2
} | {
"line": 976,
"column": 40
} | {
"line": 977,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹³ : DivisionRing K\ninst✝¹² : LinearOrder K\ninst✝¹¹ : IsOrderedRing K\ninst✝¹⁰ : Archimedean K\nM : Type u_2\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : LinearOrder M\ninst✝⁷ : IsOrderedAddMonoid M\ninst✝⁶ : Module K M\ninst✝⁵ : IsOrderedModule K M\nR : Type u_3\ninst✝⁴ : Add... | [
"case refine_2\nK : Type u_1\ninst✝¹³ : DivisionRing K\ninst✝¹² : LinearOrder K\ninst✝¹¹ : IsOrderedRing K\ninst✝¹⁰ : Archimedean K\nM : Type u_2\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : LinearOrder M\ninst✝⁷ : IsOrderedAddMonoid M\ninst✝⁶ : Module K M\ninst✝⁵ : IsOrderedModule K M\nR : Type u_3\ninst✝⁴ : AddCommGroup R\... | · apply hpartial.strictMono.comp
intro _ _ h
simpa [← Subtype.coe_lt_coe] using h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 100,
"column": 63
} | {
"line": 100,
"column": 74
} | {
"line": 100,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]\nn₁ : ℕ\nx✝⁴ : n₁ ∈ Finset.range (p.natDegree + 1)\nx✝³ : p.mirror.coeff n₁ ≠ 0\nx✝² : ℕ\nx✝¹ : x✝² ∈ Finset.range (p.natDegree + 1)\nx✝ : p.mirror.coeff x✝² ≠ 0\nh : (revAt (p.natDegree + p.natTrailingDegree)) n₁ = (revAt (p.natDegree + p.natTrailingDegree)) ... | [
"R : Type u_1\ninst✝ : Semiring R\np : R[X]\nn₁ : ℕ\nx✝⁴ : n₁ ∈ Finset.range (p.natDegree + 1)\nx✝³ : p.mirror.coeff n₁ ≠ 0\nx✝² : ℕ\nx✝¹ : x✝² ∈ Finset.range (p.natDegree + 1)\nx✝ : p.mirror.coeff x✝² ≠ 0\nh : (revAt (p.natDegree + p.natTrailingDegree)) n₁ = (revAt (p.natDegree + p.natTrailingDegree)) x✝²\n⊢ x✝² =... | revAt_invol | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 109,
"column": 25
} | {
"line": 109,
"column": 36
} | {
"line": 109,
"column": 36
} | [
{
"pp": "case h.refine_2\nR : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nhn : n ∈ Finset.range (p.natDegree + 1)\nhp : p.coeff n ≠ 0\n⊢ p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) ((revAt (p.natDegree + p.natTrailingDegree)) n)) ≠ 0",
"ppTerm": "?h.refine_2",
"assigned": true,
"usedCons... | [
"case h.refine_2\nR : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nhn : n ∈ Finset.range (p.natDegree + 1)\nhp : p.coeff n ≠ 0\n⊢ p.coeff n ≠ 0"
] | revAt_invol | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 114,
"column": 80
} | {
"line": 114,
"column": 91
} | {
"line": 114,
"column": 91
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) ((revAt (p.natDegree + p.natTrailingDegree)) n)) = p.coeff n",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.revAt",
"congrArg",
... | [
"R : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ p.coeff n = p.coeff n"
] | revAt_invol | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 47
} | {
"line": 218,
"column": 4
} | [
{
"pp": "case isUnit_or_isUnit.inr.inr.inl\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\nf : R[X]\nh1 : ¬IsUnit f\nh2 : ∀ (k : R[X]), f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror\nh3 : IsRelPrime f f.mirror\ng h : R[X]\nfgh : f = g * h\nk : R[X] := g * h.mirror\nke... | [
"case isUnit_or_isUnit.inr.inr.inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\nf : R[X]\nh1 : ¬IsUnit f\nh2 : ∀ (k : R[X]), f * f.mirror = k * k.mirror → k = f ∨ k = -f ∨ k = f.mirror ∨ k = -f.mirror\nh3 : IsRelPrime f f.mirror\ng h : R[X]\nfgh : f = g * h\nk : R[X] := g * h.mirror\nkey : f * f.mi... | · exact Or.inl (h3 g_dvd_f (by rwa [← hk])) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 261,
"column": 4
} | {
"line": 262,
"column": 37
} | {
"line": 264,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : Semiring R\np : R[X]\nS : Type u_2\ninst✝⁴ : NonAssocSemiring S\ninst✝³ : Module R S\ninst✝² : Pow S ℕ\nx : S\ninst✝¹ : NatPowAssoc S\ninst✝ : IsScalarTower R S S\nn : ℕ\n⊢ (p * X ^ (n + 1)).smeval x = p.smeval x * x ^ (n + 1)",
"ppTerm": "?m.57",
"assigned": true,
"u... | [] | rw [npow_add, ← mul_assoc, npow_one, smeval_mul_X, smeval_mul_X_pow n, npow_add,
← smeval_assoc_X_pow, npow_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 261,
"column": 4
} | {
"line": 262,
"column": 37
} | {
"line": 264,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : Semiring R\np : R[X]\nS : Type u_2\ninst✝⁴ : NonAssocSemiring S\ninst✝³ : Module R S\ninst✝² : Pow S ℕ\nx : S\ninst✝¹ : NatPowAssoc S\ninst✝ : IsScalarTower R S S\nn : ℕ\n⊢ (p * X ^ (n + 1)).smeval x = p.smeval x * x ^ (n + 1)",
"ppTerm": "?m.57",
"assigned": true,
"u... | [] | rw [npow_add, ← mul_assoc, npow_one, smeval_mul_X, smeval_mul_X_pow n, npow_add,
← smeval_assoc_X_pow, npow_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 261,
"column": 4
} | {
"line": 262,
"column": 37
} | {
"line": 264,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : Semiring R\np : R[X]\nS : Type u_2\ninst✝⁴ : NonAssocSemiring S\ninst✝³ : Module R S\ninst✝² : Pow S ℕ\nx : S\ninst✝¹ : NatPowAssoc S\ninst✝ : IsScalarTower R S S\nn : ℕ\n⊢ (p * X ^ (n + 1)).smeval x = p.smeval x * x ^ (n + 1)",
"ppTerm": "?m.57",
"assigned": true,
"u... | [] | rw [npow_add, ← mul_assoc, npow_one, smeval_mul_X, smeval_mul_X_pow n, npow_add,
← smeval_assoc_X_pow, npow_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 141,
"column": 9
} | {
"line": 141,
"column": 34
} | {
"line": 141,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\np : R[X]\nq : ℕ\nr : A\np' : A[X]\nhp : map (algebraMap R A) p = (X - C r) ^ q * p'\nx : ℕ\nh : 1 ≤ x\nh' : x ≤ q\n⊢ 1 ≤ q - (q - x)",
"ppTerm": "?m.159",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\np : R[X]\nq : ℕ\nr : A\np' : A[X]\nhp : map (algebraMap R A) p = (X - C r) ^ q * p'\nx : ℕ\nh : 1 ≤ x\nh' : x ≤ q\n⊢ 1 ≤ x"
] | tsub_tsub_cancel_of_le h' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 167,
"column": 49
} | {
"line": 167,
"column": 64
} | {
"line": 167,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\np : R[X]\nr : A\n⊢ eval₂ (algebraMap R A) r (sumIDeriv p) = eval₂ ((RingHom.id A).comp (algebraMap R A)) r (sumIDeriv p)",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"R : Type u_1\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\np : R[X]\nr : A\n⊢ eval₂ (algebraMap R A) r (sumIDeriv p) = eval₂ (algebraMap R A) r (sumIDeriv p)"
] | RingHom.id_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 258,
"column": 4
} | {
"line": 258,
"column": 46
} | {
"line": 258,
"column": 47
} | [
{
"pp": "p q : ℤ[X]\nk m m' n : ℕ\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w x z : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nhq : q = trinomial k m' n ↑x ↑v ↑z\nh : p * p.mirror = q * q.mirror\nhmul : ↑w * (trinomial k m n ↑u ↑v ↑w).trailingCoeff = ↑z * (trinomial k m' n ↑x ↑v ↑z).trailingCoeff... | [
"p q : ℤ[X]\nk m m' n : ℕ\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w x z : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nhq : q = trinomial k m' n ↑x ↑v ↑z\nh : p * p.mirror = q * q.mirror\nhmul : ↑w * ↑u = ↑z * (trinomial k m' n ↑x ↑v ↑z).trailingCoeff\n⊢ q = p ∨ q = p.mirror"
] | trinomial_trailingCoeff hkm hmn u.ne_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.QuadraticAlgebra.Defs | {
"line": 109,
"column": 79
} | {
"line": 110,
"column": 22
} | {
"line": 112,
"column": 0
} | [
{
"pp": "R : Type u_1\na b : R\ninst✝ : Zero R\nr : R\n⊢ QuadraticAlgebra.C r = 0 ↔ r = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"QuadraticAlgebra",
"congrArg",
"QuadraticAlgebra.C",
"Iff.rfl",
"id",
"QuadraticAlgebra.instZero",
... | [] | by
rw [← C_zero, C_inj] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.QuadraticAlgebra.Defs | {
"line": 350,
"column": 28
} | {
"line": 350,
"column": 68
} | {
"line": 351,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\na b r : R\nx y : QuadraticAlgebra R a b\ninst✝ : NonUnitalNonAssocSemiring R\nx✝² x✝¹ x✝ : QuadraticAlgebra R a b\n⊢ (x✝² + x✝¹) * x✝ = x✝² * x✝ + x✝¹ * x✝",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"add_mul",
"Distrib.lef... | [] | ext <;> simp [mul_add, add_mul] <;> abel | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.QuadraticAlgebra.Defs | {
"line": 350,
"column": 28
} | {
"line": 350,
"column": 68
} | {
"line": 351,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\na b r : R\nx y : QuadraticAlgebra R a b\ninst✝ : NonUnitalNonAssocSemiring R\nx✝² x✝¹ x✝ : QuadraticAlgebra R a b\n⊢ (x✝² + x✝¹) * x✝ = x✝² * x✝ + x✝¹ * x✝",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"add_mul",
"Distrib.lef... | [] | ext <;> simp [mul_add, add_mul] <;> abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.QuadraticAlgebra.Defs | {
"line": 350,
"column": 28
} | {
"line": 350,
"column": 68
} | {
"line": 351,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\na b r : R\nx y : QuadraticAlgebra R a b\ninst✝ : NonUnitalNonAssocSemiring R\nx✝² x✝¹ x✝ : QuadraticAlgebra R a b\n⊢ (x✝² + x✝¹) * x✝ = x✝² * x✝ + x✝¹ * x✝",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"add_mul",
"Distrib.lef... | [] | ext <;> simp [mul_add, add_mul] <;> abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.QuadraticAlgebra.Defs | {
"line": 499,
"column": 4
} | {
"line": 499,
"column": 62
} | {
"line": 501,
"column": 0
} | [
{
"pp": "case h\nR : Type u_1\na b : R\ninst✝ : CommSemiring R\nr✝ : R\nz : QuadraticAlgebra R a b\nr : R\nhr : z.re = r✝ * r\ni : R\nhi : z.im = r✝ * i\n⊢ z = (algebraMap R (QuadraticAlgebra R a b)) r✝ * { re := r, im := i }",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"QuadraticAlg... | [] | simp [QuadraticAlgebra.ext_iff, hr, hi, ← C_eq_algebraMap] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 314,
"column": 10
} | {
"line": 314,
"column": 58
} | {
"line": 314,
"column": 59
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\ng : ι → R[X]\nhg : ∀ i ∈ s, (g i).Monic\nhgg : (↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)\nn : ι → ℕ\nq₁ q₂ : R[X]\nr₁ r₂ : (i : ι) → Fin (n i) → R[X]\nhr₁ : ∀ i ∈ s, ∀ (j : Fin (n i)), (r₁ i j).degr... | [
"case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\ng : ι → R[X]\nhg : ∀ i ∈ s, (g i).Monic\nhgg : (↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)\nn : ι → ℕ\nq₁ q₂ : R[X]\nr₁ r₂ : (i : ι) → Fin (n i) → R[X]\nhr₁ : ∀ i ∈ s, ∀ (j : Fin (n i)), (r₁ i j).degree < (g i).d... | degree_eq_natDegree ((hg i hi).pow j.1).ne_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 351,
"column": 2
} | {
"line": 367,
"column": 79
} | {
"line": 369,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : Algebra R[X] K\ninst✝ : Nontrivial R\nι : Type u_3\ns : Finset ι\nf : R[X]\ng : ι → R[X]\nhg : ∀ i ∈ s, (g i).Monic\nhgg : (↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)\nn : ι → ℕ\ngi : ι → K\nhgi : ∀ i ∈ s, gi i * (algebraM... | [] | obtain ⟨q, r, hr, hf⟩ := eq_quo_mul_prod_pow_add_sum_rem_mul_prod_pow f hg hgg n
refine ⟨q, fun i j => r i j.rev, fun i hi j => hr i hi j.rev, ?_⟩
rw [hf, map_add, map_mul, map_prod, add_mul, mul_assoc, ← Finset.prod_mul_distrib]
have hc (x : ι) (hx : x ∈ s) : (algebraMap R[X] K) (g x ^ n x) * gi x ^ n x = 1 := b... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 351,
"column": 2
} | {
"line": 367,
"column": 79
} | {
"line": 369,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : Algebra R[X] K\ninst✝ : Nontrivial R\nι : Type u_3\ns : Finset ι\nf : R[X]\ng : ι → R[X]\nhg : ∀ i ∈ s, (g i).Monic\nhgg : (↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)\nn : ι → ℕ\ngi : ι → K\nhgi : ∀ i ∈ s, gi i * (algebraM... | [] | obtain ⟨q, r, hr, hf⟩ := eq_quo_mul_prod_pow_add_sum_rem_mul_prod_pow f hg hgg n
refine ⟨q, fun i j => r i j.rev, fun i hi j => hr i hi j.rev, ?_⟩
rw [hf, map_add, map_mul, map_prod, add_mul, mul_assoc, ← Finset.prod_mul_distrib]
have hc (x : ι) (hx : x ∈ s) : (algebraMap R[X] K) (g x ^ n x) * gi x ^ n x = 1 := b... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.QuaternionBasis | {
"line": 161,
"column": 11
} | {
"line": 161,
"column": 31
} | {
"line": 161,
"column": 31
} | [
{
"pp": "case a.inl\nR : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nc₁ c₂ c₃ : R\nB : Basis A c₁ c₂ c₃\n⊢ B.i ∈ ↑B.liftHom.range",
"ppTerm": "?a.inl✝",
"assigned": true,
"usedConstants": [
"QuaternionAlgebra.Basis.self",
"RingHom",
"Quaternio... | [
"case h\nR : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nc₁ c₂ c₃ : R\nB : Basis A c₁ c₂ c₃\n⊢ B.liftHom.toRingHom (Basis.self R).i = B.i"
] | use (Basis.self R).i | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Ring.CentroidHom | {
"line": 441,
"column": 4
} | {
"line": 441,
"column": 45
} | {
"line": 442,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u_5\ninst✝ : NonUnitalNonAssocSemiring α\nT : AddMonoid.End α\nh : T ∈ Subsemiring.centralizer (Set.range ⇑L ∪ Set.range ⇑R)\n⊢ T ∈ (toEndRingHom α).rangeS",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"AddMonoid.End.mulLeft",
"Subsemiring... | [
"case refine_2\nα : Type u_5\ninst✝ : NonUnitalNonAssocSemiring α\nT : AddMonoid.End α\nh : ∀ g ∈ Set.range ⇑L ∪ Set.range ⇑R, g * T = T * g\n⊢ T ∈ (toEndRingHom α).rangeS"
] | rw [Subsemiring.mem_centralizer_iff] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.SkewMonoidAlgebra.Support | {
"line": 107,
"column": 2
} | {
"line": 108,
"column": 24
} | {
"line": 110,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝³ : Monoid G\ninst✝² : Semiring k\ninst✝¹ : MulSemiringAction G k\nf : SkewMonoidAlgebra k G\ninst✝ : DecidableEq G\nr : k\nx : G\nrx : IsRightRegular x\nhrx : ∀ (g : G) (y : k), y * g • r = 0 ↔ y = 0\ny : G\nyf : y ∈ f.support\nhy : y * x ∈ image (fun x_1 ↦ x_1 * x) f.... | [] | simp [coeff_mul, mem_support_iff.mp yf, hrx, mem_support_iff, sum_single_index, mul_zero,
ite_self, rx.eq_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 514,
"column": 29
} | {
"line": 514,
"column": 46
} | {
"line": 514,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\na : R\nn : ℕ\ninst✝ : MulSemiringAction (Multiplicative ℕ) R\n⊢ a • (monomial n) 1 = a • X ^ n",
"ppTerm": "?m.99",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"Semiring.toM... | [
"R : Type u_1\ninst✝¹ : Semiring R\na : R\nn : ℕ\ninst✝ : MulSemiringAction (Multiplicative ℕ) R\n⊢ a • (monomial n) 1 = a • (monomial n) 1"
] | X_pow_eq_monomial | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 710,
"column": 95
} | {
"line": 711,
"column": 39
} | {
"line": 713,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : SkewPolynomial R\nn : ℕ\na : R\n⊢ (p.update n a).coeff n = a",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"SkewPolynomial.coeff_update_apply",
"if_pos",
"Nat",
"SkewPolyno... | [] | by
rw [p.coeff_update_apply, if_pos rfl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 731,
"column": 64
} | {
"line": 731,
"column": 94
} | {
"line": 733,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : SkewPolynomial R\nn : ℕ\na : R\nha : a ≠ 0\n⊢ (p.update n a).support = insert n p.support",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SkewPolynomial.support_update",
"congrArg",
"Finset",
"Classical... | [] | rw [support_update, if_neg ha] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 731,
"column": 64
} | {
"line": 731,
"column": 94
} | {
"line": 733,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : SkewPolynomial R\nn : ℕ\na : R\nha : a ≠ 0\n⊢ (p.update n a).support = insert n p.support",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SkewPolynomial.support_update",
"congrArg",
"Finset",
"Classical... | [] | rw [support_update, if_neg ha] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 731,
"column": 64
} | {
"line": 731,
"column": 94
} | {
"line": 733,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : SkewPolynomial R\nn : ℕ\na : R\nha : a ≠ 0\n⊢ (p.update n a).support = insert n p.support",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SkewPolynomial.support_update",
"congrArg",
"Finset",
"Classical... | [] | rw [support_update, if_neg ha] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 522,
"column": 11
} | {
"line": 522,
"column": 27
} | {
"line": 522,
"column": 28
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : AddCommMonoid k\nG' : Type u_3\nf : G → G'\nv : SkewMonoidAlgebra k G\nR : Type u_5\ninst✝¹ : Monoid R\ninst✝ : DistribMulAction R k\nb : R\n⊢ (mapDomain f) (b • v) = b • (mapDomain f) v",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq... | [
"k : Type u_1\nG : Type u_2\ninst✝² : AddCommMonoid k\nG' : Type u_3\nf : G → G'\nv : SkewMonoidAlgebra k G\nR : Type u_5\ninst✝¹ : Monoid R\ninst✝ : DistribMulAction R k\nb : R\n⊢ ((mapDomain f) (b • v)).toFinsupp = (b • (mapDomain f) v).toFinsupp"
] | ← toFinsupp_inj, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Tropical.Basic | {
"line": 489,
"column": 17
} | {
"line": 489,
"column": 56
} | {
"line": 495,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\nx : Tropical R\nn : ℕ\nIH : (n + 1) • x = x\n⊢ (n + 1 + 1) • x = x",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Tropical.instAddCommMonoidTropical",
"congrArg",
... | [] | rw [add_nsmul, IH, one_nsmul, add_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Tropical.Basic | {
"line": 489,
"column": 17
} | {
"line": 489,
"column": 56
} | {
"line": 495,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\nx : Tropical R\nn : ℕ\nIH : (n + 1) • x = x\n⊢ (n + 1 + 1) • x = x",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Tropical.instAddCommMonoidTropical",
"congrArg",
... | [] | rw [add_nsmul, IH, one_nsmul, add_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Tropical.Basic | {
"line": 489,
"column": 17
} | {
"line": 489,
"column": 56
} | {
"line": 495,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\nx : Tropical R\nn : ℕ\nIH : (n + 1) • x = x\n⊢ (n + 1 + 1) • x = x",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Tropical.instAddCommMonoidTropical",
"congrArg",
... | [] | rw [add_nsmul, IH, one_nsmul, add_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Star.LinearMap | {
"line": 177,
"column": 2
} | {
"line": 179,
"column": 58
} | {
"line": 180,
"column": 2
} | [
{
"pp": "R : Type u_5\nA : Type u_6\nC : Type u_7\ninst✝¹² : CommSemiring R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : NonUnitalNonAssocSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\ninst✝⁶ : StarRing A\ninst✝⁵ : StarModule R A\ninst✝⁴ : AddCommMonoid C\ninst✝³ : Module R C\n... | [
"R : Type u_5\nA : Type u_6\nC : Type u_7\ninst✝¹² : CommSemiring R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : NonUnitalNonAssocSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\ninst✝⁶ : StarRing A\ninst✝⁵ : StarModule R A\ninst✝⁴ : AddCommMonoid C\ninst✝³ : Module R C\ninst✝² : Sta... | simp_rw [convMul_def, intrinsicStar_comp', intrinsicStar_mul', intrinsicStar_map,
h, comp_assoc, ← comp_assoc _ _ (map _ _), map_comp_comm_eq,
← comp_assoc _ (TensorProduct.comm R A A).toLinearMap] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace | {
"line": 159,
"column": 10
} | {
"line": 159,
"column": 23
} | {
"line": 161,
"column": 0
} | [
{
"pp": "X✝ X Y : LocallyRingedSpace\nf : X.toSheafedSpace ⟶ Y.toSheafedSpace\nh : ∀ (x : ↑X.toTopCat), IsLocalHom (CommRingCat.Hom.hom (PresheafedSpace.Hom.stalkMap f.hom x))\n⊢ ∀ (x : ↑↑X.toPresheafedSpace), IsLocalHom (CommRingCat.Hom.hom (PresheafedSpace.Hom.stalkMap f.hom x))",
"ppTerm": "?m.35",
"... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 95,
"column": 47
} | {
"line": 95,
"column": 60
} | {
"line": 97,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y : LocallyRingedSpace\nf : X ⟶ Y\ninst✝ : LocallyRingedSpace.IsOpenImmersion f\n⊢ PresheafedSpace.IsOpenImmersion f.toHom",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"inst✝"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 49
} | {
"line": 452,
"column": 2
} | [
{
"pp": "case right\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ng : Y ⟶ Z\ns✝ s : PullbackCone f g\nm : s.pt ⟶ (pullbackConeOfLeft f g).pt\na✝ : m ≫ (pullbackConeOfLeft f g).fst... | [
"case right\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ng : Y ⟶ Z\ns✝ s : PullbackCone f g\nm : s.pt ⟶ (pullbackConeOfLeft f g).pt\na✝ : m ≫ (pullbackConeOfLeft f g).fst = s.fst\nh₂... | rw [← cancel_mono (pullbackConeOfLeft f g).snd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 470,
"column": 4
} | {
"line": 470,
"column": 19
} | {
"line": 471,
"column": 4
} | [
{
"pp": "case refine_1\nX Y : Scheme\nf : X ⟶ Y\n⊢ Epi f.base ∧ IsOpenEmbedding ⇑f → IsIso f.base",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"CategoryTheory.IsIso",
"CategoryTheory.Epi",
"... | [
"case refine_1\nX Y : Scheme\nf : X ⟶ Y\nh₁ : Epi f.base\nh₂ : IsOpenEmbedding ⇑f\n⊢ IsIso f.base"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.