module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Nat.Squarefree | {
"line": 224,
"column": 39
} | {
"line": 224,
"column": 45
} | {
"line": 224,
"column": 45
} | [
{
"pp": "n d : ℕ\nh : n.minSqFac = some d\nm : ℕ\nm2 : 2 ≤ m\nmd : m * m ∣ n\nthis : n.MinSqFacProp (some d)\nfd : m.minFac ∣ m\n⊢ 0 < 2",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Squarefree | {
"line": 224,
"column": 39
} | {
"line": 224,
"column": 45
} | {
"line": 224,
"column": 45
} | [
{
"pp": "n d : ℕ\nh : n.minSqFac = some d\nm : ℕ\nm2 : 2 ≤ m\nmd : m * m ∣ n\nthis : n.MinSqFacProp (some d)\nfd : m.minFac ∣ m\n⊢ 0 < 2",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Squarefree | {
"line": 224,
"column": 39
} | {
"line": 224,
"column": 45
} | {
"line": 224,
"column": 45
} | [
{
"pp": "n d : ℕ\nh : n.minSqFac = some d\nm : ℕ\nm2 : 2 ≤ m\nmd : m * m ∣ n\nthis : n.MinSqFacProp (some d)\nfd : m.minFac ∣ m\n⊢ 0 < 2",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 490,
"column": 2
} | {
"line": 490,
"column": 6
} | {
"line": 491,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : ℕ\ncop : m.Coprime n\n⊢ ∑ x ∈ (m * n).divisorsAntidiagonal, f x.1 * g x.2 =\n ∑ x ∈ m.divisorsAntidiagonal ×ˢ n.divisorsAntidiagonal, f x.1.1 * g x.1.2 * (f x.2.1 * g x.2.2)",
... | [
"R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : ℕ\ncop : m.Coprime n\n⊢ ∑ x ∈ m.divisorsAntidiagonal ×ˢ n.divisorsAntidiagonal, f x.1.1 * g x.1.2 * (f x.2.1 * g x.2.2) =\n ∑ x ∈ (m * n).divisorsAntidiagonal, f x.1 * g x.2"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Data.Nat.Factorization.PrimePow | {
"line": 130,
"column": 4
} | {
"line": 131,
"column": 23
} | {
"line": 132,
"column": 4
} | [
{
"pp": "a b : ℕ\nhab : a.Coprime b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\n⊢ a.factorization p = 0 ∨ b.factorization p = 0",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroCl... | [
"a b : ℕ\nhab : a.Coprime b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\n⊢ p ∉ a.factorization.support ∩ b.factorization.support"
] | rw [← Finsupp.notMem_support_iff, ← Finsupp.notMem_support_iff, ← not_and_or, ←
Finset.mem_inter] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 84,
"column": 39
} | {
"line": 95,
"column": 64
} | {
"line": 97,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nn : ℕ\nhn : Squarefree n\n⊢ (ArithmeticFunction.prodPrimeFactors fun p ↦ (f + g) p) n = (f * g) n",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"UniqueFactor... | [] | by
rw [prodPrimeFactors_apply hn.ne_zero]
simp_rw [add_apply (f := f) (g := g)]
rw [prod_add, mul_apply, sum_divisorsAntidiagonal (f · * g ·),
← divisors_filter_squarefree_of_squarefree hn, sum_divisors_filter_squarefree hn.ne_zero,
factors_eq]
apply sum_congr rfl
intro t ht
rw [t.prod_val, Function... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Factorization.PrimePow | {
"line": 137,
"column": 92
} | {
"line": 147,
"column": 30
} | {
"line": 149,
"column": 0
} | [
{
"pp": "a b : ℕ\nhab : a.Coprime b\n⊢ {d ∈ (a * b).divisors | IsPrimePow d} = {d ∈ a.divisors ∪ b.divisors | IsPrimePow d}",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Data.Nat.Factorization.PrimePow.0.Nat.mul_divisors_filter_prime_pow._simp_1_5",
"Eq.mpr"... | [] | by
rcases eq_or_ne a 0 with (rfl | ha)
· simp only [Nat.coprime_zero_left] at hab
simp [hab, Finset.filter_singleton, not_isPrimePow_one]
rcases eq_or_ne b 0 with (rfl | hb)
· simp only [Nat.coprime_zero_right] at hab
simp [hab, Finset.filter_singleton, not_isPrimePow_one]
ext n
simp only [ha, hb, F... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Antidiag.Nat | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 37
} | {
"line": 142,
"column": 4
} | [
{
"pp": "case mpr\nd : ℕ\ni : Fin d\nhd : d ≠ 1\nk r : ℕ\nhn : ¬k * r = 0\nhs : Nontrivial (Fin d)\n⊢ ∃ a, (∏ i, a i = k * r ∧ k * r ≠ 0) ∧ a i = k",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Finset.univ",
"Exists",
"exists_ne",
"Ne",
"i... | [
"case mpr\nd : ℕ\ni : Fin d\nhd : d ≠ 1\nk r : ℕ\nhn : ¬k * r = 0\nhs : Nontrivial (Fin d)\ni' : Fin d\nhi_ne : i' ≠ i\n⊢ ∃ a, (∏ i, a i = k * r ∧ k * r ≠ 0) ∧ a i = k"
] | obtain ⟨i', hi_ne⟩ := exists_ne i | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Order.Antidiag.Nat | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 77
} | {
"line": 159,
"column": 2
} | [
{
"pp": "d n p : ℕ\nhn : Squarefree n\nhp : p ∈ n.primeFactorsList\nf : Fin d → ℕ\nhf : ∏ i, f i = n ∧ n ≠ 0\n⊢ ∃! i, p ∣ f i",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Nat.Prime",
"Dvd.dvd",
"Finset.univ",
"congrArg",
... | [
"d n p : ℕ\nhn : Squarefree n\nf : Fin d → ℕ\nhp : Prime p ∧ ∃ a ∈ univ, p ∣ f a\nhf : ∏ i, f i = n ∧ n ≠ 0\n⊢ ∃! i, p ∣ f i"
] | rw [mem_primeFactorsList hf.2, ← hf.1, hp.1.prime.dvd_finsetProd_iff] at hp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Archimedean.IndicatorCard | {
"line": 60,
"column": 14
} | {
"line": 60,
"column": 17
} | {
"line": 61,
"column": 6
} | [
{
"pp": "R : 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\nn' : ℕ\nhn' : n < n' • r\n... | [
"R : 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\nn' : ℕ\nhn' : n < n' • r\nt : Finset ℕ... | i_t | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 1146,
"column": 2
} | {
"line": 1146,
"column": 32
} | {
"line": 1147,
"column": 2
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\nι : Type u_3\nm : MonomialOrder ι\ni : ι\nr : R\n⊢ m.Monic (X i - C r)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"RingHom.instRingHomClass",
"Nat.instMulZeroClass",
"RingHom... | [
"R : Type u_2\ninst✝ : CommRing R\nι : Type u_3\nm : MonomialOrder ι\ni : ι\nr : R\n⊢ m.Monic (X i + C (-r))"
] | rw [sub_eq_add_neg, ← map_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 534,
"column": 2
} | {
"line": 538,
"column": 8
} | {
"line": 540,
"column": 0
} | [
{
"pp": "case h\nR : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : ℕ\ncop : m.Coprime n\n⊢ ∀ a ∈ m.divisorsAntidiagonal ×ˢ n.divisorsAntidiagonal,\n f a.1.1 * g a.1.2 * (f a.2.1 * g a.2.2) =\n f\n (match a with\n ... | [] | · simp only [mem_divisorsAntidiagonal, mem_product]
rintro ⟨⟨a1, a2⟩, ⟨b1, b2⟩⟩ ⟨⟨rfl, ha⟩, ⟨rfl, hb⟩⟩
rw [hf.map_mul_of_coprime cop.coprime_mul_right.coprime_mul_right_right,
hg.map_mul_of_coprime cop.coprime_mul_left.coprime_mul_left_right]
ring | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.Chebyshev | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 33
} | {
"line": 178,
"column": 0
} | [
{
"pp": "case hbc\nι : Type u_1\nα : Type u_2\ninst✝³ : Semifield α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : ExistsAddOfLE α\ns : Finset ι\nf : ι → α\nhs : s.Nonempty\n⊢ (∑ i ∈ s, f i) ^ 2 ≤ ↑(#s) * ∑ i ∈ s, f i ^ 2",
"ppTerm": "?hbc",
"assigned": true,
"usedConstants": [
... | [] | exact sq_sum_le_card_mul_sum_sq | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 356,
"column": 6
} | {
"line": 356,
"column": 16
} | {
"line": 356,
"column": 16
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nh : mk (a * b) < min (mk a) (mk b)\n⊢ False",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
... | [
"M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nh : mk (a * b) < mk a ∧ mk (a * b) < mk b\n⊢ False"
] | lt_min_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 376,
"column": 2
} | {
"line": 376,
"column": 73
} | {
"line": 378,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nhab : mk a ≤ mk b\n⊢ mk a ≤ mk (a / b)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"DivisionCo... | [] | simpa [div_eq_mul_inv, hab] using mk_left_le_mk_mul (a := a) (b := b⁻¹) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 376,
"column": 2
} | {
"line": 376,
"column": 73
} | {
"line": 378,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nhab : mk a ≤ mk b\n⊢ mk a ≤ mk (a / b)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"DivisionCo... | [] | simpa [div_eq_mul_inv, hab] using mk_left_le_mk_mul (a := a) (b := b⁻¹) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 376,
"column": 2
} | {
"line": 376,
"column": 73
} | {
"line": 378,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nhab : mk a ≤ mk b\n⊢ mk a ≤ mk (a / b)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"DivisionCo... | [] | simpa [div_eq_mul_inv, hab] using mk_left_le_mk_mul (a := a) (b := b⁻¹) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 31,
"column": 48
} | {
"line": 35,
"column": 11
} | {
"line": 36,
"column": 4
} | [
{
"pp": "s : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nr : Finset ℤ := Ioc (c - ↑(#s)) c\n⊢ ∑ x ∈ s, x ≤ ∑ x ∈ s ∩ r, x + #(s \\ r) • (c - ↑(#s))",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",
"le_refl",
"instHSMul",
"congrArg",... | [] | by
rw [← sum_inter_add_sum_sdiff s r _]
gcongr
apply sum_le_card_nsmul
grind | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 43,
"column": 21
} | {
"line": 43,
"column": 28
} | {
"line": 43,
"column": 28
} | [
{
"pp": "case h₂\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nr : Finset ℤ := Ioc (c - ↑(#s)) c\nx : ℤ\nmx : x ∈ r ∧ x ∉ s\n⊢ c - ↑(#s) ≤ x",
"ppTerm": "?h₂",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"HSub.hSub... | [
"case h₂\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nr : Finset ℤ := Ioc (c - ↑(#s)) c\nx : ℤ\nmx : (c - ↑(#s) < x ∧ x ≤ c) ∧ x ∉ s\n⊢ c - ↑(#s) ≤ x"
] | mem_Ioc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 50,
"column": 20
} | {
"line": 50,
"column": 27
} | {
"line": 50,
"column": 27
} | [
{
"pp": "case refine_1\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nx : ℕ\nmx : x ∈ range #s\n⊢ c - ↑x ∈ Ioc (c - ↑(#s)) c",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"HSub... | [
"case refine_1\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nx : ℕ\nmx : x ∈ range #s\n⊢ c - ↑(#s) < c - ↑x ∧ c - ↑x ≤ c"
] | mem_Ioc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 53,
"column": 17
} | {
"line": 53,
"column": 24
} | {
"line": 53,
"column": 24
} | [
{
"pp": "case refine_3\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nx : ℤ\nmx : x ∈ Ioc (c - ↑(#s)) c\n⊢ ∃ x_1 < #s, c - ↑x_1 = x",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"HSub.hSub",
... | [
"case refine_3\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nx : ℤ\nmx : c - ↑(#s) < x ∧ x ≤ c\n⊢ ∃ x_1 < #s, c - ↑x_1 = x"
] | mem_Ioc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Interval.Set.SuccPred | {
"line": 136,
"column": 97
} | {
"line": 137,
"column": 70
} | {
"line": 139,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : One α\ninst✝¹ : Sub α\ninst✝ : PredSubOrder α\na : α\nha : ¬IsMin a\nb : α\n⊢ Ioo (a - 1) b = Ico a b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Set.Ioo_pred_left_eq_Ioc_of_not_isMin",
"PredSubOrder.toPredOrder",
... | [] | by
simpa [pred_eq_sub_one] using Ioo_pred_left_eq_Ioc_of_not_isMin ha b | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.HahnSeries.Addition | {
"line": 296,
"column": 59
} | {
"line": 301,
"column": 35
} | {
"line": 303,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : AddMonoid R\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nx y : R⟦Γ⟧\n⊢ embDomain f (x + y) = embDomain f x + embDomain f y",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"HahnSeries.embDomain",
"Ad... | [] | by
ext g
by_cases hg : g ∈ Set.range f
· obtain ⟨a, rfl⟩ := hg
simp
· simp [embDomain_notin_range hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.HahnSeries.Addition | {
"line": 536,
"column": 47
} | {
"line": 541,
"column": 35
} | {
"line": 543,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Semiring R\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nr : R\nx : R⟦Γ⟧\n⊢ embDomain f (r • x) = r • embDomain f x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne"... | [] | by
ext g
by_cases hg : g ∈ Set.range f
· obtain ⟨a, rfl⟩ := hg
simp
· simp [embDomain_notin_range hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Module.PositiveLinearMap | {
"line": 199,
"column": 17
} | {
"line": 203,
"column": 22
} | {
"line": 203,
"column": 23
} | [
{
"pp": "R : Type u_1\nE₁ : Type u_2\nE₂ : Type u_3\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommGroup E₁\ninst✝⁶ : PartialOrder E₁\ninst✝⁵ : IsOrderedAddMonoid E₁\ninst✝⁴ : AddCommGroup E₂\ninst✝³ : PartialOrder E₂\ninst✝² : IsOrderedAddMonoid E₂\ninst✝¹ : Module R E₁\ninst✝ : Module R E₂\nf : E₁ →ₗ[R] E₂\nhf : ∀ (x ... | [] | by
intro a b hab
rw [← sub_nonneg] at hab ⊢
have : 0 ≤ f (b - a) := hf _ hab
simpa using this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 60,
"column": 27
} | {
"line": 60,
"column": 44
} | {
"line": 60,
"column": 45
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝² : LinearOrder Γ\ninst✝¹ : Zero R\ninst✝ : LinearOrder R\na b : Lex R⟦Γ⟧\nhab✝ : a ≠ b\nhab : ∃ a_1, a.coeff a_1 ≠ b.coeff a_1\nu : Set Γ := {i | (ofLex a).coeff i ≠ 0} ∪ {i | (ofLex b).coeff i ≠ 0}\nv : Set Γ := {i | (ofLex a).coeff i ≠ (ofLex b).coeff i}\ni : Γ\nh : ... | [
"Γ : Type u_1\nR : Type u_2\ninst✝² : LinearOrder Γ\ninst✝¹ : Zero R\ninst✝ : LinearOrder R\na b : Lex R⟦Γ⟧\nhab✝ : a ≠ b\nhab : ∃ a_1, a.coeff a_1 ≠ b.coeff a_1\nu : Set Γ := {i | (ofLex a).coeff i ≠ 0} ∪ {i | (ofLex b).coeff i ≠ 0}\nv : Set Γ := {i | (ofLex a).coeff i ≠ (ofLex b).coeff i}\ni : Γ\nh : i ∈ v\n⊢ (of... | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 235,
"column": 6
} | {
"line": 235,
"column": 27
} | {
"line": 235,
"column": 28
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_5\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : VAdd Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\ninst✝² : AddCommMonoid V\ninst✝¹ : Zero R\ninst✝ : SMulZeroClass R V\nx : R⟦Γ⟧\ny : HahnModule Γ' R V\na : Γ'\ns : Set Γ'\nhs : s.IsPWO\nhys... | [
"Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_5\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : VAdd Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\ninst✝² : AddCommMonoid V\ninst✝¹ : Zero R\ninst✝ : SMulZeroClass R V\nx : R⟦Γ⟧\ny : HahnModule Γ' R V\na : Γ'\ns : Set Γ'\nhs : s.IsPWO\nhys : ((of R).s... | hb.2 hb.1.1 hb.1.2.2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 8
} | {
"line": 145,
"column": 4
} | [
{
"pp": "case neg\nK : 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 : Finit... | [
"case neg\nK : 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 : FiniteArchimedean... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Order.Ring.Archimedean | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 31
} | {
"line": 187,
"column": 2
} | [
{
"pp": "case inl\nS : Type u_3\ninst✝² : LinearOrder S\ninst✝¹ : CommRing S\ninst✝ : IsStrictOrderedRing S\nn : ℤ\nh : n ≠ 0\nh✝ : Subsingleton S\n⊢ mk ↑n = 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Int.cast",
"CommSemiring.toSemiring",
"ArchimedeanClass.instSubs... | [
"case inr\nS : Type u_3\ninst✝² : LinearOrder S\ninst✝¹ : CommRing S\ninst✝ : IsStrictOrderedRing S\nn : ℤ\nh : n ≠ 0\nh✝ : Nontrivial S\n⊢ mk ↑n = 0"
] | · exact Subsingleton.allEq .. | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 575,
"column": 62
} | {
"line": 579,
"column": 10
} | {
"line": 581,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝⁴ : AddCommMonoid Γ\ninst✝³ : LinearOrder Γ\ninst✝² : IsOrderedCancelAddMonoid Γ\ninst✝¹ : NonUnitalNonAssocSemiring R\nx y : R⟦Γ⟧\ninst✝ : NoZeroDivisors R\n⊢ (x * y).leadingCoeff = x.leadingCoeff * y.leadingCoeff",
"ppTerm": "?m.30",
"assigned": true,
"use... | [] | by
by_cases hx : x = 0; · simp [hx]
by_cases hy : y = 0; · simp [hy]
apply leadingCoeff_mul_of_ne_zero
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Ring.Archimedean | {
"line": 279,
"column": 4
} | {
"line": 286,
"column": 27
} | {
"line": 288,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : LinearOrder R\ninst✝² : LinearOrder S\ninst✝¹ : Field R\ninst✝ : IsOrderedRing R\nx y : R\nh : mk x = mk y\n⊢ mk x⁻¹ = mk y⁻¹",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.... | [] | obtain rfl | hx := eq_or_ne x 0
· simp_all
obtain rfl | hy := eq_or_ne y 0
· simp_all
have hx' : mk x ≠ ⊤ := by simpa using hx
apply add_left_cancel_of_ne_top hx'
nth_rw 2 [h]
simp [← mk_mul, hx, hy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Archimedean | {
"line": 279,
"column": 4
} | {
"line": 286,
"column": 27
} | {
"line": 288,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : LinearOrder R\ninst✝² : LinearOrder S\ninst✝¹ : Field R\ninst✝ : IsOrderedRing R\nx y : R\nh : mk x = mk y\n⊢ mk x⁻¹ = mk y⁻¹",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.... | [] | obtain rfl | hx := eq_or_ne x 0
· simp_all
obtain rfl | hy := eq_or_ne y 0
· simp_all
have hx' : mk x ≠ ⊤ := by simpa using hx
apply add_left_cancel_of_ne_top hx'
nth_rw 2 [h]
simp [← mk_mul, hx, hy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 798,
"column": 17
} | {
"line": 798,
"column": 73
} | {
"line": 799,
"column": 2
} | [
{
"pp": "case zero\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\na : Γ\nr : R\nx✝ : Γ\n⊢ ((single a) r ^ 0).coeff x✝ = ((single (0 • a)) (r ^ 0)).coeff x✝",
"ppTerm": "?zero",
"assigned": true,
"usedConstants"... | [] | simp only [pow_zero, coeff_one, zero_smul, coeff_single] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.UpperLower | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 21
} | {
"line": 83,
"column": 2
} | [
{
"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... | [
"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⁻¹)"
] | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.UpperLower | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 21
} | {
"line": 88,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : Preorder α\ninst✝ : IsOrderedMonoid α\ns t : Set α\nhs : IsUpperSet s\n⊢ IsUpperSet (s / t)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Division... | [
"α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : Preorder α\ninst✝ : IsOrderedMonoid α\ns t : Set α\nhs : IsUpperSet s\n⊢ IsUpperSet (s * t⁻¹)"
] | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 666,
"column": 10
} | {
"line": 666,
"column": 33
} | {
"line": 666,
"column": 34
} | [
{
"pp": "case pos\nK : Type u\ninst✝ : Field K\nA B : ValuationSubring K\nh : B.principalUnitGroup ≤ A.principalUnitGroup\nx : K\nhx : x ∈ A\nh_1 : ¬x = 0\nh_2 : x⁻¹ + 1 = 0\n⊢ x ∈ B",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.t... | [
"case pos\nK : Type u\ninst✝ : Field K\nA B : ValuationSubring K\nh : B.principalUnitGroup ≤ A.principalUnitGroup\nx : K\nhx : x ∈ A\nh_1 : ¬x = 0\nh_2 : x⁻¹ = -1\n⊢ x ∈ B"
] | add_eq_zero_iff_eq_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 428,
"column": 2
} | {
"line": 429,
"column": 34
} | {
"line": 431,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx : K\nf : ℝ →+*o K\nr : ℝ\nhx : 0 ≤ mk x\nh : stdPart x < r\n⊢ x < f r",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"neg_lt_neg_i... | [] | rw [← neg_lt_neg_iff, ← map_neg]
apply lt_of_lt_stdPart <;> simpa | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 428,
"column": 2
} | {
"line": 429,
"column": 34
} | {
"line": 431,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx : K\nf : ℝ →+*o K\nr : ℝ\nhx : 0 ≤ mk x\nh : stdPart x < r\n⊢ x < f r",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"neg_lt_neg_i... | [] | rw [← neg_lt_neg_iff, ← map_neg]
apply lt_of_lt_stdPart <;> simpa | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 813,
"column": 4
} | {
"line": 813,
"column": 78
} | {
"line": 814,
"column": 4
} | [
{
"pp": "case neg.inl\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... | [
"case neg.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 : AddCommGroup R\n... | · rw [HahnSeries.coe_truncLTLinearMap, HahnSeries.coeff_truncLT_of_lt hdc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.CoeffList | {
"line": 153,
"column": 29
} | {
"line": 153,
"column": 51
} | {
"line": 153,
"column": 52
} | [
{
"pp": "case neg\nR : 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\... | [
"case neg\nR : 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\nhdk... | List.getElem?_reverse, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 83
} | {
"line": 77,
"column": 4
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nm n : ℕ\nhp : p.IsMonicOfDegree m\nhq : q.IsMonicOfDegree n\nH : Subsingleton R\n⊢ (p * q).IsMonicOfDegree (m + n)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Eq.mp",
"id",
... | [
"case inl\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nm n : ℕ\nH : Subsingleton R\nhp : m = 0\nhq : n = 0\n⊢ m = 0 ∧ n = 0"
] | simp only [isMonicOfDegree_iff_of_subsingleton, Nat.add_eq_zero_iff] at hp hq ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Eval.Irreducible | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 11
} | {
"line": 55,
"column": 2
} | [
{
"pp": "case h\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nh_mon : f.leadingCoeff = 1\nh_irr : Irreducible (map φ f)\na b : R[X]\nh : f = a * b\nq : a.leadingCoeff * b.leadingCoeff = 1\n⊢ a.leadingCoeff * ?refine_1.b = 1",
... | [
"case h\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nh_mon : f.leadingCoeff = 1\nh_irr : Irreducible (map φ f)\na b : R[X]\nh : f = a * b\nq : a.leadingCoeff * b.leadingCoeff = 1\n⊢ b.leadingCoeff * ?refine_2.b = 1",
"case refin... | · exact q | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 32
} | {
"line": 284,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\np : R[X]\nn : ℕ\nhp : p.IsMonicOfDegree n\nr : R\n⊢ ((aeval (X - C r)) p).IsMonicOfDegree n",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"RingHom.instRingHomClass",
"RingHomClass.toAddMonoidHomC... | [
"R : Type u_1\ninst✝ : CommRing R\np : R[X]\nn : ℕ\nhp : p.IsMonicOfDegree n\nr : R\n⊢ ((aeval (X + C (-r))) p).IsMonicOfDegree n"
] | rw [sub_eq_add_neg, ← map_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 65,
"column": 7
} | {
"line": 65,
"column": 26
} | {
"line": 65,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]\nhp : ¬p = 0\na✝ : Nontrivial R\n⊢ p.reverse.leadingCoeff * (X ^ p.natTrailingDegree).leadingCoeff ≠ 0",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
... | [
"R : Type u_1\ninst✝ : Semiring R\np : R[X]\nhp : ¬p = 0\na✝ : Nontrivial R\n⊢ p.reverse.leadingCoeff * 1 ≠ 0"
] | leadingCoeff_X_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 84,
"column": 35
} | {
"line": 84,
"column": 59
} | {
"line": 84,
"column": 60
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nh2 : n ≤ p.natDegree\nh3 : p.natTrailingDegree ≤ n\n⊢ p.mirror.coeff n = p.coeff (p.natDegree - n + p.natTrailingDegree)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAd... | [
"case pos\nR : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nh2 : n ≤ p.natDegree\nh3 : p.natTrailingDegree ≤ n\n⊢ p.mirror.coeff n = p.coeff (p.natDegree - (n - p.natTrailingDegree))"
] | ← tsub_tsub_assoc h2 h3, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 163,
"column": 22
} | {
"line": 163,
"column": 45
} | {
"line": 163,
"column": 46
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : Semiring R\np : R[X]\ninst✝ : NoZeroDivisors R\nhp : p = 0\n⊢ natTrailingDegree 0 = 2 * natTrailingDegree 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"instMulNat",
"... | [
"case pos\nR : Type u_1\ninst✝¹ : Semiring R\np : R[X]\ninst✝ : NoZeroDivisors R\nhp : p = 0\n⊢ 0 = 2 * 0"
] | natTrailingDegree_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Destutter | {
"line": 288,
"column": 6
} | {
"line": 288,
"column": 66
} | {
"line": 290,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : DecidableEq α\nr : α → α → Prop\ninst✝ : Std.Antisymm r\nx y : α\nxs : List α\nhxy : x ≠ y\nh : (r x y ∧ ∀ (a : α), a ∈ xs → r x a) ∧ (∀ (a' : α), a' ∈ xs → r y a') ∧ Pairwise r xs\nthis : ¬x ∈ xs\n⊢ (if x ≠ y then x :: (y :: xs).dedup else destutter (fun x1 x2 ↦ x1 ≠ x... | [] | rw [if_pos hxy, dedup_cons_of_notMem (a := x) (by simp [*])] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 128,
"column": 12
} | {
"line": 128,
"column": 54
} | {
"line": 129,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝ : CommRing R\ng : R[X]\nhg : g.Monic\nq₁ q₂ : R[X]\nr₁ r₂ : Fin 0 → R[X]\nhr₁ : ∀ (i : Fin 0), (r₁ i).degree < g.degree\nhr₂ : ∀ (i : Fin 0), (r₂ i).degree < g.degree\nhf : q₁ * g ^ 0 + ∑ i, r₁ i * g ^ ↑i = q₂ * g ^ 0 + ∑ i, r₂ i * g ^ ↑i\n⊢ q₁ = q₂ ∧ r₁ = r₂",
"ppTer... | [] | exact ⟨by simpa using hf, funext Fin.rec0⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 128,
"column": 12
} | {
"line": 128,
"column": 54
} | {
"line": 129,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝ : CommRing R\ng : R[X]\nhg : g.Monic\nq₁ q₂ : R[X]\nr₁ r₂ : Fin 0 → R[X]\nhr₁ : ∀ (i : Fin 0), (r₁ i).degree < g.degree\nhr₂ : ∀ (i : Fin 0), (r₂ i).degree < g.degree\nhf : q₁ * g ^ 0 + ∑ i, r₁ i * g ^ ↑i = q₂ * g ^ 0 + ∑ i, r₂ i * g ^ ↑i\n⊢ q₁ = q₂ ∧ r₁ = r₂",
"ppTer... | [] | exact ⟨by simpa using hf, funext Fin.rec0⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 128,
"column": 12
} | {
"line": 128,
"column": 54
} | {
"line": 129,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝ : CommRing R\ng : R[X]\nhg : g.Monic\nq₁ q₂ : R[X]\nr₁ r₂ : Fin 0 → R[X]\nhr₁ : ∀ (i : Fin 0), (r₁ i).degree < g.degree\nhr₂ : ∀ (i : Fin 0), (r₂ i).degree < g.degree\nhf : q₁ * g ^ 0 + ∑ i, r₁ i * g ^ ↑i = q₂ * g ^ 0 + ∑ i, r₂ i * g ^ ↑i\n⊢ q₁ = q₂ ∧ r₁ = r₂",
"ppTer... | [] | exact ⟨by simpa using hf, funext Fin.rec0⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 51
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : Semiring R\nm n : ℕ\nu v w : R\nhmn : m < n\nhw : w ≠ 0\ni : ℕ\nhkm : i < m\n⊢ ↑i ≤ ↑n",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"WithBot.some",
"WithBot",
"Lattice.toSemilatticeSup",
"PartialOrder.toPre... | [] | exact WithBot.coe_le_coe.mpr (hkm.trans hmn).le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 51
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : Semiring R\nm n : ℕ\nu v w : R\nhmn : m < n\nhw : w ≠ 0\ni : ℕ\nhkm : i < m\n⊢ ↑i ≤ ↑n",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"WithBot.some",
"WithBot",
"Lattice.toSemilatticeSup",
"PartialOrder.toPre... | [] | exact WithBot.coe_le_coe.mpr (hkm.trans hmn).le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 51
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : Semiring R\nm n : ℕ\nu v w : R\nhmn : m < n\nhw : w ≠ 0\ni : ℕ\nhkm : i < m\n⊢ ↑i ≤ ↑n",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"WithBot.some",
"WithBot",
"Lattice.toSemilatticeSup",
"PartialOrder.toPre... | [] | exact WithBot.coe_le_coe.mpr (hkm.trans hmn).le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 10
} | {
"line": 192,
"column": 2
} | [
{
"pp": "case refine_1\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\n⊢ 1 + (1 + 1) = 3",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"of_decide_eq_true",
"Monoid.toMulOneClass",
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZero",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 227,
"column": 8
} | {
"line": 227,
"column": 21
} | {
"line": 227,
"column": 22
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Nontrivial A\ninst✝ : NoZeroDivisors A\np : R[X]\nq : ℕ\nhq : 0 < q\ninj_amap : Function.Injective ⇑(algebraMap R A)\np0 : p ≠ 0\nc : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choose else 0\nc_le : ∀... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Nontrivial A\ninst✝ : NoZeroDivisors A\np : R[X]\nq : ℕ\nhq : 0 < q\ninj_amap : Function.Injective ⇑(algebraMap R A)\np0 : p ≠ 0\nc : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choose else 0\nc_le : ∀ (k : ℕ), (c... | range_eq_Ico, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 227,
"column": 22
} | {
"line": 227,
"column": 35
} | {
"line": 227,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Nontrivial A\ninst✝ : NoZeroDivisors A\np : R[X]\nq : ℕ\nhq : 0 < q\ninj_amap : Function.Injective ⇑(algebraMap R A)\np0 : p ≠ 0\nc : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choose else 0\nc_le : ∀... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Nontrivial A\ninst✝ : NoZeroDivisors A\np : R[X]\nq : ℕ\nhq : 0 < q\ninj_amap : Function.Injective ⇑(algebraMap R A)\np0 : p ≠ 0\nc : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choose else 0\nc_le : ∀ (k : ℕ), (c... | range_eq_Ico, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 269,
"column": 4
} | {
"line": 269,
"column": 58
} | {
"line": 270,
"column": 2
} | [
{
"pp": "case inl\np q : ℤ[X]\nk m m' n : ℕ\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nh : p * p.mirror = q * q.mirror\nhq : q = trinomial k m' n ↑u ↑v ↑w\nhmul : ↑w * ↑u = ↑w * ↑u\n⊢ q = p ∨ q = p.mirror",
"ppTerm": "?inl",
"assigned": true,
... | [] | exact irreducible_aux2 hkm hmn hkm' hmn' u v w hp hq h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 269,
"column": 4
} | {
"line": 269,
"column": 58
} | {
"line": 270,
"column": 2
} | [
{
"pp": "case inl\np q : ℤ[X]\nk m m' n : ℕ\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nh : p * p.mirror = q * q.mirror\nhq : q = trinomial k m' n ↑u ↑v ↑w\nhmul : ↑w * ↑u = ↑w * ↑u\n⊢ q = p ∨ q = p.mirror",
"ppTerm": "?inl",
"assigned": true,
... | [] | exact irreducible_aux2 hkm hmn hkm' hmn' u v w hp hq h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 269,
"column": 4
} | {
"line": 269,
"column": 58
} | {
"line": 270,
"column": 2
} | [
{
"pp": "case inl\np q : ℤ[X]\nk m m' n : ℕ\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nh : p * p.mirror = q * q.mirror\nhq : q = trinomial k m' n ↑u ↑v ↑w\nhmul : ↑w * ↑u = ↑w * ↑u\n⊢ q = p ∨ q = p.mirror",
"ppTerm": "?inl",
"assigned": true,
... | [] | exact irreducible_aux2 hkm hmn hkm' hmn' u v w hp hq h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 139,
"column": 6
} | {
"line": 139,
"column": 10
} | {
"line": 139,
"column": 11
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsOrderedRing R\nP : R[X]\nhsc : (⇑SignType.sign ∘ fun x ↦ -x) = (fun x ↦ -x) ∘ ⇑SignType.sign\nh_neg_destutter :\n ∀ (l : List SignType),\n List.map (fun x ↦ -x) (List.destutter (fun x1 x2 ↦ ¬x1 = x2) l) =\n List.destutter (fun x1... | [
"R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsOrderedRing R\nP : R[X]\nhsc : (⇑SignType.sign ∘ fun x ↦ -x) = (fun x ↦ -x) ∘ ⇑SignType.sign\nh_neg_destutter :\n ∀ (l : List SignType),\n List.map (fun x ↦ -x) (List.destutter (fun x1 x2 ↦ ¬x1 = x2) l) =\n List.destutter (fun x1 x2 ↦ ¬x1 = ... | hsc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 250,
"column": 8
} | {
"line": 250,
"column": 21
} | {
"line": 250,
"column": 22
} | [
{
"pp": "case inr.refine_2\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Nontrivial A\ninst✝ : NoZeroDivisors A\np : R[X]\nq : ℕ\nhq : 0 < q\ninj_amap : Function.Injective ⇑(algebraMap R A)\np0 : p ≠ 0\nc : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choo... | [
"case inr.refine_2\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Nontrivial A\ninst✝ : NoZeroDivisors A\np : R[X]\nq : ℕ\nhq : 0 < q\ninj_amap : Function.Injective ⇑(algebraMap R A)\np0 : p ≠ 0\nc : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choose else 0\nc... | range_eq_Ico, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 112,
"column": 8
} | {
"line": 112,
"column": 12
} | {
"line": 113,
"column": 8
} | [
{
"pp": "K : Type u_1\nR : Type u_2\na b : R\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nu : { u // u * u = a • 1 + b • u }\nz w : QuadraticAlgebra R a b\n⊢ (z * w).re • 1 + (z * w).im • ↑u = (z.re • 1 + z.im • ↑u) * (w.re • 1 + w.im • ↑u)",
"ppTerm": "?m.97",
"assigned... | [
"K : Type u_1\nR : Type u_2\na b : R\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nu : { u // u * u = a • 1 + b • u }\nz w : QuadraticAlgebra R a b\n⊢ (z.re • 1 + z.im • ↑u) * (w.re • 1 + w.im • ↑u) = (z * w).re • 1 + (z * w).im • ↑u"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 325,
"column": 6
} | {
"line": 325,
"column": 60
} | {
"line": 326,
"column": 6
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝ : Field K\na b : K\nHab : Fact (∀ (r : K), r ^ 2 ≠ a + b * r)\nz : QuadraticAlgebra K a b\nhz : z.re * z.re + b * z.re * z.im - a * z.im * z.im = 0\nh : ¬z.im = 0\n⊢ False",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMo... | [
"case neg\nK : Type u_1\ninst✝ : Field K\na b : K\nHab : Fact (∀ (r : K), r ^ 2 ≠ a + b * r)\nz : QuadraticAlgebra K a b\nhz : z.re ^ 2 = a * z.im * z.im - b * z.re * z.im\nh : ¬z.im = 0\n⊢ False"
] | rw [← pow_two, sub_eq_zero, ← eq_sub_iff_add_eq] at hz | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Quandle | {
"line": 671,
"column": 8
} | {
"line": 671,
"column": 17
} | {
"line": 671,
"column": 18
} | [
{
"pp": "case unit\nR : Type u_1\ninst✝¹ : Rack R\nG : Type u_2\ninst✝ : Group G\nF : EnvelGroup R →* G\nx : EnvelGroup R\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))\n ⟦unit⟧ =\n F ⟦uni... | [] | | unit => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.RingQuot | {
"line": 359,
"column": 17
} | {
"line": 359,
"column": 55
} | {
"line": 360,
"column": 4
} | [
{
"pp": "R : Type uR\ninst✝⁴ : Semiring R\nS : Type uS\ninst✝³ : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr✝ : R → R → Prop\ninst✝ : Semiring T\nr : R → R → Prop\nf : R →+* T\nh : ∀ ⦃x y : R⦄, r x y → f x = f y\n⊢ Quot.lift ⇑f ⋯ (toQuot 0) = 0",
"ppTerm": "?m.172"... | [] | by simp only [← zero_quot, f.map_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 244,
"column": 2
} | {
"line": 244,
"column": 52
} | {
"line": 246,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\nf g : SkewPolynomial R\ninst✝ : MulSemiringAction (Multiplicative ℕ) R\na✝ : Multiplicative ℕ\n⊢ (SkewMonoidAlgebra.sum f fun a₁ b₁ ↦ SkewMonoidAlgebra.sum g fun a₂ b₂ ↦ if a₁ * a₂ = a✝ then b₁ * a₁ • b₂ else 0) =\n f.coeff.sum fun a₁ b ↦ g.coeff.sum fun a₁_1 b_1 ↦... | [] | simp [SkewMonoidAlgebra.sum, Finsupp.single_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Symmetrized | {
"line": 299,
"column": 6
} | {
"line": 299,
"column": 30
} | {
"line": 300,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Ring α\ninst✝ : Invertible 2\na✝ b : αˢʸᵐ\na : α\nthis : Commute 2 a\n⊢ ?m.16 a",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOne... | [] | exact this.invOf_left.eq | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 1022,
"column": 66
} | {
"line": 1023,
"column": 51
} | {
"line": 1024,
"column": 4
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Group G\ninst✝ : MulSemiringAction G k\nf g : SkewMonoidAlgebra k G\nx : G\n⊢ (f * g).coeff x = g.sum fun a b ↦ (f * single a b).coeff x",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq... | [] | by
rw [← coeff_sum, ← mul_sum f g, g.sum_single] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace | {
"line": 332,
"column": 8
} | {
"line": 332,
"column": 46
} | {
"line": 332,
"column": 47
} | [
{
"pp": "case inr\nX : LocallyRingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nn : ℕ\nhn : f ^ n = 0\nh : n > 0\n⊢ X.toRingedSpace.basicOpen f = ⊥",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopological... | [
"case inr\nX : LocallyRingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nn : ℕ\nhn : f ^ n = 0\nh : n > 0\n⊢ X.toRingedSpace.basicOpen (f ^ n) = ⊥"
] | ← X.toRingedSpace.basicOpen_pow f n h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 98,
"column": 2
} | {
"line": 101,
"column": 9
} | {
"line": 103,
"column": 2
} | [
{
"pp": "X : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nh : ∀ (x : ↑↑X.toPresheafedSpace) (hx : x ∈ U), IsUnit ((ConcreteCategory.hom (X.presheaf.germ U x hx)) f)\nV : ↥U → Opens ↑↑X.toPresheafedSpace\niVU : (x : ↥U) → V x ⟶ U\nm : ∀ (x : ↥U), ↑x ∈ V x\nh_unit : ∀ (x : ↥U), IsUn... | [
"X : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nh : ∀ (x : ↑↑X.toPresheafedSpace) (hx : x ∈ U), IsUnit ((ConcreteCategory.hom (X.presheaf.germ U x hx)) f)\nV : ↥U → Opens ↑↑X.toPresheafedSpace\niVU : (x : ↥U) → V x ⟶ U\nm : ∀ (x : ↥U), ↑x ∈ V x\nh_unit : ∀ (x : ↥U), IsUnit ((Concret... | have hcover : U ≤ iSup V := by
intro x hxU
simp only [Opens.mem_iSup]
tauto | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace | {
"line": 453,
"column": 2
} | {
"line": 457,
"column": 42
} | {
"line": 458,
"column": 2
} | [
{
"pp": "case mp\nX Y : LocallyRingedSpace\nf : X ⟶ Y\nU : Opens ↑Y.toTopCat\ns : ↑(Y.presheaf.obj (op U))\nx : ↑↑X.toPresheafedSpace\n⊢ x ∈ ↑((Opens.map f.base).obj (Y.toRingedSpace.basicOpen s)) →\n x ∈ ↑(X.toRingedSpace.basicOpen ((ConcreteCategory.hom (f.c.app (op U))) s))",
"ppTerm": "?mp",
"ass... | [
"case mpr\nX Y : LocallyRingedSpace\nf : X ⟶ Y\nU : Opens ↑Y.toTopCat\ns : ↑(Y.presheaf.obj (op U))\nx : ↑↑X.toPresheafedSpace\n⊢ x ∈ ↑(X.toRingedSpace.basicOpen ((ConcreteCategory.hom (f.c.app (op U))) s)) →\n x ∈ ↑((Opens.map f.base).obj (Y.toRingedSpace.basicOpen s))"
] | · rintro ⟨hxU, hx⟩
rw [SetLike.mem_coe, X.toRingedSpace.mem_basicOpen _ _ hxU]
delta toRingedSpace
rw [← stalkMap_germ_apply]
exact (f.stalkMap _).hom.isUnit_map hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 210,
"column": 25
} | {
"line": 215,
"column": 31
} | {
"line": 217,
"column": 0
} | [
{
"pp": "X : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nhf : IsUnit f\n⊢ X.basicOpen f = U",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Opposite",
"Co... | [] | by
apply le_antisymm
· exact X.basicOpen_le f
intro x hx
rw [X.mem_basicOpen f x hx]
exact RingHom.isUnit_map _ hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Spec | {
"line": 287,
"column": 45
} | {
"line": 287,
"column": 49
} | {
"line": 288,
"column": 2
} | [
{
"pp": "R S : CommRingCat\nf : R ⟶ S\nx✝ : ↑R\nx' : ↥(unop (op ⊤))\n⊢ ↑((CommRingCat.Hom.hom (f ≫ toSpecΓ S)) x✝) x' =\n ↑((CommRingCat.Hom.hom (toSpecΓ R ≫ Γ.map (Spec.toLocallyRingedSpace.map f.op).op)) x✝) x'",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.S... | [
"R S : CommRingCat\nf : R ⟶ S\nx✝ : ↑R\nx' : ↥(unop (op ⊤))\n⊢ ↑((CommRingCat.Hom.hom (toSpecΓ R ≫ Γ.map (Spec.toLocallyRingedSpace.map f.op).op)) x✝) x' =\n ↑((CommRingCat.Hom.hom (f ≫ toSpecΓ S)) x✝) x'"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 8
} | {
"line": 120,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nH : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\n⊢ (TopCat.Presheaf.pushforward C (Iso.refl ↑X).hom).obj X.presheaf ≅ (Y.restrict ⋯).presheaf",
"ppTerm": "?m.27",
"assig... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nH : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\n⊢ (Y.restrict ⋯).presheaf ≅ (TopCat.Presheaf.pushforward C (Iso.refl ↑X).hom).obj X.presheaf"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 301,
"column": 69
} | {
"line": 303,
"column": 16
} | {
"line": 305,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nH : IsOpenImmersion f\ninst✝ : HasColimits C\nx : ↑↑X\n⊢ IsIso (Hom.stalkMap f x)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsIso",
"AlgebraicGeometry.Pres... | [] | by
rw [← H.isoRestrict_hom_ofRestrict, PresheafedSpace.stalkMap.comp]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 870,
"column": 67
} | {
"line": 870,
"column": 80
} | {
"line": 870,
"column": 80
} | [
{
"pp": "X : Scheme\nU V : X.Opens\nh : U = V\nW : (Spec Γ(X, V)).Opens\n⊢ Γ(Spec Γ(X, V), W) = Γ(Spec Γ(X, U), Spec.map (X.presheaf.map (eqToHom h).op) ⁻¹ᵁ W)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"AlgebraicGeometry.Spec",
... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 870,
"column": 67
} | {
"line": 870,
"column": 80
} | {
"line": 870,
"column": 80
} | [
{
"pp": "X : Scheme\nU V : X.Opens\nh : U = V\nW : (Spec Γ(X, V)).Opens\n⊢ Γ(Spec Γ(X, V), W) = Γ(Spec Γ(X, U), Spec.map (X.presheaf.map (eqToHom h).op) ⁻¹ᵁ W)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"AlgebraicGeometry.Spec",
... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.JointlySurjective | {
"line": 46,
"column": 50
} | {
"line": 46,
"column": 68
} | {
"line": 46,
"column": 68
} | [
{
"pp": "X Y : Type u\nf : X ⟶ Y\nhf : Function.Surjective ⇑(ConcreteCategory.hom f)\nx : Y\n⊢ x ∈ Set.range ⇑(ConcreteCategory.hom f)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
"Set.mem_univ._simp_1",
"Set.uni... | [] | simp [hf.range_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.JointlySurjective | {
"line": 46,
"column": 50
} | {
"line": 46,
"column": 68
} | {
"line": 46,
"column": 68
} | [
{
"pp": "X Y : Type u\nf : X ⟶ Y\nhf : Function.Surjective ⇑(ConcreteCategory.hom f)\nx : Y\n⊢ x ∈ Set.range ⇑(ConcreteCategory.hom f)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
"Set.mem_univ._simp_1",
"Set.uni... | [] | simp [hf.range_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.JointlySurjective | {
"line": 46,
"column": 50
} | {
"line": 46,
"column": 68
} | {
"line": 46,
"column": 68
} | [
{
"pp": "X Y : Type u\nf : X ⟶ Y\nhf : Function.Surjective ⇑(ConcreteCategory.hom f)\nx : Y\n⊢ x ∈ Set.range ⇑(ConcreteCategory.hom f)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
"Set.mem_univ._simp_1",
"Set.uni... | [] | simp [hf.range_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 33
} | {
"line": 159,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nH : IsOpenImmersion f\nU : X.Opens\n⊢ f ''ᵁ U ≤ opensRange f",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeome... | [] | simpa using f.image_mono le_top | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 33
} | {
"line": 159,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nH : IsOpenImmersion f\nU : X.Opens\n⊢ f ''ᵁ U ≤ opensRange f",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeome... | [] | simpa using f.image_mono le_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 33
} | {
"line": 159,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nH : IsOpenImmersion f\nU : X.Opens\n⊢ f ''ᵁ U ≤ opensRange f",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeome... | [] | simpa using f.image_mono le_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 487,
"column": 29
} | {
"line": 487,
"column": 33
} | {
"line": 488,
"column": 6
} | [
{
"pp": "case w.none\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : IsOpenImmersion f\ng : Y ⟶ Z\ns : PullbackCone f g\n⊢ (limit.cone (cospan f.base g.base)).π.app none =\n ((forget C).mapCone (pullbackConeOfLeft f g)).π.app none ≫ (diagramIsoCospan (cospan f g ⋙ forget C)... | [
"case w.none\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : IsOpenImmersion f\ng : Y ⟶ Z\ns : PullbackCone f g\n⊢ ((forget C).mapCone (pullbackConeOfLeft f g)).π.app none ≫ (diagramIsoCospan (cospan f g ⋙ forget C)).hom.app none =\n (limit.cone (cospan f.base g.base)).π.app no... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 487,
"column": 29
} | {
"line": 487,
"column": 33
} | {
"line": 488,
"column": 6
} | [
{
"pp": "case w.some.left\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : IsOpenImmersion f\ng : Y ⟶ Z\ns : PullbackCone f g\n⊢ (limit.cone (cospan f.base g.base)).π.app (some WalkingPair.left) =\n ((forget C).mapCone (pullbackConeOfLeft f g)).π.app (some WalkingPair.left) ... | [
"case w.some.left\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : IsOpenImmersion f\ng : Y ⟶ Z\ns : PullbackCone f g\n⊢ ((forget C).mapCone (pullbackConeOfLeft f g)).π.app (some WalkingPair.left) ≫\n (diagramIsoCospan (cospan f g ⋙ forget C)).hom.app (some WalkingPair.left) =... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 487,
"column": 29
} | {
"line": 487,
"column": 33
} | {
"line": 488,
"column": 6
} | [
{
"pp": "case w.some.right\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : IsOpenImmersion f\ng : Y ⟶ Z\ns : PullbackCone f g\n⊢ (limit.cone (cospan f.base g.base)).π.app (some WalkingPair.right) =\n ((forget C).mapCone (pullbackConeOfLeft f g)).π.app (some WalkingPair.righ... | [
"case w.some.right\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : IsOpenImmersion f\ng : Y ⟶ Z\ns : PullbackCone f g\n⊢ ((forget C).mapCone (pullbackConeOfLeft f g)).π.app (some WalkingPair.right) ≫\n (diagramIsoCospan (cospan f g ⋙ forget C)).hom.app (some WalkingPair.right... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 380,
"column": 25
} | {
"line": 387,
"column": 65
} | {
"line": 389,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : PresheafedSpace CommRingCat\nY : Scheme\nf : X ⟶ Y.toPresheafedSpace\nH : IsOpenImmersion f\n⊢ Scheme",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"AlgebraicGeometry.Spec",
... | [] | by
apply LocallyRingedSpace.IsOpenImmersion.scheme (toLocallyRingedSpace _ f)
intro x
obtain ⟨R, i, _, h₁, h₂⟩ :=
Scheme.exists_affine_mem_range_and_range_subset (U := ⟨_, H.base_open.isOpen_range⟩) ⟨x, rfl⟩
refine ⟨R, LocallyRingedSpace.IsOpenImmersion.lift (toLocallyRingedSpaceHom _ f) _ h₂, ?_, ?_⟩
· r... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 888,
"column": 2
} | {
"line": 888,
"column": 32
} | {
"line": 889,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimits C\nι : Type v\nF : Discrete ι ⥤ SheafedSpace C\ninst✝ : HasColimit F\ni : Discrete ι\nthis :\n colimit.ι (F ⋙ forget C) i ≫ (HasColimit.isoOfNatIso Discrete.natIsoFunctor).hom =\n Discrete.natIsoFunctor.hom.app i ≫ colimit.ι (Discrete.funct... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimits C\nι : Type v\nF : Discrete ι ⥤ SheafedSpace C\ninst✝ : HasColimit F\ni : Discrete ι\nthis :\n colimit.ι (F ⋙ forget C) i =\n (Discrete.natIsoFunctor.hom.app i ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) i) ≫\n (HasColimit.is... | rw [← Iso.eq_comp_inv] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 892,
"column": 2
} | {
"line": 892,
"column": 32
} | {
"line": 893,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimits C\nι : Type v\nF : Discrete ι ⥤ SheafedSpace C\ninst✝ : HasColimit F\ni : Discrete ι\nthis✝ :\n colimit.ι (F ⋙ forget C) i =\n (Discrete.natIsoFunctor.hom.app i ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) i) ≫\n (... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimits C\nι : Type v\nF : Discrete ι ⥤ SheafedSpace C\ninst✝ : HasColimit F\ni : Discrete ι\nthis✝ :\n colimit.ι (F ⋙ forget C) i =\n (Discrete.natIsoFunctor.hom.app i ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) i) ≫\n (HasColimit.i... | rw [← Iso.eq_comp_inv] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 677,
"column": 2
} | {
"line": 682,
"column": 23
} | {
"line": 684,
"column": 0
} | [
{
"pp": "case neg\nX Y U V : Scheme\nf : X ⟶ Y\nf' : U ⟶ V\niU : U ⟶ X\niV : V ⟶ Y\ninst✝¹ : IsOpenImmersion iV\ninst✝ : IsOpenImmersion iU\nH : IsPullback f' iU iV f\nW : V.Opens\nx : ↥X\nhx : x ∉ Set.range ⇑iU\n⊢ x ∈ ↑(iU ''ᵁ f' ⁻¹ᵁ W) ↔ x ∈ ↑(f ⁻¹ᵁ iV ''ᵁ W)",
"ppTerm": "?neg✝",
"assigned": true,
... | [] | · constructor
· rintro ⟨x, hx, rfl⟩; cases hx ⟨x, rfl⟩
· rintro ⟨y, hy, e : iV y = f x⟩
obtain ⟨x, rfl⟩ := (IsOpenImmersion.range_pullbackSnd iV f).ge ⟨y, e⟩
rw [← H.isoPullback_inv_snd] at hx
cases hx ⟨_, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 1242,
"column": 18
} | {
"line": 1245,
"column": 36
} | {
"line": 1245,
"column": 36
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : LocallyRingedSpace\nf : X ⟶ Y\nH : IsOpenImmersion f\nU : Opens ↑X.toTopCat\n⊢ (Opens.map f.base).op.obj (op ((opensFunctor f).obj U)) = op U",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"ContinuousMap.continuous",
"Categ... | [] | by
have := Set.preimage_image_eq U.1 H.base_open.injective
dsimp at this
simp [Opens.map_def, this] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 424,
"column": 2
} | {
"line": 424,
"column": 56
} | {
"line": 426,
"column": 0
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : R\nf : M\nn : ℕ\n⊢ const f 1 (basicOpen s) ⋯ = const (s ^ n • f) (s ^ n) (basicOpen s) ⋯",
"ppTerm": "?m.177",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"PrimeSpectrum.bas... | [] | exact const_eq_const_of_smul_eq_smul (H := by simp) .. | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 702,
"column": 2
} | {
"line": 702,
"column": 6
} | {
"line": 703,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nX Y U : Scheme\nf : X ⟶ Y\ng : U ⟶ Y\ninst✝ : IsOpenImmersion g\nV : Y.Opens := Scheme.Hom.opensRange g\ne : U ≅ ↑V := IsOpenImmersion.isoOfRangeEq g V.ι ⋯\nt : pullback f g ⟶ pullback f V.ι := pullback.map f g f V.ι (𝟙 X) e.hom (𝟙 Y) ⋯ ⋯\n⊢ Arrow.mk (f ∣_ Sc... | [
"C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nX Y U : Scheme\nf : X ⟶ Y\ng : U ⟶ Y\ninst✝ : IsOpenImmersion g\nV : Y.Opens := ⋯\ne : U ≅ ↑V := ⋯\nt : pullback f g ⟶ pullback f V.ι := ⋯\n⊢ Arrow.mk (pullback.snd f g) ≅ Arrow.mk (f ∣_ Scheme.Hom.opensRange g)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 684,
"column": 2
} | {
"line": 684,
"column": 56
} | {
"line": 686,
"column": 0
} | [
{
"pp": "case e_6\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : ↑(PrimeSpectrum.Top R)\nf : M\ns : ↥x.asIdeal.primeCompl\n⊢ const f 1 (basicOpen ↑s) ⋯ = const (↑s • f) (↑s) (basicOpen ↑s) ⋯",
"ppTerm": "?e_6",
"assigned": true,
"usedConstants": [
"instHSM... | [] | exact const_eq_const_of_smul_eq_smul (H := by simp) .. | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 8
} | {
"line": 179,
"column": 4
} | [
{
"pp": "case e'_3\nX : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X))\nr s : (InducedCategory (Opens (PrimeSpectrum ↑(Γ.obj (op X)))) basicOpen)ᵒᵖ\nf : r ⟶ s\ne_1✝ :\n (CommRingCat.of ↑(Γ.obj (op X)) ⟶\n ((inducedFunctor basicOpen).op ⋙ ((TopCat.Sheaf.pushforward CommRingCat X.toΓSpecBase).obj X.𝒪).obj).obj s... | [
"case e'_3\nX : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X))\nr s : (InducedCategory (Opens (PrimeSpectrum ↑(Γ.obj (op X)))) basicOpen)ᵒᵖ\nf : r ⟶ s\ne_1✝ :\n (CommRingCat.of ↑(Γ.obj (op X)) ⟶\n ((inducedFunctor basicOpen).op ⋙ ((TopCat.Sheaf.pushforward CommRingCat X.toΓSpecBase).obj X.𝒪).obj).obj s) =\n (Co... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 246,
"column": 4
} | {
"line": 246,
"column": 8
} | {
"line": 247,
"column": 4
} | [
{
"pp": "X : LocallyRingedSpace\ns : Set ↑(X.presheaf.obj (op ⊤))\ni : ↑(Γ.obj (op X))\nx✝ : i ∈ s\n⊢ (↑(X.toRingedSpace.basicOpen i))ᶜ = ⇑(ConcreteCategory.hom X.toΓSpec.base) ⁻¹' (basicOpen i).carrierᶜ",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor.op",
... | [
"X : LocallyRingedSpace\ns : Set ↑(X.presheaf.obj (op ⊤))\ni : ↑(Γ.obj (op X))\nx✝ : i ∈ s\n⊢ ⇑(ConcreteCategory.hom X.toΓSpec.base) ⁻¹' (basicOpen i).carrierᶜ = (↑(X.toRingedSpace.basicOpen i))ᶜ"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.GlueData | {
"line": 103,
"column": 55
} | {
"line": 110,
"column": 18
} | {
"line": 112,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nD : GlueData C\ni j : D.J\n⊢ D.t i j ≫ D.t j i = 𝟙 (D.V (i, j))",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.pullbackSymmetry",
"CategoryTheory.GlueData.t",
"CategoryTheory.Category.assoc",
... | [] | by
have eq : (pullbackSymmetry (D.f i i) (D.f i j)).hom =
pullback.snd _ _ ≫ inv (pullback.fst _ _) := by simp
have := D.cocycle i j i
rw [D.t'_iij, D.t'_jii, D.t'_iji, fst_eq_snd_of_mono_eq, eq] at this
simp only [Category.assoc, IsIso.inv_hom_id_assoc] at this
rw [← IsIso.eq_inv_comp, ← Category.assoc... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 466,
"column": 2
} | {
"line": 473,
"column": 5
} | {
"line": 475,
"column": 0
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nV : X.Opens\nhV : IsAffineOpen V\nf : op U ⟶ op V\n⊢ Spec.map (X.presheaf.map f) ≫ hU.fromSpec = hV.fromSpec",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
"Alge... | [] | have : IsAffine U := hU
haveI : IsAffine _ := hV
conv_rhs =>
rw [fromSpec, ← X.homOfLE_ι (V := U) f.unop.le, isoSpec_inv, Category.assoc,
← Scheme.isoSpec_inv_naturality_assoc,
← Spec.map_comp_assoc, Scheme.homOfLE_appTop, ← Functor.map_comp]
rw [fromSpec, isoSpec_inv, Category.assoc, ← Spec.map_c... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 466,
"column": 2
} | {
"line": 473,
"column": 5
} | {
"line": 475,
"column": 0
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nV : X.Opens\nhV : IsAffineOpen V\nf : op U ⟶ op V\n⊢ Spec.map (X.presheaf.map f) ≫ hU.fromSpec = hV.fromSpec",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
"Alge... | [] | have : IsAffine U := hU
haveI : IsAffine _ := hV
conv_rhs =>
rw [fromSpec, ← X.homOfLE_ι (V := U) f.unop.le, isoSpec_inv, Category.assoc,
← Scheme.isoSpec_inv_naturality_assoc,
← Spec.map_comp_assoc, Scheme.homOfLE_appTop, ← Functor.map_comp]
rw [fromSpec, isoSpec_inv, Category.assoc, ← Spec.map_c... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 288,
"column": 4
} | {
"line": 288,
"column": 8
} | {
"line": 289,
"column": 4
} | [
{
"pp": "X Y : LocallyRingedSpace\nf : X ⟶ Y\n⊢ (𝟭 LocallyRingedSpace).map f ≫ Y.toΓSpec = X.toΓSpec ≫ (Γ.rightOp ⋙ Spec.toLocallyRingedSpace).map f",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [
"X Y : LocallyRingedSpace\nf : X ⟶ Y\n⊢ X.toΓSpec ≫ (Γ.rightOp ⋙ Spec.toLocallyRingedSpace).map f = (𝟭 LocallyRingedSpace).map f ≫ Y.toΓSpec"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
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