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 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.MvPolynomial.Polynomial | {
"line": 36,
"column": 4
} | {
"line": 37,
"column": 8
} | {
"line": 38,
"column": 2
} | [
{
"pp": "R : Type u_1\nn : ℕ\nx : Fin n → R\ninst✝ : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n| (eval x).comp ((Polynomial.evalRingHom q).comp (Polynomial.C.comp C))",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
... | [
"R : Type u_1\nn : ℕ\nx : Fin n → R\ninst✝ : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\nr : R\n| r"
] | refine @RingHom.ext _ _ _ _ _ (RingHom.id _) fun r => ?_
simp | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Algebra.MvPolynomial.Polynomial | {
"line": 36,
"column": 4
} | {
"line": 37,
"column": 8
} | {
"line": 38,
"column": 2
} | [
{
"pp": "R : Type u_1\nn : ℕ\nx : Fin n → R\ninst✝ : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n| (eval x).comp ((Polynomial.evalRingHom q).comp (Polynomial.C.comp C))",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
... | [
"R : Type u_1\nn : ℕ\nx : Fin n → R\ninst✝ : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\nr : R\n| r"
] | refine @RingHom.ext _ _ _ _ _ (RingHom.id _) fun r => ?_
simp | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Data.DFinsupp.WellFounded | {
"line": 100,
"column": 8
} | {
"line": 100,
"column": 28
} | {
"line": 101,
"column": 8
} | [
{
"pp": "case pos\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j\nhp : i ∉ p\nhs : s i (x ... | [
"case pos.hnc\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j\nhp : i ∉ p\nhs : s i (x i) (x₂ i... | rw [hr j h₂, if_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 83,
"column": 2
} | {
"line": 84,
"column": 62
} | {
"line": 86,
"column": 0
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nf g : MvPolynomial σ R\nhf : f ≠ 0\nhg : g ≠ 0\n⊢ (f * g).totalDegree = f.totalDegree + g.totalDegree",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Nat.instMul... | [] | cases exists_wellFoundedGT σ
simp [← degree_degLexDegree, MonomialOrder.degree_mul hf hg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 83,
"column": 2
} | {
"line": 84,
"column": 62
} | {
"line": 86,
"column": 0
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nf g : MvPolynomial σ R\nhf : f ≠ 0\nhg : g ≠ 0\n⊢ (f * g).totalDegree = f.totalDegree + g.totalDegree",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Nat.instMul... | [] | cases exists_wellFoundedGT σ
simp [← degree_degLexDegree, MonomialOrder.degree_mul hf hg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 160,
"column": 6
} | {
"line": 160,
"column": 46
} | {
"line": 160,
"column": 46
} | [
{
"pp": "case inr\nR : Type u_1\nσ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\np : MvPolynomial σ R\nn : σ →₀ ℕ\nhR : Nontrivial R\n⊢ p ∣ (monomial n) 1 ↔ ∃ m u, m ≤ n ∧ IsUnit u ∧ p = (monomial m) u",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finsu... | [
"case inr\nR : Type u_1\nσ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\np : MvPolynomial σ R\nn : σ →₀ ℕ\nhR : Nontrivial R\n⊢ (∃ m b, m ≤ n ∧ b ∣ 1 ∧ p = (monomial m) b) ↔ ∃ m u, m ≤ n ∧ IsUnit u ∧ p = (monomial m) u"
] | dvd_monomial_iff_exists (one_ne_zero' R) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 406,
"column": 6
} | {
"line": 406,
"column": 47
} | {
"line": 407,
"column": 4
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf g : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : m.toSyn (m.degree f + m.degree g) < m.toSyn c\nd e : σ →₀ ℕ\nhde : d + e = c\nhd : ¬m.toSyn (m.degree f) < m.toSyn d\nthis : m.toSyn (m.degree g) < m.toSyn e\n⊢ coeff d f * coeff e g = 0",
... | [] | rw [m.coeff_eq_zero_of_lt this, mul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 406,
"column": 6
} | {
"line": 406,
"column": 47
} | {
"line": 407,
"column": 4
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf g : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : m.toSyn (m.degree f + m.degree g) < m.toSyn c\nd e : σ →₀ ℕ\nhde : d + e = c\nhd : ¬m.toSyn (m.degree f) < m.toSyn d\nthis : m.toSyn (m.degree g) < m.toSyn e\n⊢ coeff d f * coeff e g = 0",
... | [] | rw [m.coeff_eq_zero_of_lt this, mul_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 406,
"column": 6
} | {
"line": 406,
"column": 47
} | {
"line": 407,
"column": 4
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf g : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : m.toSyn (m.degree f + m.degree g) < m.toSyn c\nd e : σ →₀ ℕ\nhde : d + e = c\nhd : ¬m.toSyn (m.degree f) < m.toSyn d\nthis : m.toSyn (m.degree g) < m.toSyn e\n⊢ coeff d f * coeff e g = 0",
... | [] | rw [m.coeff_eq_zero_of_lt this, mul_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Sym.Card | {
"line": 107,
"column": 47
} | {
"line": 109,
"column": 44
} | {
"line": 111,
"column": 0
} | [
{
"pp": "α : Type u_2\nk : ℕ\ninst✝¹ : Fintype α\ninst✝ : Fintype (Sym α k)\n⊢ Fintype.card (Sym α k) = (Fintype.card α).multichoose k",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Fintype.card_congr",
"Eq.mpr",
"congrArg",
"Sym.instFintype",
"instDecidableE... | [] | by
rw [← card_sym_fin_eq_multichoose]
exact card_congr (equivCongr (equivFin α)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 418,
"column": 2
} | {
"line": 435,
"column": 27
} | {
"line": 436,
"column": 2
} | [
{
"pp": "case h₀\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf g : MvPolynomial σ R\na b : σ →₀ ℕ\nha : m.toSyn (m.degree f) ≤ m.toSyn a\nhb : m.toSyn (m.degree g) ≤ m.toSyn b\n⊢ ∀ b_1 ∈ Finset.antidiagonal (a + b), b_1 ≠ (a, b) → coeff b_1.1 f * coeff b_1.2 g = 0",
"ppTerm": "... | [
"case h₁\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf g : MvPolynomial σ R\na b : σ →₀ ℕ\nha : m.toSyn (m.degree f) ≤ m.toSyn a\nhb : m.toSyn (m.degree g) ≤ m.toSyn b\n⊢ (a, b) ∉ Finset.antidiagonal (a + b) → coeff (a, b).1 f * coeff (a, b).2 g = 0"
] | · rintro ⟨c, d⟩ hcd h
simp only [Finset.mem_antidiagonal] at hcd
by_cases hf : m.degree f ≺[m] c
· rw [m.coeff_eq_zero_of_lt hf, zero_mul]
· suffices m.degree g ≺[m] d by
rw [coeff_eq_zero_of_lt this, mul_zero]
rw [not_lt] at hf
rw [← not_le]
intro hf'
apply h
suffi... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.MvPolynomial.SchwartzZippel | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 33
} | {
"line": 87,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\nn : ℕ\np : MvPolynomial (Fin (n + 1)) R\nhp : p ≠ 0\nS : Fin (n + 1) → Finset R\np' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p\nhp' : p' = (finSuccEquiv R n) p\n⊢ ↑(#({x ∈ piFinset fun i ↦ S i | (eval x) p... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\nn : ℕ\np : MvPolynomial (Fin (n + 1)) R\nhp : p ≠ 0\nS : Fin (n + 1) → Finset R\np' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p\nhp' : p' = (finSuccEquiv R n) p\nk : ℕ := p'.natDegree\nhk : k = p'.natDegree\n⊢ ↑(#({x ∈... | set k := p'.natDegree with hk | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 631,
"column": 4
} | {
"line": 631,
"column": 54
} | {
"line": 632,
"column": 4
} | [
{
"pp": "case insert\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nι : Type u_3\nP : ι → MvPolynomial σ R\na : ι\ns : Finset ι\nhas : a ∉ s\nhrec : m.toSyn (m.degree (∏ i ∈ s, P i)) ≤ m.toSyn (∑ i ∈ s, m.degree (P i))\n⊢ m.toSyn (m.degree (∏ i ∈ insert a s, P i)) ≤ m.toSyn (∑ i ∈ ins... | [
"case insert\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nι : Type u_3\nP : ι → MvPolynomial σ R\na : ι\ns : Finset ι\nhas : a ∉ s\nhrec : m.toSyn (m.degree (∏ i ∈ s, P i)) ≤ m.toSyn (∑ i ∈ s, m.degree (P i))\n⊢ m.toSyn (m.degree (P a * ∏ x ∈ s, P x)) ≤ m.toSyn (m.degree (P a) + ∑ x ∈ s... | rw [Finset.prod_insert has, Finset.sum_insert has] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Nat.Squarefree | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 26
} | {
"line": 101,
"column": 0
} | [
{
"pp": "n k : ℕ\nhn : n ≠ 1\nhk : k ≠ 0\nh : Squarefree (n ^ k)\nh₁ : ¬k = 1\nthis : 2 ≤ k\n⊢ n ^ 2 ∣ n ^ k",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"Nat.instMonoid",
"pow_dvd_pow",
"instOfNatNat",
"Nat",
"OfNat.ofNat"
],
"usedFVars": [
... | [] | exact pow_dvd_pow _ this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 875,
"column": 84
} | {
"line": 877,
"column": 51
} | {
"line": 879,
"column": 0
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf : MvPolynomial σ R\n⊢ m.degree f = WithBot.unbotD 0 (m.withBotDegree f)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"WithBot.some",... | [] | by
classical
by_cases h : f = 0 <;> simp [withBotDegree_eq, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.SchwartzZippel | {
"line": 162,
"column": 16
} | {
"line": 162,
"column": 42
} | {
"line": 162,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\nn : ℕ\np : MvPolynomial (Fin (n + 1)) R\nhp : p ≠ 0\nS : Fin (n + 1) → Finset R\np' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p\nhp' : p' = (finSuccEquiv R n) p\nk : ℕ := p'.natDegree\nhk : k = p'.natDegree... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\nn : ℕ\np : MvPolynomial (Fin (n + 1)) R\nhp : p ≠ 0\nS : Fin (n + 1) → Finset R\np' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p\nhp' : p' = (finSuccEquiv R n) p\nk : ℕ := p'.natDegree\nhk : k = p'.natDegree\npₖ : MvPol... | ← Polynomial.leadingCoeff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.SchwartzZippel | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 17
} | {
"line": 168,
"column": 6
} | [
{
"pp": "case calc_2\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\nn : ℕ\np : MvPolynomial (Fin (n + 1)) R\nhp : p ≠ 0\nS : Fin (n + 1) → Finset R\np' : Polynomial (MvPolynomial (Fin n) R) := ⋯\nhp' : p' = (finSuccEquiv R n) p\nk : ℕ := ⋯\nhk : k = p'.natDegree\npₖ : MvPolynomi... | [
"case calc_2\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\nn : ℕ\np : MvPolynomial (Fin (n + 1)) R\nhp : p ≠ 0\nS : Fin (n + 1) → Finset R\np' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p\nhp' : p' = (finSuccEquiv R n) p\nk : ℕ := p'.natDegree\nhk : k = p'.natDegre... | rintro i hi | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Data.Nat.Factorization.PrimePow | {
"line": 82,
"column": 6
} | {
"line": 82,
"column": 59
} | {
"line": 82,
"column": 59
} | [
{
"pp": "n : ℕ\nhn : n ≠ 1\np : ℕ\npp : Nat.Prime p\nh : n / p ^ n.factorization p = 1\n⊢ ∃ p k, Nat.Prime p ∧ 0 < k ∧ p ^ k = n",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"Nat.Prime",
"instHDiv",
... | [
"n p : ℕ\nhn : p ^ n.factorization p ≠ 1\npp : Nat.Prime p\nh : n / p ^ n.factorization p = 1\n⊢ ∃ p_1 k, Nat.Prime p_1 ∧ 0 < k ∧ p_1 ^ k = p ^ n.factorization p"
] | ← Nat.eq_of_dvd_of_div_eq_one (Nat.ordProj_dvd n p) h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Squarefree | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 25
} | {
"line": 375,
"column": 2
} | [
{
"pp": "s : Finset ℕ\nhs : ∀ p ∈ s, Prime p\nhn : ∏ p ∈ s, p ≠ 0\np : ℕ\n⊢ (Prime p ∧ ∃ a ∈ s, p ∣ a) → p ∈ s",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"Dvd.dvd",
"Finset",
"semigroupDvd",
"Membership.mem",
"Exists",
"SemigroupW... | [
"s : Finset ℕ\nhs : ∀ p ∈ s, Prime p\nhn : ∏ p ∈ s, p ≠ 0\np : ℕ\nhp : Prime p\nq : ℕ\nhq : q ∈ s\nhpq : p ∣ q\n⊢ p ∈ s"
] | rintro ⟨hp, q, hq, hpq⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 469,
"column": 4
} | {
"line": 469,
"column": 38
} | {
"line": 469,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nf : ArithmeticFunction R\nh_mult : f.IsMultiplicative\nl : ℕ\nhl : Squarefree l\n⊢ f (∏ a ∈ l.primeFactors, a) = f l",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ArithmeticFunction.instFunLikeNat",
"congrA... | [
"R : Type u_1\ninst✝ : CommMonoidWithZero R\nf : ArithmeticFunction R\nh_mult : f.IsMultiplicative\nl : ℕ\nhl : Squarefree l\n⊢ f l = f l"
] | prod_primeFactors_of_squarefree hl | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 253,
"column": 6
} | {
"line": 253,
"column": 24
} | {
"line": 253,
"column": 25
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\n⊢ ∑ d ∈ n.divisors, d = ∏ p ∈ n.primeFactors, ∑ k ∈ range (n.factorization p + 1), p ^ k",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"... | [
"n : ℕ\nhn : n ≠ 0\n⊢ (σ 1) n = ∏ p ∈ n.primeFactors, ∑ k ∈ range (n.factorization p + 1), p ^ k"
] | ← sigma_one_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Antidiag.Nat | {
"line": 295,
"column": 6
} | {
"line": 295,
"column": 37
} | {
"line": 296,
"column": 6
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\nb : ℕ × ℕ\nhb : (b.1 ∈ n.divisors ∧ b.2 ∈ n.divisors) ∧ b.1.lcm b.2 = n\ng : ℕ := b.1.gcd b.2\na : Fin (succ 0).succ.succ → ℕ := ![g, b.1 / g, b.2 / g]\n⊢ a 0 * a 1 * a 2 = n",
"ppTerm": "?m.114",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Monoid.toMul... | [
"n : ℕ\nhn : n ≠ 0\nb : ℕ × ℕ\nhb : (b.1 ∈ n.divisors ∧ b.2 ∈ n.divisors) ∧ b.1.lcm b.2 = n\ng : ℕ := b.1.gcd b.2\na : Fin (succ 0).succ.succ → ℕ := ![g, b.1 / g, b.2 / g]\n⊢ g * (b.1 / g) * (b.2 / g) = n"
] | dsimp only [a, Matrix.cons_val] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Order.Antidiag.Nat | {
"line": 299,
"column": 2
} | {
"line": 299,
"column": 16
} | {
"line": 299,
"column": 17
} | [
{
"pp": "case h\nn : ℕ\nhn : n ≠ 0\nb : ℕ × ℕ\nhb : b ∈ {x ∈ n.divisors ×ˢ n.divisors | x.1.lcm x.2 = n}\ng : ℕ := ⋯\na : Fin (succ 0).succ.succ → ℕ := ⋯\nha : a ∈ finMulAntidiag 3 n\n⊢ f a ha = b",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Prod.ext",
"Nat",
"_private.M... | [
"case h.fst\nn : ℕ\nhn : n ≠ 0\nb : ℕ × ℕ\nhb : b ∈ {x ∈ n.divisors ×ˢ n.divisors | x.1.lcm x.2 = n}\ng : ℕ := ⋯\na : Fin (succ 0).succ.succ → ℕ := ⋯\nha : a ∈ finMulAntidiag 3 n\n⊢ (f a ha).1 = b.1",
"case h.snd\nn : ℕ\nhn : n ≠ 0\nb : ℕ × ℕ\nhb : b ∈ {x ∈ n.divisors ×ˢ n.divisors | x.1.lcm x.2 = n}\ng : ℕ := ⋯\... | apply Prod.ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Order.Antidiag.Nat | {
"line": 299,
"column": 21
} | {
"line": 299,
"column": 52
} | {
"line": 300,
"column": 4
} | [
{
"pp": "case h.fst\nn : ℕ\nhn : n ≠ 0\nb : ℕ × ℕ\nhb : b ∈ {x ∈ n.divisors ×ˢ n.divisors | x.1.lcm x.2 = n}\ng : ℕ := ⋯\na : Fin (succ 0).succ.succ → ℕ := ⋯\nha : a ∈ finMulAntidiag 3 n\n⊢ (f a ha).1 = b.1",
"ppTerm": "?h.fst",
"assigned": true,
"usedConstants": [
"id",
"Prod.fst",
... | [
"case h.fst\nn : ℕ\nhn : n ≠ 0\nb : ℕ × ℕ\nhb : b ∈ {x ∈ n.divisors ×ˢ n.divisors | x.1.lcm x.2 = n}\ng : ℕ := b.1.gcd b.2\na : Fin (succ 0).succ.succ → ℕ := ![g, b.1 / g, b.2 / g]\nha : a ∈ finMulAntidiag 3 n\n⊢ g * (b.1 / g) = b.1"
] | dsimp only [a, Matrix.cons_val] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Order.Antidiag.Nat | {
"line": 299,
"column": 21
} | {
"line": 299,
"column": 52
} | {
"line": 300,
"column": 4
} | [
{
"pp": "case h.snd\nn : ℕ\nhn : n ≠ 0\nb : ℕ × ℕ\nhb : b ∈ {x ∈ n.divisors ×ˢ n.divisors | x.1.lcm x.2 = n}\ng : ℕ := ⋯\na : Fin (succ 0).succ.succ → ℕ := ⋯\nha : a ∈ finMulAntidiag 3 n\n⊢ (f a ha).2 = b.2",
"ppTerm": "?h.snd",
"assigned": true,
"usedConstants": [
"id",
"Nat",
"_p... | [
"case h.snd\nn : ℕ\nhn : n ≠ 0\nb : ℕ × ℕ\nhb : b ∈ {x ∈ n.divisors ×ˢ n.divisors | x.1.lcm x.2 = n}\ng : ℕ := b.1.gcd b.2\na : Fin (succ 0).succ.succ → ℕ := ![g, b.1 / g, b.2 / g]\nha : a ∈ finMulAntidiag 3 n\n⊢ g * (b.2 / g) = b.2"
] | dsimp only [a, Matrix.cons_val] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 1281,
"column": 8
} | {
"line": 1281,
"column": 48
} | {
"line": 1282,
"column": 6
} | [
{
"pp": "case e_a\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nι : Type u_3\ng : ι → MvPolynomial σ R\nb : ι\nB : Finset ι\nhb : b ∉ B\nhb0 : ¬g b = 0\nisunit_gb : IsUnit (m.leadingCoeff (g b))\nhfd : m.toSyn (m.degree (g b + ∑ b ∈ B, g b)) < m.toSyn (m.degree (g b))\nh :\n (∀ b_1 ∈ B,... | [] | rcases hd b' hb' with h | h <;> simp [h] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 320,
"column": 10
} | {
"line": 320,
"column": 12
} | {
"line": 320,
"column": 13
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na : M\n⊢ a ∈ Set.Ici 1 → ∀ ⦃b : M⦄, b ∈ Set.Ici 1 → a ≤ b → mk b ≤ mk a",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
"Set.Ici",
"DivisionCommMonoid.toDi... | [
"M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na : M\nha : a ∈ Set.Ici 1\n⊢ ∀ ⦃b : M⦄, b ∈ Set.Ici 1 → a ≤ b → mk b ≤ mk a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 329,
"column": 10
} | {
"line": 329,
"column": 12
} | {
"line": 329,
"column": 13
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na : M\n⊢ a ∈ Set.Iic 1 → ∀ ⦃b : M⦄, b ∈ Set.Iic 1 → a ≤ b → mk a ≤ mk b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
... | [
"M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na : M\nha : a ∈ Set.Iic 1\n⊢ ∀ ⦃b : M⦄, b ∈ Set.Iic 1 → a ≤ b → mk a ≤ mk b"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Order.Field.GeomSum | {
"line": 49,
"column": 9
} | {
"line": 49,
"column": 24
} | {
"line": 49,
"column": 24
} | [
{
"pp": "K : Type u_1\ninst✝⁵ : Semifield K\ninst✝⁴ : LinearOrder K\ninst✝³ : IsStrictOrderedRing K\ninst✝² : CanonicallyOrderedAdd K\ninst✝¹ : Sub K\ninst✝ : OrderedSub K\nx : K\nh0 : 0 < x\nh1 : x < 1\nn : ℕ\n⊢ 0 < 1 - x",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"K : Type u_1\ninst✝⁵ : Semifield K\ninst✝⁴ : LinearOrder K\ninst✝³ : IsStrictOrderedRing K\ninst✝² : CanonicallyOrderedAdd K\ninst✝¹ : Sub K\ninst✝ : OrderedSub K\nx : K\nh0 : 0 < x\nh1 : x < 1\nn : ℕ\n⊢ x < 1"
] | tsub_pos_iff_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Rearrangement | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 46
} | {
"line": 220,
"column": 4
} | [
{
"pp": "case e'_6\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁸ : Semiring α\ninst✝⁷ : LinearOrder α\ninst✝⁶ : IsStrictOrderedRing α\ninst✝⁵ : ExistsAddOfLE α\ninst✝⁴ : AddCommMonoid β\ninst✝³ : LinearOrder β\ninst✝² : IsOrderedCancelAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulStrictMono α β\ns : Fins... | [] | rw [comp_assoc, self_comp_symm, comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Rearrangement | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 46
} | {
"line": 220,
"column": 4
} | [
{
"pp": "case e'_6\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁸ : Semiring α\ninst✝⁷ : LinearOrder α\ninst✝⁶ : IsStrictOrderedRing α\ninst✝⁵ : ExistsAddOfLE α\ninst✝⁴ : AddCommMonoid β\ninst✝³ : LinearOrder β\ninst✝² : IsOrderedCancelAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulStrictMono α β\ns : Fins... | [] | rw [comp_assoc, self_comp_symm, comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Rearrangement | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 46
} | {
"line": 220,
"column": 4
} | [
{
"pp": "case e'_6\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁸ : Semiring α\ninst✝⁷ : LinearOrder α\ninst✝⁶ : IsStrictOrderedRing α\ninst✝⁵ : ExistsAddOfLE α\ninst✝⁴ : AddCommMonoid β\ninst✝³ : LinearOrder β\ninst✝² : IsOrderedCancelAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulStrictMono α β\ns : Fins... | [] | rw [comp_assoc, self_comp_symm, comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 480,
"column": 2
} | {
"line": 480,
"column": 29
} | {
"line": 481,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nh : mk a < mk b\nhpos : 1 ≤ a\n⊢ b < a",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"MulArchimedeanClass.mk_lt_mk",
"PartialOrder.toPreorder",
... | [
"M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nh✝ : mk a < mk b\nhpos : 1 ≤ a\nh : |b|ₘ ^ 1 < |a|ₘ\n⊢ b < a"
] | obtain h := mk_lt_mk.mp h 1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 486,
"column": 2
} | {
"line": 486,
"column": 29
} | {
"line": 487,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nh : mk a < mk b\nhneg : a ≤ 1\n⊢ a < b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"MulArchimedeanClass.mk_lt_mk",
"PartialOrder.toPreorder",
... | [
"M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nh✝ : mk a < mk b\nhneg : a ≤ 1\nh : |b|ₘ ^ 1 < |a|ₘ\n⊢ a < b"
] | obtain h := mk_lt_mk.mp h 1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 543,
"column": 2
} | {
"line": 543,
"column": 35
} | {
"line": 543,
"column": 35
} | [
{
"pp": "M : Type u_1\ninst✝⁵ : CommGroup M\ninst✝⁴ : LinearOrder M\ninst✝³ : IsOrderedMonoid M\nN : Type u_2\ninst✝² : CommGroup N\ninst✝¹ : LinearOrder N\ninst✝ : IsOrderedMonoid N\nf : M →*o N\nh : Function.Injective ⇑f\na b : MulArchimedeanClass M\n⊢ (orderHom f) a = (orderHom f) b → a = b",
"ppTerm": "... | [
"case mk\nM : Type u_1\ninst✝⁵ : CommGroup M\ninst✝⁴ : LinearOrder M\ninst✝³ : IsOrderedMonoid M\nN : Type u_2\ninst✝² : CommGroup N\ninst✝¹ : LinearOrder N\ninst✝ : IsOrderedMonoid N\nf : M →*o N\nh : Function.Injective ⇑f\nb : MulArchimedeanClass M\na : M\n⊢ (orderHom f) (mk a) = (orderHom f) b → mk a = b"
] | induction a using ind with | mk a
=> _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 692,
"column": 2
} | {
"line": 692,
"column": 67
} | {
"line": 694,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na✝ b✝ : MulArchimedeanClass M\nh : a✝ ≤ b✝\n⊢ b✝.ballSubgroup ≤ a✝.ballSubgroup",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"UpperSet",
"StrictMono.monotone",
"PartialOrder... | [] | exact subgroup_antitone <| (UpperSet.Ioi_strictMono _).monotone h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Group.Cone | {
"line": 56,
"column": 25
} | {
"line": 56,
"column": 73
} | {
"line": 58,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : CommGroup G\np q : GroupCone G\nh : (fun C ↦ C.carrier) p = (fun C ↦ C.carrier) q\n⊢ p = q",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"Mo... | [] | by cases p; cases q; congr; exact SetLike.ext' h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Group.Cyclic | {
"line": 45,
"column": 30
} | {
"line": 45,
"column": 32
} | {
"line": 45,
"column": 32
} | [
{
"pp": "case right\nG : Type u_1\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\nH : Subgroup G\ninst✝ : Nontrivial ↥H\nhH : IsCyclic ↥H\na : G\nha : Subgroup.zpowers a = H\nha1 : 1 < a\n⊢ Subgroup.zpowers a = H",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
... | [
"case right\nG : Type u_1\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\nH : Subgroup G\ninst✝ : Nontrivial ↥H\nhH : IsCyclic ↥H\na : G\nha : Subgroup.zpowers a = H\nha1 : 1 < a\n⊢ H = H"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Interval.Set.SuccPred | {
"line": 130,
"column": 98
} | {
"line": 131,
"column": 71
} | {
"line": 133,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : One α\ninst✝¹ : Sub α\ninst✝ : PredSubOrder α\nb : α\nhb : ¬IsMin b\na : α\n⊢ Icc a (b - 1) = Ico a b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"PredSubOrder.toPredOrder",
"congrArg",
"PartialOrder.toPreorder",... | [] | by
simpa [pred_eq_sub_one] using Icc_pred_right_eq_Ico_of_not_isMin hb a | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Basic | {
"line": 617,
"column": 14
} | {
"line": 617,
"column": 16
} | {
"line": 618,
"column": 6
} | [
{
"pp": "case left\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝¹ : CompleteLattice α\ninst✝ : DecidableLE α\nx✝ : Set (Interval α)\ns : Interval α\n⊢ s ∈ x✝ →\n s ≤\n if h : x✝ ⊆ {⊥} then ⊥\n else ↑{ fst := ⨅ s, ⨅ (_ : ↑s ∈ x✝), s.toProd.1, snd := ⨆ s, ⨆ (_ : ↑s ... | [
"case left\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝¹ : CompleteLattice α\ninst✝ : DecidableLE α\nx✝ : Set (Interval α)\ns : Interval α\nha : s ∈ x✝\n⊢ s ≤\n if h : x✝ ⊆ {⊥} then ⊥\n else ↑{ fst := ⨅ s, ⨅ (_ : ↑s ∈ x✝), s.toProd.1, snd := ⨆ s, ⨆ (_ : ↑s ∈ x✝), s.toProd.2... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.Interval.Basic | {
"line": 617,
"column": 6
} | {
"line": 629,
"column": 42
} | {
"line": 630,
"column": 4
} | [
{
"pp": "case left\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝¹ : CompleteLattice α\ninst✝ : DecidableLE α\nx✝ : Set (Interval α)\n⊢ (if h : x✝ ⊆ {⊥} then ⊥\n else ↑{ fst := ⨅ s, ⨅ (_ : ↑s ∈ x✝), s.toProd.1, snd := ⨆ s, ⨆ (_ : ↑s ∈ x✝), s.toProd.2, fst_le_snd := ⋯ }) ∈\n ... | [] | intro s ha
split_ifs with h
· exact (h ha).le
cases s
· exact bot_le
· -- Porting note: This case was
-- `exact WithBot.some_le_some.2 ⟨iInf₂_le _ ha, le_iSup₂_of_le _ ha le_rfl⟩`
-- but there seems to be a defEq-problem at `iInf₂_le` that lean cannot resolve yet.
a... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Basic | {
"line": 617,
"column": 6
} | {
"line": 629,
"column": 42
} | {
"line": 630,
"column": 4
} | [
{
"pp": "case left\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝¹ : CompleteLattice α\ninst✝ : DecidableLE α\nx✝ : Set (Interval α)\n⊢ (if h : x✝ ⊆ {⊥} then ⊥\n else ↑{ fst := ⨅ s, ⨅ (_ : ↑s ∈ x✝), s.toProd.1, snd := ⨆ s, ⨆ (_ : ↑s ∈ x✝), s.toProd.2, fst_le_snd := ⋯ }) ∈\n ... | [] | intro s ha
split_ifs with h
· exact (h ha).le
cases s
· exact bot_le
· -- Porting note: This case was
-- `exact WithBot.some_le_some.2 ⟨iInf₂_le _ ha, le_iSup₂_of_le _ ha le_rfl⟩`
-- but there seems to be a defEq-problem at `iInf₂_le` that lean cannot resolve yet.
a... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Basic | {
"line": 630,
"column": 14
} | {
"line": 630,
"column": 16
} | {
"line": 631,
"column": 6
} | [
{
"pp": "case right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝¹ : CompleteLattice α\ninst✝ : DecidableLE α\nx✝ : Set (Interval α)\ns : Interval α\n⊢ s ∈ upperBounds x✝ →\n (if h : x✝ ⊆ {⊥} then ⊥\n else ↑{ fst := ⨅ s, ⨅ (_ : ↑s ∈ x✝), s.toProd.1, snd := ⨆ s, ⨆ (_ : ... | [
"case right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝¹ : CompleteLattice α\ninst✝ : DecidableLE α\nx✝ : Set (Interval α)\ns : Interval α\nha : s ∈ upperBounds x✝\n⊢ (if h : x✝ ⊆ {⊥} then ⊥\n else ↑{ fst := ⨅ s, ⨅ (_ : ↑s ∈ x✝), s.toProd.1, snd := ⨆ s, ⨆ (_ : ↑s ∈ x✝), s.toP... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.Interval.Basic | {
"line": 651,
"column": 14
} | {
"line": 651,
"column": 16
} | {
"line": 652,
"column": 6
} | [
{
"pp": "case left\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝¹ : CompleteLattice α\ninst✝ : DecidableLE α\ns₁ : Set (Interval α)\ns : Interval α\n⊢ s ∈ s₁ →\n (if h :\n ⊥ ∉ s₁ ∧\n ∀ ⦃s : NonemptyInterval α⦄, ↑s ∈ s₁ → ∀ ⦃t : NonemptyInterval α⦄, ↑t ∈ ... | [
"case left\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝¹ : CompleteLattice α\ninst✝ : DecidableLE α\ns₁ : Set (Interval α)\ns : Interval α\nha : s ∈ s₁\n⊢ (if h :\n ⊥ ∉ s₁ ∧\n ∀ ⦃s : NonemptyInterval α⦄, ↑s ∈ s₁ → ∀ ⦃t : NonemptyInterval α⦄, ↑t ∈ s₁ → s.toProd.1 ≤... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.Interval.Basic | {
"line": 659,
"column": 14
} | {
"line": 659,
"column": 16
} | {
"line": 660,
"column": 6
} | [
{
"pp": "case right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝¹ : CompleteLattice α\ninst✝ : DecidableLE α\ns₁ : Set (Interval α)\ns : Interval α\n⊢ s ∈ lowerBounds s₁ →\n s ≤\n if h :\n ⊥ ∉ s₁ ∧\n ∀ ⦃s : NonemptyInterval α⦄, ↑s ∈ s₁ → ∀ ⦃t : Non... | [
"case right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝¹ : CompleteLattice α\ninst✝ : DecidableLE α\ns₁ : Set (Interval α)\ns : Interval α\nha : s ∈ lowerBounds s₁\n⊢ s ≤\n if h :\n ⊥ ∉ s₁ ∧\n ∀ ⦃s : NonemptyInterval α⦄, ↑s ∈ s₁ → ∀ ⦃t : NonemptyInterval α⦄, ↑... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Data.Finset.MulAntidiagonal | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 32
} | {
"line": 90,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoid α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedCancelMonoid α\ns t : Set α\nhs : s.IsPWO\nht : t.IsPWO\na : α\nx✝ : a ∈ {a | (mulAntidiagonal hs ht a).Nonempty}\nb : α × α\nhb : b ∈ mulAntidiagonal hs ht a\n⊢ a ∈ s * t",
"ppTerm": "?m.30",
"assigned": true,
"... | [
"α : Type u_1\ninst✝² : CommMonoid α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedCancelMonoid α\ns t : Set α\nhs : s.IsPWO\nht : t.IsPWO\na : α\nx✝ : a ∈ {a | (mulAntidiagonal hs ht a).Nonempty}\nb : α × α\nhb : b.1 ∈ s ∧ b.2 ∈ t ∧ b.1 * b.2 = a\n⊢ a ∈ s * t"
] | rw [mem_mulAntidiagonal] at hb | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HahnSeries.Addition | {
"line": 319,
"column": 4
} | {
"line": 320,
"column": 18
} | {
"line": 322,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nS : Type u_4\nU : Type u_5\nV : Type u_6\nα : Type u_7\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\nx y : R⟦Γ⟧\n⊢ x + y = y + x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddSemigroup",
"AddMonoid.toAddZ... | [] | ext
apply add_comm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Addition | {
"line": 319,
"column": 4
} | {
"line": 320,
"column": 18
} | {
"line": 322,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nS : Type u_4\nU : Type u_5\nV : Type u_6\nα : Type u_7\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\nx y : R⟦Γ⟧\n⊢ x + y = y + x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddSemigroup",
"AddMonoid.toAddZ... | [] | ext
apply add_comm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 28
} | {
"line": 142,
"column": 0
} | [
{
"pp": "case inr\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⟧\nhge : 0 ≤ x\n⊢ (ofLex |x|).support = (ofLex x).support",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"HahnSeries.suppor... | [] | rw [abs_eq_self.mpr hge] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 28
} | {
"line": 142,
"column": 0
} | [
{
"pp": "case inr\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⟧\nhge : 0 ≤ x\n⊢ (ofLex |x|).support = (ofLex x).support",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"HahnSeries.suppor... | [] | rw [abs_eq_self.mpr hge] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 28
} | {
"line": 142,
"column": 0
} | [
{
"pp": "case inr\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⟧\nhge : 0 ≤ x\n⊢ (ofLex |x|).support = (ofLex x).support",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"HahnSeries.suppor... | [] | rw [abs_eq_self.mpr hge] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 28
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case inr\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⟧\nhge : 0 ≤ x\n⊢ (ofLex |x|).orderTop = (ofLex x).orderTop",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
... | [] | rw [abs_eq_self.mpr hge] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 28
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case inr\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⟧\nhge : 0 ≤ x\n⊢ (ofLex |x|).orderTop = (ofLex x).orderTop",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
... | [] | rw [abs_eq_self.mpr hge] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 28
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case inr\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⟧\nhge : 0 ≤ x\n⊢ (ofLex |x|).orderTop = (ofLex x).orderTop",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
... | [] | rw [abs_eq_self.mpr hge] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 111,
"column": 10
} | {
"line": 111,
"column": 12
} | {
"line": 111,
"column": 13
} | [
{
"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... | [
"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✝¹ : LinearO... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 121,
"column": 10
} | {
"line": 121,
"column": 12
} | {
"line": 121,
"column": 13
} | [
{
"pp": "K : Type u_1\ninst✝⁸ : DivisionRing K\ninst✝⁷ : LinearOrder K\ninst✝⁶ : IsOrderedRing K\ninst✝⁵ : Archimedean K\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedAddMonoid M\ninst✝¹ : Module K M\ninst✝ : IsOrderedModule K M\nu : ArchimedeanStrata K M\nc : FiniteArchimede... | [
"K : Type u_1\ninst✝⁸ : DivisionRing K\ninst✝⁷ : LinearOrder K\ninst✝⁶ : IsOrderedRing K\ninst✝⁵ : Archimedean K\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedAddMonoid M\ninst✝¹ : Module K M\ninst✝ : IsOrderedModule K M\nu : ArchimedeanStrata K M\nc : FiniteArchimedeanClass M\na... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Order.Quantale | {
"line": 171,
"column": 43
} | {
"line": 171,
"column": 57
} | {
"line": 171,
"column": 57
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nx y z : α\ns : Set α\nf : ι → α\ninst✝² : Semigroup α\ninst✝¹ : CompleteLattice α\ninst✝ : IsQuantale α\nm✝ n₁✝ n₂✝ : α\na✝ : n₁✝ ≤ n₂✝\n⊢ m✝ * n₂✝ = m✝ * (n₂✝ ⊔ n₁✝)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul... | [
"α : Type u_1\nι : Type u_2\nx y z : α\ns : Set α\nf : ι → α\ninst✝² : Semigroup α\ninst✝¹ : CompleteLattice α\ninst✝ : IsQuantale α\nm✝ n₁✝ n₂✝ : α\na✝ : n₁✝ ≤ n₂✝\n⊢ m✝ * n₂✝ = m✝ * n₂✝",
"α : Type u_1\nι : Type u_2\nx y z : α\ns : Set α\nf : ι → α\ninst✝² : Semigroup α\ninst✝¹ : CompleteLattice α\ninst✝ : IsQu... | sup_of_le_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Quantale | {
"line": 177,
"column": 43
} | {
"line": 177,
"column": 57
} | {
"line": 177,
"column": 57
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nx y z : α\ns : Set α\nf : ι → α\ninst✝² : Semigroup α\ninst✝¹ : CompleteLattice α\ninst✝ : IsQuantale α\nm✝ n₁✝ n₂✝ : α\na✝ : n₁✝ ≤ n₂✝\n⊢ swap (fun x1 x2 ↦ x1 * x2) m✝ n₂✝ = (n₂✝ ⊔ n₁✝) * m✝",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"α : Type u_1\nι : Type u_2\nx y z : α\ns : Set α\nf : ι → α\ninst✝² : Semigroup α\ninst✝¹ : CompleteLattice α\ninst✝ : IsQuantale α\nm✝ n₁✝ n₂✝ : α\na✝ : n₁✝ ≤ n₂✝\n⊢ swap (fun x1 x2 ↦ x1 * x2) m✝ n₂✝ = n₂✝ * m✝",
"α : Type u_1\nι : Type u_2\nx y z : α\ns : Set α\nf : ι → α\ninst✝² : Semigroup α\ninst✝¹ : Comple... | sup_of_le_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 188,
"column": 18
} | {
"line": 188,
"column": 20
} | {
"line": 189,
"column": 10
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nS : Type u_4\nV : Type u_5\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : SMul R V\ninst✝⁴ : PartialOrder Γ'\ninst✝³ : VAdd Γ Γ'\ninst✝² : IsOrderedCancelVAdd Γ Γ'\ninst✝¹ : Zero R\ninst✝ : AddCommMonoid V\nx : R⟦Γ⟧\ny : HahnModule Γ' R V\na : Γ'\n⊢ a ∈ {a | ∑ ij ∈ VAddAn... | [
"Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nS : Type u_4\nV : Type u_5\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : SMul R V\ninst✝⁴ : PartialOrder Γ'\ninst✝³ : VAdd Γ Γ'\ninst✝² : IsOrderedCancelVAdd Γ Γ'\ninst✝¹ : Zero R\ninst✝ : AddCommMonoid V\nx : R⟦Γ⟧\ny : HahnModule Γ' R V\na : Γ'\nha : a ∈ {a | ∑ ij ∈ VAddAntidiagona... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Order.Ring.Cone | {
"line": 38,
"column": 25
} | {
"line": 38,
"column": 73
} | {
"line": 40,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np q : RingCone R\nh : (fun C ↦ C.carrier) p = (fun C ↦ C.carrier) q\n⊢ p = q",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"Subsemiring.instSetLike",
"Submonoid.toSubsemigroup",
"Membership.mem",
"... | [] | by cases p; cases q; congr; exact SetLike.ext' h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 57,
"column": 25
} | {
"line": 57,
"column": 73
} | {
"line": 59,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\np q : RingPreordering R\nh : (fun P ↦ (↑P).carrier) p = (fun P ↦ (↑P).carrier) q\n⊢ p = q",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"CommRing",
"Subsemiring.instSetLike",
"RingPreordering.toSubse... | [] | by cases p; cases q; congr; exact SetLike.ext' h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Ring.Archimedean | {
"line": 314,
"column": 72
} | {
"line": 314,
"column": 87
} | {
"line": 314,
"column": 88
} | [
{
"pp": "case mk\nR : Type u_1\nS : Type u_2\ninst✝³ : LinearOrder R\ninst✝² : LinearOrder S\ninst✝¹ : Field R\ninst✝ : IsOrderedRing R\nn : ℕ\nx : R\n⊢ -(mk (x ^ n) + mk x) = -(↑n • mk x + mk x)",
"ppTerm": "?mk",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"ins... | [
"case mk\nR : Type u_1\nS : Type u_2\ninst✝³ : LinearOrder R\ninst✝² : LinearOrder S\ninst✝¹ : Field R\ninst✝ : IsOrderedRing R\nn : ℕ\nx : R\n⊢ -(mk (x ^ ↑n) + mk x) = -(↑n • mk x + mk x)"
] | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 359,
"column": 41
} | {
"line": 359,
"column": 75
} | {
"line": 359,
"column": 75
} | [
{
"pp": "case refine_2.inr\nK : 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 : A... | [
"case refine_2.inr\nK : 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 ... | HahnSeries.coeff_truncLT_of_le hdc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 968,
"column": 4
} | {
"line": 968,
"column": 19
} | {
"line": 969,
"column": 4
} | [
{
"pp": "Γ : 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⟧\nhyz : (fun x_1 ↦ x * x... | [
"Γ : 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 : AddCancelCommMonoid R := { ... | contrapose! hyz | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 990,
"column": 4
} | {
"line": 990,
"column": 19
} | {
"line": 991,
"column": 4
} | [
{
"pp": "Γ : 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⟧\nhyz : (fun x_1 ↦ x_1 *... | [
"Γ : 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 : AddCancelCommMonoid R := { ... | contrapose! hyz | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.Order.Ring.Units | {
"line": 18,
"column": 2
} | {
"line": 18,
"column": 25
} | {
"line": 19,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\n⊢ ∃ a, ∀ (b : Rˣ), Xor (b * a ∈ posSubgroup R) (b ∈ posSubgroup R)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Ring.toNonAssocRing",
"Monoid.toMulOneClass",
... | [
"R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : Rˣ\n⊢ Xor (a * -1 ∈ posSubgroup R) (a ∈ posSubgroup R)"
] | refine ⟨-1, fun a ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Order.Ring.Units | {
"line": 21,
"column": 2
} | {
"line": 21,
"column": 28
} | {
"line": 22,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : Rˣ\nh : 0 < ↑a\n⊢ Xor (a * -1 ∈ posSubgroup R) (a ∈ posSubgroup R)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Left.neg_pos_iff._simp_1",
"AddGroup.toSubtractionMono... | [] | · simp [h, xor_comm, h.le] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.Star.Pi | {
"line": 25,
"column": 4
} | {
"line": 33,
"column": 37
} | {
"line": 34,
"column": 4
} | [
{
"pp": "ι : Type u_1\ninst✝⁴ : Finite ι\nA : ι → Type u_2\ninst✝³ : (i : ι) → PartialOrder (A i)\ninst✝² : (i : ι) → NonUnitalSemiring (A i)\ninst✝¹ : (i : ι) → StarRing (A i)\ninst✝ : ∀ (i : ι), StarOrderedRing (A i)\nxa xy : (i : ι) → A i\n⊢ xa ≤ xy ↔ ∃ p ∈ closure (Set.range fun s ↦ star s * s), xy = xa + p... | [
"ι : Type u_1\ninst✝⁴ : Finite ι\nA : ι → Type u_2\ninst✝³ : (i : ι) → PartialOrder (A i)\ninst✝² : (i : ι) → NonUnitalSemiring (A i)\ninst✝¹ : (i : ι) → StarRing (A i)\ninst✝ : ∀ (i : ι), StarOrderedRing (A i)\nxa xy : (i : ι) → A i\nthis : closure (Set.range fun s ↦ star s * s) = pi Set.univ fun i ↦ closure (Set.... | have : closure (Set.range fun s : Π i, A i ↦ star s * s) =
pi Set.univ fun i => (closure <| Set.range fun s : A i ↦ star s * s) := by
rw [← closure_pi fun _ => Set.mem_range.mpr ⟨0, by simp⟩]
congr
ext x
simp only [Set.mem_range, funext_iff, mul_apply, star_apply, Set.mem_pi,
Set... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 582,
"column": 59
} | {
"line": 582,
"column": 74
} | {
"line": 583,
"column": 4
} | [
{
"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... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 637,
"column": 14
} | {
"line": 637,
"column": 16
} | {
"line": 637,
"column": 17
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na b : Kˣ\n⊢ a ∈ {x | A.valuation (↑x - 1) < 1} → b ∈ {x | A.valuation (↑x - 1) < 1} → a * b ∈ {x | A.valuation (↑x - 1) < 1}",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Units.val",
"LinearOrderedCommGroupWithZer... | [
"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na b : Kˣ\nha : a ∈ {x | A.valuation (↑x - 1) < 1}\n⊢ b ∈ {x | A.valuation (↑x - 1) < 1} → a * b ∈ {x | A.valuation (↑x - 1) < 1}"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 646,
"column": 12
} | {
"line": 646,
"column": 14
} | {
"line": 647,
"column": 4
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : Kˣ\n⊢ A.valuation (↑a - 1) < 1 → A.valuation (↑a⁻¹ - 1) < 1",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"Units.val",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"Preorder.toLT",... | [
"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : Kˣ\nha : A.valuation (↑a - 1) < 1\n⊢ A.valuation (↑a⁻¹ - 1) < 1"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Pointwise.Stabilizer | {
"line": 57,
"column": 28
} | {
"line": 57,
"column": 30
} | {
"line": 57,
"column": 30
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nf : G →* H\ns : Set G\na : G\nha : a • s = s\n⊢ ⇑f '' (a • s) = ⇑f '' s",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"MonoidHom.instFunLike",
"instSMulOfMul",
"Mo... | [
"G : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nf : G →* H\ns : Set G\na : G\nha : a • s = s\n⊢ ⇑f '' s = ⇑f '' s"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 806,
"column": 48
} | {
"line": 806,
"column": 82
} | {
"line": 806,
"column": 82
} | [
{
"pp": "case pos.inr\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 : AddC... | [] | HahnSeries.coeff_truncLT_of_le hdc | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.CoeffList | {
"line": 146,
"column": 14
} | {
"line": 149,
"column": 7
} | {
"line": 150,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nP : R[X]\nh : P ≠ 0\nhdp : ¬P.natDegree = 0\nhep : ¬P.eraseLead = 0\nh₁ : P.degree.succ = P.natDegree + 1\nh₂ : P.eraseLead.degree.succ = P.eraseLead.natDegree + 1\nn : ℕ\nhn : P.natDegree = P.eraseLead.natDegree + 1 + n\nk : ℕ\nhkd : k + 1 < P.natDegree + 1\ndk : ℕ\nh... | [] | arg 1; equals P.eraseLead.coeff dk =>
rw [eraseLead_coeff_of_ne (f := P) dk (by lia)]
congr
lia | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Algebra.Polynomial.CoeffList | {
"line": 146,
"column": 14
} | {
"line": 149,
"column": 7
} | {
"line": 150,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nP : R[X]\nh : P ≠ 0\nhdp : ¬P.natDegree = 0\nhep : ¬P.eraseLead = 0\nh₁ : P.degree.succ = P.natDegree + 1\nh₂ : P.eraseLead.degree.succ = P.eraseLead.natDegree + 1\nn : ℕ\nhn : P.natDegree = P.eraseLead.natDegree + 1 + n\nk : ℕ\nhkd : k + 1 < P.natDegree + 1\ndk : ℕ\nh... | [] | arg 1; equals P.eraseLead.coeff dk =>
rw [eraseLead_coeff_of_ne (f := P) dk (by lia)]
congr
lia | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 223,
"column": 6
} | {
"line": 223,
"column": 57
} | {
"line": 223,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np q : R[X]\nn : ℕ\nhn : n ≠ 0\nhp : (p - X ^ n).natDegree < n\nhq : (q - X ^ n).natDegree < n\n⊢ (p - X ^ n - (q - X ^ n)).natDegree < n",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Nat.le_sub_one_iff_lt",
"Eq.mpr",
"AddGroupWith... | [
"R : Type u_1\ninst✝ : Ring R\np q : R[X]\nn : ℕ\nhn : n ≠ 0\nhp : (p - X ^ n).natDegree ≤ n - 1\nhq : (q - X ^ n).natDegree ≤ n - 1\n⊢ (p - X ^ n - (q - X ^ n)).natDegree ≤ n - 1"
] | ← Nat.le_sub_one_iff_lt (Nat.zero_lt_of_ne_zero hn) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 970,
"column": 2
} | {
"line": 970,
"column": 74
} | {
"line": 971,
"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... | obtain ⟨e, hdomain⟩ := HahnEmbedding.Partial.exists_domain_eq_top h.some | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Polynomial.DenomsClearable | {
"line": 90,
"column": 2
} | {
"line": 95,
"column": 25
} | {
"line": 96,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\nf : ℤ[X]\na b : ℤ\nb0 : 0 < b\nfab : eval (↑a / ↑b) (Polynomial.map (algebraMap ℤ K) f) ≠ 0\n⊢ 1 ≤ ↑b ^ f.natDegree * |eval (↑a / ↑b) (Polynomial.map (algebraMap ℤ K) f)|",
"ppTerm": "?m.54",
"assigned": true... | [
"K : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\nf : ℤ[X]\na b : ℤ\nb0 : 0 < b\nfab : eval (↑a / ↑b) (Polynomial.map (algebraMap ℤ K) f) ≠ 0\nev : ℤ\nbi : K\nbu : bi * (algebraMap ℤ K) b = 1\nhF :\n (algebraMap ℤ K) ev =\n (algebraMap ℤ K) b ^ f.natDegree * eval ((algebraM... | obtain ⟨ev, bi, bu, hF⟩ :=
denomsClearable_natDegree (b := b) (algebraMap ℤ K) f a
(by
rw [eq_intCast, one_div_mul_cancel]
rw [Int.cast_ne_zero]
exact b0.ne.symm) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.List.Destutter | {
"line": 196,
"column": 8
} | {
"line": 196,
"column": 30
} | {
"line": 197,
"column": 8
} | [
{
"pp": "α : Type u_1\nR : α → α → Prop\ninst✝ : DecidableRel R\nb : α\ni : IsTrans α Rᶜ\na c : α\nl : List α\nhba : ¬R b a\nhbc : ¬R b c\nhac : ¬R a c\n⊢ (destutter' R b l).length ≤ (if R a c then a :: destutter' R c l else destutter' R a l).length",
"ppTerm": "?m.79",
"assigned": true,
"usedConsta... | [
"α : Type u_1\nR : α → α → Prop\ninst✝ : DecidableRel R\nb : α\ni : IsTrans α Rᶜ\na c : α\nl : List α\nhba : ¬R b a\nhbc : ¬R b c\nhac : ¬R a c\n⊢ (destutter' R b l).length ≤ (destutter' R a l).length"
] | simp only [if_neg hac] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 118,
"column": 17
} | {
"line": 118,
"column": 74
} | {
"line": 120,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝³ : Semiring R\nS : Type u_2\ninst✝² : AddCommMonoid S\ninst✝¹ : Pow S ℕ\ninst✝ : Module R S\nx : S\nn : ℕ\nih : (↑n).smeval x = n • x ^ 0\n⊢ (↑(n + 1)).smeval x = (n + 1) • x ^ 0",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [] | rw [n.cast_succ, smeval_add, ih, smeval_one, ← add_nsmul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 118,
"column": 17
} | {
"line": 118,
"column": 74
} | {
"line": 120,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝³ : Semiring R\nS : Type u_2\ninst✝² : AddCommMonoid S\ninst✝¹ : Pow S ℕ\ninst✝ : Module R S\nx : S\nn : ℕ\nih : (↑n).smeval x = n • x ^ 0\n⊢ (↑(n + 1)).smeval x = (n + 1) • x ^ 0",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [] | rw [n.cast_succ, smeval_add, ih, smeval_one, ← add_nsmul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 118,
"column": 17
} | {
"line": 118,
"column": 74
} | {
"line": 120,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝³ : Semiring R\nS : Type u_2\ninst✝² : AddCommMonoid S\ninst✝¹ : Pow S ℕ\ninst✝ : Module R S\nx : S\nn : ℕ\nih : (↑n).smeval x = n • x ^ 0\n⊢ (↑(n + 1)).smeval x = (n + 1) • x ^ 0",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [] | rw [n.cast_succ, smeval_add, ih, smeval_one, ← add_nsmul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 38
} | {
"line": 89,
"column": 2
} | [
{
"pp": "case neg.nil\nR : Type u_1\ninst✝¹ : Semiring R\ninst✝ : LinearOrder R\nP : R[X]\nh : P ≠ 0\nhpz : ¬P = 0\nhsl : SignType.sign P.leadingCoeff ≠ 0\nh_eL : P.eraseLead.coeffList = []\n⊢ (List.destutter (fun x1 x2 ↦ x1 ≠ x2)\n (List.filter (fun x ↦ decide (x ≠ 0))\n (SignType.sign P.le... | [
"case neg.cons\nR : Type u_1\ninst✝¹ : Semiring R\ninst✝ : LinearOrder R\nP : R[X]\nh : P ≠ 0\nhpz : ¬P = 0\nhsl : SignType.sign P.leadingCoeff ≠ 0\nc : R\ncs : List R\nh_eL : P.eraseLead.coeffList = c :: cs\n⊢ (List.destutter (fun x1 x2 ↦ x1 ≠ x2)\n (List.filter (fun x ↦ decide (x ≠ 0))\n (Sign... | · simp [coeffList_eq_nil.mp h_eL, h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 185,
"column": 4
} | {
"line": 185,
"column": 31
} | {
"line": 186,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nP : R[X]\nη : R\nhη : 0 < η\nhP₀ : 0 < P.leadingCoeff\nhc : P.nextCoeff < 0\nd : ℕ\nhd : P.natDegree = d + 1\nhndxP : ((X - C η) * P).natDegree = P.natDegree + 1\nhndxeP : ((X - C η) * P.eraseLead).natDegree = P.natDe... | [] | · exact hd ▸ mul_pos hη hP₀ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 222,
"column": 61
} | {
"line": 222,
"column": 76
} | {
"line": 222,
"column": 76
} | [
{
"pp": "case h\np : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ 0 < n - m",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Nat.instOrderedSub",
"Preorder.to... | [
"case h\np : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ m < n"
] | tsub_pos_iff_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 261,
"column": 4
} | {
"line": 269,
"column": 73
} | {
"line": 270,
"column": 2
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nP : R[X]\nη : R\nh₁ : 0 < P.leadingCoeff\nh₃ : P ≠ 0\nh₄ : P.eraseLead.natDegree + 1 = P.natDegree\nh₅ : X - C η ≠ 0\nh₆ : P.eraseLead ≠ 0\nd : ℕ\nhd : P.natDegree = 0 + d + 1\nh₂ : P.eraseLead.leadingCoeff ... | [] | · suffices ((X - C η) * P).eraseLead.eraseLead = ((X - C η) * P.eraseLead).eraseLead by
have := leadingCoeff_cons_eraseLead h₉
have := coeffList_eraseLead (mt nextCoeff_eq_zero_of_eraseLead_eq_zero h₉)
grind [leadingCoeff_eraseLead_eq_nextCoeff]
suffices monomial P.natDegree ((X - C η) * P... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 309,
"column": 8
} | {
"line": 309,
"column": 45
} | {
"line": 309,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nη : R\nhη : 0 < η\nd : ℕ\nih : ∀ m < d, ∀ {P : R[X]}, P ≠ 0 → P.natDegree = m → P.signVariations + 1 ≤ ((X - C η) * P).signVariations\nP : R[X]\nhP : P ≠ 0\nhd : P.natDegree = d\nh_lC : 0 < P.leadingCoeff\nh_mul : (X ... | [
"R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nη : R\nhη : 0 < η\nd : ℕ\nih : ∀ m < d, ∀ {P : R[X]}, P ≠ 0 → P.natDegree = m → P.signVariations + 1 ≤ ((X - C η) * P).signVariations\nP : R[X]\nhP : P ≠ 0\nhd : P.natDegree = d\nh_lC : 0 < P.leadingCoeff\nh_mul : (X - C η) * P ≠... | natDegree_mul (X_sub_C_ne_zero η) hP, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 12
} | {
"line": 288,
"column": 2
} | [
{
"pp": "R : Type u_2\na b : R\ninst✝ : CommRing R\nz : QuadraticAlgebra R a b\nhz : ∀ (x : QuadraticAlgebra R a b), x * z = 0 → x = 0\n⊢ ∀ (x : QuadraticAlgebra R a b), x * star z = 0 → x = 0",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"QuadraticAlgebra",
... | [
"R : Type u_2\na b : R\ninst✝ : CommRing R\nz : QuadraticAlgebra R a b\nhz : ∀ (x : QuadraticAlgebra R a b), x * z = 0 → x = 0\nw : QuadraticAlgebra R a b\nhw : w * star z = 0\n⊢ w = 0"
] | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 316,
"column": 18
} | {
"line": 316,
"column": 82
} | {
"line": 317,
"column": 6
} | [
{
"pp": "R : 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).degr... | [
"R : 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).degree\nhr₂ : ∀ ... | rw [← Nat.sub_add_cancel (Nat.one_le_iff_ne_zero.2 j.neZero.ne)] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 316,
"column": 18
} | {
"line": 316,
"column": 82
} | {
"line": 317,
"column": 6
} | [
{
"pp": "R : 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).degr... | [
"R : 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).degree\nhr₂ : ∀ ... | rw [← Nat.sub_add_cancel (Nat.one_le_iff_ne_zero.2 j.neZero.ne)] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 316,
"column": 18
} | {
"line": 316,
"column": 82
} | {
"line": 317,
"column": 6
} | [
{
"pp": "R : 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).degr... | [
"R : 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).degree\nhr₂ : ∀ ... | rw [← Nat.sub_add_cancel (Nat.one_le_iff_ne_zero.2 j.neZero.ne)] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 299,
"column": 61
} | {
"line": 370,
"column": 65
} | {
"line": 372,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nP : R[X]\nη : R\nhη : 0 < η\nhP : P ≠ 0\n⊢ P.signVariations + 1 ≤ ((X - C η) * P).signVariations",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mp... | [] | by
-- do induction on the degree
generalize hd : P.natDegree = d
induction d using Nat.strong_induction_on generalizing P with | _ d ih =>
-- can assume it starts positive, otherwise negate P
wlog h_lC : 0 < leadingCoeff P generalizing P with H
· simpa using @H (-P) (by simpa) (by simpa) (by grind [leadingC... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Quandle | {
"line": 685,
"column": 10
} | {
"line": 686,
"column": 41
} | {
"line": 688,
"column": 0
} | [
{
"pp": "case inv\nR : Type u_1\ninst✝¹ : Rack R\nG : Type u_2\ninst✝ : Group G\nF : EnvelGroup R →* G\nx✝ : EnvelGroup R\nx : PreEnvelGroup R\nih_x :\n ((fun f ↦ { toFun := fun x ↦ Quotient.liftOn x (mapAux f) ⋯, map_one' := ⋯, map_mul' := ⋯ })\n ((fun F ↦ (Quandle.Conj.map F).comp (toEnvelGroup R)) F)... | [] | have hm : ⟦x.inv⟧ = @Inv.inv (EnvelGroup R) _ ⟦x⟧ := rfl
rw [hm, map_inv, map_inv, ih_x] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Quandle | {
"line": 685,
"column": 10
} | {
"line": 686,
"column": 41
} | {
"line": 688,
"column": 0
} | [
{
"pp": "case inv\nR : Type u_1\ninst✝¹ : Rack R\nG : Type u_2\ninst✝ : Group G\nF : EnvelGroup R →* G\nx✝ : EnvelGroup R\nx : PreEnvelGroup R\nih_x :\n ((fun f ↦ { toFun := fun x ↦ Quotient.liftOn x (mapAux f) ⋯, map_one' := ⋯, map_mul' := ⋯ })\n ((fun F ↦ (Quandle.Conj.map F).comp (toEnvelGroup R)) F)... | [] | have hm : ⟦x.inv⟧ = @Inv.inv (EnvelGroup R) _ ⟦x⟧ := rfl
rw [hm, map_inv, map_inv, ih_x] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.QuaternionBasis | {
"line": 172,
"column": 16
} | {
"line": 172,
"column": 41
} | {
"line": 173,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nc₁ c₂ c₃ : R\nq : Basis A c₁ c₂ c₃\nF : A →ₐ[R] B\n⊢ F q.i * F q.j = F q.k",
"ppTerm": "?m.164",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [← map_mul, q.i_mul_j] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.QuaternionBasis | {
"line": 172,
"column": 16
} | {
"line": 172,
"column": 41
} | {
"line": 173,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nc₁ c₂ c₃ : R\nq : Basis A c₁ c₂ c₃\nF : A →ₐ[R] B\n⊢ F q.i * F q.j = F q.k",
"ppTerm": "?m.164",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [← map_mul, q.i_mul_j] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.QuaternionBasis | {
"line": 172,
"column": 16
} | {
"line": 172,
"column": 41
} | {
"line": 173,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nc₁ c₂ c₃ : R\nq : Basis A c₁ c₂ c₃\nF : A →ₐ[R] B\n⊢ F q.i * F q.j = F q.k",
"ppTerm": "?m.164",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [← map_mul, q.i_mul_j] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.RingQuot | {
"line": 109,
"column": 24
} | {
"line": 109,
"column": 59
} | {
"line": 110,
"column": 6
} | [
{
"pp": "case mp.rel.mul_left\nR : Type uR\ninst✝ : Semiring R\nr : R → R → Prop\nx₁ x₂ x✝ y✝ a✝¹ b✝ c✝ : R\na✝ : Rel r a✝¹ b✝\nh : RingConGen.Rel r a✝¹ b✝\n⊢ RingConGen.Rel r (a✝¹ * c✝) (b✝ * c✝)",
"ppTerm": "?mp.rel.mul_left",
"assigned": true,
"usedConstants": [
"RingConGen.Rel.mul",
... | [] | exact h.mul (RingConGen.Rel.refl _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.RingQuot | {
"line": 109,
"column": 24
} | {
"line": 109,
"column": 59
} | {
"line": 110,
"column": 6
} | [
{
"pp": "case mp.rel.mul_left\nR : Type uR\ninst✝ : Semiring R\nr : R → R → Prop\nx₁ x₂ x✝ y✝ a✝¹ b✝ c✝ : R\na✝ : Rel r a✝¹ b✝\nh : RingConGen.Rel r a✝¹ b✝\n⊢ RingConGen.Rel r (a✝¹ * c✝) (b✝ * c✝)",
"ppTerm": "?mp.rel.mul_left",
"assigned": true,
"usedConstants": [
"RingConGen.Rel.mul",
... | [] | exact h.mul (RingConGen.Rel.refl _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.RingQuot | {
"line": 109,
"column": 24
} | {
"line": 109,
"column": 59
} | {
"line": 110,
"column": 6
} | [
{
"pp": "case mp.rel.mul_left\nR : Type uR\ninst✝ : Semiring R\nr : R → R → Prop\nx₁ x₂ x✝ y✝ a✝¹ b✝ c✝ : R\na✝ : Rel r a✝¹ b✝\nh : RingConGen.Rel r a✝¹ b✝\n⊢ RingConGen.Rel r (a✝¹ * c✝) (b✝ * c✝)",
"ppTerm": "?mp.rel.mul_left",
"assigned": true,
"usedConstants": [
"RingConGen.Rel.mul",
... | [] | exact h.mul (RingConGen.Rel.refl _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.