module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.MvPolynomial.Division | {
"line": 375,
"column": 4
} | {
"line": 377,
"column": 36
} | {
"line": 379,
"column": 0
} | [
{
"pp": "case inr.succ.refine_2\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np✝ q✝ : MvPolynomial σ R\ninst✝ : IsCancelMulZero R\nn✝ : σ →₀ ℕ\nhR : Nontrivial R\nd : ℕ\nhd :\n ∀ (n : σ →₀ ℕ),\n Finsupp.degree n = d →\n ∀ (p q : MvPolynomial σ R), p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p =... | [] | · rw [hp, ← add_tsub_cancel_of_le hmn, ← mul_one 1, ← monomial_mul, mul_one, mul_assoc]
apply mul_dvd_mul dvd_rfl
apply dvd_mul_of_dvd_right hrq | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 333,
"column": 6
} | {
"line": 333,
"column": 20
} | {
"line": 333,
"column": 21
} | [
{
"pp": "case insert\nα : Type u_1\nR : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Semiring R\nx : α → R\na : α\ns : Finset α\nha : a ∉ s\nhc : (↑(insert a s)).Pairwise (Commute on x)\nih : ∀ (n : ℕ), s.sum x ^ n = ∑ k, ↑(↑↑k).countPerms * (Multiset.map x ↑↑k).noncommProd ⋯\nn : ℕ\n⊢ (insert a s).sum x ^ n = ∑ k... | [
"case insert\nα : Type u_1\nR : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Semiring R\nx : α → R\na : α\ns : Finset α\nha : a ∉ s\nhc : (↑(insert a s)).Pairwise (Commute on x)\nih : ∀ (n : ℕ), s.sum x ^ n = ∑ k, ↑(↑↑k).countPerms * (Multiset.map x ↑↑k).noncommProd ⋯\nn : ℕ\n⊢ (x a + ∑ x_1 ∈ s, x x_1) ^ n = ∑ k, ↑(↑↑... | sum_insert ha, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 342,
"column": 33
} | {
"line": 342,
"column": 46
} | {
"line": 342,
"column": 47
} | [
{
"pp": "case insert\nα : Type u_1\nR : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Semiring R\nx : α → R\na : α\ns : Finset α\nha : a ∉ s\nhc : (↑(insert a s)).Pairwise (Commute on x)\nih : ∀ (n : ℕ), s.sum x ^ n = ∑ k, ↑(↑↑k).countPerms * (Multiset.map x ↑↑k).noncommProd ⋯\nn : ℕ\nm : ↥((insert a s).sym n)\n⊢ ↑... | [
"case insert\nα : Type u_1\nR : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Semiring R\nx : α → R\na : α\ns : Finset α\nha : a ∉ s\nhc : (↑(insert a s)).Pairwise (Commute on x)\nih : ∀ (n : ℕ), s.sum x ^ n = ∑ k, ↑(↑↑k).countPerms * (Multiset.map x ↑↑k).noncommProd ⋯\nn : ℕ\nm : ↥((insert a s).sym n)\n⊢ ↑(n.choose (M... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.GameAdd | {
"line": 61,
"column": 2
} | {
"line": 63,
"column": 41
} | {
"line": 65,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\nrα : α → α → Prop\nrβ : β → β → Prop\nx y : α × β\n⊢ rα x.1 y.1 ∧ x.2 = y.2 ∨ rβ x.2 y.2 ∧ x.1 = y.1 → GameAdd rα rβ x y",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Prod.GameAdd",
"Prod.mk",
"Prod.GameAdd.snd",
"Pro... | [] | · revert x y
rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ (⟨h, rfl : b₁ = b₂⟩ | ⟨h, rfl : a₁ = a₂⟩)
exacts [GameAdd.fst h, GameAdd.snd h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 359,
"column": 2
} | {
"line": 359,
"column": 35
} | {
"line": 361,
"column": 0
} | [
{
"pp": "case a\nα : Type u_1\nR : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : CommSemiring R\ns : Finset α\nx : α → R\nn : ℕ\nx✝ : ↥(s.sym n)\na✝ : x✝ ∈ univ\n⊢ (Multiset.map x ↑↑x✝).prod = (Multiset.map x ↑↑x✝).noncommProd ⋯",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [Multiset.noncommProd_eq_prod] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.GameAdd | {
"line": 186,
"column": 2
} | {
"line": 198,
"column": 18
} | {
"line": 200,
"column": 0
} | [
{
"pp": "α : Type u_1\nrα : α → α → Prop\na b : α\nha : Acc rα a\nhb : Acc rα b\n⊢ Acc (Sym2.GameAdd rα) s(a, b)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Prod.GameAdd.casesOn",
"Equiv.instEquivLike",
"Sym2.mk",
"congrArg",
"Prod.GameAdd"... | [] | induction ha generalizing b with | _ a _ iha
induction hb with | _ b hb ihb
refine Acc.intro _ fun s => ?_
induction s with | _ c d
rw [Sym2.GameAdd]
dsimp
rintro ((rc | rd) | (rd | rc))
· exact iha c rc ⟨b, hb⟩
· exact ihb d rd
· rw [Sym2.eq_swap]
exact iha d rd ⟨b, hb⟩
· rw [Sym2.eq_swap]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.GameAdd | {
"line": 186,
"column": 2
} | {
"line": 198,
"column": 18
} | {
"line": 200,
"column": 0
} | [
{
"pp": "α : Type u_1\nrα : α → α → Prop\na b : α\nha : Acc rα a\nhb : Acc rα b\n⊢ Acc (Sym2.GameAdd rα) s(a, b)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Prod.GameAdd.casesOn",
"Equiv.instEquivLike",
"Sym2.mk",
"congrArg",
"Prod.GameAdd"... | [] | induction ha generalizing b with | _ a _ iha
induction hb with | _ b hb ihb
refine Acc.intro _ fun s => ?_
induction s with | _ c d
rw [Sym2.GameAdd]
dsimp
rintro ((rc | rd) | (rd | rc))
· exact iha c rc ⟨b, hb⟩
· exact ihb d rd
· rw [Sym2.eq_swap]
exact iha d rd ⟨b, hb⟩
· rw [Sym2.eq_swap]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 447,
"column": 2
} | {
"line": 448,
"column": 38
} | {
"line": 450,
"column": 0
} | [
{
"pp": "case e_a\nx : ℕ\nl : List ℕ\nsuccEmb : ℕ ↪ ℕ := addRightEmbedding 1\nthis :\n (Finsupp.single 0 x + Finsupp.embDomain succEmb l.toFinsupp).update 0 0 =\n (Finsupp.embDomain succEmb l.toFinsupp).update 0 0\nh : ∀ (x : ℕ), (Finsupp.embDomain succEmb l.toFinsupp) (x + 1) = l[x]?.getD 0\n⊢ ((x :: l).to... | [] | simp [toFinsupp_cons_eq_single_add_embDomain, Finsupp.multinomial_eq,
succEmb, this, Nat.multinomial, h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.DFinsupp.WellFounded | {
"line": 80,
"column": 4
} | {
"line": 82,
"column": 18
} | {
"line": 83,
"column": 4
} | [
{
"pp": "case pos.refine_2\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 ... | [
"case pos.refine_3\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) ... | · split_ifs with hi
· rwa [hr i hi, if_pos hp] at hs
· assumption | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 51,
"column": 61
} | {
"line": 61,
"column": 52
} | {
"line": 63,
"column": 0
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\nn : σ\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nι : Type u_3\ns : Finset ι\nf : ι → MvPolynomial σ R\nh : ∀ i ∈ s, f i ≠ 0\n⊢ degreeOf n (∏ i ∈ s, f i) = ∑ i ∈ s, degreeOf n (f i)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Nontrivi... | [] | by
rcases subsingleton_or_nontrivial (MvPolynomial σ R) with nontrivial | nontrivial
· simp [Subsingleton.eq_zero]
· classical
induction s using Finset.induction_on with
| empty => simp
| insert a s a_not_mem ih =>
simp only [mem_insert, ne_eq, forall_eq_or_imp] at h
obtain ⟨ha, hs⟩ := h
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 312,
"column": 2
} | {
"line": 318,
"column": 22
} | {
"line": 320,
"column": 0
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf : MvPolynomial σ R\nhf : m.degree f = 0\nd : σ →₀ ℕ\n⊢ coeff d f = coeff d (C (coeff 0 f))",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"lt_iff_le_and_ne",
"Finsupp.instAddZeroClass",
"... | [] | classical
by_cases hd : d = 0
· simp [hd]
· rw [coeff_C, if_neg (Ne.symm hd)]
apply coeff_eq_zero_of_lt (m := m)
rw [hf, map_zero, lt_iff_le_and_ne, ne_eq, eq_comm, EmbeddingLike.map_eq_zero_iff]
exact ⟨bot_le, hd⟩ | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 608,
"column": 31
} | {
"line": 608,
"column": 44
} | {
"line": 608,
"column": 45
} | [
{
"pp": "case neg\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nr : R\nhr : r ∈ R⁰\nf : MvPolynomial σ R\nhf : ¬f = 0\n⊢ coeff (0 + m.degree f) (C r * f) ≠ 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
... | [
"case neg\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nr : R\nhr : r ∈ R⁰\nf : MvPolynomial σ R\nhf : ¬f = 0\n⊢ coeff (degree ?m.65 (C r) + m.degree f) (C r * f) ≠ 0",
"σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nr : R\nhr : r ∈ R⁰\nf : MvPolynomial σ R\nh... | ← degree_C r, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Squarefree | {
"line": 60,
"column": 4
} | {
"line": 61,
"column": 21
} | {
"line": 63,
"column": 0
} | [
{
"pp": "case neg\nn p : ℕ\nhn : ∀ (x : ℕ), emultiplicity x n ≤ 1 ∨ IsUnit x\nhn' : n ≠ 0\nhp : ¬Prime p\n⊢ n.factorization p ≤ 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg"... | [] | rw [factorization_eq_zero_of_not_prime _ hp]
exact zero_le_one | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Squarefree | {
"line": 60,
"column": 4
} | {
"line": 61,
"column": 21
} | {
"line": 63,
"column": 0
} | [
{
"pp": "case neg\nn p : ℕ\nhn : ∀ (x : ℕ), emultiplicity x n ≤ 1 ∨ IsUnit x\nhn' : n ≠ 0\nhp : ¬Prime p\n⊢ n.factorization p ≤ 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg"... | [] | rw [factorization_eq_zero_of_not_prime _ hp]
exact zero_le_one | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.SchwartzZippel | {
"line": 143,
"column": 52
} | {
"line": 143,
"column": 76
} | {
"line": 144,
"column": 8
} | [
{
"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... | [] | by simp [mul_comm, tail] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 992,
"column": 4
} | {
"line": 992,
"column": 17
} | {
"line": 993,
"column": 4
} | [
{
"pp": "case neg\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf g : MvPolynomial σ R\nhf : ¬(m.degree f = 0 ∧ m.degree g = 0)\nhs : ¬m.sPolynomial f g = 0\n⊢ m.degree (m.sPolynomial f g) ≠ m.degree f ⊔ m.degree g",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [
"case neg\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf g : MvPolynomial σ R\nhf : ¬(m.degree f = 0 ∧ m.degree g = 0)\nhs : m.degree (m.sPolynomial f g) = m.degree f ⊔ m.degree g\n⊢ m.sPolynomial f g = 0"
] | contrapose hs | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 1011,
"column": 2
} | {
"line": 1011,
"column": 54
} | {
"line": 1012,
"column": 2
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\np₁ p₂ : MvPolynomial σ R\nd₁ d₂ : σ →₀ ℕ\nc₁ c₂ : R\n⊢ (monomial (m.degree ((monomial d₁) c₁ * p₁) ⊔ m.degree ((monomial d₂) c₂ * p₂) - m.degree ((monomial d₁) c₁ * p₁)))\n (m.leadingCoeff ((monomia... | [
"case inr\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\np₁ p₂ : MvPolynomial σ R\nd₁ d₂ : σ →₀ ℕ\nc₁ c₂ : R\nthis :\n ∀ {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [inst : CommRing R] [NoZeroDivisors R]\n (p₁ p₂ : MvPolynomial σ R) (d₁ d₂ : σ →₀ ℕ) (c₁ ... | wlog! +distrib H : c₁ ≠ 0 ∧ c₂ ≠ 0 ∧ p₁ ≠ 0 ∧ p₂ ≠ 0 | Mathlib.Tactic._aux_Mathlib_Tactic_WLOG___elabRules_Mathlib_Tactic_wlog!_1 | Mathlib.Tactic.wlog! |
Mathlib.Data.Nat.Squarefree | {
"line": 366,
"column": 6
} | {
"line": 366,
"column": 25
} | {
"line": 366,
"column": 26
} | [
{
"pp": "n : ℕ\nhn : Squarefree n\n⊢ ∏ p ∈ n.primeFactors, p = n",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"id",
"List.toFinset",
"Nat.toFinset_factors",
"Finset.prod",
"Nat",
"Nat.instCommMonoid",
... | [
"n : ℕ\nhn : Squarefree n\n⊢ ∏ p ∈ n.primeFactorsList.toFinset, p = n"
] | ← toFinset_factors, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Factorization.PrimePow | {
"line": 89,
"column": 94
} | {
"line": 108,
"column": 6
} | {
"line": 110,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ IsPrimePow n ↔ ∃! p, Nat.Prime p ∧ p ∣ n",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.Prime.factorization_pos_of_dvd",
"List.replicate",
"dvd_zero",
"Nat.instMulZeroClass",
"Nat.Prime",
... | [] | by
rw [isPrimePow_nat_iff]
constructor
· rintro ⟨p, k, hp, hk, rfl⟩
refine ⟨p, ⟨hp, dvd_pow_self _ hk.ne'⟩, ?_⟩
rintro q ⟨hq, hq'⟩
exact (Nat.prime_dvd_prime_iff_eq hq hp).1 (hq.dvd_of_dvd_pow hq')
rintro ⟨p, ⟨hp, hn⟩, hq⟩
rcases eq_or_ne n 0 with (rfl | hn₀)
· cases (hq 2 ⟨Nat.prime_two, dvd_ze... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Factorization.PrimePow | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 7
} | {
"line": 203,
"column": 2
} | [
{
"pp": "p a m n : ℕ\nhp : Prime p\nhn : n ≠ 0\nk : ℕ\nh : (p ^ k) ^ n = a ^ n\nm_eq : m = n * k\n⊢ ∃ k, a = p ^ k",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Nat.instMonoid",
"NPow.toPow",
"HPow.hPow",
"Nat",
"Exists.intro",
"instHPow",
"Eq",... | [
"case h\np a m n : ℕ\nhp : Prime p\nhn : n ≠ 0\nk : ℕ\nh : (p ^ k) ^ n = a ^ n\nm_eq : m = n * k\n⊢ a = p ^ k"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 224,
"column": 2
} | {
"line": 226,
"column": 61
} | {
"line": 228,
"column": 0
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\n⊢ #n.divisors = ∏ x ∈ n.primeFactors, (n.factorization x + 1)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLik... | [] | rw [← sigma_zero_apply, isMultiplicative_sigma.multiplicative_factorization _ hn]
exact prod_congr n.support_factorization fun _ h =>
sigma_zero_apply_prime_pow <| prime_of_mem_primeFactors h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 224,
"column": 2
} | {
"line": 226,
"column": 61
} | {
"line": 228,
"column": 0
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\n⊢ #n.divisors = ∏ x ∈ n.primeFactors, (n.factorization x + 1)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLik... | [] | rw [← sigma_zero_apply, isMultiplicative_sigma.multiplicative_factorization _ hn]
exact prod_congr n.support_factorization fun _ h =>
sigma_zero_apply_prime_pow <| prime_of_mem_primeFactors h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 1034,
"column": 2
} | {
"line": 1034,
"column": 54
} | {
"line": 1035,
"column": 2
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\np₁ p₂ : MvPolynomial σ R\nd₁ d₂ : σ →₀ ℕ\nc₁ c₂ : R\n⊢ m.sPolynomial ((monomial d₁) c₁ * p₁) ((monomial d₂) c₂ * p₂) =\n (monomial (m.degree ((monomial d₁) c₁ * p₁) ⊔ m.degree ((monomial d₂) c₂ * p₂) - m.... | [
"case inr\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\np₁ p₂ : MvPolynomial σ R\nd₁ d₂ : σ →₀ ℕ\nc₁ c₂ : R\nthis :\n ∀ {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [inst : CommRing R] [NoZeroDivisors R]\n (p₁ p₂ : MvPolynomial σ R) (d₁ d₂ : σ →₀ ℕ) (c₁ ... | wlog! +distrib H : c₁ ≠ 0 ∧ c₂ ≠ 0 ∧ p₁ ≠ 0 ∧ p₂ ≠ 0 | Mathlib.Tactic._aux_Mathlib_Tactic_WLOG___elabRules_Mathlib_Tactic_wlog!_1 | Mathlib.Tactic.wlog! |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 357,
"column": 4
} | {
"line": 357,
"column": 23
} | {
"line": 357,
"column": 24
} | [
{
"pp": "n : ℕ\n⊢ n.primeFactorsList.dedup.length = 1 ↔ #n.primeFactors = 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"List.dedup",
"id",
"instOfNatNat",
"List.toFinset",
"Nat.toFinset_factors",
"If... | [
"n : ℕ\n⊢ n.primeFactorsList.dedup.length = 1 ↔ #n.primeFactorsList.toFinset = 1"
] | ← toFinset_factors, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 505,
"column": 6
} | {
"line": 507,
"column": 18
} | {
"line": 508,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : ℕ\nha : ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 ≠ 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2... | [] | · rw [← hcd.1.1, ← hcd.2.1] at cop
rw [← hcd.1.1, h.1, gcd_mul_left, cop.coprime_mul_left.coprime_mul_right_right.gcd_eq_one,
mul_one] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.Archimedean.IndicatorCard | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 14
} | {
"line": 54,
"column": 4
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝⁴ : AddCommMonoid R\ninst✝³ : PartialOrder R\ninst✝² : IsOrderedAddMonoid R\ninst✝¹ : AddLeftStrictMono R\ninst✝ : Archimedean R\nr : R\nh : 0 < r\ns : Set ℕ\nh_mono : Monotone fun n ↦ ∑ k ∈ Finset.range n, s.indicator (fun x ↦ r) k\n⊢ s.Infinite → ∀ (b : R), ∃ a, b ≤ ∑ k ∈ ... | [
"case mp\nR : Type u_1\ninst✝⁴ : AddCommMonoid R\ninst✝³ : PartialOrder R\ninst✝² : IsOrderedAddMonoid R\ninst✝¹ : AddLeftStrictMono R\ninst✝ : Archimedean R\nr : R\nh : 0 < r\ns : Set ℕ\nh_mono : Monotone fun n ↦ ∑ k ∈ Finset.range n, s.indicator (fun x ↦ r) k\nhs : s.Infinite\nn : R\n⊢ ∃ a, n ≤ ∑ k ∈ Finset.range... | intro hs n | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Order.CompleteField | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 31
} | {
"line": 139,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u_2\nβ : Type u_3\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : Field β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : Archimedean α\na b : α\nqa : ℚ\nha : qa ∈ {t | ↑t < a}\nqb : ℚ\nhb : qb ∈ {t | ↑t < b}\n⊢ qa + qb ∈ {t |... | [
"case refine_2\nα : Type u_2\nβ : Type u_3\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : Field β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : Archimedean α\na b : α\nqa : ℚ\nha : qa ∈ {t | ↑t < a}\nqb : ℚ\nhb : qb ∈ {t | ↑t < b}\n⊢ ↑qa + ↑qb < a + b"
] | rw [mem_setOf_eq, cast_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Chebyshev | {
"line": 60,
"column": 63
} | {
"line": 65,
"column": 89
} | {
"line": 67,
"column": 0
} | [
{
"pp": "ι : 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✝ : PosSMulMono α β\ns : Finset ι\nf : ι → α\n... | [] | by
classical
obtain ⟨σ, hσ, hs⟩ := s.countable_toSet.exists_cycleOn
rw [← card_range #s, sum_smul_sum_eq_sum_perm hσ]
exact sum_le_card_nsmul _ _ _ fun n _ ↦
hfg.sum_smul_comp_perm_le_sum_smul fun x hx ↦ hs fun h ↦ hx <| IsFixedPt.perm_pow h _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 601,
"column": 36
} | {
"line": 601,
"column": 65
} | {
"line": 601,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝ : CommGroupWithZero R\nf : ArithmeticFunction R\nhf : f.IsMultiplicative\nx y : ℕ\nhf_lcm : f (x.lcm y) ≠ 0\n⊢ f (x.gcd y) = f (x.lcm y) * f (x.gcd y) / f (x.lcm y)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Nat.lcm",
"Eq.mpr"... | [
"R : Type u_1\ninst✝ : CommGroupWithZero R\nf : ArithmeticFunction R\nhf : f.IsMultiplicative\nx y : ℕ\nhf_lcm : f (x.lcm y) ≠ 0\n⊢ f (x.gcd y) = f (x.gcd y)"
] | mul_div_cancel_left₀ _ hf_lcm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Floor.Div | {
"line": 193,
"column": 27
} | {
"line": 193,
"column": 78
} | {
"line": 194,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\na : ℕ\nha : a ≤ 0\nb : ℕ\n⊢ (b + a - 1) / a = 0",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.zero_le",
"instHDiv",
"congrArg",
"AddMonoid.toAddZeroClass",
"HSub.hSub",
"LE.le.anti... | [] | by simp_rw [ha.antisymm <| zero_le _, Nat.div_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 615,
"column": 6
} | {
"line": 615,
"column": 38
} | {
"line": 616,
"column": 4
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\ns : UpperSet (MulArchimedeanClass M)\nhs : ¬s = ⊤\nu : MulArchimedeanClass M\nhu : u ∈ ↑s\n⊢ mk 1 ∈ ↑s",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem... | [] | simpa using s.upper (by simp) hu | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Group.Cone | {
"line": 104,
"column": 68
} | {
"line": 104,
"column": 86
} | {
"line": 104,
"column": 86
} | [
{
"pp": "S : Type u_1\nG : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : SetLike S G\nC : S\ninst✝ : GroupConeClass S G\na b : G\nnab : b / a ∈ C\nnba : a / b ∈ C\n⊢ (b / a)⁻¹ ∈ C",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"DivisionCommMonoid.toDivis... | [] | by simpa using nba | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Basic | {
"line": 244,
"column": 25
} | {
"line": 244,
"column": 34
} | {
"line": 244,
"column": 35
} | [
{
"pp": "α : Type u_1\ninst✝ : PartialOrder α\na b : α\n⊢ b ∈ ↑(pure a) ↔ b = a",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"NonemptyInterval.pure",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"Set.instSi... | [
"α : Type u_1\ninst✝ : PartialOrder α\na b : α\n⊢ b ∈ {a} ↔ b = a"
] | coe_pure, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Basic | {
"line": 454,
"column": 14
} | {
"line": 454,
"column": 48
} | {
"line": 456,
"column": 0
} | [
{
"pp": "case coe\nα : Type u_1\ninst✝ : PartialOrder α\ns₀ : NonemptyInterval α\n⊢ ↑(dual ↑s₀) = ⇑ofDual ⁻¹' ↑↑s₀",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"NonemptyInterval.coe_dual"
],
"usedFVars": [
"α",
"inst✝",
"s₀"
],
"usedGoals": []
}
... | [] | exact NonemptyInterval.coe_dual s₀ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Interval.Basic | {
"line": 454,
"column": 14
} | {
"line": 454,
"column": 48
} | {
"line": 456,
"column": 0
} | [
{
"pp": "case coe\nα : Type u_1\ninst✝ : PartialOrder α\ns₀ : NonemptyInterval α\n⊢ ↑(dual ↑s₀) = ⇑ofDual ⁻¹' ↑↑s₀",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"NonemptyInterval.coe_dual"
],
"usedFVars": [
"α",
"inst✝",
"s₀"
],
"usedGoals": []
}
... | [] | exact NonemptyInterval.coe_dual s₀ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Basic | {
"line": 454,
"column": 14
} | {
"line": 454,
"column": 48
} | {
"line": 456,
"column": 0
} | [
{
"pp": "case coe\nα : Type u_1\ninst✝ : PartialOrder α\ns₀ : NonemptyInterval α\n⊢ ↑(dual ↑s₀) = ⇑ofDual ⁻¹' ↑↑s₀",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"NonemptyInterval.coe_dual"
],
"usedFVars": [
"α",
"inst✝",
"s₀"
],
"usedGoals": []
}
... | [] | exact NonemptyInterval.coe_dual s₀ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Basic | {
"line": 461,
"column": 67
} | {
"line": 461,
"column": 76
} | {
"line": 461,
"column": 77
} | [
{
"pp": "α : Type u_1\ninst✝ : PartialOrder α\na b : α\n⊢ b ∈ ↑(pure a) ↔ b = a",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Interval.pure",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"Set.instSingletonS... | [
"α : Type u_1\ninst✝ : PartialOrder α\na b : α\n⊢ b ∈ {a} ↔ b = a"
] | coe_pure, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 416,
"column": 2
} | {
"line": 418,
"column": 71
} | {
"line": 420,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : Zero Γ\nx : R⟦Γ⟧\n⊢ x.leadingCoeff = x.coeff x.order",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"Iff.mpr",
"Eq.mpr",
"HahnSeries.orderTop_ne_top",
... | [] | by_cases h : x = 0
· rw [h, leadingCoeff_zero, coeff_zero]
· simp [leadingCoeff_of_ne_zero, orderTop_of_ne_zero, order_of_ne, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 416,
"column": 2
} | {
"line": 418,
"column": 71
} | {
"line": 420,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : Zero Γ\nx : R⟦Γ⟧\n⊢ x.leadingCoeff = x.coeff x.order",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"Iff.mpr",
"Eq.mpr",
"HahnSeries.orderTop_ne_top",
... | [] | by_cases h : x = 0
· rw [h, leadingCoeff_zero, coeff_zero]
· simp [leadingCoeff_of_ne_zero, orderTop_of_ne_zero, order_of_ne, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 483,
"column": 2
} | {
"line": 483,
"column": 38
} | {
"line": 485,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nx✝ : Γ'\n⊢ (embDomain f 0).coeff x✝ = coeff 0 x✝",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"False",
"HahnSeries.... | [] | simp [embDomain_notin_image_support] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.HahnSeries.Addition | {
"line": 70,
"column": 2
} | {
"line": 71,
"column": 20
} | {
"line": 72,
"column": 2
} | [
{
"pp": "case pos\nΓ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\nV : Type u_8\ninst✝¹ : Zero V\ninst✝ : SMulZeroClass R V\nr : R\nx : V⟦Γ⟧\nhrx : r • x = 0\n⊢ ¬(r • x).orderTop < x.orderTop",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Pre... | [
"case neg\nΓ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\nV : Type u_8\ninst✝¹ : Zero V\ninst✝ : SMulZeroClass R V\nr : R\nx : V⟦Γ⟧\nhrx : ¬r • x = 0\n⊢ ¬(r • x).orderTop < x.orderTop"
] | · rw [hrx, orderTop_zero]
exact not_top_lt | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 18
} | {
"line": 170,
"column": 2
} | [
{
"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\n⊢ ⨆ i, u.baseDomain... | [
"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\n⊢ ∀ (i : FiniteArchimedeanClass... | apply iSup_congr | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Order.Ring.Archimedean | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 41
} | {
"line": 175,
"column": 2
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁶ : LinearOrder R\ninst✝⁵ : CommRing R\ninst✝⁴ : IsStrictOrderedRing R\nS : Type u_3\ninst✝³ : LinearOrder S\ninst✝² : CommRing S\ninst✝¹ : IsStrictOrderedRing S\ninst✝ : Archimedean S\nf : S →+*o R\nx : R\nhx : 0 < mk x\ny : S\nhy : 0 < y\n⊢ mk (f y) < mk x",
"ppTerm": "... | [] | rwa [mk_map_of_archimedean' f hy.ne'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Order.Ring.Archimedean | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 41
} | {
"line": 175,
"column": 2
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁶ : LinearOrder R\ninst✝⁵ : CommRing R\ninst✝⁴ : IsStrictOrderedRing R\nS : Type u_3\ninst✝³ : LinearOrder S\ninst✝² : CommRing S\ninst✝¹ : IsStrictOrderedRing S\ninst✝ : Archimedean S\nf : S →+*o R\nx : R\nhx : 0 < mk x\ny : S\nhy : 0 < y\n⊢ mk (f y) < mk x",
"ppTerm": "... | [] | rwa [mk_map_of_archimedean' f hy.ne'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Archimedean | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 41
} | {
"line": 175,
"column": 2
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁶ : LinearOrder R\ninst✝⁵ : CommRing R\ninst✝⁴ : IsStrictOrderedRing R\nS : Type u_3\ninst✝³ : LinearOrder S\ninst✝² : CommRing S\ninst✝¹ : IsStrictOrderedRing S\ninst✝ : Archimedean S\nf : S →+*o R\nx : R\nhx : 0 < mk x\ny : S\nhy : 0 < y\n⊢ mk (f y) < mk x",
"ppTerm": "... | [] | rwa [mk_map_of_archimedean' f hy.ne'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 317,
"column": 4
} | {
"line": 317,
"column": 65
} | {
"line": 318,
"column": 4
} | [
{
"pp": "case pos\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 : AddCommGro... | [
"case pos\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 R\ninst✝¹... | rw [ArchimedeanClass.mk_sum hsupport' (hmono.mono (by simp))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 528,
"column": 2
} | {
"line": 529,
"column": 96
} | {
"line": 531,
"column": 0
} | [
{
"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... | [] | have hnonempty : ∃ y : f.val.domain, y.val - x ∈ ball K c := ⟨y, hy⟩
simpa [evalCoeff, dif_pos hnonempty] using coeff_eq_of_mem f x hnonempty.choose_spec hy le_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 528,
"column": 2
} | {
"line": 529,
"column": 96
} | {
"line": 531,
"column": 0
} | [
{
"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... | [] | have hnonempty : ∃ y : f.val.domain, y.val - x ∈ ball K c := ⟨y, hy⟩
simpa [evalCoeff, dif_pos hnonempty] using coeff_eq_of_mem f x hnonempty.choose_spec hy le_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 600,
"column": 2
} | {
"line": 600,
"column": 59
} | {
"line": 601,
"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... | have : x - y.val = x - z.val + (z.val - y.val) := by abel | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 632,
"column": 52
} | {
"line": 632,
"column": 64
} | {
"line": 633,
"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... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.UpperLower | {
"line": 81,
"column": 62
} | {
"line": 83,
"column": 23
} | {
"line": 85,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\ns t : Set α\nht : IsUpperSet t\n⊢ IsLowerSet (s / t)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Divi... | [] | by
rw [div_eq_mul_inv]
exact ht.inv.mul_left | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 449,
"column": 2
} | {
"line": 449,
"column": 37
} | {
"line": 450,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nf : ℝ →+*o K\nx : K\n⊢ stdPart x = sInf {r | x < f r}",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Real",
"IsDomain.to_noZeroDivisors",
"Preorder.toLT",
"NonUnitalCommRing.... | [
"case inl\nK : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nf : ℝ →+*o K\nx : K\nhx : 0 ≤ mk x\n⊢ stdPart x = sInf {r | x < f r}",
"case inr\nK : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nf : ℝ →+*o K\nx : K\nhx : mk x < 0\n⊢ stdPart x = sInf {r |... | obtain hx | hx := le_or_gt 0 (mk x) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 248,
"column": 2
} | {
"line": 251,
"column": 37
} | {
"line": 253,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nf : R[X]\n⊢ f.IsMonicOfDegree 2 ↔ ∃ a b, f = X ^ 2 - C a * X + C b",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NegZeroClass.toNeg",
"RingHom.instRingHomClass",
"RingHo... | [] | refine ⟨fun H ↦ ?_, fun ⟨a, b, h⟩ ↦ h ▸ isMonicOfDegree_sub_add_two a b⟩
simp only [sub_eq_add_neg, ← neg_mul, ← map_neg]
obtain ⟨a, b, h⟩ := isMonicOfDegree_two_iff.mp H
exact ⟨-a, b, (neg_neg a).symm ▸ h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 248,
"column": 2
} | {
"line": 251,
"column": 37
} | {
"line": 253,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nf : R[X]\n⊢ f.IsMonicOfDegree 2 ↔ ∃ a b, f = X ^ 2 - C a * X + C b",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NegZeroClass.toNeg",
"RingHom.instRingHomClass",
"RingHo... | [] | refine ⟨fun H ↦ ?_, fun ⟨a, b, h⟩ ↦ h ▸ isMonicOfDegree_sub_add_two a b⟩
simp only [sub_eq_add_neg, ← neg_mul, ← map_neg]
obtain ⟨a, b, h⟩ := isMonicOfDegree_two_iff.mp H
exact ⟨-a, b, (neg_neg a).symm ▸ h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 18
} | {
"line": 279,
"column": 2
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : CommRing R\np : R[X]\nn : ℕ\nhp : p.IsMonicOfDegree n\nr : R\nH : Nontrivial R\n⊢ ((aeval (X + C r)) p).IsMonicOfDegree n",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"MulOne.toOne",
"HMul.hMul",
... | [
"case inr\nR : Type u_1\ninst✝ : CommRing R\np : R[X]\nn : ℕ\nhp : p.IsMonicOfDegree n\nr : R\nH : Nontrivial R\n⊢ ((aeval (X + C r)) p).IsMonicOfDegree (n * 1)"
] | rw [← mul_one n] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 213,
"column": 4
} | {
"line": 213,
"column": 23
} | {
"line": 214,
"column": 4
} | [
{
"pp": "case isUnit_or_isUnit\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... | [
"case isUnit_or_isUnit\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.mirror = k * k... | have hk := h2 k key | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.Homogenize | {
"line": 183,
"column": 2
} | {
"line": 185,
"column": 12
} | {
"line": 187,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\np q : R[X]\nm n : ℕ\nhm : p.natDegree ≤ m\nhn : q.natDegree ≤ n\n⊢ (p * q).homogenize (m + n) = p.homogenize m * q.homogenize n",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Polynomial.instOne",
"Nat.instMulZeroClass",
"Ad... | [] | apply homogenize_eq_of_isHomogeneous
· apply_rules [MvPolynomial.IsHomogeneous.mul, isHomogeneous_homogenize]
· simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Homogenize | {
"line": 183,
"column": 2
} | {
"line": 185,
"column": 12
} | {
"line": 187,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\np q : R[X]\nm n : ℕ\nhm : p.natDegree ≤ m\nhn : q.natDegree ≤ n\n⊢ (p * q).homogenize (m + n) = p.homogenize m * q.homogenize n",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Polynomial.instOne",
"Nat.instMulZeroClass",
"Ad... | [] | apply homogenize_eq_of_isHomogeneous
· apply_rules [MvPolynomial.IsHomogeneous.mul, isHomogeneous_homogenize]
· simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 70,
"column": 2
} | {
"line": 71,
"column": 5
} | {
"line": 73,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nr : R\np : R[X]\n⊢ eval r p = p.smeval r",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"MonoidWithZero.toMulActionWithZero",
"HMul.hMul",
"congrArg",
"Polynomial.sum",
"id",
... | [] | rw [eval_eq_sum, smeval_eq_sum]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 70,
"column": 2
} | {
"line": 71,
"column": 5
} | {
"line": 73,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nr : R\np : R[X]\n⊢ eval r p = p.smeval r",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"MonoidWithZero.toMulActionWithZero",
"HMul.hMul",
"congrArg",
"Polynomial.sum",
"id",
... | [] | rw [eval_eq_sum, smeval_eq_sum]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 174,
"column": 4
} | {
"line": 177,
"column": 46
} | {
"line": 178,
"column": 4
} | [
{
"pp": "case cons.refine_1\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : Nontrivial R\nι : Type u_2\ninst✝ : DecidableEq ι\nf : R[X]\ng : ι → R[X]\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n (∀ i ∈ s, (g i).Monic) →\n ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →\n ∃ q r, (∀ i ∈ s, (r i).degree < (g i... | [
"case cons.refine_2\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : Nontrivial R\nι : Type u_2\ninst✝ : DecidableEq ι\nf : R[X]\ng : ι → R[X]\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n (∀ i ∈ s, (g i).Monic) →\n ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →\n ∃ q r, (∀ i ∈ s, (r i).degree < (g i).degree) ∧ ... | · rw [Finset.forall_mem_cons, Function.update_self]
refine ⟨degree_modByMonic_lt _ hg.1, fun j hj => ?_⟩
rw [Function.update_of_ne (hjs hj).symm]
exact degree_modByMonic_lt _ (hg.2 j hj) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 196,
"column": 69
} | {
"line": 196,
"column": 82
} | {
"line": 196,
"column": 83
} | [
{
"pp": "case monomial\nS : Type u_2\ninst✝² : NonAssocSemiring S\ninst✝¹ : Pow S ℕ\ninst✝ : NatPowAssoc S\nn✝ a n : ℕ\n⊢ ↑a * ↑n ^ n✝ = ↑(a * n ^ n✝)",
"ppTerm": "?monomial",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
... | [
"case monomial\nS : Type u_2\ninst✝² : NonAssocSemiring S\ninst✝¹ : Pow S ℕ\ninst✝ : NatPowAssoc S\nn✝ a n : ℕ\n⊢ ↑a * ↑n ^ n✝ = ↑a * ↑(n ^ n✝)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 210,
"column": 60
} | {
"line": 215,
"column": 49
} | {
"line": 217,
"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✝ : SMulCommClass R S S\n⊢ (X * p).smeval x = x * p.smeval x",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Dis... | [] | by
induction p using Polynomial.induction_on' with
| add p q ph qh => simp only [smeval_add, ph, qh, mul_add]
| monomial n a =>
rw [← monomial_one_one_eq_X, monomial_mul_monomial, smeval_monomial, one_mul, npow_add,
npow_one, ← mul_smul_comm, smeval_monomial] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 81,
"column": 55
} | {
"line": 82,
"column": 89
} | {
"line": 84,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\na : R\n⊢ sumIDeriv (C a) = C a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Polynomial.C",
"Polynomial.sumIDeriv",
"Semiring.toModule",
"congrArg",
"AddMonoid.toAddZe... | [] | by
rw [sumIDeriv_apply, natDegree_C, zero_add, sum_range_one, Function.iterate_zero_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 162,
"column": 39
} | {
"line": 162,
"column": 51
} | {
"line": 162,
"column": 52
} | [
{
"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\np' : R[X]\nk : ℕ\nhk : q ≤ q + k\np'_le : p'.natDegree ≤ p.natDegree - (q + k)\nhp' : (⇑derivative)^[q + k] p = (q + k)! • p'\nr : A\n⊢ (aeval r) ↑(q + k)! * (aeval r) p' = ↑q ! * ((aeval r) ... | [
"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\np' : R[X]\nk : ℕ\nhk : q ≤ q + k\np'_le : p'.natDegree ≤ p.natDegree - (q + k)\nhp' : (⇑derivative)^[q + k] p = (q + k)! • p'\nr : A\n⊢ ↑(q + k)! * (aeval r) p' = ↑q ! * (↑((q + k).descFactorial k) * (ae... | map_natCast, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 292,
"column": 2
} | {
"line": 292,
"column": 10
} | {
"line": 293,
"column": 2
} | [
{
"pp": "p : ℤ[X]\nh : IsRelPrime p p.mirror\nq : ℤ[X]\nhpq : p * p.mirror = q * q.mirror\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nm' n' : ℕ\nhmn' : m' < n'\nx y z : ℤˣ\nhn : n = n'\nhkm' : k < m'\nhq : q = trinomial k m' n' ↑x ↑y ↑z\n⊢ q = p ∨ q = -p ∨ q = p.mirror ∨... | [
"p : ℤ[X]\nh : IsRelPrime p p.mirror\nq : ℤ[X]\nhpq : p * p.mirror = q * q.mirror\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nm' : ℕ\nx y z : ℤˣ\nhkm' : k < m'\nhmn' : m' < n\nhq : q = trinomial k m' n ↑x ↑y ↑z\n⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror"
] | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 40
} | {
"line": 185,
"column": 4
} | [
{
"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... | [] | grind [nextCoeff_of_natDegree_pos] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 40
} | {
"line": 185,
"column": 4
} | [
{
"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... | [] | grind [nextCoeff_of_natDegree_pos] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 40
} | {
"line": 185,
"column": 4
} | [
{
"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... | [] | grind [nextCoeff_of_natDegree_pos] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.QuadraticAlgebra.Defs | {
"line": 331,
"column": 60
} | {
"line": 331,
"column": 65
} | {
"line": 331,
"column": 65
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\na b r : R\nx y : QuadraticAlgebra R a b\ninst✝ : AddCommGroupWithOne R\nn : ℕ\n⊢ QuadraticAlgebra.C (-↑(↑n + 1)) = -↑(n + 1)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\na b r : R\nx y : QuadraticAlgebra R a b\ninst✝ : AddCommGroupWithOne R\nn : ℕ\n⊢ -QuadraticAlgebra.C ↑(↑n + 1) = -↑(n + 1)"
] | C_neg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 241,
"column": 4
} | {
"line": 242,
"column": 31
} | {
"line": 243,
"column": 4
} | [
{
"pp": "case mpr\nR : Type u_2\na b : R\ninst✝ : CommRing R\nx : QuadraticAlgebra R a b\nr : R\nhr : norm x * r = 1\nhr' : r * norm x = 1\n⊢ ∃ b_1, x * b_1 = 1 ∧ b_1 * x = 1",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"RingHom.instRingHomClass",
"MulOne.toOne",
"Semig... | [
"case mpr\nR : Type u_2\na b : R\ninst✝ : CommRing R\nx : QuadraticAlgebra R a b\nr : R\nhr : x * (star x * (algebraMap R (QuadraticAlgebra R a b)) r) = 1\nhr' : r * norm x = 1\n⊢ ∃ b_1, x * b_1 = 1 ∧ b_1 * x = 1"
] | rw [← C_inj (R := R) (a := a) (b := b), C_mul, C_eq_algebraMap, algebraMap_norm_eq_mul_star,
mul_assoc, map_one] at hr | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.QuaternionBasis | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 27
} | {
"line": 67,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nc₁ c₂ c₃ : R\nq₁ q₂ : Basis A c₁ c₂ c₃\nhi : q₁.i = q₂.i\nhj : q₁.j = q₂.j\n⊢ q₁ = q₂",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"instHSMul",
"HMul.hMul",
"AddGroupWithOne... | [] | cases q₁; cases q₂; grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.QuaternionBasis | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 27
} | {
"line": 67,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nc₁ c₂ c₃ : R\nq₁ q₂ : Basis A c₁ c₂ c₃\nhi : q₁.i = q₂.i\nhj : q₁.j = q₂.j\n⊢ q₁ = q₂",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"instHSMul",
"HMul.hMul",
"AddGroupWithOne... | [] | cases q₁; cases q₂; grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Ext | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 38
} | {
"line": 75,
"column": 2
} | [
{
"pp": "case h_add\nR : Type u\na₁✝ a₂✝ : NonUnitalNonAssocSemiring R\nh : a₁✝.toDistrib = a₂✝.toDistrib\nx y : R\n⊢ x + y = x + y",
"ppTerm": "?h_add",
"assigned": true,
"usedConstants": [
"congrArg",
"Distrib.toAdd",
"Add.add",
"Distrib",
"NonUnitalNonAssocSemiring.t... | [
"case h_mul\nR : Type u\na₁✝ a₂✝ : NonUnitalNonAssocSemiring R\nh : a₁✝.toDistrib = a₂✝.toDistrib\nx y : R\n⊢ x * y = x * y"
] | · exact congrArg (·.toAdd.add x y) h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Ring.Ext | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 38
} | {
"line": 180,
"column": 2
} | [
{
"pp": "case h_add\nR : Type u\na₁✝ a₂✝ : NonUnitalNonAssocRing R\nh : a₁✝.toNonUnitalNonAssocSemiring = a₂✝.toNonUnitalNonAssocSemiring\nx y : R\n⊢ x + y = x + y",
"ppTerm": "?h_add",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddSemigroup",
"congrArg",
"NonUnitalNonAssocR... | [
"case h_mul\nR : Type u\na₁✝ a₂✝ : NonUnitalNonAssocRing R\nh : a₁✝.toNonUnitalNonAssocSemiring = a₂✝.toNonUnitalNonAssocSemiring\nx y : R\n⊢ x * y = x * y"
] | · exact congrArg (·.toAdd.add x y) h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Ring.Ext | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 38
} | {
"line": 202,
"column": 2
} | [
{
"pp": "case h_add\nR : Type u\na₁✝ a₂✝ : NonUnitalRing R\nh : a₁✝.toNonUnitalSemiring = a₂✝.toNonUnitalSemiring\nx y : R\n⊢ x + y = x + y",
"ppTerm": "?h_add",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddSemigroup",
"congrArg",
"NonUnitalSemiring.toNonUnitalNonAssocSemir... | [
"case h_mul\nR : Type u\na₁✝ a₂✝ : NonUnitalRing R\nh : a₁✝.toNonUnitalSemiring = a₂✝.toNonUnitalSemiring\nx y : R\n⊢ x * y = x * y"
] | · exact congrArg (·.toAdd.add x y) h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Ring.Ext | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 38
} | {
"line": 266,
"column": 2
} | [
{
"pp": "case h_add\nR : Type u\na₁✝ a₂✝ : NonAssocRing R\nh : a₁✝.toNonAssocSemiring = a₂✝.toNonAssocSemiring\nx y : R\n⊢ x + y = x + y",
"ppTerm": "?h_add",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddSemigroup",
"congrArg",
"Add.add",
"NonUnitalNonAssocSemiring.to... | [
"case h_mul\nR : Type u\na₁✝ a₂✝ : NonAssocRing R\nh : a₁✝.toNonAssocSemiring = a₂✝.toNonAssocSemiring\nx y : R\n⊢ x * y = x * y"
] | · exact congrArg (·.toAdd.add x y) h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Ring.Ext | {
"line": 299,
"column": 2
} | {
"line": 299,
"column": 38
} | {
"line": 300,
"column": 2
} | [
{
"pp": "case h_add\nR : Type u\na₁✝ a₂✝ : Semiring R\nh : a₁✝.toNonUnitalSemiring = a₂✝.toNonUnitalSemiring\nx y : R\n⊢ x + y = x + y",
"ppTerm": "?h_add",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddSemigroup",
"congrArg",
"NonUnitalSemiring.toNonUnitalNonAssocSemiring",... | [
"case h_mul\nR : Type u\na₁✝ a₂✝ : Semiring R\nh : a₁✝.toNonUnitalSemiring = a₂✝.toNonUnitalSemiring\nx y : R\n⊢ x * y = x * y"
] | · exact congrArg (·.toAdd.add x y) h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Ring.Ext | {
"line": 306,
"column": 2
} | {
"line": 306,
"column": 38
} | {
"line": 307,
"column": 2
} | [
{
"pp": "case h_add\nR : Type u\na₁✝ a₂✝ : Semiring R\nh : a₁✝.toNonAssocSemiring = a₂✝.toNonAssocSemiring\nx y : R\n⊢ x + y = x + y",
"ppTerm": "?h_add",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddSemigroup",
"congrArg",
"Add.add",
"NonUnitalNonAssocSemiring.toAddC... | [
"case h_mul\nR : Type u\na₁✝ a₂✝ : Semiring R\nh : a₁✝.toNonAssocSemiring = a₂✝.toNonAssocSemiring\nx y : R\n⊢ x * y = x * y"
] | · exact congrArg (·.toAdd.add x y) h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Ring.Ext | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 38
} | {
"line": 337,
"column": 2
} | [
{
"pp": "case h_add\nR : Type u\na₁✝ a₂✝ : Ring R\nh : toNonUnitalRing = toNonUnitalRing\nx y : R\n⊢ x + y = x + y",
"ppTerm": "?h_add",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddSemigroup",
"congrArg",
"NonUnitalNonAssocRing.toAddCommGroup",
"AddCommGroup.toAddGro... | [
"case h_mul\nR : Type u\na₁✝ a₂✝ : Ring R\nh : toNonUnitalRing = toNonUnitalRing\nx y : R\n⊢ x * y = x * y"
] | · exact congrArg (·.toAdd.add x y) h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Ring.Ext | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 38
} | {
"line": 344,
"column": 2
} | [
{
"pp": "case h_add\nR : Type u\na₁✝ a₂✝ : Ring R\nh : toNonAssocRing = toNonAssocRing\nx y : R\n⊢ x + y = x + y",
"ppTerm": "?h_add",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddSemigroup",
"Ring.toNonAssocRing",
"congrArg",
"NonUnitalNonAssocRing.toAddCommGroup",
... | [
"case h_mul\nR : Type u\na₁✝ a₂✝ : Ring R\nh : toNonAssocRing = toNonAssocRing\nx y : R\n⊢ x * y = x * y"
] | · exact congrArg (·.toAdd.add x y) h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Ring.Ext | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 38
} | {
"line": 351,
"column": 2
} | [
{
"pp": "case h_add\nR : Type u\na₁✝ a₂✝ : Ring R\nh : a₁✝.toSemiring = a₂✝.toSemiring\nx y : R\n⊢ x + y = x + y",
"ppTerm": "?h_add",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddSemigroup",
"congrArg",
"Add.add",
"AddSemigroup.toAdd",
"Semiring",
"AddCom... | [
"case h_mul\nR : Type u\na₁✝ a₂✝ : Ring R\nh : a₁✝.toSemiring = a₂✝.toSemiring\nx y : R\n⊢ x * y = x * y"
] | · exact congrArg (·.toAdd.add x y) h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Ring.CentroidHom | {
"line": 513,
"column": 4
} | {
"line": 513,
"column": 36
} | {
"line": 513,
"column": 37
} | [
{
"pp": "α : Type u_5\ninst✝ : NonUnitalNonAssocCommSemiring α\na : α\n⊢ AddMonoid.End.mulRight a = L a ∧ L a ∈ Subsemiring.centralizer (Set.range ⇑L ∪ Set.range ⇑AddMonoid.End.mulRight) ↔\n ∀ (b : α), Commute (L b) (L a)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"AddMonoid.... | [
"α : Type u_5\ninst✝ : NonUnitalNonAssocCommSemiring α\na : α\n⊢ (AddMonoid.End.mulRight a = L a ∧ ∀ g ∈ Set.range ⇑L ∪ Set.range ⇑AddMonoid.End.mulRight, g * L a = L a * g) ↔\n ∀ (b : α), Commute (L b) (L a)"
] | Subsemiring.mem_centralizer_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.CentroidHom | {
"line": 526,
"column": 12
} | {
"line": 526,
"column": 85
} | {
"line": 526,
"column": 85
} | [
{
"pp": "F : Type u_1\nM : Type u_2\nN : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝ : NonAssocSemiring α\nT : CentroidHom α\n⊢ T 1 ∈ Subsemiring.center α",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"_private.Mathlib.Algebra.Ring.C... | [] | by constructor <;> simp [commute_iff_eq, ← map_mul_left, ← map_mul_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Star.CentroidHom | {
"line": 124,
"column": 10
} | {
"line": 124,
"column": 83
} | {
"line": 124,
"column": 83
} | [
{
"pp": "α : Type u_1\ninst✝¹ : NonAssocSemiring α\ninst✝ : StarRing α\nT : CentroidHom α\n⊢ T 1 ∈ StarSubsemiring.center α",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
"Commute",
"_private... | [] | by constructor <;> simp [commute_iff_eq, ← map_mul_left, ← map_mul_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 517,
"column": 28
} | {
"line": 517,
"column": 46
} | {
"line": 517,
"column": 47
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝ : AddCommMonoid k\nG' : Type u_3\nf : G → G'\nv : SkewMonoidAlgebra k G\n⊢ (v.sum fun a ↦ single (f a)).coeff = Finsupp.mapDomain f v.coeff",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Finsupp.mapDomain",
"AddMonoid.toAddZeroClass"... | [
"k : Type u_1\nG : Type u_2\ninst✝ : AddCommMonoid k\nG' : Type u_3\nf : G → G'\nv : SkewMonoidAlgebra k G\n⊢ (v.sum fun a ↦ single (f a)).coeff = v.coeff.sum fun a ↦ Finsupp.single (f a)"
] | Finsupp.mapDomain, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 721,
"column": 6
} | {
"line": 721,
"column": 21
} | {
"line": 721,
"column": 22
} | [
{
"pp": "case single.single\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Monoid G\ninst✝ : MulSemiringAction G k\nh : SkewMonoidAlgebra k G\nx : G\na : k\ny : G\nb : k\n⊢ single x a * single y b * h = single x a * (single y b * h)",
"ppTerm": "?single.single",
"assigned": true,
"usedCo... | [] | | single y b => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 891,
"column": 2
} | {
"line": 891,
"column": 26
} | {
"line": 893,
"column": 0
} | [
{
"pp": "case e_g.e_g\nk : Type u_1\nG : Type u_2\ninst✝³ : Semiring k\ninst✝² : Mul G\ninst✝¹ : SMulZeroClass G k\ninst✝ : DecidableEq G\nf g : SkewMonoidAlgebra k G\nx x✝³ : G\nx✝² : k\nx✝¹ : G\nx✝ : k\n⊢ (single (x✝³ * x✝¹) (x✝² * x✝³ • x✝)).coeff x = if x✝³ * x✝¹ = x then x✝² * x✝³ • x✝ else 0",
"ppTerm... | [] | exact coeff_single_apply | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Symmetrized | {
"line": 253,
"column": 6
} | {
"line": 254,
"column": 27
} | {
"line": 255,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Invertible 2\nx✝ : αˢʸᵐ\n⊢ 0 * x✝ = 0",
"ppTerm": "?m.291",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Equiv.instEquivLike",
"HMul.hMul",
"SymAlg.sym_zero",
"MulZer... | [] | rw [mul_def, unsym_zero, zero_mul, mul_zero, add_zero,
mul_zero, sym_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Symmetrized | {
"line": 253,
"column": 6
} | {
"line": 254,
"column": 27
} | {
"line": 255,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Invertible 2\nx✝ : αˢʸᵐ\n⊢ 0 * x✝ = 0",
"ppTerm": "?m.291",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Equiv.instEquivLike",
"HMul.hMul",
"SymAlg.sym_zero",
"MulZer... | [] | rw [mul_def, unsym_zero, zero_mul, mul_zero, add_zero,
mul_zero, sym_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Symmetrized | {
"line": 253,
"column": 6
} | {
"line": 254,
"column": 27
} | {
"line": 255,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Invertible 2\nx✝ : αˢʸᵐ\n⊢ 0 * x✝ = 0",
"ppTerm": "?m.291",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Equiv.instEquivLike",
"HMul.hMul",
"SymAlg.sym_zero",
"MulZer... | [] | rw [mul_def, unsym_zero, zero_mul, mul_zero, add_zero,
mul_zero, sym_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Symmetrized | {
"line": 256,
"column": 6
} | {
"line": 257,
"column": 27
} | {
"line": 258,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Invertible 2\nx✝ : αˢʸᵐ\n⊢ x✝ * 0 = 0",
"ppTerm": "?m.323",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Equiv.instEquivLike",
"HMul.hMul",
"SymAlg.sym_zero",
"MulZer... | [] | rw [mul_def, unsym_zero, zero_mul, mul_zero, add_zero,
mul_zero, sym_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Symmetrized | {
"line": 256,
"column": 6
} | {
"line": 257,
"column": 27
} | {
"line": 258,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Invertible 2\nx✝ : αˢʸᵐ\n⊢ x✝ * 0 = 0",
"ppTerm": "?m.323",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Equiv.instEquivLike",
"HMul.hMul",
"SymAlg.sym_zero",
"MulZer... | [] | rw [mul_def, unsym_zero, zero_mul, mul_zero, add_zero,
mul_zero, sym_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Symmetrized | {
"line": 256,
"column": 6
} | {
"line": 257,
"column": 27
} | {
"line": 258,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Invertible 2\nx✝ : αˢʸᵐ\n⊢ x✝ * 0 = 0",
"ppTerm": "?m.323",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Equiv.instEquivLike",
"HMul.hMul",
"SymAlg.sym_zero",
"MulZer... | [] | rw [mul_def, unsym_zero, zero_mul, mul_zero, add_zero,
mul_zero, sym_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 1091,
"column": 20
} | {
"line": 1091,
"column": 37
} | {
"line": 1091,
"column": 38
} | [
{
"pp": "α : Type u_3\nα₂ : Type u_4\nβ : Type u_5\nF : Type u_6\ninst✝⁴ : Semiring β\ninst✝³ : Monoid α\ninst✝² : Monoid α₂\ninst✝¹ : FunLike F α α₂\ninst✝ : MonoidHomClass F α α₂\nf : F\n⊢ (mapDomain ⇑f) (single 1 1) = single 1 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.... | [
"α : Type u_3\nα₂ : Type u_4\nβ : Type u_5\nF : Type u_6\ninst✝⁴ : Semiring β\ninst✝³ : Monoid α\ninst✝² : Monoid α₂\ninst✝¹ : FunLike F α α₂\ninst✝ : MonoidHomClass F α α₂\nf : F\n⊢ single (f 1) 1 = single 1 1"
] | mapDomain_single, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 1103,
"column": 13
} | {
"line": 1103,
"column": 30
} | {
"line": 1103,
"column": 31
} | [
{
"pp": "α : Type u_3\nα₂ : Type u_4\nβ : Type u_5\nF : Type u_6\ninst✝⁶ : Semiring β\ninst✝⁵ : Monoid α\ninst✝⁴ : Monoid α₂\ninst✝³ : FunLike F α α₂\ninst✝² : MulSemiringAction α β\ninst✝¹ : MulSemiringAction α₂ β\ninst✝ : MulHomClass F α α₂\nf : F\nx y : SkewMonoidAlgebra β α\nhf : ∀ (a : α) (x : β), a • x = ... | [
"α : Type u_3\nα₂ : Type u_4\nβ : Type u_5\nF : Type u_6\ninst✝⁶ : Semiring β\ninst✝⁵ : Monoid α\ninst✝⁴ : Monoid α₂\ninst✝³ : FunLike F α α₂\ninst✝² : MulSemiringAction α β\ninst✝¹ : MulSemiringAction α₂ β\ninst✝ : MulHomClass F α α₂\nf : F\nx y : SkewMonoidAlgebra β α\nhf : ∀ (a : α) (x : β), a • x = f a • x\n⊢ (... | mapDomain_single, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Tropical.Lattice | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 19
} | {
"line": 72,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder R\ns : Set (Tropical R)\nhs : ¬BddAbove s\nthis : Set.range untrop = Set.univ\n⊢ ¬BddAbove (untrop '' s)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"PartialOrder.toPre... | [
"R : Type u_1\nS : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder R\ns : Set (Tropical R)\nthis : Set.range untrop = Set.univ\nhs : BddAbove (untrop '' s)\n⊢ BddAbove s"
] | contrapose hs | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.Algebra.Tropical.Lattice | {
"line": 79,
"column": 6
} | {
"line": 79,
"column": 19
} | {
"line": 80,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder R\ns : Set (Tropical R)\nhs : ¬BddBelow s\nthis : Set.range untrop = Set.univ\n⊢ ¬BddBelow (untrop '' s)",
"ppTerm": "?m.91",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"PartialOrder.toPre... | [
"R : Type u_1\nS : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder R\ns : Set (Tropical R)\nthis : Set.range untrop = Set.univ\nhs : BddBelow (untrop '' s)\n⊢ BddBelow s"
] | contrapose hs | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.Geometry.RingedSpace.SheafedSpace | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 100
} | {
"line": 261,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (Cate... | [
"C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (CategoryTheory.f... | erw [← PresheafedSpace.stalkMap_germ_apply ⟨f, fc⟩, ← PresheafedSpace.stalkMap_germ_apply ⟨f, gc⟩] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.Geometry.RingedSpace.SheafedSpace | {
"line": 272,
"column": 82
} | {
"line": 272,
"column": 99
} | {
"line": 273,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (Cate... | [
"C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (CategoryTheory.f... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.