module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.SetTheory.Cardinal.Order | {
"line": 309,
"column": 6
} | {
"line": 309,
"column": 62
} | {
"line": 309,
"column": 63
} | [
{
"pp": "α✝ β✝ : Type u\na b : Cardinal.{u}\nα β : Type u\n⊢ #α * #β = 0 → #α = 0 ∨ #β = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Cardinal",
"congrArg",
"Cardinal.mk",
"_private.Mathlib.SetTheory.Cardinal.Order.0.Cardinal... | [
"α✝ β✝ : Type u\na b : Cardinal.{u}\nα β : Type u\n⊢ IsEmpty α ∨ IsEmpty β → IsEmpty α ∨ IsEmpty β"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Order | {
"line": 330,
"column": 80
} | {
"line": 334,
"column": 28
} | {
"line": 336,
"column": 0
} | [
{
"pp": "a b : Cardinal.{u_1}\nhb : 1 ≤ b\n⊢ a ≤ a ^ b",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"zero_le",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Cardinal.instOne",
"Cardinal.instPowCardinal",
"Cardinal",
"CommSemiring.toSemir... | [] | by
rcases eq_or_ne a 0 with (rfl | ha)
· exact zero_le
· convert! power_le_power_left ha hb
exact (power_one a).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Order | {
"line": 370,
"column": 6
} | {
"line": 370,
"column": 31
} | {
"line": 370,
"column": 32
} | [
{
"pp": "a : Cardinal.{u}\nh : ¬Acc (fun x1 x2 ↦ x1 < x2) a\nι : Type (max 0 (u + 1)) := { c // ¬Acc (fun x1 x2 ↦ x1 < x2) c }\nf : ι → Cardinal.{u} := Subtype.val\nhι : Nonempty ι\nc : Cardinal.{u}\nhc : ¬Acc (fun x1 x2 ↦ x1 < x2) c\nh_1 : (j : ι) → Quotient.out (f ⟨c, hc⟩) ↪ Quotient.out (f j)\nj : Cardinal.{... | [
"a : Cardinal.{u}\nh : ¬Acc (fun x1 x2 ↦ x1 < x2) a\nι : Type (max 0 (u + 1)) := { c // ¬Acc (fun x1 x2 ↦ x1 < x2) c }\nf : ι → Cardinal.{u} := Subtype.val\nhι : Nonempty ι\nc : Cardinal.{u}\nhc : ¬Acc (fun x1 x2 ↦ x1 < x2) c\nh_1 : (j : ι) → Quotient.out (f ⟨c, hc⟩) ↪ Quotient.out (f j)\nj : Cardinal.{u}\nh' : j <... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 422,
"column": 56
} | {
"line": 422,
"column": 79
} | {
"line": 422,
"column": 79
} | [
{
"pp": "α : Type u_2\nγ : Type u_4\nβ : α → Type u_6\ninst✝¹ : (x : α) → OmegaCompletePartialOrder (β x)\ninst✝ : OmegaCompletePartialOrder γ\nf : γ → (x : α) → β x\nhf : ∀ (a : α), ωScottContinuous fun x ↦ f x a\nc : Chain γ\na : α\n⊢ f (ωSup c) a = ωSup (c.map { toFun := f, monotone' := ⋯ }) a",
"ppTerm"... | [] | apply (hf a).map_ωSup c | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.SetTheory.Cardinal.Order | {
"line": 478,
"column": 48
} | {
"line": 478,
"column": 59
} | {
"line": 478,
"column": 60
} | [
{
"pp": "ι : Type u\nf : ι → Cardinal.{v}\ni : ι\na b : ULift.{u, v} (Quotient.out (f i))\nh : (fun a ↦ ⟨i, a.down⟩) a = (fun a ↦ ⟨i, a.down⟩) b\n⊢ a = b",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u\nf : ι → Cardinal.{v}\ni : ι\na b : ULift.{u, v} (Quotient.out (f i))\nh : (fun a ↦ ⟨i, a.down⟩) a = (fun a ↦ ⟨i, a.down⟩) b\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Order | {
"line": 481,
"column": 2
} | {
"line": 481,
"column": 27
} | {
"line": 481,
"column": 28
} | [
{
"pp": "ι : Type u\nf : ι → Cardinal.{max u v}\ni : ι\n⊢ f i ≤ sum f",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u\nf : ι → Cardinal.{max u v}\ni : ι\n⊢ f i ≤ sum f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.SchroederBernstein | {
"line": 125,
"column": 10
} | {
"line": 125,
"column": 34
} | {
"line": 125,
"column": 35
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\nI : Nonempty ι\ns : Set ((i : ι) → β i)\nhs : Maximal (fun x ↦ x ∈ sets β) s\nh : ¬∃ i, Surjective fun x ↦ ↑x i\n⊢ ∀ (i : ι), ∃ y, ∀ x ∈ s, x i ≠ y",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Exists",
"id",
... | [
"ι : Type u\nβ : ι → Type v\nI : Nonempty ι\ns : Set ((i : ι) → β i)\nhs : Maximal (fun x ↦ x ∈ sets β) s\nh : ¬∃ i, Surjective fun x ↦ ↑x i\n⊢ ∀ (i : ι), ∃ y, ∀ x ∈ s, ¬x i = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Order | {
"line": 601,
"column": 4
} | {
"line": 601,
"column": 27
} | {
"line": 602,
"column": 4
} | [
{
"pp": "ι : Type u_1\nf g : ι → Cardinal.{u_2}\nH : ∀ (i : ι), f i < g i\nx✝ : prod g ≤ sum f\nF : ((i : ι) → Quotient.out (g i)) ↪ (i : ι) × Quotient.out (f i)\nthis : Inhabited ((i : ι) → Quotient.out (g i))\nG : (i : ι) × Quotient.out (f i) → (i : ι) → Quotient.out (g i) := invFun ⇑F\nsG : Surjective G\nC :... | [
"ι : Type u_1\nf g : ι → Cardinal.{u_2}\nH : ∀ (i : ι), f i < g i\nx✝ : prod g ≤ sum f\nF : ((i : ι) → Quotient.out (g i)) ↪ (i : ι) × Quotient.out (f i)\nthis : Inhabited ((i : ι) → Quotient.out (g i))\nG : (i : ι) × Quotient.out (f i) → (i : ι) → Quotient.out (g i) := invFun ⇑F\nsG : Surjective G\nC : (i : ι) → Q... | let ⟨⟨i, a⟩, h⟩ := sG C | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.SetTheory.Cardinal.Order | {
"line": 615,
"column": 2
} | {
"line": 615,
"column": 13
} | {
"line": 615,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\n⊢ ℵ₀ ≤ lift.{v, u} c ↔ ℵ₀ ≤ c",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\n⊢ ℵ₀ ≤ lift.{v, u} c ↔ ℵ₀ ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Order | {
"line": 619,
"column": 2
} | {
"line": 619,
"column": 13
} | {
"line": 619,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\n⊢ lift.{v, u} c ≤ ℵ₀ ↔ c ≤ ℵ₀",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\n⊢ lift.{v, u} c ≤ ℵ₀ ↔ c ≤ ℵ₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Order | {
"line": 623,
"column": 2
} | {
"line": 623,
"column": 13
} | {
"line": 623,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\n⊢ ℵ₀ < lift.{v, u} c ↔ ℵ₀ < c",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\n⊢ ℵ₀ < lift.{v, u} c ↔ ℵ₀ < c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Order | {
"line": 627,
"column": 2
} | {
"line": 627,
"column": 13
} | {
"line": 627,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\n⊢ lift.{v, u} c < ℵ₀ ↔ c < ℵ₀",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\n⊢ lift.{v, u} c < ℵ₀ ↔ c < ℵ₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Order | {
"line": 631,
"column": 2
} | {
"line": 631,
"column": 13
} | {
"line": 631,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\n⊢ ℵ₀ = lift.{v, u} c ↔ ℵ₀ = c",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\n⊢ ℵ₀ = lift.{v, u} c ↔ ℵ₀ = c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Order | {
"line": 685,
"column": 2
} | {
"line": 685,
"column": 13
} | {
"line": 685,
"column": 14
} | [
{
"pp": "a : Cardinal.{u}\n⊢ lift.{v, u} a ≤ 1 ↔ a ≤ 1",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : Cardinal.{u}\n⊢ lift.{v, u} a ≤ 1 ↔ a ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Order | {
"line": 699,
"column": 2
} | {
"line": 699,
"column": 13
} | {
"line": 699,
"column": 14
} | [
{
"pp": "a : Cardinal.{u}\n⊢ 1 ≤ lift.{v, u} a ↔ 1 ≤ a",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : Cardinal.{u}\n⊢ 1 ≤ lift.{v, u} a ↔ 1 ≤ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Order | {
"line": 722,
"column": 2
} | {
"line": 722,
"column": 13
} | {
"line": 722,
"column": 14
} | [
{
"pp": "a : Cardinal.{u}\n⊢ 0 < lift.{v, u} a ↔ 0 < a",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : Cardinal.{u}\n⊢ 0 < lift.{v, u} a ↔ 0 < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Order | {
"line": 727,
"column": 2
} | {
"line": 727,
"column": 13
} | {
"line": 727,
"column": 14
} | [
{
"pp": "a : Cardinal.{u}\n⊢ 1 < lift.{v, u} a ↔ 1 < a",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : Cardinal.{u}\n⊢ 1 < lift.{v, u} a ↔ 1 < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.ENat | {
"line": 206,
"column": 8
} | {
"line": 206,
"column": 37
} | {
"line": 207,
"column": 8
} | [
{
"pp": "case inr.inr\nx y : Cardinal.{u}\nhle : x ≤ y\nhy : ℵ₀ ≤ y\nhx : x ≠ 0\n⊢ x.toENatAux ≠ 0",
"ppTerm": "?inr.inr✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"id",
"Ne",
"ENat",
"Cardinal.... | [
"case inr.inr\nx y : Cardinal.{u}\nhle : x ≤ y\nhy : ℵ₀ ≤ y\nhx : x ≠ 0\n⊢ ℵ₀ ≤ x * y"
] | · rwa [Ne, toENatAux_eq_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 696,
"column": 8
} | {
"line": 696,
"column": 50
} | {
"line": 697,
"column": 6
} | [
{
"pp": "case a.h₁\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : OmegaCompletePartialOrder α\ninst✝¹ : OmegaCompletePartialOrder β\ninst✝ : OmegaCompletePartialOrder γ\nf : α → β →𝒄 γ\nhf : ωScottContinuous f\ng : α → β\nhg : ωScottContinuous g\nc : Chain α\ni j : ℕ\n⊢ f (c i) ≤ f (c.toOrderHom (max i j)... | [] | apply hf.monotone (c.monotone le_sup_left) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 696,
"column": 8
} | {
"line": 696,
"column": 50
} | {
"line": 697,
"column": 6
} | [
{
"pp": "case a.h₁\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : OmegaCompletePartialOrder α\ninst✝¹ : OmegaCompletePartialOrder β\ninst✝ : OmegaCompletePartialOrder γ\nf : α → β →𝒄 γ\nhf : ωScottContinuous f\ng : α → β\nhg : ωScottContinuous g\nc : Chain α\ni j : ℕ\n⊢ f (c i) ≤ f (c.toOrderHom (max i j)... | [] | apply hf.monotone (c.monotone le_sup_left) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 696,
"column": 8
} | {
"line": 696,
"column": 50
} | {
"line": 697,
"column": 6
} | [
{
"pp": "case a.h₁\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : OmegaCompletePartialOrder α\ninst✝¹ : OmegaCompletePartialOrder β\ninst✝ : OmegaCompletePartialOrder γ\nf : α → β →𝒄 γ\nhf : ωScottContinuous f\ng : α → β\nhg : ωScottContinuous g\nc : Chain α\ni j : ℕ\n⊢ f (c i) ≤ f (c.toOrderHom (max i j)... | [] | apply hf.monotone (c.monotone le_sup_left) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 13
} | {
"line": 149,
"column": 14
} | [
{
"pp": "α β : Type u\na x : Cardinal.{u}\nhx : x ∈ Iic #(Quotient.out a)\n⊢ ∃ a_1, (fun x ↦ ⟨#↑x, ⋯⟩) a_1 = ⟨x, hx⟩",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Cardinal.mk",
"Membership.m... | [
"α β : Type u\na x : Cardinal.{u}\nhx : x ∈ Iic #(Quotient.out a)\n⊢ ∃ a_1, #↑a_1 = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 15
} | {
"line": 163,
"column": 16
} | [
{
"pp": "case h\ns : Set Cardinal.{u}\nι : Type u\ne : ↑s ≃ ι\na : Cardinal.{u}\nha : a ∈ s\n⊢ a ≤ sum fun x ↦ ↑(e.symm x)",
"ppTerm": "?h",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case h\ns : Set Cardinal.{u}\nι : Type u\ne : ↑s ≃ ι\na : Cardinal.{u}\nha : a ∈ s\n⊢ a ≤ sum fun x ↦ ↑(e.symm x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 27
} | {
"line": 205,
"column": 28
} | [
{
"pp": "ι : Type u\nf : ι → Cardinal.{max u v}\n⊢ sum f ≤ lift.{v, u} #ι * ⨆ i, f i",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Cardinal",
"congrArg",
"iSup",
"Cardinal.lift",
"Cardinal.mk",
"_private.Mathlib.SetT... | [
"ι : Type u\nf : ι → Cardinal.{max u v}\n⊢ sum f ≤ lift.{max u v, u} #ι * ⨆ i, f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 13
} | {
"line": 208,
"column": 14
} | [
{
"pp": "ι : Type u\nf : ι → Cardinal.{u}\n⊢ sum f ≤ #ι * ⨆ i, f i",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u\nf : ι → Cardinal.{u}\n⊢ sum f ≤ #ι * ⨆ i, f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 301,
"column": 72
} | {
"line": 301,
"column": 83
} | {
"line": 301,
"column": 84
} | [
{
"pp": "α : Type u_1\nn : ℕ\nh : ↑n ≤ #α\nhα✝ : Finite α\nhα : Fintype α := Fintype.ofFinite α\n⊢ n ≤ Finset.univ.card",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Finset.univ",
"id",
"LE.le",
"instLENat",
"Nat",
"Finset.card"
],
"usedFVars"... | [
"α : Type u_1\nn : ℕ\nh : ↑n ≤ #α\nhα✝ : Finite α\nhα : Fintype α := Fintype.ofFinite α\n⊢ n ≤ Fintype.card α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Hom.CompTypeclasses | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 40
} | {
"line": 96,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nφ : M →* N\nψ : N →* P\nχ : M →* P\nh : φ.CompTriple ψ χ\nx : M\n⊢ ψ (φ x) = χ x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.instFunLike",
"... | [] | rw [← h.comp_eq, MonoidHom.comp_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Group.Hom.CompTypeclasses | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 40
} | {
"line": 96,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nφ : M →* N\nψ : N →* P\nχ : M →* P\nh : φ.CompTriple ψ χ\nx : M\n⊢ ψ (φ x) = χ x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.instFunLike",
"... | [] | rw [← h.comp_eq, MonoidHom.comp_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Hom.CompTypeclasses | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 40
} | {
"line": 96,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nφ : M →* N\nψ : N →* P\nχ : M →* P\nh : φ.CompTriple ψ χ\nx : M\n⊢ ψ (φ x) = χ x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.instFunLike",
"... | [] | rw [← h.comp_eq, MonoidHom.comp_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 13
} | {
"line": 314,
"column": 14
} | [
{
"pp": "α : Type u\nn : ℕ\nH : ↑n < #α\n⊢ ↑(n + 1) ≤ #α",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal.natCast_add_one_le_iff._simp_1",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Preorder.toLT",
"AddMonoid.toAddSemigroup",
"Cardina... | [
"α : Type u\nn : ℕ\nH : ↑n < #α\n⊢ ↑n < #α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 13
} | {
"line": 328,
"column": 14
} | [
{
"pp": "c : Cardinal.{u_1}\n⊢ c < 1 ↔ c = 0",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u_1}\n⊢ c < 1 ↔ c = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 334,
"column": 2
} | {
"line": 334,
"column": 13
} | {
"line": 334,
"column": 14
} | [
{
"pp": "c : Cardinal.{u_1}\n⊢ c ≤ 1 ↔ c = 0 ∨ c = 1",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u_1}\n⊢ c ≤ 1 ↔ c = 0 ∨ c = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 351,
"column": 37
} | {
"line": 351,
"column": 48
} | {
"line": 351,
"column": 49
} | [
{
"pp": "⊢ 1 < ℵ₀",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ 1 < ℵ₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 482,
"column": 12
} | {
"line": 482,
"column": 23
} | {
"line": 482,
"column": 24
} | [
{
"pp": "a : Cardinal.{u_1}\n⊢ 0 • a < ℵ₀ ↔ 0 = 0 ∨ a < ℵ₀",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"Preorder.toLT",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"... | [
"a : Cardinal.{u_1}\n⊢ 0 < ℵ₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 502,
"column": 4
} | {
"line": 502,
"column": 53
} | {
"line": 503,
"column": 4
} | [
{
"pp": "case neg\na b : Cardinal.{u_1}\nh : a * b < ℵ₀\nha : ¬a = 0\nhb : ¬b = 0\n⊢ a < ℵ₀ ∧ b < ℵ₀",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Cardinal.instOne",
"Cardinal",
"congrArg",
"Eq.mp",
"Cardinal.one_le_iff_ne_zero",
"Ne",
"LE.le",
... | [
"case neg\na b : Cardinal.{u_1}\nh : a * b < ℵ₀\nha : 1 ≤ a\nhb : 1 ≤ b\n⊢ a < ℵ₀ ∧ b < ℵ₀"
] | rw [← Ne, ← Cardinal.one_le_iff_ne_zero] at ha hb | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 663,
"column": 6
} | {
"line": 663,
"column": 21
} | {
"line": 663,
"column": 22
} | [
{
"pp": "α : Type u\ns : Set α\n⊢ #↑s = 0 ↔ s = ∅",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Cardinal.mk",
"Set.Elem",
"id",
"IsEmpty",
"Cardinal.mk_eq_zero_iff",
"Iff",
"propext",
"Set... | [
"α : Type u\ns : Set α\n⊢ IsEmpty ↑s ↔ s = ∅"
] | mk_eq_zero_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 740,
"column": 2
} | {
"line": 740,
"column": 13
} | {
"line": 740,
"column": 14
} | [
{
"pp": "α β : Type u\nf : α ↪ β\ns : Set α\n⊢ #↑(⇑f '' s) = #↑s",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α β : Type u\nf : α ↪ β\ns : Set α\n⊢ #↑(⇑f '' s) = #↑s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 802,
"column": 4
} | {
"line": 802,
"column": 15
} | {
"line": 802,
"column": 16
} | [
{
"pp": "case mp\nα : Type u_1\nn : ℕ\ns : Finset α\nh : #↑↑s = ↑n\n⊢ ∃ t, ↑t = ↑s ∧ t.card = n",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"SetLike.coe_set_eq._simp_1",
"Exists",
"id",
"funext",
"And",
... | [
"case mp\nα : Type u_1\nn : ℕ\ns : Finset α\nh : #↑↑s = ↑n\n⊢ s.card = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 853,
"column": 26
} | {
"line": 853,
"column": 37
} | {
"line": 853,
"column": 38
} | [
{
"pp": "α : Type u\nn : ℕ\nt : Set α\nH : #↑t ≤ ↑n\ns : Finset α\nhs : ↑s ⊆ t\n⊢ s.card ≤ n",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nn : ℕ\nt : Set α\nH : #↑t ≤ ↑n\ns : Finset α\nhs : ↑s ⊆ t\n⊢ s.card ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 867,
"column": 2
} | {
"line": 867,
"column": 13
} | {
"line": 867,
"column": 14
} | [
{
"pp": "α : Type u\nA B : Set α\nhfin : B.Finite\nhlt : A ⊂ B\nthis✝ : Fintype ↑A\nthis : Fintype ↑B\n⊢ #↑A < #↑B",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Preorder.toLT",
"IsOrderedRing.toZeroLEOneClass",... | [
"α : Type u\nA B : Set α\nhfin : B.Finite\nhlt : A ⊂ B\nthis✝ : Fintype ↑A\nthis : Fintype ↑B\n⊢ Fintype.card ↑A < Fintype.card ↑B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 900,
"column": 2
} | {
"line": 900,
"column": 39
} | {
"line": 900,
"column": 40
} | [
{
"pp": "α : Type u\nS : Set α\nh : #↑S < #↑univ\n⊢ Sᶜ.Nonempty",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Set.univ",
"Set.compl_eq_univ_sdiff",
"id",
"Set.instCompl",
"SDiff.sdiff",
"Set.Nonem... | [
"α : Type u\nS : Set α\nh : #↑S < #↑univ\n⊢ (univ \\ S).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 939,
"column": 2
} | {
"line": 939,
"column": 13
} | {
"line": 939,
"column": 14
} | [
{
"pp": "α β : Type u\nf : α ≃ β\ns : Set β\n⊢ #↑(⇑f ⁻¹' s) = #↑s",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α β : Type u\nf : α ≃ β\ns : Set β\n⊢ #↑(⇑f ⁻¹' s) = #↑s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 989,
"column": 25
} | {
"line": 989,
"column": 36
} | {
"line": 989,
"column": 37
} | [
{
"pp": "α : Type u\nx y : α\nhne : x ≠ y\nht : ↑{x, y} = univ\n⊢ {x, y} = univ",
"ppTerm": "?m.73",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nx y : α\nhne : x ≠ y\nht : ↑{x, y} = univ\n⊢ {x, y} = univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 991,
"column": 22
} | {
"line": 991,
"column": 33
} | {
"line": 991,
"column": 34
} | [
{
"pp": "α : Type u\nx y : α\nhne : x ≠ y\nh : {x, y} = univ\n⊢ ↑{x, y} = univ",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.coe_singleton",
"congrArg",
"Finset",
"Set.univ",
"Classical.propDecidable",
"Set.instSingletonSet",
... | [
"α : Type u\nx y : α\nhne : x ≠ y\nh : {x, y} = univ\n⊢ {x, y} = univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 1013,
"column": 29
} | {
"line": 1013,
"column": 40
} | {
"line": 1013,
"column": 41
} | [
{
"pp": "α : Type u_1\nh : 3 ≤ #α\nx y : α\n⊢ ↑3 ≤ #α",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Cardinal",
"Cardinal.mk",
"id",
"instOfNatNat",
"LE.le",
"Nat.cast",
"Cardinal.instLE",
"Nat",
"OfNat.ofNat",
"Cardinal.instNat... | [
"α : Type u_1\nh : 3 ≤ #α\nx y : α\n⊢ 3 ≤ #α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 1016,
"column": 2
} | {
"line": 1016,
"column": 22
} | {
"line": 1016,
"column": 23
} | [
{
"pp": "α : Type u_1\nh : 3 ≤ #α\nx y : α\nthis✝¹ : ↑3 ≤ #α\nthis✝ : ↑2 < #α\nthis : ∃ z, z ∉ [x, y]\n⊢ ∃ z, z ≠ x ∧ z ≠ y",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Exists",
"id",
"Ne",
"And"
],
"usedFVars": [
"α",
"x",
"y"
],
... | [
"α : Type u_1\nh : 3 ≤ #α\nx y : α\nthis✝¹ : ↑3 ≤ #α\nthis✝ : ↑2 < #α\nthis : ∃ z, z ∉ [x, y]\n⊢ ∃ z, ¬z = x ∧ ¬z = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Action.Prod | {
"line": 68,
"column": 39
} | {
"line": 68,
"column": 57
} | {
"line": 70,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\nE : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁵ : SMul M α\ninst✝⁴ : SMul M β\ninst✝³ : SMul N α\ninst✝² : SMul N β\na✝ : M\nx : α × β\ninst✝¹ : FaithfulSMul M α\ninst✝ : Nonempty β\nm₁✝ m₂✝ : M\nh : ∀ (a : α × β), m₁✝ • a = m₂✝ • a\nb : β\na : α\n⊢ m₁✝ • a = ... | [] | injection h (a, b) | Lean.Elab.Tactic.evalInjection | Lean.Parser.Tactic.injection |
Mathlib.Algebra.Group.Action.Prod | {
"line": 68,
"column": 39
} | {
"line": 68,
"column": 57
} | {
"line": 70,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\nE : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁵ : SMul M α\ninst✝⁴ : SMul M β\ninst✝³ : SMul N α\ninst✝² : SMul N β\na✝ : M\nx : α × β\ninst✝¹ : FaithfulSMul M α\ninst✝ : Nonempty β\nm₁✝ m₂✝ : M\nh : ∀ (a : α × β), m₁✝ • a = m₂✝ • a\nb : β\na : α\n⊢ m₁✝ • a = ... | [] | injection h (a, b) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Action.Prod | {
"line": 68,
"column": 39
} | {
"line": 68,
"column": 57
} | {
"line": 70,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\nE : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁵ : SMul M α\ninst✝⁴ : SMul M β\ninst✝³ : SMul N α\ninst✝² : SMul N β\na✝ : M\nx : α × β\ninst✝¹ : FaithfulSMul M α\ninst✝ : Nonempty β\nm₁✝ m₂✝ : M\nh : ∀ (a : α × β), m₁✝ • a = m₂✝ • a\nb : β\na : α\n⊢ m₁✝ • a = ... | [] | injection h (a, b) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Action.Prod | {
"line": 74,
"column": 39
} | {
"line": 74,
"column": 57
} | {
"line": 76,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\nE : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁵ : SMul M α\ninst✝⁴ : SMul M β\ninst✝³ : SMul N α\ninst✝² : SMul N β\na✝ : M\nx : α × β\ninst✝¹ : Nonempty α\ninst✝ : FaithfulSMul M β\nm₁✝ m₂✝ : M\nh : ∀ (a : α × β), m₁✝ • a = m₂✝ • a\na : α\nb : β\n⊢ m₁✝ • b = ... | [] | injection h (a, b) | Lean.Elab.Tactic.evalInjection | Lean.Parser.Tactic.injection |
Mathlib.Algebra.Group.Action.Prod | {
"line": 74,
"column": 39
} | {
"line": 74,
"column": 57
} | {
"line": 76,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\nE : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁵ : SMul M α\ninst✝⁴ : SMul M β\ninst✝³ : SMul N α\ninst✝² : SMul N β\na✝ : M\nx : α × β\ninst✝¹ : Nonempty α\ninst✝ : FaithfulSMul M β\nm₁✝ m₂✝ : M\nh : ∀ (a : α × β), m₁✝ • a = m₂✝ • a\na : α\nb : β\n⊢ m₁✝ • b = ... | [] | injection h (a, b) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Action.Prod | {
"line": 74,
"column": 39
} | {
"line": 74,
"column": 57
} | {
"line": 76,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\nE : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁵ : SMul M α\ninst✝⁴ : SMul M β\ninst✝³ : SMul N α\ninst✝² : SMul N β\na✝ : M\nx : α × β\ninst✝¹ : Nonempty α\ninst✝ : FaithfulSMul M β\nm₁✝ m₂✝ : M\nh : ∀ (a : α × β), m₁✝ • a = m₂✝ • a\na : α\nb : β\n⊢ m₁✝ • b = ... | [] | injection h (a, b) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Action.Basic | {
"line": 110,
"column": 40
} | {
"line": 110,
"column": 51
} | {
"line": 110,
"column": 52
} | [
{
"pp": "G : Type u_1\nG₀ : Type u_2\ninst✝⁴ : Group G\ninst✝³ : GroupWithZero G₀\ninst✝² : MulAction G G₀\ninst✝¹ : IsScalarTower G G₀ G₀\ninst✝ : SMulCommClass G G₀ G₀\ng h : G\na b : G₀\nx : G\n⊢ x • 0 = 0",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"G : Type u_1\nG₀ : Type u_2\ninst✝⁴ : Group G\ninst✝³ : GroupWithZero G₀\ninst✝² : MulAction G G₀\ninst✝¹ : IsScalarTower G G₀ G₀\ninst✝ : SMulCommClass G G₀ G₀\ng h : G\na b : G₀\nx : G\n⊢ x • 0 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.DomAct.Basic | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 33
} | {
"line": 188,
"column": 34
} | [
{
"pp": "M : Type u_1\nβ : Type u_2\nα : Type u_3\nN : Type u_4\ninst✝² : SMul M α\ninst✝¹ : FaithfulSMul M α\ninst✝ : Nontrivial β\nc₁ c₂ : Mᵈᵐᵃ\na : α\nx y : β\nhne : mk.symm c₁ • a ≠ mk.symm c₂ • a\nthis : DecidableEq α\nh : (c₁ • update (const α x) (mk.symm c₂ • a) y) a = (c₂ • update (const α x) (mk.symm c... | [
"M : Type u_1\nβ : Type u_2\nα : Type u_3\nN : Type u_4\ninst✝² : SMul M α\ninst✝¹ : FaithfulSMul M α\ninst✝ : Nontrivial β\nc₁ c₂ : Mᵈᵐᵃ\na : α\nx y : β\nhne : mk.symm c₁ • a ≠ mk.symm c₂ • a\nthis : DecidableEq α\nh : (c₁ • update (const α x) (mk.symm c₂ • a) y) a = (c₂ • update (const α x) (mk.symm c₂ • a) y) a\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.Hom | {
"line": 334,
"column": 2
} | {
"line": 334,
"column": 13
} | {
"line": 334,
"column": 14
} | [
{
"pp": "M : Type u_2\nN : Type u_3\nφ : M → N\nX : Type u_5\ninst✝¹ : SMul M X\nY : Type u_6\ninst✝ : SMul N Y\nφ' : N → M\nf : X →ₑ[φ] Y\ng : Y → X\nk₁ : Function.LeftInverse φ' φ\nk₂ : Function.RightInverse φ' φ\nh₁ : Function.LeftInverse g ⇑f\nh₂ : Function.RightInverse g ⇑f\nx✝ : X\n⊢ ((f.inverse' g k₂ h₁ ... | [
"M : Type u_2\nN : Type u_3\nφ : M → N\nX : Type u_5\ninst✝¹ : SMul M X\nY : Type u_6\ninst✝ : SMul N Y\nφ' : N → M\nf : X →ₑ[φ] Y\ng : Y → X\nk₁ : Function.LeftInverse φ' φ\nk₂ : Function.RightInverse φ' φ\nh₁ : Function.LeftInverse g ⇑f\nh₂ : Function.RightInverse g ⇑f\nx✝ : X\n⊢ g (f x✝) = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.Hom | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 13
} | {
"line": 342,
"column": 14
} | [
{
"pp": "M : Type u_2\nN : Type u_3\nφ : M → N\nX : Type u_5\ninst✝¹ : SMul M X\nY : Type u_6\ninst✝ : SMul N Y\nφ' : N → M\nf : X →ₑ[φ] Y\ng : Y → X\nk₂ : Function.RightInverse φ' φ\nh₁ : Function.LeftInverse g ⇑f\nh₂ : Function.RightInverse g ⇑f\nx✝ : Y\n⊢ (f.comp (f.inverse' g k₂ h₁ h₂)) x✝ = (MulActionHom.i... | [
"M : Type u_2\nN : Type u_3\nφ : M → N\nX : Type u_5\ninst✝¹ : SMul M X\nY : Type u_6\ninst✝ : SMul N Y\nφ' : N → M\nf : X →ₑ[φ] Y\ng : Y → X\nk₂ : Function.RightInverse φ' φ\nh₁ : Function.LeftInverse g ⇑f\nh₂ : Function.RightInverse g ⇑f\nx✝ : Y\n⊢ f (g x✝) = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.LinearMap.Defs | {
"line": 566,
"column": 53
} | {
"line": 566,
"column": 69
} | {
"line": 566,
"column": 69
} | [
{
"pp": "R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nS : Type u_5\nS₃ : Type u_6\nT : Type u_7\nM : Type u_8\nM₁ : Type u_9\nM₂ : Type u_10\nM₃ : Type u_11\nN₂ : Type u_12\nN₃ : Type u_13\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : AddC... | [
"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nS : Type u_5\nS₃ : Type u_6\nT : Type u_7\nM : Type u_8\nM₁ : Type u_9\nM₂ : Type u_10\nM₃ : Type u_11\nN₂ : Type u_12\nN₃ : Type u_13\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : AddCommMonoid M₃... | ← h₁ (g x + g y) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.LinearMap.Defs | {
"line": 562,
"column": 20
} | {
"line": 569,
"column": 19
} | {
"line": 571,
"column": 0
} | [
{
"pp": "R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nS : Type u_5\nS₃ : Type u_6\nT : Type u_7\nM : Type u_8\nM₁ : Type u_9\nM₂ : Type u_10\nM₃ : Type u_11\nN₂ : Type u_12\nN₃ : Type u_13\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : AddC... | [] | by
dsimp [LeftInverse, Function.RightInverse] at h₁ h₂
exact
{ toFun := g
map_add' := fun x y ↦ by rw [← h₁ (g (x + y)), ← h₁ (g x + g y)]; simp [h₂]
map_smul' := fun a b ↦ by
rw [← h₁ (g (a • b)), ← h₁ (σ' a • g b)]
simp [h₂] } | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Torsion.Free | {
"line": 56,
"column": 62
} | {
"line": 56,
"column": 80
} | {
"line": 56,
"column": 81
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : IsTorsionFree R N\nf : M → N\nhf : Injective f\nsmul : ∀ (r : R) (m : M), f (r • m) = r • f m\nr : R\nhr : IsRegular r\nm₁ m₂ : M\nhm : (f... | [
"R : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : IsTorsionFree R N\nf : M → N\nhf : Injective f\nsmul : ∀ (r : R) (m : M), f (r • m) = r • f m\nr : R\nhr : IsRegular r\nm₁ m₂ : M\nhm : (fun x ↦ r • x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.Free | {
"line": 66,
"column": 50
} | {
"line": 66,
"column": 85
} | {
"line": 66,
"column": 86
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nr : R\nm m₁ m₂ : M\ninst✝ : IsAddTorsionFree M\nn : ℕ\nhn : IsRegular n\n⊢ n ≠ 0",
"pp... | [
"R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nr : R\nm m₁ m₂ : M\ninst✝ : IsAddTorsionFree M\nn : ℕ\nhn : IsRegular n\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.Free | {
"line": 90,
"column": 58
} | {
"line": 90,
"column": 69
} | {
"line": 90,
"column": 70
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝¹⁰ : Semiring R\ninst✝⁹ : Semiring S\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : Module S M\ninst✝⁵ : IsTorsionFree R M\ninst✝⁴ : Module S R\ninst✝³ : IsTorsionFree S R\ninst✝² : IsScalarTower S R R\ninst✝¹ : SMulCommClass S R R\ninst✝ : IsSca... | [
"R : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝¹⁰ : Semiring R\ninst✝⁹ : Semiring S\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : Module S M\ninst✝⁵ : IsTorsionFree R M\ninst✝⁴ : Module S R\ninst✝³ : IsTorsionFree S R\ninst✝² : IsScalarTower S R R\ninst✝¹ : SMulCommClass S R R\ninst✝ : IsScalarTower S R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.Free | {
"line": 91,
"column": 48
} | {
"line": 91,
"column": 59
} | {
"line": 91,
"column": 60
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝¹⁰ : Semiring R\ninst✝⁹ : Semiring S\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : Module S M\ninst✝⁵ : IsTorsionFree R M\ninst✝⁴ : Module S R\ninst✝³ : IsTorsionFree S R\ninst✝² : IsScalarTower S R R\ninst✝¹ : SMulCommClass S R R\ninst✝ : IsSca... | [
"R : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝¹⁰ : Semiring R\ninst✝⁹ : Semiring S\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : Module S M\ninst✝⁵ : IsTorsionFree R M\ninst✝⁴ : Module S R\ninst✝³ : IsTorsionFree S R\ninst✝² : IsScalarTower S R R\ninst✝¹ : SMulCommClass S R R\ninst✝ : IsScalarTower S R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.Free | {
"line": 92,
"column": 43
} | {
"line": 92,
"column": 54
} | {
"line": 92,
"column": 55
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝¹⁰ : Semiring R\ninst✝⁹ : Semiring S\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : Module S M\ninst✝⁵ : IsTorsionFree R M\ninst✝⁴ : Module S R\ninst✝³ : IsTorsionFree S R\ninst✝² : IsScalarTower S R R\ninst✝¹ : SMulCommClass S R R\ninst✝ : IsSca... | [
"R : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝¹⁰ : Semiring R\ninst✝⁹ : Semiring S\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : Module S M\ninst✝⁵ : IsTorsionFree R M\ninst✝⁴ : Module S R\ninst✝³ : IsTorsionFree S R\ninst✝² : IsScalarTower S R R\ninst✝¹ : SMulCommClass S R R\ninst✝ : IsScalarTower S R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.Free | {
"line": 144,
"column": 50
} | {
"line": 144,
"column": 85
} | {
"line": 144,
"column": 86
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nm : M\ninst✝ : IsAddTorsionFree M\nn : ℤ\nhn : IsRegular n\n⊢ n ≠ 0",
"ppTerm": "?m.20",
"assigned": true,
... | [
"R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nm : M\ninst✝ : IsAddTorsionFree M\nn : ℤ\nhn : IsRegular n\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.Free | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 41
} | {
"line": 161,
"column": 42
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nh : ∀ (r : R) (m : M), r • m = 0 → r = 0 ∨ m = 0\nr : R\nhr : IsRegular r\nm₁ m₂ : M\nhm : (fun x ↦ r • x) m₁ = (fun x ↦ r • x) m₂\n⊢ m₁ = m₂",
"ppTerm": "?m.29",
"assigned": false,
... | [
"R : Type u_1\nM : Type u_3\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nh : ∀ (r : R) (m : M), r • m = 0 → r = 0 ∨ m = 0\nr : R\nhr : IsRegular r\nm₁ m₂ : M\nhm : (fun x ↦ r • x) m₁ = (fun x ↦ r • x) m₂\n⊢ m₁ = m₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.Free | {
"line": 161,
"column": 60
} | {
"line": 161,
"column": 95
} | {
"line": 161,
"column": 96
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nh : ∀ (r : R) (m : M), r • m = 0 → r = 0 ∨ m = 0\nr : R\nhr : IsRegular r\nm₁ m₂ : M\nhm : (fun x ↦ r • x) m₁ = (fun x ↦ r • x) m₂\n⊢ r • (m₁ - m₂) = 0",
"ppTerm": "?m.35",
"assigned... | [
"R : Type u_1\nM : Type u_3\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nh : ∀ (r : R) (m : M), r • m = 0 → r = 0 ∨ m = 0\nr : R\nhr : IsRegular r\nm₁ m₂ : M\nhm : (fun x ↦ r • x) m₁ = (fun x ↦ r • x) m₂\n⊢ r • m₁ = r • m₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.LinearMap.Basic | {
"line": 145,
"column": 15
} | {
"line": 145,
"column": 26
} | {
"line": 145,
"column": 27
} | [
{
"pp": "R : Type u_6\nA : Type u_7\ninst✝³ : Semiring R\ninst✝² : NonAssocSemiring A\ninst✝¹ : Module R A\ninst✝ : SMulCommClass R A A\na b : A\nh : mulLeft R a = mulLeft R b\n⊢ a = b",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_6\nA : Type u_7\ninst✝³ : Semiring R\ninst✝² : NonAssocSemiring A\ninst✝¹ : Module R A\ninst✝ : SMulCommClass R A A\na b : A\nh : mulLeft R a = mulLeft R b\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.LinearMap.Basic | {
"line": 149,
"column": 15
} | {
"line": 149,
"column": 26
} | {
"line": 149,
"column": 27
} | [
{
"pp": "R : Type u_6\nA : Type u_7\ninst✝³ : Semiring R\ninst✝² : NonAssocSemiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\na b : A\nh : mulRight R a = mulRight R b\n⊢ a = b",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_6\nA : Type u_7\ninst✝³ : Semiring R\ninst✝² : NonAssocSemiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\na b : A\nh : mulRight R a = mulRight R b\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.Free | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 13
} | {
"line": 191,
"column": 14
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Nontrivial M\ninst✝ : IsAddTorsionFree M\nn m : ℕ\nx : M\nhx : x ≠ 0\nh : ↑n • x = ↑m • x\n⊢ n = m",
"ppTerm": "?m.39",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"u... | [
"R : Type u_1\nM : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Nontrivial M\ninst✝ : IsAddTorsionFree M\nn m : ℕ\nx : M\nhx : x ≠ 0\nh : ↑n • x = ↑m • x\n⊢ n = m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Basic | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 13
} | {
"line": 157,
"column": 14
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Ring R\ninst✝² : IsDomain R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : M\nN : Submodule R M\northo : ∀ (c : R), ∀ y ∈ N, c • x + y = 0 → c = 0\nhx : x ∈ N\n⊢ False",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"... | [
"R : Type u\nM : Type v\ninst✝³ : Ring R\ninst✝² : IsDomain R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : M\nN : Submodule R M\northo : ∀ (c : R), ∀ y ∈ N, c • x + y = 0 → c = 0\nhx : x ∈ N\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Defs | {
"line": 83,
"column": 29
} | {
"line": 83,
"column": 51
} | {
"line": 83,
"column": 52
} | [
{
"pp": "G : Type u''\nS : Type u'\nR : Type u\nM : Type v\nι : Type w\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nC : Set M\nnonempty : C.Nonempty\nlinearComb : ∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C\nx y : M\nhx : x ∈ C\nhy : y ∈ C\n⊢ x + y ∈ C",
"ppTerm": "?m.47",
"a... | [
"G : Type u''\nS : Type u'\nR : Type u\nM : Type v\nι : Type w\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nC : Set M\nnonempty : C.Nonempty\nlinearComb : ∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C\nx y : M\nhx : x ∈ C\nhy : y ∈ C\n⊢ x + y ∈ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Defs | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 37
} | {
"line": 82,
"column": 38
} | [
{
"pp": "G : Type u''\nS : Type u'\nR : Type u\nM : Type v\nι : Type w\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nC : Set M\nlinearComb : ∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C\nx : M\nhx : x ∈ C\n⊢ 0 ∈ C",
"ppTerm": "?m.68",
"assigned": false,
"usedConstants": [],... | [
"G : Type u''\nS : Type u'\nR : Type u\nM : Type v\nι : Type w\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nC : Set M\nlinearComb : ∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C\nx : M\nhx : x ∈ C\n⊢ 0 ∈ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Defs | {
"line": 84,
"column": 25
} | {
"line": 84,
"column": 36
} | {
"line": 84,
"column": 37
} | [
{
"pp": "G : Type u''\nS : Type u'\nR : Type u\nM : Type v\nι : Type w\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nC : Set M\nnonempty : C.Nonempty\nlinearComb : ∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C\nc : R\nx : M\nhx : x ∈ C\n⊢ c • x ∈ C",
"ppTerm": "?m.83",
"assigned... | [
"G : Type u''\nS : Type u'\nR : Type u\nM : Type v\nι : Type w\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nC : Set M\nnonempty : C.Nonempty\nlinearComb : ∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C\nc : R\nx : M\nhx : x ∈ C\n⊢ c • x ∈ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Defs | {
"line": 135,
"column": 18
} | {
"line": 135,
"column": 34
} | {
"line": 135,
"column": 35
} | [
{
"pp": "G : Type u''\nS : Type u'\nR : Type u\nM : Type v\nι : Type w\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np✝ q p : Submodule R M\ns : Set M\nhs : s = ↑p\n⊢ ∀ (c : R) {x : M}, x ∈ s → c • x ∈ s",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"G : Type u''\nS : Type u'\nR : Type u\nM : Type v\nι : Type w\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np✝ q p : Submodule R M\ns : Set M\nhs : s = ↑p\n⊢ ∀ (c : R) {x : M}, x ∈ p → c • x ∈ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.LinearMap | {
"line": 288,
"column": 2
} | {
"line": 289,
"column": 78
} | {
"line": 290,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ng : Module.End R ↥N\nG : Module.End R M\nh : G ∘ₗ N.subtype = N.subtype ∘ₗ g\nk : ℕ\nhG : G ^ k = 0\nm : ↥N\n⊢ ↑((g ^ k) m) = ↑(0 m)",
"ppTerm": "?m.130",
"assigned": true,
"use... | [
"R : Type u_1\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ng : Module.End R ↥N\nG : Module.End R M\nh : G ∘ₗ N.subtype = N.subtype ∘ₗ g\nk : ℕ\nhG : G ^ k = 0\nm : ↥N\nhg : (N.subtype ∘ₗ g ^ k) m = 0\n⊢ ↑((g ^ k) m) = ↑(0 m)"
] | have hg : N.subtype.comp (g ^ k) m = 0 := by
rw [← Module.End.commute_pow_left_of_commute h, hG, zero_comp, zero_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Module.Submodule.LinearMap | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 13
} | {
"line": 290,
"column": 14
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ng : Module.End R ↥N\nG : Module.End R M\nh : G ∘ₗ N.subtype = N.subtype ∘ₗ g\nk : ℕ\nhG : G ^ k = 0\nm : ↥N\nhg : (N.subtype ∘ₗ g ^ k) m = 0\n⊢ ↑((g ^ k) m) = ↑(0 m)",
"ppTerm": "?m.164... | [
"R : Type u_1\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ng : Module.End R ↥N\nG : Module.End R M\nh : G ∘ₗ N.subtype = N.subtype ∘ₗ g\nk : ℕ\nhG : G ^ k = 0\nm : ↥N\nhg : (N.subtype ∘ₗ g ^ k) m = 0\n⊢ (g ^ k) m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 135,
"column": 15
} | {
"line": 135,
"column": 26
} | {
"line": 135,
"column": 27
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nS : Type u_3\ninst✝³ : Monoid R\ninst✝² : MulAction R M\ninst✝¹ : SetLike S M\ninst✝ : SMulMemClass S R M\nN : S\nx : M\nh : ∀ (a : R), a • x ∈ N\n⊢ x ∈ N",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nM : Type u_2\nS : Type u_3\ninst✝³ : Monoid R\ninst✝² : MulAction R M\ninst✝¹ : SetLike S M\ninst✝ : SMulMemClass S R M\nN : S\nx : M\nh : ∀ (a : R), a • x ∈ N\n⊢ x ∈ N"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 13
} | {
"line": 142,
"column": 14
} | [
{
"pp": "S : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝² : SetLike S M\ninst✝¹ : SMul R M\ninst✝ : SMulMemClass S R M\nr : R\ns : S\nx : M\nhx : x ∈ ↑s\n⊢ (fun x ↦ r • x) x ∈ ↑s",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"instHSM... | [
"S : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝² : SetLike S M\ninst✝¹ : SMul R M\ninst✝ : SMulMemClass S R M\nr : R\ns : S\nx : M\nhx : x ∈ ↑s\n⊢ r • x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 598,
"column": 2
} | {
"line": 600,
"column": 36
} | {
"line": 602,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nα : Type u_2\ninst✝ : MulAction G α\nH : Subgroup G\nhH : H.Normal\ng : G\na : α\nha : a ∈ MulAction.fixedPoints (↥H) α\n⊢ g • a ∈ MulAction.fixedPoints (↥H) α",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv... | [] | intro h
rw [Subgroup.smul_def, ← inv_smul_eq_iff, smul_smul, smul_smul]
exact ha ⟨_, hH.conj_mem' _ h.2 _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 598,
"column": 2
} | {
"line": 600,
"column": 36
} | {
"line": 602,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nα : Type u_2\ninst✝ : MulAction G α\nH : Subgroup G\nhH : H.Normal\ng : G\na : α\nha : a ∈ MulAction.fixedPoints (↥H) α\n⊢ g • a ∈ MulAction.fixedPoints (↥H) α",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv... | [] | intro h
rw [Subgroup.smul_def, ← inv_smul_eq_iff, smul_smul, smul_smul]
exact ha ⟨_, hH.conj_mem' _ h.2 _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Submodule.Lattice | {
"line": 116,
"column": 26
} | {
"line": 116,
"column": 37
} | {
"line": 116,
"column": 38
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nh : ∀ (x y : ↥p), x = y\nx : M\nhx : x ∈ p\n⊢ x = 0",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nh : ∀ (x y : ↥p), x = y\nx : M\nhx : x ∈ p\n⊢ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Lattice | {
"line": 250,
"column": 30
} | {
"line": 250,
"column": 66
} | {
"line": 250,
"column": 67
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Sort u_4\ninst✝ : Nonempty ι\np : ι → Submodule R M\nq : Submodule R M\n⊢ ↑(q ⊓ ⨅ i, p i) = ↑(⨅ i, q ⊓ p i)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Su... | [
"R : Type u_1\nM : Type u_3\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Sort u_4\ninst✝ : Nonempty ι\np : ι → Submodule R M\nq : Submodule R M\n⊢ ↑q ∩ ⋂ i, ↑(p i) = ⋂ i, ↑q ∩ ↑(p i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Hom.CompleteLattice | {
"line": 663,
"column": 17
} | {
"line": 663,
"column": 28
} | {
"line": 663,
"column": 29
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nι : Sort u_6\nκ : ι → Sort u_7\ninst✝ : FunLike F α β\ne : α ≃ β\na✝ b✝ : Set α\nh :\n { toFun := fun s ↦ ⇑e '' s, invFun := fun s ↦ ⇑e.symm '' s, left_inv := ⋯, right_inv := ⋯ } a✝ ⊆\n { toFun := fun s ↦ ⇑e '' s, invFun := fun s... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nι : Sort u_6\nκ : ι → Sort u_7\ninst✝ : FunLike F α β\ne : α ≃ β\na✝ b✝ : Set α\nh :\n { toFun := fun s ↦ ⇑e '' s, invFun := fun s ↦ ⇑e.symm '' s, left_inv := ⋯, right_inv := ⋯ } a✝ ⊆\n { toFun := fun s ↦ ⇑e '' s, invFun := fun s ↦ ⇑e.symm '... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Lattice | {
"line": 306,
"column": 18
} | {
"line": 306,
"column": 46
} | {
"line": 306,
"column": 47
} | [
{
"pp": "case zero\nR : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set (Submodule R M)\nt : R\nm : M\n⊢ t • 0 ∈ ⨆ a, ↑a",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"AddSubmon... | [
"case zero\nR : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set (Submodule R M)\nt : R\nm : M\n⊢ 0 ∈ ⨆ a, ↑a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.RestrictScalars | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 35
} | {
"line": 171,
"column": 36
} | [
{
"pp": "S : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Semiring S\ninst✝³ : Module S M\ninst✝² : Module R M\ninst✝¹ : SMul S R\ninst✝ : IsScalarTower S R M\ns t : Submodule R M\n⊢ restrictScalars S (s ⊔ t) = restrictScalars S s ⊔ restrictScalars S t",
"ppT... | [
"S : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Semiring S\ninst✝³ : Module S M\ninst✝² : Module R M\ninst✝¹ : SMul S R\ninst✝ : IsScalarTower S R M\ns t : Submodule R M\n⊢ restrictScalars S (s ⊔ t) = restrictScalars S s ⊔ restrictScalars S t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Ker | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 35
} | {
"line": 108,
"column": 36
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_7\ninst✝⁵ : Semiring R\ninst✝⁴ : Semiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nf : M →ₛₗ[τ₁₂] M₂\n⊢ f.ker = ⊥ ↔ ∀ (m : M), f m = 0 → m = 0",
"ppTerm": "?m.49",
"... | [
"R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_7\ninst✝⁵ : Semiring R\ninst✝⁴ : Semiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nf : M →ₛₗ[τ₁₂] M₂\n⊢ f.ker = ⊥ ↔ ∀ (m : M), f m = 0 → m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Ker | {
"line": 112,
"column": 28
} | {
"line": 112,
"column": 90
} | {
"line": 114,
"column": 0
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nτ₂₁ : R₂ →+* R\ninst✝ : RingHomInvPair τ₁₂ τ₂₁\nf : M →ₛₗ[τ₁₂] M₂\ng : M₂ →ₛₗ[τ₂₁] M\nh :... | [] | by rw [← id_apply (R := R) m, ← h, comp_apply, hm, g.map_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Submodule.Ker | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 62
} | {
"line": 154,
"column": 63
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_7\ninst✝⁵ : Semiring R\ninst✝⁴ : Semiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nf : M →ₛₗ[τ₁₂] M₂\nhf : Injective ⇑f\nx : M\nhx : x ∈ f.ker\n⊢ x ∈ ⊥",
"ppTerm": "?m.5... | [
"R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_7\ninst✝⁵ : Semiring R\ninst✝⁴ : Semiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nf : M →ₛₗ[τ₁₂] M₂\nhf : Injective ⇑f\nx : M\nhx : x ∈ f.ker\n⊢ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Ker | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 35
} | {
"line": 194,
"column": 36
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_7\ninst✝⁵ : Ring R\ninst✝⁴ : Ring R₂\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nf : M →ₛₗ[τ₁₂] M₂\n⊢ f.ker = ⊥ ↔ Injective ⇑f",
"ppTerm": "?m.40",
"assigned": false,
"u... | [
"R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_7\ninst✝⁵ : Ring R\ninst✝⁴ : Ring R₂\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nf : M →ₛₗ[τ₁₂] M₂\n⊢ f.ker = ⊥ ↔ Injective ⇑f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Ker | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 24
} | {
"line": 225,
"column": 25
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_7\ninst✝⁵ : Semiring R\ninst✝⁴ : CommSemiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nf : M →ₛₗ[τ₁₂] M₂\nc : R₂\n⊢ f.ker ≤ (c • f).ker",
"ppTerm": "?m.43",
"assigne... | [
"R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_7\ninst✝⁵ : Semiring R\ninst✝⁴ : CommSemiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nf : M →ₛₗ[τ₁₂] M₂\nc : R₂\n⊢ Submodule.comap f ⊥ ≤ Submodule.comap (c • f) ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Submodule.Map | {
"line": 131,
"column": 30
} | {
"line": 131,
"column": 66
} | {
"line": 131,
"column": 67
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_3\nM : Type u_5\nM₂ : Type u_7\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring R₂\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\nσ₁₂ : R →+* R₂\ninst✝¹ : RingHomSurjective σ₁₂\nι : Sort u_9\ninst✝ : Nonempty ι\np : ι → Submodule R M\n... | [
"R : Type u_1\nR₂ : Type u_3\nM : Type u_5\nM₂ : Type u_7\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring R₂\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\nσ₁₂ : R →+* R₂\ninst✝¹ : RingHomSurjective σ₁₂\nι : Sort u_9\ninst✝ : Nonempty ι\np : ι → Submodule R M\nf : M →ₛₗ[σ₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.CharZero | {
"line": 84,
"column": 43
} | {
"line": 84,
"column": 59
} | {
"line": 84,
"column": 60
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nS : Type u_3\nn✝ : ℕ\ninst✝² : Semiring R\ninst✝¹ : CharZero R\ninst✝ : IsCancelMulZero R\nn : ℕ\nhn : n ≠ 0\na b : R\nhab : (fun a ↦ n • a) a = (fun a ↦ n • a) b\n⊢ a = b",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"α : Type u_1\nR : Type u_2\nS : Type u_3\nn✝ : ℕ\ninst✝² : Semiring R\ninst✝¹ : CharZero R\ninst✝ : IsCancelMulZero R\nn : ℕ\nhn : n ≠ 0\na b : R\nhab : (fun a ↦ n • a) a = (fun a ↦ n • a) b\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.CharZero | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 17
} | {
"line": 104,
"column": 18
} | [
{
"pp": "R : Type u_2\ninst✝² : NonAssocRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\na b : R\nh : ↑n * a = ↑n * b\nw : n ≠ 0\n⊢ a = b",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_2\ninst✝² : NonAssocRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\na b : R\nh : ↑n * a = ↑n * b\nw : n ≠ 0\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NonUnitalSubsemiring.Defs | {
"line": 34,
"column": 2
} | {
"line": 34,
"column": 13
} | {
"line": 34,
"column": 14
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : Mul R\ninst✝² : HasDistribNeg R\ninst✝¹ : SetLike S R\ninst✝ : MulMemClass S R\ns : S\nx y : R\nhx : -x ∈ s\nhy : y ∈ s\n⊢ -(x * y) ∈ s",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nS : Type u_2\ninst✝³ : Mul R\ninst✝² : HasDistribNeg R\ninst✝¹ : SetLike S R\ninst✝ : MulMemClass S R\ns : S\nx y : R\nhx : -x ∈ s\nhy : y ∈ s\n⊢ -(x * y) ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NonUnitalSubsemiring.Defs | {
"line": 40,
"column": 2
} | {
"line": 40,
"column": 13
} | {
"line": 40,
"column": 14
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : Mul R\ninst✝² : HasDistribNeg R\ninst✝¹ : SetLike S R\ninst✝ : MulMemClass S R\ns : S\nx y : R\nhx : x ∈ s\nhy : -y ∈ s\n⊢ -(x * y) ∈ s",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nS : Type u_2\ninst✝³ : Mul R\ninst✝² : HasDistribNeg R\ninst✝¹ : SetLike S R\ninst✝ : MulMemClass S R\ns : S\nx y : R\nhx : x ∈ s\nhy : -y ∈ s\n⊢ -(x * y) ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Equiv.Basic | {
"line": 557,
"column": 4
} | {
"line": 558,
"column": 72
} | {
"line": 559,
"column": 2
} | [
{
"pp": "R : Type u_1\nR₂✝ : Type u_2\nK : Type u_3\nS : Type u_4\nM : Type u_5\nM₁✝ : Type u_6\nM₂✝ : Type u_7\nM₃ : Type u_8\nR₁ : Type u_9\nR₂ : Type u_10\nR₁' : Type u_11\nR₂' : Type u_12\nM₁ : Type u_13\nM₂ : Type u_14\nM₁' : Type u_15\nM₂' : Type u_16\ninst✝¹⁹ : Semiring R₁\ninst✝¹⁸ : Semiring R₂\ninst✝¹⁷... | [] | ext x
simp only [symm_apply_apply, Function.comp_apply, coe_comp, coe_coe] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Equiv.Basic | {
"line": 557,
"column": 4
} | {
"line": 558,
"column": 72
} | {
"line": 559,
"column": 2
} | [
{
"pp": "R : Type u_1\nR₂✝ : Type u_2\nK : Type u_3\nS : Type u_4\nM : Type u_5\nM₁✝ : Type u_6\nM₂✝ : Type u_7\nM₃ : Type u_8\nR₁ : Type u_9\nR₂ : Type u_10\nR₁' : Type u_11\nR₂' : Type u_12\nM₁ : Type u_13\nM₂ : Type u_14\nM₁' : Type u_15\nM₂' : Type u_16\ninst✝¹⁹ : Semiring R₁\ninst✝¹⁸ : Semiring R₂\ninst✝¹⁷... | [] | ext x
simp only [symm_apply_apply, Function.comp_apply, coe_comp, coe_coe] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Center | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 30
} | {
"line": 52,
"column": 2
} | [
{
"pp": "case inl\nG₀ : Type u_2\ninst✝ : GroupWithZero G₀\ns : Set G₀\nha : 0 ∈ s.centralizer\n⊢ 0 ∈ s.centralizer",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Set.zero_mem_centralizer",
"MonoidWithZero.toMulZeroOneClass",
"MulZ... | [] | exact zero_mem_centralizer | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.GroupWithZero.Center | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 35
} | {
"line": 57,
"column": 36
} | [
{
"pp": "G₀ : Type u_2\ninst✝ : GroupWithZero G₀\ns : Set G₀\na b : G₀\nha : a ∈ s.centralizer\nhb : b ∈ s.centralizer\n⊢ a / b ∈ s.centralizer",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"DivInvMonoid.toInv",
"instHDi... | [
"G₀ : Type u_2\ninst✝ : GroupWithZero G₀\ns : Set G₀\na b : G₀\nha : a ∈ s.centralizer\nhb : b ∈ s.centralizer\n⊢ a * b⁻¹ ∈ s.centralizer"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subsemigroup.Membership | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 55
} | {
"line": 54,
"column": 56
} | [
{
"pp": "M : Type u_2\ninst✝ : Mul M\nι : Sort u_3\nS : ι → Subsemigroup M\nhS : Directed (fun x1 x2 ↦ x1 ≤ x2) S\nx : M\nthis : x ∈ closure (⋃ i, ↑(S i)) → ∃ i, x ∈ S i\n⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i",
"ppTerm": "?m.58",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"M : Type u_2\ninst✝ : Mul M\nι : Sort u_3\nS : ι → Subsemigroup M\nhS : Directed (fun x1 x2 ↦ x1 ≤ x2) S\nx : M\nthis : x ∈ closure (⋃ i, ↑(S i)) → ∃ i, x ∈ S i\n⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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