module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.GroupTheory.Perm.Closure | {
"line": 121,
"column": 40
} | {
"line": 121,
"column": 45
} | {
"line": 121,
"column": 46
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nh0 : Nat.Prime (Fintype.card α)\nh1 : σ.IsCycle\nh2 : σ.support = univ\nx y : α\nh4 : x ≠ y\nh5 : τ = swap x y\ni m : ℕ\nhi : ((σ ^ #univ) ^ m) x = y\nhm : i = Fintype.card α * m\n⊢ x = y",
"ppTerm": "?m.143",
"assigned": tr... | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nh0 : Nat.Prime (Fintype.card α)\nh1 : σ.IsCycle\nh2 : σ.support = univ\nx y : α\nh4 : x ≠ y\nh5 : τ = swap x y\ni m : ℕ\nhi : ((σ ^ #σ.support) ^ m) x = y\nhm : i = Fintype.card α * m\n⊢ x = y"
] | ← h2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Iterate | {
"line": 59,
"column": 60
} | {
"line": 59,
"column": 70
} | {
"line": 59,
"column": 70
} | [
{
"pp": "α : Type u_1\nf : α → α\na : α\nm n : ℕ\n⊢ map (fun x ↦ f^[x] a) (take m (range n)) = map (fun x ↦ f^[x] a) (range (min m n))",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"List.take_range",
"congrArg",
"List.map",
"id",
"List.range"... | [
"α : Type u_1\nf : α → α\na : α\nm n : ℕ\n⊢ map (fun x ↦ f^[x] a) (range (min m n)) = map (fun x ↦ f^[x] a) (range (min m n))"
] | take_range | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.NoncommPiCoprod | {
"line": 163,
"column": 4
} | {
"line": 167,
"column": 8
} | {
"line": 168,
"column": 2
} | [
{
"pp": "case a\nM : Type u_1\ninst✝² : Monoid M\nι : Type u_2\ninst✝¹ : Fintype ι\nN : ι → Type u_3\ninst✝ : (i : ι) → Monoid (N i)\nϕ : (i : ι) → N i →* M\nhcomm : Pairwise fun i j ↦ ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)\nthis : DecidableEq ι := ⋯\n⊢ mrange (noncommPiCoprod ϕ hcomm) ≤ ⨆ i, mrange... | [] | rintro x ⟨f, rfl⟩
refine Submonoid.noncommProd_mem _ _ _ (fun _ _ _ _ h => hcomm h _ _) (fun i _ => ?_)
apply Submonoid.mem_sSup_of_mem
· use i
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.NoncommPiCoprod | {
"line": 163,
"column": 4
} | {
"line": 167,
"column": 8
} | {
"line": 168,
"column": 2
} | [
{
"pp": "case a\nM : Type u_1\ninst✝² : Monoid M\nι : Type u_2\ninst✝¹ : Fintype ι\nN : ι → Type u_3\ninst✝ : (i : ι) → Monoid (N i)\nϕ : (i : ι) → N i →* M\nhcomm : Pairwise fun i j ↦ ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)\nthis : DecidableEq ι := ⋯\n⊢ mrange (noncommPiCoprod ϕ hcomm) ≤ ⨆ i, mrange... | [] | rintro x ⟨f, rfl⟩
refine Submonoid.noncommProd_mem _ _ _ (fun _ _ _ _ h => hcomm h _ _) (fun i _ => ?_)
apply Submonoid.mem_sSup_of_mem
· use i
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 36
} | {
"line": 51,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_2\nf : Perm α\ninst✝ : DecidableRel f.SameCycle\nx y : α\nh : ¬f.SameCycle x y\n⊢ (ofSubtype (f.subtypePerm ⋯)) y = y",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Equiv.Perm.sameCycle_apply_right",
"Equiv.Perm.SameCycle",
"Equiv.Perm.subt... | [
"case neg\nα : Type u_2\nf : Perm α\ninst✝ : DecidableRel f.SameCycle\nx y : α\nh : ¬f.SameCycle x y\n⊢ ¬f.SameCycle x y"
] | apply ofSubtype_apply_of_not_mem | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 55,
"column": 21
} | {
"line": 57,
"column": 70
} | {
"line": 59,
"column": 0
} | [
{
"pp": "α : Type u_2\nf : Perm α\ninst✝ : DecidableRel f.SameCycle\nx y : α\n⊢ (f.cycleOf x)⁻¹ y = (f⁻¹.cycleOf x) y",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Equiv.apply_symm_apply",
"Equiv.instEquivLike",
"Equiv.Perm.instInv",
... | [] | by
rw [inv_eq_iff_eq, cycleOf_apply, cycleOf_apply]
split_ifs <;> simp_all [sameCycle_inv, sameCycle_symm_apply_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 149,
"column": 20
} | {
"line": 149,
"column": 38
} | {
"line": 149,
"column": 38
} | [
{
"pp": "case pos\nα : Type u_2\nf : Perm α\nx : α\ninst✝¹ : DecidableRel f.SameCycle\ninst✝ : DecidableEq α\nhf : f.IsCycle\nhx : f x = x\n⊢ f.cycleOf x = 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"Equiv.Perm.instOne",
"Equi... | [
"case pos\nα : Type u_2\nf : Perm α\nx : α\ninst✝¹ : DecidableRel f.SameCycle\ninst✝ : DecidableEq α\nhf : f.IsCycle\nhx : f x = x\n⊢ f x = x"
] | cycleOf_eq_one_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Cycle.Basic | {
"line": 1009,
"column": 6
} | {
"line": 1016,
"column": 20
} | {
"line": 1017,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_2\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng c : Perm α\nhc : c.IsCycle\nhc' : ∀ (x : α), g x ∈ c.support ↔ x ∈ c.support\ni : ℤ\nhi : (fun x ↦ c.subtypePermOfSupport ^ x) i = g.subtypePerm hc'\nx : α\nhx : x ∈ c.support\n⊢ g (c x) = c (g x)",
"ppTerm": "?pos✝",
"assigne... | [] | suffices hi' : ∀ x ∈ c.support, g x = (c ^ i) x by
rw [hi' x hx, hi' (c x) (apply_mem_support.mpr hx)]
simp only [← mul_apply, ← zpow_add_one, ← zpow_one_add, add_comm]
intro x hx
have hix := Perm.congr_fun hi ⟨x, hx⟩
simp only [← Subtype.coe_inj, subtypePermOfSupport, subtypePerm_appl... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Cycle.Basic | {
"line": 1009,
"column": 6
} | {
"line": 1016,
"column": 20
} | {
"line": 1017,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_2\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng c : Perm α\nhc : c.IsCycle\nhc' : ∀ (x : α), g x ∈ c.support ↔ x ∈ c.support\ni : ℤ\nhi : (fun x ↦ c.subtypePermOfSupport ^ x) i = g.subtypePerm hc'\nx : α\nhx : x ∈ c.support\n⊢ g (c x) = c (g x)",
"ppTerm": "?pos✝",
"assigne... | [] | suffices hi' : ∀ x ∈ c.support, g x = (c ^ i) x by
rw [hi' x hx, hi' (c x) (apply_mem_support.mpr hx)]
simp only [← mul_apply, ← zpow_add_one, ← zpow_one_add, add_comm]
intro x hx
have hix := Perm.congr_fun hi ⟨x, hx⟩
simp only [← Subtype.coe_inj, subtypePermOfSupport, subtypePerm_appl... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Cycle.Basic | {
"line": 1027,
"column": 2
} | {
"line": 1027,
"column": 53
} | {
"line": 1028,
"column": 2
} | [
{
"pp": "α : Type u_2\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng c : Perm α\nhc : c.IsCycle\n⊢ Commute g c ↔ ∃ (hc' : ∀ (x : α), g x ∈ c.support ↔ x ∈ c.support), ofSubtype (g.subtypePerm hc') ∈ Subgroup.zpowers c",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"E... | [
"α : Type u_2\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng c : Perm α\nhc : c.IsCycle\n⊢ (∃ (h : ∀ (x : α), g x ∈ c.support ↔ x ∈ c.support), ∃ k, c.subtypePermOfSupport ^ k = g.subtypePerm ⋯) ↔\n ∃ (h : ∀ (x : α), g x ∈ c.support ↔ x ∈ c.support), ∃ k, c ^ k = ofSubtype (g.subtypePerm ⋯)"
] | simp_rw [hc.commute_iff', Subgroup.mem_zpowers_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.GroupTheory.NoncommPiCoprod | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 14
} | {
"line": 273,
"column": 2
} | [
{
"pp": "case intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : Pairwise fun i j ↦ ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j ↦ (Fintype.c... | [
"case intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : Pairwise fun i j ↦ ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j ↦ (Fintype.card (H i)).C... | change f = 1 | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.GroupTheory.NoncommPiCoprod | {
"line": 253,
"column": 2
} | {
"line": 280,
"column": 18
} | {
"line": 282,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\nι : Type u_2\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : Pairwise fun i j ↦ ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j ↦ (Fintype.card (H i)).C... | [] | cases nonempty_fintype ι
let := Classical.decEq ι
rintro i
rw [disjoint_iff_inf_le]
rintro f ⟨hxi, hxp⟩
dsimp at hxi hxp
rw [iSup_subtype', ← noncommPiCoprod_range] at hxp
rotate_left
· intro _ _ hj
apply hcomm
exact hj ∘ Subtype.ext
obtain ⟨g, hgf⟩ := hxp
obtain ⟨g', hg'f⟩ := hxi
have hxi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.NoncommPiCoprod | {
"line": 253,
"column": 2
} | {
"line": 280,
"column": 18
} | {
"line": 282,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\nι : Type u_2\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nhcomm : Pairwise fun i j ↦ ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j ↦ (Fintype.card (H i)).C... | [] | cases nonempty_fintype ι
let := Classical.decEq ι
rintro i
rw [disjoint_iff_inf_le]
rintro f ⟨hxi, hxp⟩
dsimp at hxi hxp
rw [iSup_subtype', ← noncommPiCoprod_range] at hxp
rotate_left
· intro _ _ hj
apply hcomm
exact hj ∘ Subtype.ext
obtain ⟨g, hgf⟩ := hxp
obtain ⟨g', hg'f⟩ := hxi
have hxi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.NoncommPiCoprod | {
"line": 342,
"column": 2
} | {
"line": 345,
"column": 31
} | {
"line": 347,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nι : Type u_2\nH : ι → Subgroup G\ninst✝ : Fintype ι\nhcomm : Pairwise fun i j ↦ ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y\nhind : iSupIndep H\n⊢ Function.Injective ⇑(noncommPiCoprod hcomm)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq... | [] | apply MonoidHom.injective_noncommPiCoprod_of_iSupIndep
· simpa using hind
· intro i
exact Subtype.coe_injective | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.NoncommPiCoprod | {
"line": 342,
"column": 2
} | {
"line": 345,
"column": 31
} | {
"line": 347,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nι : Type u_2\nH : ι → Subgroup G\ninst✝ : Fintype ι\nhcomm : Pairwise fun i j ↦ ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y\nhind : iSupIndep H\n⊢ Function.Injective ⇑(noncommPiCoprod hcomm)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq... | [] | apply MonoidHom.injective_noncommPiCoprod_of_iSupIndep
· simpa using hind
· intro i
exact Subtype.coe_injective | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Option | {
"line": 29,
"column": 4
} | {
"line": 29,
"column": 23
} | {
"line": 30,
"column": 4
} | [
{
"pp": "case some\nα : Type u_1\ninst✝ : DecidableEq α\nx y i a✝ : α\n⊢ (optionCongr (swap x y)) (some i) = some a✝ ↔ (swap (some x) (some y)) (some i) = some a✝",
"ppTerm": "?some",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"Option.instDecidableEq",
"Equiv.swap"... | [
"case pos\nα : Type u_1\ninst✝ : DecidableEq α\nx y i a✝ : α\nhx : i = x\n⊢ (optionCongr (swap x y)) (some i) = some a✝ ↔ (swap (some x) (some y)) (some i) = some a✝",
"case neg\nα : Type u_1\ninst✝ : DecidableEq α\nx y i a✝ : α\nhx : ¬i = x\n⊢ (optionCongr (swap x y)) (some i) = some a✝ ↔ (swap (some x) (some y)... | by_cases hx : i = x | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.GroupTheory.Perm.Cycle.Type | {
"line": 359,
"column": 4
} | {
"line": 359,
"column": 36
} | {
"line": 359,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nhσ : σ ^ 2 = 1\n⊢ (if Odd 2 then 1 else (-1) ^ ((Fintype.card α - Fintype.card ↑(Function.fixedPoints ⇑σ)) / 2)) =\n (-1) ^ ((Fintype.card α - Fintype.card ↑(Function.fixedPoints ⇑σ)) / 2)",
"ppTerm": "?m.60",
"assigned": t... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nhσ : σ ^ 2 = 1\n⊢ (-1) ^ ((Fintype.card α - Fintype.card ↑(Function.fixedPoints ⇑σ)) / 2) =\n (-1) ^ ((Fintype.card α - Fintype.card ↑(Function.fixedPoints ⇑σ)) / 2)"
] | if_neg (Nat.not_odd_iff.mpr rfl) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Cycle.Type | {
"line": 417,
"column": 36
} | {
"line": 417,
"column": 41
} | {
"line": 417,
"column": 42
} | [
{
"pp": "α : Type u_1\ninst✝ : Fintype α\nσ : Perm α\nh1 : Nat.Prime (Fintype.card α)\nh2 : orderOf σ = Fintype.card α\n⊢ 1 * Fintype.card α < 2 * orderOf σ",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"congrArg",
"Nat... | [
"α : Type u_1\ninst✝ : Fintype α\nσ : Perm α\nh1 : Nat.Prime (Fintype.card α)\nh2 : orderOf σ = Fintype.card α\n⊢ 1 * orderOf σ < 2 * orderOf σ"
] | ← h2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 569,
"column": 62
} | {
"line": 569,
"column": 80
} | {
"line": 569,
"column": 80
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\nx : α\n⊢ ¬g.cycleOf x = 1 ↔ g x ≠ x",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Equiv.Perm.instDecidableRelSameCycle",
"Eq.mpr",
"Equiv.instEquivLike",
"Equiv.Perm.instOne",
"Eq... | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\nx : α\n⊢ ¬g x = x ↔ g x ≠ x"
] | cycleOf_eq_one_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Cycle.Type | {
"line": 639,
"column": 23
} | {
"line": 639,
"column": 35
} | {
"line": 639,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nh : σ.IsThreeCycle\n⊢ σ.cycleType.sum = 3",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Eq.mpr",
"Equiv.Perm.cycleType",
"congrArg",
"Multiset",
"id",
... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nh : σ.IsThreeCycle\n⊢ {3}.sum = 3"
] | h.cycleType, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Cycle.Type | {
"line": 656,
"column": 31
} | {
"line": 656,
"column": 43
} | {
"line": 656,
"column": 44
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nh : σ.IsThreeCycle\n⊢ σ.cycleType.card = 1",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.cycleType",
"congrArg",
"Multiset",
"id",
"instOfNatNat",
... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nh : σ.IsThreeCycle\n⊢ {3}.card = 1"
] | h.cycleType, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.Ring.Colimits | {
"line": 246,
"column": 20
} | {
"line": 246,
"column": 40
} | {
"line": 247,
"column": 4
} | [
{
"pp": "case a.add_zero\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ RingCat\ns : Cocone F\nx✝ y x : Prequotient F\n⊢ descFunLift F s (x.add zero) = descFunLift F s x",
"ppTerm": "?a.add_zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddMonoid.toAddZeroClass",... | [] | dsimp; rw [add_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.Ring.Colimits | {
"line": 246,
"column": 20
} | {
"line": 246,
"column": 40
} | {
"line": 247,
"column": 4
} | [
{
"pp": "case a.add_zero\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ RingCat\ns : Cocone F\nx✝ y x : Prequotient F\n⊢ descFunLift F s (x.add zero) = descFunLift F s x",
"ppTerm": "?a.add_zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddMonoid.toAddZeroClass",... | [] | dsimp; rw [add_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.Ring.Colimits | {
"line": 546,
"column": 20
} | {
"line": 546,
"column": 40
} | {
"line": 547,
"column": 4
} | [
{
"pp": "case a.add_zero\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ CommRingCat\ns : Cocone F\nx✝ y x : Prequotient F\n⊢ descFunLift F s (x.add zero) = descFunLift F s x",
"ppTerm": "?a.add_zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CommRingCat.carrier",
"congrArg",
... | [] | dsimp; rw [add_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.Ring.Colimits | {
"line": 546,
"column": 20
} | {
"line": 546,
"column": 40
} | {
"line": 547,
"column": 4
} | [
{
"pp": "case a.add_zero\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ CommRingCat\ns : Cocone F\nx✝ y x : Prequotient F\n⊢ descFunLift F s (x.add zero) = descFunLift F s x",
"ppTerm": "?a.add_zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CommRingCat.carrier",
"congrArg",
... | [] | dsimp; rw [add_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Determinant.Basic | {
"line": 259,
"column": 31
} | {
"line": 259,
"column": 45
} | {
"line": 259,
"column": 46
} | [
{
"pp": "m : Type u_1\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\ne e' : m ≃ n\nM : Matrix m m R\n⊢ ((reindex (e.trans e'.symm) (Equiv.refl m)) M).det = ↑↑(sign (e'.trans e.symm)) * M.det",
"ppTerm": "?m.78",
"ass... | [
"m : Type u_1\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\ne e' : m ≃ n\nM : Matrix m m R\n⊢ (M.submatrix ⇑(e.trans e'.symm).symm ⇑(Equiv.refl m).symm).det = ↑↑(sign (e'.trans e.symm)) * M.det"
] | reindex_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Category.Grp.FilteredColimits | {
"line": 100,
"column": 11
} | {
"line": 105,
"column": 11
} | {
"line": 107,
"column": 0
} | [
{
"pp": "J : Type v\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ GrpCat\nx : ↑(G F)\n⊢ ↑(G F)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"GrpCat.instConcreteCategoryMonoidHomCarrier",
"GrpCat.hasForgetToMonCat",
"GrpCat",
"MonoidHom.instFunLike",
... | [] | by
refine Quot.lift (colimitInvAux.{v, u} F) ?_ x
intro x y h
apply colimitInvAux_eq_of_rel
apply Types.FilteredColimit.rel_of_colimitTypeRel
exact h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.MonCat.FilteredColimits | {
"line": 118,
"column": 2
} | {
"line": 120,
"column": 68
} | {
"line": 121,
"column": 2
} | [
{
"pp": "J : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : ⟨j₁, x⟩.fst ⟶ l\ng : ⟨j₃, x'⟩.fst ⟶ l\nhfg : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map g)) x'\n⊢ colimitMulAux F ⟨j₁... | [
"J : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : ⟨j₁, x⟩.fst ⟶ l\ng : ⟨j₃, x'⟩.fst ⟶ l\nhfg : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map g)) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶... | obtain ⟨s, α, β, γ, h₁, h₂, h₃⟩ :=
IsFiltered.tulip (IsFiltered.leftToMax j₁ j₂) (IsFiltered.rightToMax j₁ j₂)
(IsFiltered.rightToMax j₃ j₂) (IsFiltered.leftToMax j₃ j₂) f g | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Category.MonCat.FilteredColimits | {
"line": 257,
"column": 10
} | {
"line": 258,
"column": 35
} | {
"line": 258,
"column": 35
} | [
{
"pp": "J : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nt : Cocone F\ni : J\nx : ↑(F.obj i)\nj : J\ny : ↑(F.obj j)\n⊢ (F ⋙ forget MonCat).descColimitType ((F ⋙ forget MonCat).coconeTypesEquiv.symm ((forget MonCat).mapCocone t))\n (M.mk F ⟨i, x⟩ * M.mk F ⟨j, y⟩) =\n (F ⋙ forg... | [
"J : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nt : Cocone F\ni : J\nx : ↑(F.obj i)\nj : J\ny : ↑(F.obj j)\n⊢ (F ⋙ forget MonCat).descColimitType ((F ⋙ forget MonCat).coconeTypesEquiv.symm ((forget MonCat).mapCocone t))\n (M.mk F\n ⟨IsFiltered.max i j,\n (ConcreteCa... | colimit_mul_mk_eq F ⟨i, x⟩ ⟨j, y⟩ (max' i j) (IsFiltered.leftToMax i j)
(IsFiltered.rightToMax i j) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Functor.Currying | {
"line": 45,
"column": 29
} | {
"line": 45,
"column": 50
} | {
"line": 45,
"column": 50
} | [
{
"pp": "case a.a\nB : Type u₁\ninst✝⁴ : Category.{v₁, u₁} B\nC : Type u₂\ninst✝³ : Category.{v₂, u₂} C\nD : Type u₃\ninst✝² : Category.{v₃, u₃} D\nE : Type u₄\ninst✝¹ : Category.{v₄, u₄} E\nH : Type u₅\ninst✝ : Category.{v₅, u₅} H\nF : C ⥤ D ⥤ E\nX✝ Y✝ Z✝ : C × D\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n| (F.map g.1).app X✝... | [
"case a.a\nB : Type u₁\ninst✝⁴ : Category.{v₁, u₁} B\nC : Type u₂\ninst✝³ : Category.{v₂, u₂} C\nD : Type u₃\ninst✝² : Category.{v₃, u₃} D\nE : Type u₄\ninst✝¹ : Category.{v₄, u₄} E\nH : Type u₅\ninst✝ : Category.{v₅, u₅} H\nF : C ⥤ D ⥤ E\nX✝ Y✝ Z✝ : C × D\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n| (F.obj Y✝.1).map f.2 ≫ (F.map ... | ← NatTrans.naturality | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Limits.FunctorCategory.Basic | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 41
} | {
"line": 175,
"column": 6
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\nJ : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\ninst✝ : HasColimitsOfShape J C\nF : J ⥤ K ⥤ C\nX Y : J\nf : X ⟶ Y\nx : K\n⊢ (F.map f).app x ≫ colimit.ι (F.flip.obj x) Y = colimit.ι (... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\nJ : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\ninst✝ : HasColimitsOfShape J C\nF : J ⥤ K ⥤ C\nX Y : J\nf : X ⟶ Y\nx : K\n⊢ (F.flip.obj x).map f ≫ colimit.ι (F.flip.obj x) Y = colimit.ι (F.flip.... | change (F.flip.obj x).map f ≫ _ = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.CategoryTheory.Monoidal.Transport | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 25
} | {
"line": 119,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategoryStruct D\nF : D ⥤ C\ninst✝ : F.Faithful\nfData : InducingFunctorData F\n⊢ F.CoreMonoidal",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategoryStruct D\nF : D ⥤ C\ninst✝ : F.Faithful\nfData : InducingFunctorData F\nthis : MonoidalCategory D := induced F fData\n⊢ F.CoreMonoidal"
] | letI := induced F fData | Mathlib.Linter.HaveILetI._aux_Mathlib_Tactic_Linter_HaveILetI___elabRules_Mathlib_Linter_HaveILetI_tacticLetI___1 | Mathlib.Linter.HaveILetI.tacticLetI__ |
Mathlib.CategoryTheory.Linear.LinearFunctor | {
"line": 104,
"column": 73
} | {
"line": 108,
"column": 76
} | {
"line": 110,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : Semiring R\nC : Type u_2\nD : Type u_3\ninst✝⁶ : Category.{v_1, u_2} C\ninst✝⁵ : Category.{v_2, u_3} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\ninst✝² : CategoryTheory.Linear R C\ninst✝¹ : CategoryTheory.Linear R D\nF G : C ⥤ D\ne : F ≅ G\ninst✝ : Linear R F\n⊢ Linear R G... | [] | by
exact
{ map_smul := fun f r => by
simp only [← NatIso.naturality_1 e (r • f), F.map_smul, Linear.smul_comp,
NatTrans.naturality, Linear.comp_smul, Iso.inv_hom_id_app_assoc] } | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Preadditive | {
"line": 202,
"column": 2
} | {
"line": 203,
"column": 82
} | {
"line": 205,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Finite J\nX : C\nf : J → C\nj : J\n⊢ biproduct.ι (fun j ↦ X ⊗ f j) j ≫ (leftDistributor X f).inv = X ◁ biproduct.ι f j",
... | [] | cases nonempty_fintype J
simp [leftDistributor_inv, Preadditive.comp_sum, biproduct.ι_π_assoc, dite_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Preadditive | {
"line": 202,
"column": 2
} | {
"line": 203,
"column": 82
} | {
"line": 205,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Finite J\nX : C\nf : J → C\nj : J\n⊢ biproduct.ι (fun j ↦ X ⊗ f j) j ≫ (leftDistributor X f).inv = X ◁ biproduct.ι f j",
... | [] | cases nonempty_fintype J
simp [leftDistributor_inv, Preadditive.comp_sum, biproduct.ι_π_assoc, dite_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.CategoryTheory.Monoidal.Datatypes | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 13
} | {
"line": 128,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf g h : C\nη : g ⟶ h\nη' : g ≅ h\nih_η : η'.hom = η\n⊢ (f ◁ᵢ η').hom = f ◁ η",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheory.Monoida... | [] | simp [ih_η] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Tactic.CategoryTheory.Monoidal.Datatypes | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 13
} | {
"line": 128,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf g h : C\nη : g ⟶ h\nη' : g ≅ h\nih_η : η'.hom = η\n⊢ (f ◁ᵢ η').hom = f ◁ η",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheory.Monoida... | [] | simp [ih_η] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.CategoryTheory.Monoidal.Datatypes | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 13
} | {
"line": 128,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf g h : C\nη : g ⟶ h\nη' : g ≅ h\nih_η : η'.hom = η\n⊢ (f ◁ᵢ η').hom = f ◁ η",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheory.Monoida... | [] | simp [ih_η] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.CategoryTheory.Monoidal.Datatypes | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 13
} | {
"line": 133,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf g h : C\nη : f ⟶ g\nη' : f ≅ g\nih_η : η'.hom = η\n⊢ (η' ▷ᵢ h).hom = η ▷ h",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | [] | simp [ih_η] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Tactic.CategoryTheory.Monoidal.Datatypes | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 13
} | {
"line": 133,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf g h : C\nη : f ⟶ g\nη' : f ≅ g\nih_η : η'.hom = η\n⊢ (η' ▷ᵢ h).hom = η ▷ h",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | [] | simp [ih_η] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.CategoryTheory.Monoidal.Datatypes | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 13
} | {
"line": 133,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf g h : C\nη : f ⟶ g\nη' : f ≅ g\nih_η : η'.hom = η\n⊢ (η' ▷ᵢ h).hom = η ▷ h",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | [] | simp [ih_η] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.CategoryTheory.Monoidal.Normalize | {
"line": 52,
"column": 17
} | {
"line": 53,
"column": 23
} | {
"line": 55,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nf g h : C\nη η' : f ⟶ g\nθ θ' : g ⟶ h\nι : f ⟶ h\ne_η : η = η'\ne_θ : θ = θ'\ne_ηθ : η' ≫ θ' = ι\n⊢ η ≫ θ = ι",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg"... | [] | by
simp [e_η, e_θ, e_ηθ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 880,
"column": 2
} | {
"line": 880,
"column": 28
} | {
"line": 881,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\ninst✝⁶ : MonoidalCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\nE : Type u₃\ninst✝³ : Category.{v₃, u₃} E\ninst✝² : MonoidalCategory E\nF : C ⥤ D\nG : C ⥤ E\ninst✝¹ : F.OplaxMonoidal\ninst✝ : G.OplaxMonoidal\n⊢ (η (F.prod' ... | [
"C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\ninst✝⁶ : MonoidalCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\nE : Type u₃\ninst✝³ : Category.{v₃, u₃} E\ninst✝² : MonoidalCategory E\nF : C ⥤ D\nG : C ⥤ E\ninst✝¹ : F.OplaxMonoidal\ninst✝ : G.OplaxMonoidal\n⊢ G.map (𝟙 ((diag C).obj ... | change G.map (𝟙 _) ≫ _ = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 892,
"column": 2
} | {
"line": 892,
"column": 28
} | {
"line": 893,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\ninst✝⁶ : MonoidalCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\nE : Type u₃\ninst✝³ : Category.{v₃, u₃} E\ninst✝² : MonoidalCategory E\nF : C ⥤ D\nG : C ⥤ E\ninst✝¹ : F.OplaxMonoidal\ninst✝ : G.OplaxMonoidal\nX Y : C\n⊢ (δ ... | [
"C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\ninst✝⁶ : MonoidalCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\nE : Type u₃\ninst✝³ : Category.{v₃, u₃} E\ninst✝² : MonoidalCategory E\nF : C ⥤ D\nG : C ⥤ E\ninst✝¹ : F.OplaxMonoidal\ninst✝ : G.OplaxMonoidal\nX Y : C\n⊢ G.map (𝟙 ((dia... | change G.map (𝟙 _) ≫ _ = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.CategoryTheory.Limits.Opposites | {
"line": 336,
"column": 88
} | {
"line": 337,
"column": 33
} | {
"line": 339,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\nF : Jᵒᵖ ⥤ C\ninst✝ : HasColimit F\nj : J\n⊢ (limitRightOpIsoOpColimit F).inv ≫ limit.π F.rightOp j = (colimit.ι F (op j)).op",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"CategoryTheory.... | [] | by
simp [limitRightOpIsoOpColimit] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 973,
"column": 58
} | {
"line": 973,
"column": 81
} | {
"line": 973,
"column": 81
} | [
{
"pp": "C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\ninst✝⁶ : MonoidalCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\nE : Type u₃\ninst✝³ : Category.{v₃, u₃} E\ninst✝² : MonoidalCategory E\nC' : Type u₁'\ninst✝¹ : Category.{v₁', u₁'} C'\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst... | [
"C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\ninst✝⁶ : MonoidalCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\nE : Type u₃\ninst✝³ : Category.{v₃, u₃} E\ninst✝² : MonoidalCategory E\nC' : Type u₁'\ninst✝¹ : Category.{v₁', u₁'} C'\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝ : F.OplaxM... | ← δ_natural_right_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.MorphismProperty.Concrete | {
"line": 74,
"column": 4
} | {
"line": 76,
"column": 9
} | {
"line": 77,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type u_2\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝ : ConcreteCategory C FC\nX : C\n⊢ MorphismProperty.bijective C (𝟙 X)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | delta MorphismProperty.bijective
convert! bijective_id
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.Concrete | {
"line": 74,
"column": 4
} | {
"line": 76,
"column": 9
} | {
"line": 77,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type u_2\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝ : ConcreteCategory C FC\nX : C\n⊢ MorphismProperty.bijective C (𝟙 X)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | delta MorphismProperty.bijective
convert! bijective_id
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Braided.Basic | {
"line": 641,
"column": 2
} | {
"line": 641,
"column": 42
} | {
"line": 642,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C\nf₁ : X₁ ⟶ Y₁\nf₂ : X₂ ⟶ Y₂\ng₁ : U₁ ⟶ V₁\ng₂ : U₂ ⟶ V₂\n⊢ ((f₁ ⊗ₘ f₂) ⊗ₘ g₁ ⊗ₘ g₂) ≫\n (α_ Y₁ Y₂ (V₁ ⊗ V₂)).hom ≫\n Y₁ ◁ (α_ Y₂ V₁ V₂).inv ≫ Y₁ ◁ (β_ Y₂ V₁).hom ▷ ... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C\nf₁ : X₁ ⟶ Y₁\nf₂ : X₂ ⟶ Y₂\ng₁ : U₁ ⟶ V₁\ng₂ : U₂ ⟶ V₂\n⊢ ((f₁ ⊗ₘ f₂) ⊗ₘ g₁ ⊗ₘ g₂) ≫\n (α_ Y₁ Y₂ (V₁ ⊗ V₂)).hom ≫\n (𝟙 Y₁ ⊗ₘ (α_ Y₂ V₁ V₂).inv) ≫\n (𝟙 Y₁ ⊗ₘ (β_ Y₂ ... | simp_rw [← id_tensorHom, ← tensorHom_id] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 51
} | {
"line": 184,
"column": 2
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nC : Type u₃\nD : Type u₄\nE : Type u₅\nF : Type u₆\ninst✝⁵ : Category.{v₁, u₁} A\ninst✝⁴ : Category.{v₂, u₂} B\ninst✝³ : Category.{v₃, u₃} C\ninst✝² : Category.{v₄, u₄} D\ninst✝¹ : Category.{v₅, u₅} E\ninst✝ : Category.{v₆, u₆} F\nG₁ : A ⥤ C\nG₂ : C ⥤ E\nH₁ : B ⥤ D\nH₂ : D ⥤ F... | [
"A : Type u₁\nB : Type u₂\nC : Type u₃\nD : Type u₄\nE : Type u₅\nF : Type u₆\ninst✝⁵ : Category.{v₁, u₁} A\ninst✝⁴ : Category.{v₂, u₂} B\ninst✝³ : Category.{v₃, u₃} C\ninst✝² : Category.{v₄, u₄} D\ninst✝¹ : Category.{v₅, u₅} E\ninst✝ : Category.{v₆, u₆} F\nG₁ : A ⥤ C\nG₂ : C ⥤ E\nH₁ : B ⥤ D\nH₂ : D ⥤ F\nL₁ : A ⥤ B... | rw [(L₂ ⋙ H₂).map_comp, R₃.map_comp, R₃.map_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 369,
"column": 27
} | {
"line": 369,
"column": 55
} | {
"line": 369,
"column": 55
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nα : R₁ ⟶ R₂\nβ : R₂ ⟶ R₃\n⊢ (conjugateEquiv adj₁ adj₃) ((conjugateEquiv adj₂ adj₃).symm β ≫ (conjugateEquiv adj₁ adj₂).symm α) = α ≫ β... | [
"C : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nα : R₁ ⟶ R₂\nβ : R₂ ⟶ R₃\n⊢ (conjugateEquiv adj₁ adj₂) ((conjugateEquiv adj₁ adj₂).symm α) ≫\n (conjugateEquiv adj₂ adj₃) ((conjugateEquiv ad... | ← conjugateEquiv_comp _ adj₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 1064,
"column": 2
} | {
"line": 1065,
"column": 37
} | {
"line": 1066,
"column": 2
} | [
{
"pp": "R₁ R₂ : Type u₁\ninst✝¹ : CommRing R₁\ninst✝ : CommRing R₂\nf₁₂ : R₁ →+* R₂\nM : ModuleCat R₁\nm : ↑M\n⊢ (Hom.hom ((extendScalarsId R₂).hom.app ((extendScalars f₁₂).obj M)))\n ((Hom.hom ((extendScalarsComp f₁₂ (RingHom.id R₂)).hom.app M)) (1 ⊗ₜ[R₁] m)) =\n 1 ⊗ₜ[R₁] m",
"ppTerm": "?m.86",
... | [
"R₁ R₂ : Type u₁\ninst✝¹ : CommRing R₁\ninst✝ : CommRing R₂\nf₁₂ : R₁ →+* R₂\nM : ModuleCat R₁\nm : ↑M\n⊢ 1 ⊗ₜ[R₁] m = 1 ⊗ₜ[R₁] m"
] | erw [extendScalarsComp_hom_app_one_tmul f₁₂ (RingHom.id R₂) M m,
extendScalarsId_hom_app_one_tmul] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.Algebra.Ring.BooleanRing | {
"line": 93,
"column": 40
} | {
"line": 93,
"column": 60
} | {
"line": 94,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝ : BooleanRing α\na b : α\n⊢ a * a + a * b + (b * a + b * b) = a + a * b + (b * a + b)",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"BooleanRing.mul_self",
"congrArg",
"Distrib.toAdd",
"BooleanRing.toRing",
"i... | [] | simp only [mul_self] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Ring.BooleanRing | {
"line": 93,
"column": 40
} | {
"line": 93,
"column": 60
} | {
"line": 94,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝ : BooleanRing α\na b : α\n⊢ a * a + a * b + (b * a + b * b) = a + a * b + (b * a + b)",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"BooleanRing.mul_self",
"congrArg",
"Distrib.toAdd",
"BooleanRing.toRing",
"i... | [] | simp only [mul_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.BooleanRing | {
"line": 93,
"column": 40
} | {
"line": 93,
"column": 60
} | {
"line": 94,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝ : BooleanRing α\na b : α\n⊢ a * a + a * b + (b * a + b * b) = a + a * b + (b * a + b)",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"BooleanRing.mul_self",
"congrArg",
"Distrib.toAdd",
"BooleanRing.toRing",
"i... | [] | simp only [mul_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Bialgebra.TensorProduct | {
"line": 155,
"column": 96
} | {
"line": 157,
"column": 44
} | {
"line": 159,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Semiring A\ninst✝³ : Bialgebra S A\ninst✝² : Algebra R A\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R S A\n⊢ (TensorProduct.rid R S A).toAlgEquiv = Algebra.TensorProduct.rid R S A",
"ppTerm": "?m.... | [] | by
ext x
exact coalgebra_rid_eq_algebra_rid_apply x | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Coalgebra.CoassocSimps | {
"line": 656,
"column": 62
} | {
"line": 656,
"column": 97
} | {
"line": 656,
"column": 97
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nM' : Type u_6\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Coalgebra R M\nf : M →ₗ[R] M'\n⊢ LinearMap.lTensor R f ∘ₗ LinearMap.rTensor M ε ∘ₗ δ = LinearMap.id ⊗ₘ f ∘ₗ λ⁻¹",
"ppTerm": "?m... | [
"R : Type u_1\nM : Type u_3\nM' : Type u_6\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Coalgebra R M\nf : M →ₗ[R] M'\n⊢ LinearMap.lTensor R f ∘ₗ (TensorProduct.mk R R M) 1 = LinearMap.id ⊗ₘ f ∘ₗ λ⁻¹"
] | Coalgebra.rTensor_counit_comp_comul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Comon_ | {
"line": 432,
"column": 19
} | {
"line": 432,
"column": 61
} | {
"line": 433,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹⁰ : Category.{v₁, u₁} C\ninst✝⁹ : MonoidalCategory C\nM N O : C\ninst✝⁸ : ComonObj M\ninst✝⁷ : ComonObj N\ninst✝⁶ : ComonObj O\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\nF : C ⥤ D\ninst✝³ : F.OplaxMonoidal\nX Y : C\ninst✝² : ComonObj X\ninst✝¹ : ComonObj... | [] | dsimp; rw [← F.map_comp_assoc, hom_counit] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Comon_ | {
"line": 432,
"column": 19
} | {
"line": 432,
"column": 61
} | {
"line": 433,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹⁰ : Category.{v₁, u₁} C\ninst✝⁹ : MonoidalCategory C\nM N O : C\ninst✝⁸ : ComonObj M\ninst✝⁷ : ComonObj N\ninst✝⁶ : ComonObj O\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\nF : C ⥤ D\ninst✝³ : F.OplaxMonoidal\nX Y : C\ninst✝² : ComonObj X\ninst✝¹ : ComonObj... | [] | dsimp; rw [← F.map_comp_assoc, hom_counit] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 272,
"column": 4
} | {
"line": 273,
"column": 18
} | {
"line": 274,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹⁴ : CommSemiring R\nA : Type u_5\ninst✝¹³ : CommSemiring A\ninst✝¹² : Algebra R A\nB : Type u_6\ninst✝¹¹ : CommSemiring B\ninst✝¹⁰ : Algebra R B\nAₚ : Type u_7\ninst✝⁹ : CommSemiring Aₚ\ninst✝⁸ : Algebra A Aₚ\ninst✝⁷ : Algebra R Aₚ\ninst✝⁶ : IsScalarTower R A Aₚ\nBₚ : Type u_8\ninst... | [] | simp only [AlgHom.toRingHom_eq_coe, Submonoid.map_powers, RingHom.coe_coe]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 272,
"column": 4
} | {
"line": 273,
"column": 18
} | {
"line": 274,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹⁴ : CommSemiring R\nA : Type u_5\ninst✝¹³ : CommSemiring A\ninst✝¹² : Algebra R A\nB : Type u_6\ninst✝¹¹ : CommSemiring B\ninst✝¹⁰ : Algebra R B\nAₚ : Type u_7\ninst✝⁹ : CommSemiring Aₚ\ninst✝⁸ : Algebra A Aₚ\ninst✝⁷ : Algebra R Aₚ\ninst✝⁶ : IsScalarTower R A Aₚ\nBₚ : Type u_8\ninst... | [] | simp only [AlgHom.toRingHom_eq_coe, Submonoid.map_powers, RingHom.coe_coe]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 285,
"column": 4
} | {
"line": 286,
"column": 67
} | {
"line": 287,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝⁸ : CommSemiring R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nT : Type u_5\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra S T\ninst✝³ : Algebra R T\ninst✝² : IsScalarTower R S T\nx y : R\ninst✝¹ : Away x S\ninst✝ : Away ((algebraMap R S) y) T\n⊢ IsUnit (... | [] | simp only [map_mul, IsUnit.mul_iff, IsScalarTower.algebraMap_apply R S T]
exact ⟨algebraMap_isUnit _, IsUnit.map _ (algebraMap_isUnit x)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 285,
"column": 4
} | {
"line": 286,
"column": 67
} | {
"line": 287,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝⁸ : CommSemiring R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nT : Type u_5\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra S T\ninst✝³ : Algebra R T\ninst✝² : IsScalarTower R S T\nx y : R\ninst✝¹ : Away x S\ninst✝ : Away ((algebraMap R S) y) T\n⊢ IsUnit (... | [] | simp only [map_mul, IsUnit.mul_iff, IsScalarTower.algebraMap_apply R S T]
exact ⟨algebraMap_isUnit _, IsUnit.map _ (algebraMap_isUnit x)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 417,
"column": 8
} | {
"line": 417,
"column": 22
} | {
"line": 417,
"column": 23
} | [
{
"pp": "R : Type u_4\nS : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\ne : R\nhe : IsIdempotentElem e\nH : RingHom.ker (algebraMap R S) = Ideal.span {1 - e}\nH' : Function.Surjective ⇑(algebraMap R S)\nx y : R\n⊢ (algebraMap R S) x = (algebraMap R S) y ↔ e * x = e * y",
"ppTerm"... | [
"R : Type u_4\nS : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\ne : R\nhe : IsIdempotentElem e\nH : RingHom.ker (algebraMap R S) = Ideal.span {1 - e}\nH' : Function.Surjective ⇑(algebraMap R S)\nx y : R\n⊢ (algebraMap R S) x - (algebraMap R S) y = 0 ↔ e * x = e * y"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 453,
"column": 8
} | {
"line": 453,
"column": 22
} | {
"line": 453,
"column": 23
} | [
{
"pp": "case refine_2\nS : Type u_4\nT : Type u_5\ninst✝² : CommRing S\ninst✝¹ : CommRing T\ninst✝ : Algebra S T\nh₁ : Function.Surjective ⇑(algebraMap S T)\nr : S\nhr : IsUnit ((algebraMap S T) r)\nn : ℕ\nhn : r ^ n • RingHom.ker (algebraMap S T) ≤ ⊥\nx y : S\nh : (algebraMap S T) x = (algebraMap S T) y\n⊢ r ... | [
"case refine_2\nS : Type u_4\nT : Type u_5\ninst✝² : CommRing S\ninst✝¹ : CommRing T\ninst✝ : Algebra S T\nh₁ : Function.Surjective ⇑(algebraMap S T)\nr : S\nhr : IsUnit ((algebraMap S T) r)\nn : ℕ\nhn : r ^ n • RingHom.ker (algebraMap S T) ≤ ⊥\nx y : S\nh : (algebraMap S T) x = (algebraMap S T) y\n⊢ r ^ n * x - r ... | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Jacobson.Ideal | {
"line": 185,
"column": 4
} | {
"line": 189,
"column": 44
} | {
"line": 190,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI : Ideal R\nf : R →+* S\nhf : Function.Surjective ⇑f\nh : RingHom.ker f ≤ I\nthis : ∀ J ∈ {J | I ≤ J ∧ J.IsMaximal}, RingHom.ker f ≤ J\n⊢ sInf (map f '' {J | I ≤ J ∧ J.IsMaximal}) ≤ sInf {J | map f I ≤ J ∧ J.IsMaximal}",
"ppTe... | [] | refine
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, ?_⟩, map_comap_of_surjective f hf J⟩⟩
have : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Jacobson.Ideal | {
"line": 185,
"column": 4
} | {
"line": 189,
"column": 44
} | {
"line": 190,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI : Ideal R\nf : R →+* S\nhf : Function.Surjective ⇑f\nh : RingHom.ker f ≤ I\nthis : ∀ J ∈ {J | I ≤ J ∧ J.IsMaximal}, RingHom.ker f ≤ J\n⊢ sInf (map f '' {J | I ≤ J ∧ J.IsMaximal}) ≤ sInf {J | map f I ≤ J ∧ J.IsMaximal}",
"ppTe... | [] | refine
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, ?_⟩, map_comap_of_surjective f hf J⟩⟩
have : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Jacobson.Ideal | {
"line": 239,
"column": 12
} | {
"line": 239,
"column": 17
} | {
"line": 240,
"column": 4
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : Ring R\ninst✝¹ : Ring S\nI✝ I : Ideal R\ninst✝ : I.IsTwoSided\nx r : R\nxJ : x ∈ I.jacobson\n𝔪 : Ideal R\n⊢ 𝔪 ∈ {J | I ≤ J ∧ J.IsMaximal} → x * r ∈ 𝔪",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Set.ofPred",
... | [
"R : Type u\nS : Type v\ninst✝² : Ring R\ninst✝¹ : Ring S\nI✝ I : Ideal R\ninst✝ : I.IsTwoSided\nx r : R\nxJ : x ∈ I.jacobson\n𝔪 : Ideal R\n𝔪_mem : 𝔪 ∈ {J | I ≤ J ∧ J.IsMaximal}\n⊢ x * r ∈ 𝔪"
] | 𝔪_mem | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Jacobson.Ideal | {
"line": 292,
"column": 36
} | {
"line": 292,
"column": 50
} | {
"line": 292,
"column": 51
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nx : R\nhx : x ∈ ⊥.jacobson\ny z : R\nhz : z * y * x + z - 1 ∈ ⊥\n⊢ x * y * z + z = 1",
"ppTerm": "?m.112",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulO... | [
"R : Type u\ninst✝ : CommRing R\nx : R\nhx : x ∈ ⊥.jacobson\ny z : R\nhz : z * y * x + z - 1 ∈ ⊥\n⊢ x * y * z + z - 1 = 0"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic | {
"line": 72,
"column": 75
} | {
"line": 74,
"column": 29
} | {
"line": 76,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsLocalRing R\nJ : Ideal R\nhJ : J ≠ ⊤\n⊢ J ≤ maximalIdeal R",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder"... | [] | by
rcases Ideal.exists_le_maximal J hJ with ⟨M, hM1, hM2⟩
rwa [← eq_maximalIdeal hM1] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.Ring.Instances | {
"line": 27,
"column": 96
} | {
"line": 29,
"column": 33
} | {
"line": 31,
"column": 0
} | [
{
"pp": "R : CommRingCat\n⊢ IsIso (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away 1)))",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"CommRing",
"CategoryTheory.IsIso",
"CommRingCat.carrier",
"Algebra.algebraMap",
"OreLocalization.instAlgebra",
... | [] | by
cases R
exact localization_unit_isIso _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 716,
"column": 4
} | {
"line": 716,
"column": 57
} | {
"line": 717,
"column": 4
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝³ : CommSemiring R\nx : R\nB : Type u_2\ninst✝² : CommSemiring B\ninst✝¹ : Algebra R B\ninst✝ : IsLocalization.Away x B\na : R\nb : B\nm d : ℤ\nh : selfZPow x B m * mk' B a 1 * selfZPow x B (-d) = b\n⊢ selfZPow x B m * mk' B a 1 = selfZPow x B d * b",
"ppTerm": "?mp",
... | [
"case mp\nR : Type u_1\ninst✝³ : CommSemiring R\nx : R\nB : Type u_2\ninst✝² : CommSemiring B\ninst✝¹ : Algebra R B\ninst✝ : IsLocalization.Away x B\na : R\nb : B\nm d : ℤ\nh : selfZPow x B m * mk' B a 1 * selfZPow x B (-d) = b\nthis : selfZPow x B m * mk' B a 1 * selfZPow x B (-d) * selfZPow x B d = b * selfZPow x... | have := congr_arg (fun s : B => s * selfZPow x B d) h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic | {
"line": 92,
"column": 38
} | {
"line": 92,
"column": 47
} | {
"line": 92,
"column": 47
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\nh : IsPullback fst snd f g\nP' X' Y' Z' : C\nfst' : P' ⟶ X'\nsnd' : P' ⟶ Y'\nf' : X' ⟶ Z'\ng' : Y' ⟶ Z'\ne₁ : P ≅ P'\ne₂ : X ≅ X'\ne₃ : Y ≅ Y'\ne₄ : Z ≅ Z'\ncommfst : fst ≫ e₂.hom = e₁.hom ≫ fst'\ncom... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\nh : IsPullback fst snd f g\nP' X' Y' Z' : C\nfst' : P' ⟶ X'\nsnd' : P' ⟶ Y'\nf' : X' ⟶ Z'\ng' : Y' ⟶ Z'\ne₁ : P ≅ P'\ne₂ : X ≅ X'\ne₃ : Y ≅ Y'\ne₄ : Z ≅ Z'\ncommfst : fst ≫ e₂.hom = e₁.hom ≫ fst'\ncommsnd : snd ≫... | h.w_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic | {
"line": 474,
"column": 38
} | {
"line": 474,
"column": 47
} | {
"line": 474,
"column": 47
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nh : IsPushout f g inl inr\nZ' X' Y' P' : C\nf' : Z' ⟶ X'\ng' : Z' ⟶ Y'\ninl' : X' ⟶ P'\ninr' : Y' ⟶ P'\ne₁ : Z ≅ Z'\ne₂ : X ≅ X'\ne₃ : Y ≅ Y'\ne₄ : P ≅ P'\ncommf : f ≫ e₂.hom = e₁.hom ≫ f'\ncommg : g ... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nh : IsPushout f g inl inr\nZ' X' Y' P' : C\nf' : Z' ⟶ X'\ng' : Z' ⟶ Y'\ninl' : X' ⟶ P'\ninr' : Y' ⟶ P'\ne₁ : Z ≅ Z'\ne₂ : X ≅ X'\ne₃ : Y ≅ Y'\ne₄ : P ≅ P'\ncommf : f ≫ e₂.hom = e₁.hom ≫ f'\ncommg : g ≫ e₃.hom = e... | h.w_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 246,
"column": 6
} | {
"line": 246,
"column": 30
} | {
"line": 247,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝¹⁰ : CommSemiring R\nS✝ : Submonoid R\nM : Type v\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nT : Type u_1\ninst✝⁷ : CommSemiring T\ninst✝⁶ : Algebra R T\ninst✝⁵ : IsLocalization S✝ T\nT' : Type u_2\ninst✝⁴ : CommSemiring T'\ninst✝³ : Algebra R T'\ninst✝² : IsLocalization S✝ T'\nA ... | [
"case mk.mk\nR : Type u\ninst✝¹⁰ : CommSemiring R\nS✝ : Submonoid R\nM : Type v\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nT : Type u_1\ninst✝⁷ : CommSemiring T\ninst✝⁶ : Algebra R T\ninst✝⁵ : IsLocalization S✝ T\nT' : Type u_2\ninst✝⁴ : CommSemiring T'\ninst✝³ : Algebra R T'\ninst✝² : IsLocalization S✝ T'\nA ... | rintro ⟨a₁, s₁⟩ ⟨a₂, s₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic | {
"line": 868,
"column": 2
} | {
"line": 868,
"column": 39
} | {
"line": 869,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\nh : IsPullback fst snd f g\n⊢ PreservesLimit (cospan f g) F ↔ IsPullback (F.map fst) (F.map snd) (F.map f) (F.map g)",
"ppTerm": "?m.66",
... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\nh : IsPullback fst snd f g\nhF : IsPullback (F.map fst) (F.map snd) (F.map f) (F.map g)\n⊢ PreservesLimit (cospan f g) F"
] | refine ⟨fun _ ↦ h.map _, fun hF ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic | {
"line": 876,
"column": 2
} | {
"line": 876,
"column": 39
} | {
"line": 877,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nP X Y Z : C\ninl : X ⟶ P\ninr : Y ⟶ P\nf : Z ⟶ X\ng : Z ⟶ Y\nh : IsPushout f g inl inr\n⊢ PreservesColimit (span f g) F ↔ IsPushout (F.map f) (F.map g) (F.map inl) (F.map inr)",
"ppTerm": "?m.66",
"a... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nP X Y Z : C\ninl : X ⟶ P\ninr : Y ⟶ P\nf : Z ⟶ X\ng : Z ⟶ Y\nh : IsPushout f g inl inr\nhF : IsPushout (F.map f) (F.map g) (F.map inl) (F.map inr)\n⊢ PreservesColimit (span f g) F"
] | refine ⟨fun _ ↦ h.map _, fun hF ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Monad.Basic | {
"line": 313,
"column": 4
} | {
"line": 314,
"column": 27
} | {
"line": 315,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nT✝ : Monad C\nG : Comonad C\nF : C ⥤ C\nT : Monad C\ni : T.toFunctor ≅ F\nX : C\n⊢ (T.η ≫ i.hom).app (F.obj X) ≫ ((i.inv ◫ i.inv) ≫ T.μ ≫ i.hom).app X = 𝟙 ((𝟭 C).obj (F.obj X))",
"ppTerm": "?m.212",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nT✝ : Monad C\nG : Comonad C\nF : C ⥤ C\nT : Monad C\ni : T.toFunctor ≅ F\nX : C\n⊢ T.η.app (F.obj X) ≫ T.map (i.inv.app X) ≫ T.μ.app X ≫ i.hom.app X = 𝟙 (F.obj X)"
] | simp only [Functor.id_obj, NatTrans.comp_app, comp_obj, NatTrans.hcomp_app, Category.assoc,
hom_inv_id_app_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 1218,
"column": 6
} | {
"line": 1218,
"column": 20
} | {
"line": 1218,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommSemiring R\nS : Submonoid R\nM : Type u_6\nM' : Type u_7\nN : Type u_8\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\ninst✝² : IsLocalizedModule S f\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\ng : M' →ₗ[R] N\nH... | [
"R : Type u_1\ninst✝⁷ : CommSemiring R\nS : Submonoid R\nM : Type u_6\nM' : Type u_7\nN : Type u_8\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\ninst✝² : IsLocalizedModule S f\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\ng : M' →ₗ[R] N\nH : ∀ (m : M)... | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 1352,
"column": 8
} | {
"line": 1352,
"column": 22
} | {
"line": 1352,
"column": 23
} | [
{
"pp": "case exists_of_eq\nR : Type u_6\nS : Type u_7\nS' : Type u_8\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring S\ninst✝² : Ring S'\ninst✝¹ : Algebra R S\ninst✝ : Algebra R S'\nM : Submonoid R\nf : S →ₐ[R] S'\nh₁ : ∀ x ∈ M, IsUnit ((algebraMap R S') x)\nh₂ : ∀ (y : S'), ∃ x, x.2 • y = f x.1\nh₃ : ∀ (x : S), f x =... | [
"case exists_of_eq\nR : Type u_6\nS : Type u_7\nS' : Type u_8\ninst✝⁴ : CommSemiring R\ninst✝³ : Ring S\ninst✝² : Ring S'\ninst✝¹ : Algebra R S\ninst✝ : Algebra R S'\nM : Submonoid R\nf : S →ₐ[R] S'\nh₁ : ∀ x ∈ M, IsUnit ((algebraMap R S') x)\nh₂ : ∀ (y : S'), ∃ x, x.2 • y = f x.1\nh₃ : ∀ (x : S), f x = 0 ↔ ∃ m, m ... | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.WithTerminal.Cone | {
"line": 99,
"column": 26
} | {
"line": 99,
"column": 70
} | {
"line": 99,
"column": 71
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nJ : Type w\ninst✝ : Category.{w', w} J\nX : C\nK : J ⥤ Over X\nF : C ⥤ D\nt✝ : Cone K\nt : Cone (liftFromOver.obj K)\nx✝¹ x✝ : J\nf : x✝¹ ⟶ x✝\n⊢ ((Functor.const J).obj (Over.mk (t.π.app star))).map f ≫ Over.homMk (t.... | [] | by ext; simpa using! (t.w (incl.map f)).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Types.Products | {
"line": 178,
"column": 2
} | {
"line": 179,
"column": 46
} | {
"line": 181,
"column": 0
} | [
{
"pp": "case refine_2\nX✝ Y✝ : Type u\nx✝ : X✝ ⟶ Y✝\n⊢ binaryProductFunctor.map x✝ ≫\n (NatIso.ofComponents (fun Y ↦ ((limit.isLimit (pair Y✝ Y)).conePointUniqueUpToIso (binaryProductLimit Y✝ Y)).symm)\n ⋯).hom =\n (NatIso.ofComponents (fun Y ↦ ((limit.isLimit (pair X✝ Y)).conePointUniqueUpToI... | [] | · ext : 2
apply Limits.prod.hom_ext <;> simp <;> rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.FinitePresentation | {
"line": 217,
"column": 2
} | {
"line": 220,
"column": 37
} | {
"line": 221,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : IsScalarTower R A B\ninst✝¹ : FinitePresentation R B\ninst✝ : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R... | [
"case refine_2\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : IsScalarTower R A B\ninst✝¹ : FinitePresentation R B\ninst✝ : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] B\nhf... | · rw [← AlgHom.range_eq_top, ← Algebra.adjoin_range_eq_range_aeval,
Set.range_comp f MvPolynomial.X, eq_top_iff, ← @adjoin_adjoin_of_tower R A B,
adjoin_image, adjoin_range_X, Algebra.map_top, (AlgHom.range_eq_top _).mpr hf]
exact fun {x} => subset_adjoin ⟨⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.MorphismProperty.Comma | {
"line": 573,
"column": 4
} | {
"line": 573,
"column": 12
} | {
"line": 574,
"column": 2
} | [
{
"pp": "case h₁\nT : Type u_1\ninst✝² : Category.{v_1, u_1} T\nP Q W : MorphismProperty T\ninst✝¹ : Q.IsMultiplicative\ninst✝ : W.IsMultiplicative\nA B : P.Arrow Q W\nf g : A ⟶ B\nhl : f.left = g.left\nhr : f.right = g.right\n⊢ (Comma.Hom.hom f).left = (Comma.Hom.hom g).left",
"ppTerm": "?h₁",
"assigne... | [] | exact hl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.MorphismProperty.Comma | {
"line": 573,
"column": 4
} | {
"line": 573,
"column": 12
} | {
"line": 574,
"column": 2
} | [
{
"pp": "case h₁\nT : Type u_1\ninst✝² : Category.{v_1, u_1} T\nP Q W : MorphismProperty T\ninst✝¹ : Q.IsMultiplicative\ninst✝ : W.IsMultiplicative\nA B : P.Arrow Q W\nf g : A ⟶ B\nhl : f.left = g.left\nhr : f.right = g.right\n⊢ (Comma.Hom.hom f).left = (Comma.Hom.hom g).left",
"ppTerm": "?h₁",
"assigne... | [] | exact hl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.Comma | {
"line": 573,
"column": 4
} | {
"line": 573,
"column": 12
} | {
"line": 574,
"column": 2
} | [
{
"pp": "case h₁\nT : Type u_1\ninst✝² : Category.{v_1, u_1} T\nP Q W : MorphismProperty T\ninst✝¹ : Q.IsMultiplicative\ninst✝ : W.IsMultiplicative\nA B : P.Arrow Q W\nf g : A ⟶ B\nhl : f.left = g.left\nhr : f.right = g.right\n⊢ (Comma.Hom.hom f).left = (Comma.Hom.hom g).left",
"ppTerm": "?h₁",
"assigne... | [] | exact hl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FinitePresentation | {
"line": 244,
"column": 65
} | {
"line": 244,
"column": 79
} | {
"line": 244,
"column": 79
} | [
{
"pp": "case inl\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : IsScalarTower R A B\ninst✝¹ : FinitePresentation R B\ninst✝ : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R... | [
"case inl\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : IsScalarTower R A B\ninst✝¹ : FinitePresentation R B\ninst✝ : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] B\nhf : Su... | ← aeval_unique | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FinitePresentation | {
"line": 265,
"column": 12
} | {
"line": 265,
"column": 37
} | {
"line": 266,
"column": 12
} | [
{
"pp": "case hy\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : IsScalarTower R A B\ninst✝¹ : FinitePresentation R B\ninst✝ : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R]... | [
"case hy\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : IsScalarTower R A B\ninst✝¹ : FinitePresentation R B\ninst✝ : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] B\nhf : Sur... | apply Set.mem_union_right | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Grothendieck | {
"line": 259,
"column": 4
} | {
"line": 260,
"column": 72
} | {
"line": 261,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u₁\ninst✝ : Category.{v₁, u₁} D\nF G : C ⥤ Cat\nα : F ⟶ G\nX Y Z : Grothendieck F\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ eqToHom ⋯ ≫\n { base := (f ≫ g).base,\n fiber := (eqToHom ⋯).toNatTrans.app X.fiber ≫ (α.app Z.base).toFunctor.map (f ≫ g).fiber }.fi... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u₁\ninst✝ : Category.{v₁, u₁} D\nF G : C ⥤ Cat\nα : F ⟶ G\nX Y Z : Grothendieck F\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ eqToHom ⋯ ≫\n (eqToHom ⋯).toNatTrans.app X.fiber ≫\n (α.app Z.base).toFunctor.map (eqToHom ⋯) ≫\n (eqToHom ⋯ ≫ (α.app Y.base ≫ G.map g.bas... | simp only [comp_fiber, map_comp, ← Cat.Hom.comp_map,
Functor.congr_hom congr($(α.naturality g.base).toFunctor) f.fiber] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Final | {
"line": 462,
"column": 4
} | {
"line": 462,
"column": 22
} | {
"line": 463,
"column": 4
} | [
{
"pp": "case rel\nC : Type v\ninst✝¹ : Category.{v, v} C\nD : Type u₁\ninst✝ : Category.{v, u₁} D\nF : C ⥤ D\nd : D\nf₁ f₂ x y : (X : C) × (d ⟶ F.obj X)\nr : (F ⋙ coyoneda.obj (op d)).ColimitTypeRel x y\n⊢ Zigzag (StructuredArrow.mk x.snd) (StructuredArrow.mk y.snd)",
"ppTerm": "?rel",
"assigned": true... | [
"case rel\nC : Type v\ninst✝¹ : Category.{v, v} C\nD : Type u₁\ninst✝ : Category.{v, u₁} D\nF : C ⥤ D\nd : D\nf₁ f₂ x y : (X : C) × (d ⟶ F.obj X)\nf : x.fst ⟶ y.fst\nw : y.snd = (ConcreteCategory.hom ((F ⋙ coyoneda.obj (op d)).map f)) x.snd\n⊢ Zigzag (StructuredArrow.mk x.snd) (StructuredArrow.mk y.snd)"
] | obtain ⟨f, w⟩ := r | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.FinitePresentation | {
"line": 319,
"column": 8
} | {
"line": 319,
"column": 22
} | {
"line": 319,
"column": 22
} | [
{
"pp": "R : Type w₁\nA : Type w₂\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ⇑f\ninst✝ : FinitePresentation R A\nm : ℕ\nf' : MvPolynomial (Fin m) R →ₐ[R] A\nhf' : Surjective ⇑f'\ns : Finset (MvPolynomial (Fin m) R)\nhs : Ideal.span... | [
"R : Type w₁\nA : Type w₂\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ⇑f\ninst✝ : FinitePresentation R A\nm : ℕ\nf' : MvPolynomial (Fin m) R →ₐ[R] A\nhf' : Surjective ⇑f'\ns : Finset (MvPolynomial (Fin m) R)\nhs : Ideal.span ↑s = RingHo... | ← aeval_unique | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Equalizers | {
"line": 205,
"column": 82
} | {
"line": 206,
"column": 33
} | {
"line": 208,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y : C\nf g : X ⟶ Y\nc : Cofork f g\n⊢ c.op.unop.π = c.π",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Limits.WalkingParallelPai... | [] | by
simp [Fork.unop_π, Cofork.op_ι] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Equalizers | {
"line": 231,
"column": 82
} | {
"line": 232,
"column": 33
} | {
"line": 234,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y : Cᵒᵖ\nf g : X ⟶ Y\nc : Fork f g\n⊢ c.unop.op.ι = c.ι",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Opposite",
"Quiver.opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.... | [] | by
simp [Fork.unop_π, Cofork.op_ι] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.FinitePresentation | {
"line": 367,
"column": 4
} | {
"line": 367,
"column": 29
} | {
"line": 368,
"column": 4
} | [
{
"pp": "R : Type w₁\nA : Type w₂\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ⇑f\ninst✝ : FinitePresentation R A\nm : ℕ\nf' : MvPolynomial (Fin m) R →ₐ[R] A\nhf' : Surjective ⇑f'\ns : Finset (MvPolynomial (Fin m) R)\nhs : Ideal.span... | [
"R : Type w₁\nA : Type w₂\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ⇑f\ninst✝ : FinitePresentation R A\nm : ℕ\nf' : MvPolynomial (Fin m) R →ₐ[R] A\nhf' : Surjective ⇑f'\ns : Finset (MvPolynomial (Fin m) R)\nhs : Ideal.span ↑s = RingHo... | apply Set.mem_union_right | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.RingHomProperties | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 14
} | {
"line": 146,
"column": 2
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nh₁ : RespectsIso P\nh₂ :\n ∀ ⦃R S T : Type u⦄ [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n [inst_4 : Algebra R T], P (algebraMap R T) → P (algebraMap S (S ⊗[R] T))\n⊢ IsS... | [
"P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nh₁ : RespectsIso P\nh₂ :\n ∀ ⦃R S T : Type u⦄ [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n [inst_4 : Algebra R T], P (algebraMap R T) → P (algebraMap S (S ⊗[R] T))\nR S R' S' : Type ... | introv R h H | Mathlib.Tactic.evalIntrov | Mathlib.Tactic.introv |
Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts | {
"line": 202,
"column": 4
} | {
"line": 211,
"column": 32
} | {
"line": 213,
"column": 0
} | [
{
"pp": "J : Type v\ninst✝² : SmallCategory J\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nn : ℕ\nf : Fin (n + 1) → C\nc₁ : Cofan fun i ↦ f i.succ\nc₂ : BinaryCofan (f 0) c₁.pt\nt₁ : IsColimit c₁\nt₂ : IsColimit c₂\ns : Cocone (Discrete.functor f)\nm : (extendCofan c₁ c₂).p... | [] | apply BinaryCofan.IsColimit.hom_ext t₂
· rw [(BinaryCofan.IsColimit.desc' t₂ _ _).2.1]
apply w ⟨0⟩
· rw [(BinaryCofan.IsColimit.desc' t₂ _ _).2.2]
apply t₁.uniq ⟨_, _⟩
rintro ⟨j⟩
dsimp only [Discrete.natTrans_app]
rw [← w ⟨j.succ⟩]
dsimp only [extendCofan_ι_app]
rw [Fin... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts | {
"line": 202,
"column": 4
} | {
"line": 211,
"column": 32
} | {
"line": 213,
"column": 0
} | [
{
"pp": "J : Type v\ninst✝² : SmallCategory J\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nn : ℕ\nf : Fin (n + 1) → C\nc₁ : Cofan fun i ↦ f i.succ\nc₂ : BinaryCofan (f 0) c₁.pt\nt₁ : IsColimit c₁\nt₂ : IsColimit c₂\ns : Cocone (Discrete.functor f)\nm : (extendCofan c₁ c₂).p... | [] | apply BinaryCofan.IsColimit.hom_ext t₂
· rw [(BinaryCofan.IsColimit.desc' t₂ _ _).2.1]
apply w ⟨0⟩
· rw [(BinaryCofan.IsColimit.desc' t₂ _ _).2.2]
apply t₁.uniq ⟨_, _⟩
rintro ⟨j⟩
dsimp only [Discrete.natTrans_app]
rw [← w ⟨j.succ⟩]
dsimp only [extendCofan_ι_app]
rw [Fin... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 61
} | {
"line": 620,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nW : MorphismProperty C\nJ : Type u_1\ninst✝¹ : Category.{v_1, u_1} J\ninst✝ : W.IsStableUnderColimitsOfShape J\n⊢ W.colimitsOfShape J ≤ W",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty"... | [] | rwa [← isStableUnderColimitsOfShape_iff_colimitsOfShape_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 61
} | {
"line": 620,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nW : MorphismProperty C\nJ : Type u_1\ninst✝¹ : Category.{v_1, u_1} J\ninst✝ : W.IsStableUnderColimitsOfShape J\n⊢ W.colimitsOfShape J ≤ W",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty"... | [] | rwa [← isStableUnderColimitsOfShape_iff_colimitsOfShape_le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 61
} | {
"line": 620,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nW : MorphismProperty C\nJ : Type u_1\ninst✝¹ : Category.{v_1, u_1} J\ninst✝ : W.IsStableUnderColimitsOfShape J\n⊢ W.colimitsOfShape J ≤ W",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty"... | [] | rwa [← isStableUnderColimitsOfShape_iff_colimitsOfShape_le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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