module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Basis.Cardinality | {
"line": 103,
"column": 4
} | {
"line": 106,
"column": 65
} | {
"line": 107,
"column": 4
} | [
{
"pp": "case some\nR : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Nontrivial R\ninst✝ : Module R M\nι : Type w\nb : Basis ι R M\nκ : Type w'\nv : κ → M\nind : LinearIndependent R v\nm : ind.Maximal\ni : ι\nw : ∀ (x : κ), (b.repr (v x)) i = 0\nrepr_eq_zero : ∀ (l : κ →₀ R), (b.r... | [
"case some\nR : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Nontrivial R\ninst✝ : Module R M\nι : Type w\nb : Basis ι R M\nκ : Type w'\nv : κ → M\nind : LinearIndependent R v\nm : ind.Maximal\ni : ι\nw : ∀ (x : κ), (b.repr (v x)) i = 0\nrepr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linear... | have l₁ : l.some = l'.some := ind <| b.repr.injective <| ext fun j ↦ by
obtain rfl | ne := eq_or_ne i j
· simp_rw [repr_eq_zero]
classical simpa [single_apply, ne] using congr(b.repr $z j) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 553,
"column": 30
} | {
"line": 553,
"column": 44
} | {
"line": 553,
"column": 45
} | [
{
"pp": "κ μ : Cardinal.{u}\nH1✝ : ℵ₀ ≤ κ\nH2 : μ < ℵ₀\nn✝ : ℕ\nH3 : μ = ↑n✝\nα : Type u\nH1 : ℵ₀ ≤ ⟦α⟧\nn : ℕ\nih : ⟦α⟧ ^ ↑n ≤ ⟦α⟧\n⊢ ⟦α⟧ ^ ↑n.succ ≤ ⟦α⟧",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.cast_succ",... | [
"κ μ : Cardinal.{u}\nH1✝ : ℵ₀ ≤ κ\nH2 : μ < ℵ₀\nn✝ : ℕ\nH3 : μ = ↑n✝\nα : Type u\nH1 : ℵ₀ ≤ ⟦α⟧\nn : ℕ\nih : ⟦α⟧ ^ ↑n ≤ ⟦α⟧\n⊢ ⟦α⟧ ^ (↑n + 1) ≤ ⟦α⟧"
] | Nat.cast_succ, | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 569,
"column": 8
} | {
"line": 569,
"column": 19
} | {
"line": 569,
"column": 20
} | [
{
"pp": "case a\nι : Type u\ninst✝ : Infinite ι\nc : ι → Cardinal.{v}\nh₁ : ∀ (i : ι), 2 ≤ c i\nh₂ : ∀ (i : ι), lift.{u, v} (c i) ≤ lift.{v, u} #ι\n⊢ (prod fun i ↦ lift.{v, u} #ι) = lift.{v, u} (2 ^ #ι)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal.instPowCa... | [
"case a\nι : Type u\ninst✝ : Infinite ι\nc : ι → Cardinal.{v}\nh₁ : ∀ (i : ι), 2 ≤ c i\nh₂ : ∀ (i : ι), lift.{u, v} (c i) ≤ lift.{v, u} #ι\n⊢ lift.{u, max u v} (lift.{v, u} #ι) ^ lift.{max u v, u} #ι = lift.{v, u} (2 ^ #ι)"
] | prod_const, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 599,
"column": 4
} | {
"line": 600,
"column": 30
} | {
"line": 601,
"column": 2
} | [
{
"pp": "case a\nc : Cardinal.{u_1}\nh : ℵ₀ ≤ c\n⊢ c ^< ℵ₀ ≤ c",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal.instPowCardinal",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Cardinal.pow_le",
"Preorde... | [] | rw [powerlt_le]
exact fun _ a ↦ pow_le h a | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 599,
"column": 4
} | {
"line": 600,
"column": 30
} | {
"line": 601,
"column": 2
} | [
{
"pp": "case a\nc : Cardinal.{u_1}\nh : ℵ₀ ≤ c\n⊢ c ^< ℵ₀ ≤ c",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal.instPowCardinal",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Cardinal.pow_le",
"Preorde... | [] | rw [powerlt_le]
exact fun _ a ↦ pow_le h a | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.DFinsupp.Defs | {
"line": 489,
"column": 4
} | {
"line": 489,
"column": 34
} | {
"line": 490,
"column": 4
} | [
{
"pp": "case mpr\nι : Type u\nβ : ι → Type v\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : DecidableEq ι\ni j : ι\nxi : β i\nxj : β j\n⊢ i = j ∧ xi ≍ xj ∨ xi = 0 ∧ xj = 0 → single i xi = single j xj",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"DFinsupp.single",
"Or.casesOn",
... | [
"case mpr.inl\nι : Type u\nβ : ι → Type v\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : DecidableEq ι\ni : ι\nxi xj : β i\nhxi : xi ≍ xj\n⊢ single i xi = single i xj",
"case mpr.inr\nι : Type u\nβ : ι → Type v\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : DecidableEq ι\ni j : ι\nxi : β i\nxj : β j\nhi : xi = 0\nhj : xj = 0\n... | rintro (⟨rfl, hxi⟩ | ⟨hi, hj⟩) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Data.DFinsupp.Defs | {
"line": 649,
"column": 2
} | {
"line": 649,
"column": 20
} | {
"line": 651,
"column": 0
} | [
{
"pp": "case inr\nι : Type u\ninst✝¹ : DecidableEq ι\nβ : ι → Type u_1\ninst✝ : (i : ι) → AddZeroClass (β i)\nf : Π₀ (i : ι), β i\ni : ι\nb : β i\nj : ι\nh : i ≠ j\n⊢ (f.update i b) j = (single i b + erase i f) j",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.recOn",
"Fals... | [] | · simp [h, h.symm] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.DFinsupp.Defs | {
"line": 656,
"column": 2
} | {
"line": 656,
"column": 20
} | {
"line": 658,
"column": 0
} | [
{
"pp": "case inr\nι : Type u\ninst✝¹ : DecidableEq ι\nβ : ι → Type u_1\ninst✝ : (i : ι) → AddZeroClass (β i)\nf : Π₀ (i : ι), β i\ni : ι\nb : β i\nj : ι\nh : i ≠ j\n⊢ (f.update i b) j = (erase i f + single i b) j",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.recOn",
"Fals... | [] | · simp [h, h.symm] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 805,
"column": 2
} | {
"line": 805,
"column": 67
} | {
"line": 806,
"column": 2
} | [
{
"pp": "α : Type u\nc : Cardinal.{u}\n⊢ #{ t // #↑t ≤ c } ≤ max #α ℵ₀ ^ c",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Sum.infinite_of_left",
"Lattice.toSemilatticeSup",
"Cardinal.instPowCardinal",
"Cardinal",
"PartialOrder.toPreorder",
"ULift",
... | [
"α : Type u\nc : Cardinal.{u}\n⊢ #(ULift.{u, 0} ℕ ⊕ α) ^ c ≤ max #α ℵ₀ ^ c"
] | apply (mk_bounded_set_le_of_infinite ((ULift.{u} ℕ) ⊕ α) c).trans | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.DFinsupp.Defs | {
"line": 753,
"column": 2
} | {
"line": 753,
"column": 93
} | {
"line": 754,
"column": 2
} | [
{
"pp": "case h.cons\nι : Type u\nβ : ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\ni : ι\ns : Multiset ι\nih : ∀ (f : (i : ι) → β i) (H : ∀ (i : ι), i ... | [
"case h.cons\nι : Type u\nβ : ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\ni : ι\ns : Multiset ι\nih : ∀ (f : (i : ι) → β i) (H : ∀ (i : ι), i ∈ s ∨ f i = ... | have H3 : single i _ + _ = (⟨f, Trunc.mk ⟨i ::ₘ s, H⟩⟩ : Π₀ i, β i) := single_add_erase _ _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.DFinsupp.Defs | {
"line": 780,
"column": 40
} | {
"line": 780,
"column": 57
} | {
"line": 782,
"column": 0
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns : Finset ι\ni : ι\n⊢ (if H : i ∈ s then 0 ⟨i, H⟩ else 0) = 0 i",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Membership.mem",
"DFins... | [] | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.DFinsupp.Defs | {
"line": 806,
"column": 82
} | {
"line": 815,
"column": 86
} | {
"line": 817,
"column": 0
} | [
{
"pp": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\n⊢ ∀ (a b : { s // ∀ (i : ι), i ∈ s ∨ f.toFun i = 0 }), {i ∈ (↑a).toFinset | f i ≠ 0} = {i ∈ (↑b).toFin... | [] | by
rintro ⟨sx, hx⟩ ⟨sy, hy⟩
dsimp only [Subtype.coe_mk, toFun_eq_coe] at *
ext i; constructor
· intro H
rcases Finset.mem_filter.1 H with ⟨_, h⟩
exact Finset.mem_filter.2 ⟨Multiset.mem_toFinset.2 <| (hy i).resolve_right h, h⟩
· intro H
rcases Finset.mem_filter.1 H with ⟨_, h⟩
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.DFinsupp.Sigma | {
"line": 93,
"column": 6
} | {
"line": 93,
"column": 46
} | {
"line": 94,
"column": 6
} | [
{
"pp": "case inl.inr\nι : Type u\nα : ι → Type u_2\nδ : (i : ι) → α i → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → DecidableEq (α i)\ninst✝ : (i : ι) → (j : α i) → Zero (δ i j)\ni : ι\nj : α i\nx : δ ⟨i, j⟩.fst ⟨i, j⟩.snd\nj' : α i\nhj : j' ≠ j\n⊢ (single ⟨i, j⟩ x) ⟨i, j'⟩ = (single j x) j'",
"ppTe... | [
"case inl.inr\nι : Type u\nα : ι → Type u_2\nδ : (i : ι) → α i → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → DecidableEq (α i)\ninst✝ : (i : ι) → (j : α i) → Zero (δ i j)\ni : ι\nj : α i\nx : δ ⟨i, j⟩.fst ⟨i, j⟩.snd\nj' : α i\nhj : j' ≠ j\n⊢ ⟨i, j'⟩ ≠ ⟨i, j⟩"
] | rw [single_eq_of_ne, single_eq_of_ne hj] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.DFinsupp.Defs | {
"line": 982,
"column": 90
} | {
"line": 987,
"column": 38
} | {
"line": 989,
"column": 0
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\ni : ι\nb : β i\ninst✝ : Decidable (b = 0)\n⊢ (f.update i b).support = if b = 0 then (erase i f).support else insert i f.support",
"ppTerm": "?m.4... | [] | by
ext j
split_ifs with hb
· subst hb
simp [update_eq_erase, support_erase]
· rw [support_update_ne_zero f _ hb] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.DFinsupp.Defs | {
"line": 1002,
"column": 35
} | {
"line": 1002,
"column": 60
} | {
"line": 1004,
"column": 0
} | [
{
"pp": "case pos\nι : Type u\nβ : ι → Type v\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\np : ι → Prop\ninst✝ : DecidablePred p\nf : Π₀ (i : ι), β i\ni : Subtype p\nh2 : f ↑i ≠ 0\n⊢ (subtypeDomain p f) i = (mk (Finset.subtype p f.support) fun i ↦ f ↑... | [] | try simp at h2; simp [h2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.Data.DFinsupp.Defs | {
"line": 1002,
"column": 35
} | {
"line": 1002,
"column": 60
} | {
"line": 1004,
"column": 0
} | [
{
"pp": "case neg\nι : Type u\nβ : ι → Type v\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\np : ι → Prop\ninst✝ : DecidablePred p\nf : Π₀ (i : ι), β i\ni : Subtype p\nh2 : ¬f ↑i ≠ 0\n⊢ (subtypeDomain p f) i = (mk (Finset.subtype p f.support) fun i ↦ f ... | [] | try simp at h2; simp [h2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.Data.DFinsupp.BigOperators | {
"line": 436,
"column": 55
} | {
"line": 439,
"column": 61
} | {
"line": 441,
"column": 0
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\ninst✝³ : DecidableEq ι\nγ : Type w\nα : Type x\ninst✝² : (i : ι) → AddCommMonoid (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\ns : Finset α\ng : α → Π₀ (i : ι), β i\nh : (i : ι) → β i → γ\nh_zero : ∀ (i : ι), h i 0 = 1\nh_add : ∀ (i : ι) (b₁ ... | [] | by
classical
exact Finset.induction_on s (by simp [prod_zero_index])
(by simp +contextual [prod_add_index, h_zero, h_add]) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dual.Defs | {
"line": 503,
"column": 81
} | {
"line": 504,
"column": 52
} | {
"line": 506,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nM' : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nf : M →ₗ[R] M'\n⊢ f.dualMap.ker = (Dual.eval R M' ∘ₗ f).range.dualCoannihilator",
"ppTerm": "?m.76",
"assigned": true,
"usedCons... | [] | by
ext x; simp [LinearMap.ext_iff (f := dualMap f x)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 302,
"column": 41
} | {
"line": 302,
"column": 52
} | {
"line": 302,
"column": 53
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nx y : M\nS : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Module S R\ninst✝³ : Module S M\ninst✝² : SMulCommClass S R M\ninst✝¹ : IsScalarTower S R M\ninst✝ : IsTorsionFree S R\nu : S\nhu : u ≠ 0\... | [
"R : Type u_2\nM : Type u_4\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nx y : M\nS : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Module S R\ninst✝³ : Module S M\ninst✝² : SMulCommClass S R M\ninst✝¹ : IsScalarTower S R M\ninst✝ : IsTorsionFree S R\nu : S\nhu : u ≠ 0\nh : ∀ (s t ... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 302,
"column": 53
} | {
"line": 302,
"column": 64
} | {
"line": 302,
"column": 65
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nx y : M\nS : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Module S R\ninst✝³ : Module S M\ninst✝² : SMulCommClass S R M\ninst✝¹ : IsScalarTower S R M\ninst✝ : IsTorsionFree S R\nu : S\nhu : u ≠ 0\... | [
"R : Type u_2\nM : Type u_4\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nx y : M\nS : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Module S R\ninst✝³ : Module S M\ninst✝² : SMulCommClass S R M\ninst✝¹ : IsScalarTower S R M\ninst✝ : IsTorsionFree S R\nu : S\nhu : u ≠ 0\nh : ∀ (s t ... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Tactic.Module | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 17
} | {
"line": 170,
"column": 0
} | [
{
"pp": "case e_f\nR : Type u_2\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : Semiring R\ninst✝ : Module R M\nl : NF R M\nx : M\nh :\n x =\n (map\n (fun x ↦\n match x with\n | (r, x) => r • x)\n l).sum\nr : R\np : R × M\n⊢ ((fun x ↦ x.1 • x.2) ∘ fun x ↦ (r * x.1, x.2)) p ... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Pointwise.Set.ListOfFn | {
"line": 30,
"column": 21
} | {
"line": 30,
"column": 36
} | {
"line": 30,
"column": 37
} | [
{
"pp": "case zero\nα : Type u_1\ninst✝ : Monoid α\nn : ℕ\na : α\ns : Fin 0 → Set α\n⊢ a ∈ (List.ofFn s).prod ↔ ∃ f, (List.ofFn fun i ↦ ↑(f i)).prod = a",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
... | [
"case zero\nα : Type u_1\ninst✝ : Monoid α\nn : ℕ\na : α\ns : Fin 0 → Set α\n⊢ a ∈ [].prod ↔ ∃ f, [].prod = a"
] | List.ofFn_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Finset.NAry | {
"line": 237,
"column": 4
} | {
"line": 238,
"column": 34
} | {
"line": 240,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\ninst✝ : DecidableEq γ\nf : α → β → γ\ns : Finset α\nt : Finset β\n⊢ ↑(t.biUnion fun b ↦ image (fun a ↦ f a b) s) = ↑(image₂ f s t)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.coe_biUnion",
"Iff.of_eq... | [] | push_cast
exact Set.iUnion_image_right _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.NAry | {
"line": 237,
"column": 4
} | {
"line": 238,
"column": 34
} | {
"line": 240,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\ninst✝ : DecidableEq γ\nf : α → β → γ\ns : Finset α\nt : Finset β\n⊢ ↑(t.biUnion fun b ↦ image (fun a ↦ f a b) s) = ↑(image₂ f s t)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.coe_biUnion",
"Iff.of_eq... | [] | push_cast
exact Set.iUnion_image_right _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 823,
"column": 2
} | {
"line": 868,
"column": 43
} | {
"line": 869,
"column": 2
} | [
{
"pp": "K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set V\nt : Finset V\nhs : LinearIndepOn K id s\nhst : s ⊆ ↑(span K ↑t)\n⊢ ∃ t', ↑t' ⊆ s ∪ ↑t ∧ s ⊆ ↑t' ∧ t'.card = t.card",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"subs... | [
"K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set V\nt : Finset V\nhs : LinearIndepOn K id s\nhst : s ⊆ ↑(span K ↑t)\nthis :\n ∀ (t s' : Finset V),\n ↑s' ⊆ s → s ∩ ↑t = ∅ → s ⊆ ↑(span K ↑(s' ∪ t)) → ∃ t', ↑t' ⊆ s ∪ ↑t ∧ s ⊆ ↑t' ∧ t'.card = (s' ∪ t).card\n⊢ ... | have :
∀ t : Finset V,
∀ s' : Finset V,
↑s' ⊆ s →
s ∩ ↑t = ∅ →
s ⊆ (span K ↑(s' ∪ t) : Submodule K V) →
∃ t' : Finset V, ↑t' ⊆ s ∪ ↑t ∧ s ⊆ ↑t' ∧ t'.card = (s' ∪ t).card :=
fun t =>
Finset.induction_on t
(fun s' hs' _ hss' =>
have : s = ↑s' := ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Finset.NAry | {
"line": 500,
"column": 4
} | {
"line": 501,
"column": 38
} | {
"line": 503,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → α → β\ns t : Finset α\nhf : ∀ (a b : α), f a b = f b a\n⊢ ↑(image₂ f (s ∪ t) (s ∩ t)) ⊆ ↑(image₂ f s t)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.instUnion",
... | [] | push_cast
exact image2_union_inter_subset hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.NAry | {
"line": 500,
"column": 4
} | {
"line": 501,
"column": 38
} | {
"line": 503,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → α → β\ns t : Finset α\nhf : ∀ (a b : α), f a b = f b a\n⊢ ↑(image₂ f (s ∪ t) (s ∩ t)) ⊆ ↑(image₂ f s t)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.instUnion",
... | [] | push_cast
exact image2_union_inter_subset hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors | {
"line": 113,
"column": 13
} | {
"line": 113,
"column": 21
} | {
"line": 113,
"column": 22
} | [
{
"pp": "case H.inr\nR : Type u_1\nA : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : IsCancelAdd R\ninst✝² : IsLeftCancelMulZero R\ninst✝¹ : Mul A\ninst✝ : UniqueProds A\nf : R[A]\nhf : f ≠ 0\ng₁ g₂ : R[A]\nih :\n ∀ s ∈ g₁.support ∪ g₂.support,\n ∀ {g₁_1 g₂_1 : R[A]},\n (fun x ↦ f * x) g₁_1 = (fun x ↦ f * x)... | [
"case H.inr\nR : Type u_1\nA : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : IsCancelAdd R\ninst✝² : IsLeftCancelMulZero R\ninst✝¹ : Mul A\ninst✝ : UniqueProds A\nf : R[A]\nhf : f ≠ 0\ng₁ g₂ : R[A]\nih :\n ∀ s ∈ g₁.support ∪ g₂.support,\n ∀ {g₁_1 g₂_1 : R[A]},\n (fun x ↦ f * x) g₁_1 = (fun x ↦ f * x) g₂_1 →\n ... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 471,
"column": 48
} | {
"line": 471,
"column": 88
} | {
"line": 473,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\nr₁ r₂ : R\nm₁ m₂ : M\ninst✝ : Mul M\nhm : Commute m₁ m₂\nhr : Commute r₁ r₂\n⊢ Commute (single m₁ r₁) (single m₂ r₂)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"MonoidAlgebra.instMul",... | [] | simp [Commute, SemiconjBy, hm.eq, hr.eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 471,
"column": 48
} | {
"line": 471,
"column": 88
} | {
"line": 473,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\nr₁ r₂ : R\nm₁ m₂ : M\ninst✝ : Mul M\nhm : Commute m₁ m₂\nhr : Commute r₁ r₂\n⊢ Commute (single m₁ r₁) (single m₂ r₂)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"MonoidAlgebra.instMul",... | [] | simp [Commute, SemiconjBy, hm.eq, hr.eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 471,
"column": 48
} | {
"line": 471,
"column": 88
} | {
"line": 473,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\nr₁ r₂ : R\nm₁ m₂ : M\ninst✝ : Mul M\nhm : Commute m₁ m₂\nhr : Commute r₁ r₂\n⊢ Commute (single m₁ r₁) (single m₂ r₂)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"MonoidAlgebra.instMul",... | [] | simp [Commute, SemiconjBy, hm.eq, hr.eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 369,
"column": 4
} | {
"line": 369,
"column": 51
} | {
"line": 370,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Semigroup G\ninst✝ : IsCancelMul G\nh : ∀ {A : Finset G}, A.Nonempty → ∃ a1 ∈ A, ∃ a2 ∈ A, UniqueMul A A a1 a2\nA B : Finset G\nhA : A.Nonempty\nhB : B.Nonempty\ng1 : G\nh1 : g1 ∈ B * A\ng2 : G\nh2 : g2 ∈ B * A\nhu : UniqueMul (B * A) (B * A) g1 g2\n⊢ ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMu... | [
"G : Type u_1\ninst✝¹ : Semigroup G\ninst✝ : IsCancelMul G\nh : ∀ {A : Finset G}, A.Nonempty → ∃ a1 ∈ A, ∃ a2 ∈ A, UniqueMul A A a1 a2\nA B : Finset G\nhA : A.Nonempty\nhB : B.Nonempty\ng2 : G\nh2 : g2 ∈ B * A\nb1 : G\nhb1 : b1 ∈ B\na1 : G\nha1 : a1 ∈ A\nh1 : b1 * a1 ∈ B * A\nhu : UniqueMul (B * A) (B * A) (b1 * a1... | obtain ⟨b1, hb1, a1, ha1, rfl⟩ := mem_mul.mp h1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 982,
"column": 23
} | {
"line": 982,
"column": 38
} | {
"line": 982,
"column": 39
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DivisionMonoid α\ns : Finset α\n⊢ IsUnit ↑s ↔ ∃ a, s = {a} ∧ IsUnit a",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"IsUnit",
"Exists",
"_private.Mathlib.Algebra.Grou... | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DivisionMonoid α\ns : Finset α\n⊢ (∃ a, ↑s = {a} ∧ IsUnit a) ↔ ∃ a, s = {a} ∧ IsUnit a"
] | Set.isUnit_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Finset.Sort | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 25
} | {
"line": 98,
"column": 0
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ninst✝⁴ : DecidableRel r\ninst✝³ : IsTrans α r\ninst✝² : Std.Antisymm r\ninst✝¹ : Std.Total r\ninst✝ : DecidableEq α\nl : List α\nhl : l.Nodup\nh : l.toFinset.sort r = l\n⊢ List.Pairwise r (l.toFinset.sort r)",
"ppTerm": "?m.50",
"assigned": true,
"usedConstan... | [] | exact pairwise_sort _ r | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finset.Sort | {
"line": 91,
"column": 44
} | {
"line": 96,
"column": 25
} | {
"line": 98,
"column": 0
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ninst✝⁴ : DecidableRel r\ninst✝³ : IsTrans α r\ninst✝² : Std.Antisymm r\ninst✝¹ : Std.Total r\ninst✝ : DecidableEq α\nl : List α\nhl : l.Nodup\n⊢ l.toFinset.sort r = l ↔ List.Pairwise r l",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [] | by
refine ⟨?_, ((sort_perm_toList _ r).trans (List.toFinset_toList hl)).eq_of_pairwise'
(pairwise_sort _ _)⟩
intro h
rw [← h]
exact pairwise_sort _ r | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Finsupp.Span | {
"line": 130,
"column": 9
} | {
"line": 130,
"column": 22
} | {
"line": 130,
"column": 22
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nS : Set (Submodule R M)\nm : M\nx✝ : ∃ s, m ∈ ⨆ i ∈ s, ↑i\ns : Finset (Subtype (Membership.mem S))\nhs : m ∈ ⨆ i ∈ s, ↑i\n⊢ m ∈ ⨆ i, ⨆ (hi : i ∈ S), ⨆ (_ : ⟨i, hi⟩ ∈ s), i",
"ppTerm": "?ref... | [
"case refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nS : Set (Submodule R M)\nm : M\nx✝ : ∃ s, m ∈ ⨆ i ∈ s, ↑i\ns : Finset (Subtype (Membership.mem S))\nhs : m ∈ ⨆ i ∈ s, ↑i\n⊢ m ∈ ⨆ x, ⨆ (_ : ⟨↑x, ⋯⟩ ∈ s), ↑x"
] | iSup_subtype' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Basic | {
"line": 868,
"column": 2
} | {
"line": 868,
"column": 27
} | {
"line": 869,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\n⊢ p * q = ∑ i ∈ p.support, q.sum fun j a ↦ (monomial (i + j)) (p.coeff i * a)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"HMul.hMul",
"Polynomial.sum",
"Polynomial.toFinsupp_injective",
... | [
"R : Type u\ninst✝ : Semiring R\np q : R[X]\n⊢ (p * q).toFinsupp = (∑ i ∈ p.support, q.sum fun j a ↦ (monomial (i + j)) (p.coeff i * a)).toFinsupp"
] | apply toFinsupp_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Polynomial.Basic | {
"line": 895,
"column": 2
} | {
"line": 896,
"column": 78
} | {
"line": 898,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nS : Type u_1\ninst✝ : AddCommMonoid S\np q : R[X]\nf : ℕ → R → S\nhf : ∀ (i : ℕ), f i 0 = 0\nh_add : ∀ (a : ℕ) (b₁ b₂ : R), f a (b₁ + b₂) = f a b₁ + f a b₂\n⊢ (p + q).sum f = p.sum f + q.sum f",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"... | [] | rw [show p + q = ⟨p.toFinsupp + q.toFinsupp⟩ from rfl]
exact Finsupp.sum_add_index (fun i _ ↦ hf i) (fun a _ b₁ b₂ ↦ h_add a b₁ b₂) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Basic | {
"line": 895,
"column": 2
} | {
"line": 896,
"column": 78
} | {
"line": 898,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nS : Type u_1\ninst✝ : AddCommMonoid S\np q : R[X]\nf : ℕ → R → S\nhf : ∀ (i : ℕ), f i 0 = 0\nh_add : ∀ (a : ℕ) (b₁ b₂ : R), f a (b₁ + b₂) = f a b₁ + f a b₂\n⊢ (p + q).sum f = p.sum f + q.sum f",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"... | [] | rw [show p + q = ⟨p.toFinsupp + q.toFinsupp⟩ from rfl]
exact Finsupp.sum_add_index (fun i _ ↦ hf i) (fun a _ b₁ b₂ ↦ h_add a b₁ b₂) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Basic | {
"line": 934,
"column": 93
} | {
"line": 948,
"column": 21
} | {
"line": 950,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nmotive : R[X] → Prop\np : R[X]\nC : ∀ (a : R), motive (Polynomial.C a)\nadd : ∀ (p q : R[X]), motive p → motive q → motive (p + q)\nmonomial : ∀ (n : ℕ) (a : R), motive (Polynomial.C a * X ^ n) → motive (Polynomial.C a * X ^ (n + 1))\n⊢ motive p",
"ppTerm": "?m.61",
... | [] | by
have A : ∀ {n : ℕ} {a}, motive (Polynomial.C a * X ^ n) := by
intro n a
induction n with
| zero => rw [pow_zero, mul_one]; exact C a
| succ n ih => exact monomial _ _ ih
have B : ∀ s : Finset ℕ, motive (s.sum fun n : ℕ ↦ Polynomial.C (p.coeff n) * X ^ n) := by
apply Finset.induction
· con... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Basic | {
"line": 1032,
"column": 2
} | {
"line": 1033,
"column": 38
} | {
"line": 1035,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ p.update n 0 = erase n p",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.ext",
"congrArg",
"Polynomial.coeff_erase",
"Polynomial.update",
"Polynomial.coeff_update_apply",
... | [] | ext
rw [coeff_update_apply, coeff_erase] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Basic | {
"line": 1032,
"column": 2
} | {
"line": 1033,
"column": 38
} | {
"line": 1035,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ p.update n 0 = erase n p",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.ext",
"congrArg",
"Polynomial.coeff_erase",
"Polynomial.update",
"Polynomial.coeff_update_apply",
... | [] | ext
rw [coeff_update_apply, coeff_erase] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Finiteness.Cardinality | {
"line": 34,
"column": 2
} | {
"line": 35,
"column": 27
} | {
"line": 37,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\n⊢ N.FG ↔ ∃ n f, f.range = N",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Submodule",
"RingHomSurjective.ids... | [] | simp_rw [fg_iff_exists_fin_generating_family, ← ((Pi.basisFun R _).constr ℕ).exists_congr_right]
simp [Basis.constr_range] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Finiteness.Cardinality | {
"line": 34,
"column": 2
} | {
"line": 35,
"column": 27
} | {
"line": 37,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\n⊢ N.FG ↔ ∃ n f, f.range = N",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Submodule",
"RingHomSurjective.ids... | [] | simp_rw [fg_iff_exists_fin_generating_family, ← ((Pi.basisFun R _).constr ℕ).exists_congr_right]
simp [Basis.constr_range] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 593,
"column": 34
} | {
"line": 593,
"column": 39
} | {
"line": 593,
"column": 40
} | [
{
"pp": "case refine_1\nG : Type u\nH : Type v\ninst✝⁴ : Mul G\ninst✝³ : Mul H\ninst✝² : IsRightCancelMul G\ninst✝¹ : LinearOrder G\ninst✝ : MulLeftStrictMono G\nA B : Finset G\nhc : 1 < #(A ×ˢ B)\nhA : A.Nonempty\nhB : B.Nonempty\na0 : G\nha0 : a0 ∈ A\nb0 : G\nhb0 : b0 ∈ B\nhe0 : a0 * b0 = (A * B).max' ⋯\na1 :... | [
"case refine_1\nG : Type u\nH : Type v\ninst✝⁴ : Mul G\ninst✝³ : Mul H\ninst✝² : IsRightCancelMul G\ninst✝¹ : LinearOrder G\ninst✝ : MulLeftStrictMono G\nA B : Finset G\nhc : 1 < #(A ×ˢ B)\nhA : A.Nonempty\nhB : B.Nonempty\na0 : G\nha0 : a0 ∈ A\nb0 : G\nhb0 : b0 ∈ B\na1 : G\nhe0 : a1 * b0 = (A * B).max' ⋯\nha1 : a1... | he.1, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 593,
"column": 46
} | {
"line": 593,
"column": 49
} | {
"line": 593,
"column": 49
} | [
{
"pp": "case refine_1\nG : Type u\nH : Type v\ninst✝⁴ : Mul G\ninst✝³ : Mul H\ninst✝² : IsRightCancelMul G\ninst✝¹ : LinearOrder G\ninst✝ : MulLeftStrictMono G\nA B : Finset G\nhc : 1 < #(A ×ˢ B)\nhA : A.Nonempty\nhB : B.Nonempty\na0 : G\nha0 : a0 ∈ A\nb0 : G\nhb0 : b0 ∈ B\na1 : G\nha1 : a1 ∈ A\nb1 : G\nhe0 : ... | [
"case refine_1\nG : Type u\nH : Type v\ninst✝⁴ : Mul G\ninst✝³ : Mul H\ninst✝² : IsRightCancelMul G\ninst✝¹ : LinearOrder G\ninst✝ : MulLeftStrictMono G\nA B : Finset G\nhc : 1 < #(A ×ˢ B)\nhA : A.Nonempty\nhB : B.Nonempty\na0 : G\nha0 : a0 ∈ A\nb0 : G\nhb0 : b0 ∈ B\na1 : G\nha1 : a1 ∈ A\nb1 : G\nhe0 : (A * B).min'... | he1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Submodule.Bilinear | {
"line": 139,
"column": 2
} | {
"line": 140,
"column": 35
} | {
"line": 141,
"column": 2
} | [
{
"pp": "ι : Sort uι\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\ns : ι → Submodule R M\nt : Submodule R N\n⊢ ma... | [
"ι : Sort uι\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\ns : ι → Submodule R M\nt : Submodule R N\n⊢ map₂ f (⨆ i, s... | suffices map₂ f (⨆ i, span R (s i : Set M)) (span R t) = ⨆ i, map₂ f (span R (s i)) (span R t) by
simpa only [span_eq] using this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Algebra.Module.Submodule.Bilinear | {
"line": 152,
"column": 35
} | {
"line": 152,
"column": 57
} | {
"line": 152,
"column": 58
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\nm : M\ns : Submodule R N\n⊢ span R (image2 (fun m n ↦ (f m)... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\nm : M\ns : Submodule R N\n⊢ span R ((fun n ↦ (f m) n) '' ↑s) = map (f m... | image2_singleton_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Pointwise.Set.BigOperators | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 68
} | {
"line": 95,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\ns : Set α\na : α\n⊢ a ∈ s ^ n ↔ ∃ f, (∀ (i : Fin n), f i ∈ s) ∧ ∏ i, f i = a",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Fintype.card_fin",
"Finset.univ",
"Iff.of_eq",
"congrArg",
"Finset",
"instIn... | [] | simpa using mem_finsetProd (t := .univ) (f := fun _ : Fin n ↦ s) _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Group.Pointwise.Set.BigOperators | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 68
} | {
"line": 95,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\ns : Set α\na : α\n⊢ a ∈ s ^ n ↔ ∃ f, (∀ (i : Fin n), f i ∈ s) ∧ ∏ i, f i = a",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Fintype.card_fin",
"Finset.univ",
"Iff.of_eq",
"congrArg",
"Finset",
"instIn... | [] | simpa using mem_finsetProd (t := .univ) (f := fun _ : Fin n ↦ s) _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Pointwise.Set.BigOperators | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 68
} | {
"line": 95,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : CommMonoid α\nn : ℕ\ns : Set α\na : α\n⊢ a ∈ s ^ n ↔ ∃ f, (∀ (i : Fin n), f i ∈ s) ∧ ∏ i, f i = a",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Fintype.card_fin",
"Finset.univ",
"Iff.of_eq",
"congrArg",
"Finset",
"instIn... | [] | simpa using mem_finsetProd (t := .univ) (f := fun _ : Fin n ↦ s) _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Submonoid.Pointwise | {
"line": 165,
"column": 41
} | {
"line": 165,
"column": 51
} | {
"line": 165,
"column": 51
} | [
{
"pp": "R : Type u_2\ninst✝ : NonUnitalNonAssocSemiring R\nM N : AddSubmonoid R\n⊢ M * N = M * closure ↑N",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"AddSubmonoid.mul",
"AddMonoid.toAddZeroClass",
"id",
"Add... | [
"R : Type u_2\ninst✝ : NonUnitalNonAssocSemiring R\nM N : AddSubmonoid R\n⊢ M * N = M * N"
] | closure_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Submonoid.Pointwise | {
"line": 282,
"column": 21
} | {
"line": 282,
"column": 31
} | {
"line": 282,
"column": 31
} | [
{
"pp": "R : Type u_2\ninst✝ : Semiring R\ns : AddSubmonoid R\nn : ℕ\n⊢ s ^ n = closure ↑s ^ n",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"AddMonoid.toAddZeroClass",
"id",
"AddSubmonoi... | [
"R : Type u_2\ninst✝ : Semiring R\ns : AddSubmonoid R\nn : ℕ\n⊢ s ^ n = s ^ n"
] | closure_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 24
} | {
"line": 117,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Monoid R\ninst✝³ : MulAction R M\ninst✝² : Monoid M\ninst✝¹ : IsScalarTower R M M\ninst✝ : SMulCommClass R M M\np : SubMulAction R M\n⊢ 1 ⊆ ↑1",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
... | [] | exact subset_coe_one | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Coprime.Basic | {
"line": 62,
"column": 48
} | {
"line": 62,
"column": 57
} | {
"line": 62,
"column": 58
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx : R\nx✝ : IsCoprime 0 x\na b : R\nH : a * 0 + b * x = 1\n⊢ x * b = 1",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"CommS... | [
"R : Type u\ninst✝ : CommSemiring R\nx : R\nx✝ : IsCoprime 0 x\na b : R\nH : 0 + b * x = 1\n⊢ x * b = 1"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Coprime.Basic | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 33
} | {
"line": 84,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nh : IsCoprime (0 0) (0 1)\n⊢ False",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"not_isCoprime_zero_zero"
],
"usedFVars": [
"R",
"inst✝¹",
"inst✝",
"h"
],
"usedGoals": []
... | [] | exact not_isCoprime_zero_zero h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Coprime.Basic | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 33
} | {
"line": 89,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nh : IsCoprime 0 0\n⊢ False",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"not_isCoprime_zero_zero"
],
"usedFVars": [
"R",
"inst✝¹",
"inst✝",
"h"
],
"usedGoals": []
}
] | [] | exact not_isCoprime_zero_zero h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Coprime.Basic | {
"line": 97,
"column": 24
} | {
"line": 97,
"column": 32
} | {
"line": 97,
"column": 33
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : IsCoprime x z\nH2 : x ∣ y * z\na b : R\nH : a * x + b * z = 1\n⊢ x ∣ y * (a * x + b * z)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
... | [
"R : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : IsCoprime x z\nH2 : x ∣ y * z\na b : R\nH : a * x + b * z = 1\n⊢ x ∣ y * (a * x) + y * (b * z)"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Coprime.Basic | {
"line": 384,
"column": 22
} | {
"line": 384,
"column": 32
} | {
"line": 384,
"column": 33
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nx y z : R\n⊢ IsCoprime (x + -(y * z)) y ↔ IsCoprime x y",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"AddGroupWi... | [
"R : Type u\ninst✝ : CommRing R\nx y z : R\n⊢ IsCoprime (x + y * -z) y ↔ IsCoprime x y"
] | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Coprime.Basic | {
"line": 390,
"column": 22
} | {
"line": 390,
"column": 32
} | {
"line": 390,
"column": 33
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nx y z : R\n⊢ IsCoprime x (y + -(x * z)) ↔ IsCoprime x y",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"AddGroupWi... | [
"R : Type u\ninst✝ : CommRing R\nx y z : R\n⊢ IsCoprime x (y + x * -z) ↔ IsCoprime x y"
] | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Coprime.Lemmas | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 37
} | {
"line": 111,
"column": 4
} | [
{
"pp": "case insert\nR : Type u\nI : Type v\ninst✝ : CommSemiring R\nz : R\ns : I → R\nt : Finset I\na : I\nr : Finset I\nhar : a ∉ r\nih : (↑r).Pairwise (IsCoprime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z\nHs : (↑(insert a r)).Pairwise (IsCoprime on s)\nHs1 : ∀ i ∈ insert a r, s i ∣ z\n⊢ s a * ∏ x ∈ r, s... | [
"case insert.refine_1\nR : Type u\nI : Type v\ninst✝ : CommSemiring R\nz : R\ns : I → R\nt : Finset I\na : I\nr : Finset I\nhar : a ∉ r\nih : (↑r).Pairwise (IsCoprime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z\nHs : (↑(insert a r)).Pairwise (IsCoprime on s)\nHs1 : ∀ i ∈ insert a r, s i ∣ z\n⊢ IsCoprime (s a) (∏ ... | refine IsCoprime.mul_dvd ?_ ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Coprime.Basic | {
"line": 457,
"column": 2
} | {
"line": 457,
"column": 33
} | {
"line": 459,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nha hb : 0 ^ 2 = 0\nh : IsCoprime 0 0\nh' : 0 ^ 2 + 0 ^ 2 = 0\n⊢ False",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"AddGroupWithOne.toAddMonoidWithOne",
"SemilatticeInf.toPart... | [] | exact not_isCoprime_zero_zero h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Coprime.Basic | {
"line": 552,
"column": 22
} | {
"line": 552,
"column": 32
} | {
"line": 552,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nx y z : R\n⊢ IsRelPrime (x + -(y * z)) y ↔ IsRelPrime x y",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"AddGro... | [
"R : Type u_1\ninst✝ : CommRing R\nx y z : R\n⊢ IsRelPrime (x + y * -z) y ↔ IsRelPrime x y"
] | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Coprime.Basic | {
"line": 558,
"column": 22
} | {
"line": 558,
"column": 32
} | {
"line": 558,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nx y z : R\n⊢ IsRelPrime x (y + -(x * z)) ↔ IsRelPrime x y",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"AddGro... | [
"R : Type u_1\ninst✝ : CommRing R\nx y z : R\n⊢ IsRelPrime x (y + x * -z) ↔ IsRelPrime x y"
] | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Operations | {
"line": 598,
"column": 2
} | {
"line": 598,
"column": 26
} | {
"line": 600,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM : Submodule R A\nn : ℕ\n⊢ M ^ n = span R (↑M ^ n)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
... | [] | rw [← span_pow, span_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Algebra.Operations | {
"line": 598,
"column": 2
} | {
"line": 598,
"column": 26
} | {
"line": 600,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM : Submodule R A\nn : ℕ\n⊢ M ^ n = span R (↑M ^ n)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
... | [] | rw [← span_pow, span_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Operations | {
"line": 598,
"column": 2
} | {
"line": 598,
"column": 26
} | {
"line": 600,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM : Submodule R A\nn : ℕ\n⊢ M ^ n = span R (↑M ^ n)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
... | [] | rw [← span_pow, span_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Prod | {
"line": 85,
"column": 6
} | {
"line": 86,
"column": 15
} | {
"line": 86,
"column": 15
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal R\nJ : Ideal S\nx : S\n⊢ (∃ x_1 ∈ I.prod J, (RingHom.snd R S) x_1 = x) → x ∈ J",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Submodule",
"Semiring.toModule",
"RingHom",
"Membershi... | [] | rintro ⟨x, ⟨h, rfl⟩⟩
exact h.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Prod | {
"line": 85,
"column": 6
} | {
"line": 86,
"column": 15
} | {
"line": 86,
"column": 15
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal R\nJ : Ideal S\nx : S\n⊢ (∃ x_1 ∈ I.prod J, (RingHom.snd R S) x_1 = x) → x ∈ J",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Submodule",
"Semiring.toModule",
"RingHom",
"Membershi... | [] | rintro ⟨x, ⟨h, rfl⟩⟩
exact h.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Prod | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 41
} | {
"line": 105,
"column": 4
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI✝ : Ideal R\nJ✝ : Ideal S\nI J : Ideal (R × S)\nh : map (RingHom.fst R S) I ≤ map (RingHom.fst R S) J ∧ map (RingHom.snd R S) I ≤ map (RingHom.snd R S) J\n⊢ I ≤ J",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
... | [
"R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI✝ : Ideal R\nJ✝ : Ideal S\nI J : Ideal (R × S)\nh : map (RingHom.fst R S) I ≤ map (RingHom.fst R S) J ∧ map (RingHom.snd R S) I ≤ map (RingHom.snd R S) J\n⊢ (map (RingHom.fst R S) I).prod (map (RingHom.snd R S) I) ≤ (map (RingHom.fst R S) J).prod (m... | rw [ideal_prod_eq I, ideal_prod_eq J] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Algebra.Operations | {
"line": 921,
"column": 4
} | {
"line": 923,
"column": 45
} | {
"line": 924,
"column": 4
} | [
{
"pp": "case refine_1.refine_1\nR : Type u\ninst✝⁶ : CommSemiring R\nS : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ns : Set S\nN : Submodule R M\nx : M\nx_in : x ∈ ⇑(algebraMap S R) '' s... | [
"case refine_1.refine_2\nR : Type u\ninst✝⁶ : CommSemiring R\nS : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ns : Set S\nN : Submodule R M\nx : M\nx_in : x ∈ ⇑(algebraMap S R) '' s • N\n⊢ ∀ (r... | · rintro _ x ⟨r, r_in, rfl⟩ x_in
rw [algebraMap_smul]
exact mem_set_smul_of_mem_mem r_in x_in | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.Maps | {
"line": 93,
"column": 34
} | {
"line": 93,
"column": 77
} | {
"line": 94,
"column": 2
} | [
{
"pp": "R : Type u\nF : Type u_1\ninst✝⁴ : Semiring R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : FunLike F R S\ninst✝¹ : RingHomClass F R S\nf : F\nP : Ideal R\ninst✝ : P.IsPrime\nI : Ideal S\nle : comap f I ≤ P\nx✝ : S\n⊢ x✝ ∈ ↑I → x✝ ∉ ↑(Submonoid.map f P.primeCompl)",
"ppTerm": "?m.55",
"assig... | [] | by rintro hI ⟨r, hp, rfl⟩; exact hp (le hI) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Maps | {
"line": 367,
"column": 4
} | {
"line": 367,
"column": 37
} | {
"line": 368,
"column": 4
} | [
{
"pp": "R : Type u\nS : Type v\nF : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : FunLike F R S\nf✝ : F\nI✝ J : Ideal R\nK L : Ideal S\nG : Type u_2\ninst✝³ : FunLike G S R\ninst✝² : RingHomClass F R S\nι : Sort u_3\nf : R →+* S\ninst✝¹ : RingHomSurjective f\nI : Ideal R\ninst✝ : I.IsTwoSided\na... | [
"R : Type u\nS : Type v\nF : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : FunLike F R S\nf✝ : F\nI✝ J : Ideal R\nK L : Ideal S\nG : Type u_2\ninst✝³ : FunLike G S R\ninst✝² : RingHomClass F R S\nι : Sort u_3\nf : R →+* S\ninst✝¹ : RingHomSurjective f\nI : Ideal R\ninst✝ : I.IsTwoSided\na✝ b : S\nha ... | rw [map_eq_submodule_map] at ha ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Maps | {
"line": 433,
"column": 6
} | {
"line": 434,
"column": 98
} | {
"line": 435,
"column": 6
} | [
{
"pp": "R✝ : Type u\nS : Type v\nF : Type u_1\ninst✝⁶ : Semiring R✝\ninst✝⁵ : Semiring S\ninst✝⁴ : FunLike F R✝ S\nf : F\nI✝ J : Ideal R✝\nK L : Ideal S\nG : Type u_2\ninst✝³ : FunLike G S R✝\ninst✝² : RingHomClass F R✝ S\nι✝ : Sort u_3\nι : Type u_4\nR : ι → Type u_5\ninst✝¹ : (i : ι) → Semiring (R i)\ninst✝ ... | [
"R✝ : Type u\nS : Type v\nF : Type u_1\ninst✝⁶ : Semiring R✝\ninst✝⁵ : Semiring S\ninst✝⁴ : FunLike F R✝ S\nf : F\nI✝ J : Ideal R✝\nK L : Ideal S\nG : Type u_2\ninst✝³ : FunLike G S R✝\ninst✝² : RingHomClass F R✝ S\nι✝ : Sort u_3\nι : Type u_4\nR : ι → Type u_5\ninst✝¹ : (i : ι) → Semiring (R i)\ninst✝ : Finite ι\n... | classical rw [show r = ∑ i, Pi.single i 1 * r' i from funext fun i ↦ by
rw [← (hr' _).2, Finset.sum_apply, Fintype.sum_eq_single i fun j ne ↦ by simp [ne]]; simp] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 40
} | {
"line": 145,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na✝ b✝ a b : R\nx✝¹ : a ≠ 0\nx✝ : gcd (b % a) a ∣ b % a ∧ gcd (b % a) a ∣ a\nIH₁ : gcd (b % a) a ∣ b % a\nIH₂ : gcd (b % a) a ∣ a\n⊢ gcd (b % a) a ∣ a ∧ gcd (b % a) a ∣ b",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
... | [] | exact ⟨IH₂, (dvd_mod_iff IH₂).1 IH₁⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 208,
"column": 65
} | {
"line": 208,
"column": 74
} | {
"line": 208,
"column": 75
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ a = a + b * 0",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiring",
"id",
"Distrib.toAd... | [
"R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ a = a + 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 209,
"column": 34
} | {
"line": 209,
"column": 43
} | {
"line": 209,
"column": 44
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ b = a * 0 + b",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiring",
"id",
"Distrib.toAd... | [
"R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ b = 0 + b"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 284,
"column": 60
} | {
"line": 284,
"column": 69
} | {
"line": 284,
"column": 70
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx : R\n⊢ x * 0 / gcd x 0 = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiring",
"id... | [
"R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx : R\n⊢ 0 / gcd x 0 = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 336,
"column": 6
} | {
"line": 336,
"column": 14
} | {
"line": 336,
"column": 15
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\nx y z : R\nh1 : y ≠ 0\nh2 : y ∣ x\n⊢ y * (x / y + z) = x + y * z",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"congrArg",
"CommSemiring.toSe... | [
"R : Type u\ninst✝ : EuclideanDomain R\nx y z : R\nh1 : y ≠ 0\nh2 : y ∣ x\n⊢ y * (x / y) + y * z = x + y * z"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 354,
"column": 6
} | {
"line": 354,
"column": 14
} | {
"line": 354,
"column": 15
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\nx y z : R\nh1 : z ≠ 0\nh2 : z ∣ y\n⊢ z * (x + y / z) = z * x + y",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"congrArg",
"CommSemiring.toSe... | [
"R : Type u\ninst✝ : EuclideanDomain R\nx y z : R\nh1 : z ≠ 0\nh2 : z ∣ y\n⊢ z * x + z * (y / z) = z * x + y"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 895,
"column": 9
} | {
"line": 895,
"column": 34
} | {
"line": 896,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Type u_4\ninst✝ : Nonempty ι\nr : R\n⊢ r ∈ annihilator R (ι →₀ M) ↔ r ∈ annihilator R M",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Sem... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Type u_4\ninst✝ : Nonempty ι\nr : R\n⊢ (∀ (m : ι →₀ M), r • m = 0) ↔ ∀ (m : M), r • m = 0"
] | simp_rw [mem_annihilator] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 386,
"column": 6
} | {
"line": 386,
"column": 14
} | {
"line": 386,
"column": 15
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\nx y z t : R\nh1 : y ≠ 0\nh2 : t ≠ 0\nh3 : y ∣ x\nh4 : t ∣ z\n⊢ t * y * (x / y + z / t) = t * x + y * z",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"instHDiv",
"HMul.hMul",
... | [
"R : Type u\ninst✝ : EuclideanDomain R\nx y z t : R\nh1 : y ≠ 0\nh2 : t ≠ 0\nh3 : y ∣ x\nh4 : t ∣ z\n⊢ t * y * (x / y) + t * y * (z / t) = t * x + y * z"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 440,
"column": 17
} | {
"line": 442,
"column": 26
} | {
"line": 444,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : Semiring R\ninst✝² : NoZeroDivisors R\nI J : Ideal R\ninst✝¹ : I.IsTwoSided\ninst✝ : J.IsTwoSided\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\n⊢ I ⊓ J ≠ ⊥",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"Preorder.toLT",
"Semiring.toMod... | [] | by
grw [← bot_lt_iff_ne_bot, ← mul_le_inf, bot_lt_iff_ne_bot, Ne, mul_eq_bot]
exact not_or_intro hI hJ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.Multiset | {
"line": 182,
"column": 53
} | {
"line": 182,
"column": 61
} | {
"line": 182,
"column": 61
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : s.dedup.gcd = s.gcd\nh : a ∈ s\n⊢ GCDMonoid.gcd a (fold GCDMonoid.gcd 0 (s.erase a)) =\n GCDMonoid.gcd (GCDMonoid.gcd a a) (fold GCDMonoid.gcd 0 ... | [
"case pos\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : s.dedup.gcd = s.gcd\nh : a ∈ s\n⊢ GCDMonoid.gcd a (fold GCDMonoid.gcd 0 (s.erase a)) = GCDMonoid.gcd (normalize a) (fold GCDMonoid.gcd 0 (s.erase a))"
] | gcd_same | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Nat | {
"line": 135,
"column": 57
} | {
"line": 135,
"column": 73
} | {
"line": 137,
"column": 0
} | [
{
"pp": "i j : ℤ\n⊢ 0 ≤ GCDMonoid.lcm i j",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Int.lcm",
"Int.natCast_nonneg._simp_1",
"Int",
"LE.le",
"Nat.cast",
"instOfNat",
"of_eq_true",
"instNatCastInt",
"OfNat.ofNat",
"Int.instLEI... | [] | simp [← coe_lcm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.GCDMonoid.Nat | {
"line": 135,
"column": 57
} | {
"line": 135,
"column": 73
} | {
"line": 137,
"column": 0
} | [
{
"pp": "i j : ℤ\n⊢ 0 ≤ GCDMonoid.lcm i j",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Int.lcm",
"Int.natCast_nonneg._simp_1",
"Int",
"LE.le",
"Nat.cast",
"instOfNat",
"of_eq_true",
"instNatCastInt",
"OfNat.ofNat",
"Int.instLEI... | [] | simp [← coe_lcm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GCDMonoid.Nat | {
"line": 135,
"column": 57
} | {
"line": 135,
"column": 73
} | {
"line": 137,
"column": 0
} | [
{
"pp": "i j : ℤ\n⊢ 0 ≤ GCDMonoid.lcm i j",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Int.lcm",
"Int.natCast_nonneg._simp_1",
"Int",
"LE.le",
"Nat.cast",
"instOfNat",
"of_eq_true",
"instNatCastInt",
"OfNat.ofNat",
"Int.instLEI... | [] | simp [← coe_lcm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Operations | {
"line": 532,
"column": 8
} | {
"line": 532,
"column": 22
} | {
"line": 532,
"column": 23
} | [
{
"pp": "case succ.succ\nR : Type u\ninst✝ : Semiring R\nn : ℕ\nih : n + 1 ≠ 0 → ↑(n + 1) = ⊤\nhn : n + 1 + 1 ≠ 0\n⊢ ↑(n + 1 + 1) = ⊤",
"ppTerm": "?succ.succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"Submodule.instAddCommMonoidWithOne",
"Semiring.toMo... | [
"case succ.succ\nR : Type u\ninst✝ : Semiring R\nn : ℕ\nih : n + 1 ≠ 0 → ↑(n + 1) = ⊤\nhn : n + 1 + 1 ≠ 0\n⊢ ↑(n + 1) + 1 = ⊤"
] | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 567,
"column": 4
} | {
"line": 568,
"column": 56
} | {
"line": 569,
"column": 4
} | [
{
"pp": "case neg\nR : Type u\ninst✝ : CommSemiring R\nI J : Ideal R\nn m i : ℕ\nhi : i ∈ Finset.range (n + m + 1)\nhn : ¬n ≤ i\n⊢ I ^ i * J ^ (n + m - i) * ↑((n + m).choose i) ≤ I ^ n + J ^ m",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"case neg\nR : Type u\ninst✝ : CommSemiring R\nI J : Ideal R\nn m i : ℕ\nhi : i ∈ Finset.range (n + m + 1)\nhn : ¬n ≤ i\n⊢ m ≤ n + m - i"
] | refine (Ideal.mul_le_right.trans (Ideal.mul_le_left.trans
((Ideal.pow_le_pow_right ?_).trans le_sup_right))) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 102,
"column": 45
} | {
"line": 102,
"column": 54
} | {
"line": 102,
"column": 55
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : NormalizationMonoid α\nx y : α\nhx : ¬x = 0\nhy : y = 0\n⊢ x * 0 * ↑(normUnit (x * 0)) = x * ↑(normUnit x) * (0 * ↑(normUnit 0))",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"HMul.hMul... | [
"α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : NormalizationMonoid α\nx y : α\nhx : ¬x = 0\nhy : y = 0\n⊢ 0 * ↑(normUnit 0) = x * ↑(normUnit x) * (0 * ↑(normUnit 0))"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 949,
"column": 8
} | {
"line": 949,
"column": 16
} | {
"line": 949,
"column": 17
} | [
{
"pp": "case h\nα : Type u_1\ninst✝¹ : CommRing α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : c = gcd a c * e\nf : α\nhf : a = gcd a c * f\n⊢ b = gcd a c * (e + f * d)",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq... | [
"case h\nα : Type u_1\ninst✝¹ : CommRing α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : c = gcd a c * e\nf : α\nhf : a = gcd a c * f\n⊢ b = gcd a c * e + gcd a c * (f * d)"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1008,
"column": 12
} | {
"line": 1008,
"column": 21
} | {
"line": 1008,
"column": 22
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\na b : α\na0 : a = 0\n⊢ Associated (gcd a b ... | [
"case pos\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\na b : α\na0 : a = 0\n⊢ Associated 0 (a * b)"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1061,
"column": 12
} | {
"line": 1061,
"column": 21
} | {
"line": 1061,
"column": 22
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : IsCancelMulZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\nnormalize_g... | [
"case pos\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : IsCancelMulZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\nnormalize_gcd : ∀ (a b ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1076,
"column": 13
} | {
"line": 1076,
"column": 22
} | {
"line": 1076,
"column": 23
} | [
{
"pp": "α : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : IsCancelMulZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\nnormalize_gcd : ∀ (a ... | [
"α : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : IsCancelMulZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\nnormalize_gcd : ∀ (a b : α), norm... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1025,
"column": 11
} | {
"line": 1025,
"column": 39
} | {
"line": 1025,
"column": 40
} | [
{
"pp": "R : Type u\nι : Type u_1\ninst✝ : CommSemiring R\ns : Finset ι\nx : ι → R\np : Ideal R\nhp : p.IsPrime\n⊢ ∏ i ∈ s, x i ∈ p ↔ ∃ i ∈ s, x i ∈ p",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring... | [
"R : Type u\nι : Type u_1\ninst✝ : CommSemiring R\ns : Finset ι\nx : ι → R\np : Ideal R\nhp : p.IsPrime\n⊢ span {∏ i ∈ s, x i} ≤ p ↔ ∃ i ∈ s, span {x i} ≤ p"
] | ← span_singleton_le_iff_mem, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 310,
"column": 12
} | {
"line": 311,
"column": 45
} | {
"line": 311,
"column": 45
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝ : EuclideanDomain R\nS : Ideal R\nh : ¬{x | x ∈ S ∧ x ≠ 0}.Nonempty\na : R\n⊢ a ∈ S ↔ a = 0",
"ppTerm": "?m.253",
"assigned": true,
"usedConstants": [
"Submodule",
"Semiring.toModule",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass... | [] | exact ⟨fun haS => by_contra fun ha0 => h ⟨a, ⟨haS, ha0⟩⟩,
fun h₁ => h₁.symm ▸ S.zero_mem⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1088,
"column": 55
} | {
"line": 1088,
"column": 64
} | {
"line": 1088,
"column": 65
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : DecidableEq α\nlcm : α → α → α\ndvd_lcm_left : ∀ (a b : α), a ∣ lcm a b\ndvd_lcm_right : ∀ (a b : α), b ∣ lcm a b\nlcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a\nexists_gcd : ∀ (a b : α), lcm a b ∣ a * b :... | [
"case pos\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : DecidableEq α\nlcm : α → α → α\ndvd_lcm_left : ∀ (a b : α), a ∣ lcm a b\ndvd_lcm_right : ∀ (a b : α), b ∣ lcm a b\nlcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a\nexists_gcd : ∀ (a b : α), lcm a b ∣ a * b := fun a b ↦ ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Matrix.Diagonal | {
"line": 207,
"column": 2
} | {
"line": 208,
"column": 46
} | {
"line": 210,
"column": 0
} | [
{
"pp": "n : Type u_3\nα : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nd : n → α\ni : n\n⊢ (diagonal d).col i = Pi.single i (d i)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"congrArg",
"Matrix",
"Pi.single_apply",
"Matrix.of",
... | [] | ext
simp +contextual [diagonal, Pi.single_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Matrix.Diagonal | {
"line": 207,
"column": 2
} | {
"line": 208,
"column": 46
} | {
"line": 210,
"column": 0
} | [
{
"pp": "n : Type u_3\nα : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nd : n → α\ni : n\n⊢ (diagonal d).col i = Pi.single i (d i)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"congrArg",
"Matrix",
"Pi.single_apply",
"Matrix.of",
... | [] | ext
simp +contextual [diagonal, Pi.single_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Matrix.Diagonal | {
"line": 264,
"column": 8
} | {
"line": 264,
"column": 22
} | {
"line": 264,
"column": 23
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn✝ : Type u_3\no : Type u_4\nm' : o → Type u_5\nn' : o → Type u_6\nR : Type u_7\nS : Type u_8\nα : Type v\nβ : Type w\nγ : Type u_9\ninst✝¹ : DecidableEq n✝\ninst✝ : AddMonoidWithOne α\nn : ℕ\n⊢ (diagonal fun x ↦ ↑(n + 1)) = (diagonal fun x ↦ ↑n) + 1",
"ppTerm": "?m.42",... | [
"l : Type u_1\nm : Type u_2\nn✝ : Type u_3\no : Type u_4\nm' : o → Type u_5\nn' : o → Type u_6\nR : Type u_7\nS : Type u_8\nα : Type v\nβ : Type w\nγ : Type u_9\ninst✝¹ : DecidableEq n✝\ninst✝ : AddMonoidWithOne α\nn : ℕ\n⊢ (diagonal fun x ↦ ↑n + 1) = (diagonal fun x ↦ ↑n) + 1"
] | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
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