module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.ENat.Lattice | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 9
} | {
"line": 170,
"column": 2
} | [
{
"pp": "case inl\nι : Sort u_2\nf : ι → ℕ∞\na : ℕ∞\nh₀ : a = 0 → Nonempty ι\nhι : IsEmpty ι\n⊢ a ≠ 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"False",
"CommSemiring.toSemiring",
"False.elim",
"not_isEmpty_of_nonempty._simp_1",
"Eq.mp",
"id",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Exact.Basic | {
"line": 171,
"column": 6
} | {
"line": 171,
"column": 57
} | {
"line": 172,
"column": 2
} | [
{
"pp": "M₁ : Type u_8\nM₂ : Type u_9\nM₃ : Type u_10\nN₁ : Type u_11\nN₂ : Type u_12\nN₃ : Type u_13\ninst✝⁵ : CommMonoid M₁\ninst✝⁴ : CommMonoid M₂\ninst✝³ : CommMonoid M₃\ninst✝² : CommMonoid N₁\ninst✝¹ : CommMonoid N₂\ninst✝ : CommMonoid N₃\nf : M₁ →* M₂\ng : M₂ →* M₃\nf' : N₁ →* N₂\ng' : N₂ →* N₃\nτ₁ : M₁ ... | [] | exact ⟨x₁, h₂.1 (by simpa only [comm₁₂] using hy₁)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Projection | {
"line": 776,
"column": 4
} | {
"line": 776,
"column": 40
} | {
"line": 776,
"column": 41
} | [
{
"pp": "E : Type u_1\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nf : E →ₗ[R] E\nhf : IsIdempotentElem f\nT : (E →ₗ[R] E)ˣ\n⊢ Commute f ↑((GeneralLinearGroup.generalLinearEquiv R E) T) ↔\n (f.range ∈ Module.End.invtSubmodule ↑((GeneralLinearGroup.generalLinearEquiv R E) T) ∧\... | [
"E : Type u_1\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nf : E →ₗ[R] E\nhf : IsIdempotentElem f\nT : (E →ₗ[R] E)ˣ\n⊢ Commute f ↑((GeneralLinearGroup.generalLinearEquiv R E) T) ↔\n (f.range ∈ Module.End.invtSubmodule ↑((GeneralLinearGroup.generalLinearEquiv R E) T) ∧\n f.k... | and_and_and_comm (c := (ker f ∈ _)), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Rel | {
"line": 120,
"column": 57
} | {
"line": 120,
"column": 62
} | {
"line": 121,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n⊢ ∅.dom = ∅",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"SetRel",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"Membership.mem",
"Exists",
"Prod.mk",
"SetRel.dom",
"Set... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 120,
"column": 57
} | {
"line": 120,
"column": 62
} | {
"line": 121,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n⊢ ∅.dom = ∅",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"SetRel",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"Membership.mem",
"Exists",
"Prod.mk",
"SetRel.dom",
"Set... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 120,
"column": 57
} | {
"line": 120,
"column": 62
} | {
"line": 121,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n⊢ ∅.dom = ∅",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"SetRel",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"Membership.mem",
"Exists",
"Prod.mk",
"SetRel.dom",
"Set... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 121,
"column": 57
} | {
"line": 121,
"column": 62
} | {
"line": 123,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n⊢ ∅.cod = ∅",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"SetRel.cod",
"SetRel",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"Membership.mem",
"Exists",
"Prod.mk",
"iff... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 121,
"column": 57
} | {
"line": 121,
"column": 62
} | {
"line": 123,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n⊢ ∅.cod = ∅",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"SetRel.cod",
"SetRel",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"Membership.mem",
"Exists",
"Prod.mk",
"iff... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 121,
"column": 57
} | {
"line": 121,
"column": 62
} | {
"line": 123,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n⊢ ∅.cod = ∅",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"SetRel.cod",
"SetRel",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"Membership.mem",
"Exists",
"Prod.mk",
"iff... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 131,
"column": 77
} | {
"line": 131,
"column": 82
} | {
"line": 132,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : Nonempty β\n⊢ dom Set.univ = Set.univ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel",
"congrArg",
"Set.mem_univ._simp_1",
"exists_const._simp_1",
"Set.univ",
"Membership.mem",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 131,
"column": 77
} | {
"line": 131,
"column": 82
} | {
"line": 132,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : Nonempty β\n⊢ dom Set.univ = Set.univ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel",
"congrArg",
"Set.mem_univ._simp_1",
"exists_const._simp_1",
"Set.univ",
"Membership.mem",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 131,
"column": 77
} | {
"line": 131,
"column": 82
} | {
"line": 132,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : Nonempty β\n⊢ dom Set.univ = Set.univ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel",
"congrArg",
"Set.mem_univ._simp_1",
"exists_const._simp_1",
"Set.univ",
"Membership.mem",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 132,
"column": 77
} | {
"line": 132,
"column": 82
} | {
"line": 134,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : Nonempty α\n⊢ cod Set.univ = Set.univ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel.cod",
"SetRel",
"congrArg",
"Set.mem_univ._simp_1",
"exists_const._simp_1",
"Set.univ",
"Memb... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 132,
"column": 77
} | {
"line": 132,
"column": 82
} | {
"line": 134,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : Nonempty α\n⊢ cod Set.univ = Set.univ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel.cod",
"SetRel",
"congrArg",
"Set.mem_univ._simp_1",
"exists_const._simp_1",
"Set.univ",
"Memb... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 132,
"column": 77
} | {
"line": 132,
"column": 82
} | {
"line": 134,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : Nonempty α\n⊢ cod Set.univ = Set.univ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel.cod",
"SetRel",
"congrArg",
"Set.mem_univ._simp_1",
"exists_const._simp_1",
"Set.univ",
"Memb... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 143,
"column": 50
} | {
"line": 143,
"column": 55
} | {
"line": 145,
"column": 0
} | [
{
"pp": "α : Type u_1\n⊢ SetRel.id.inv = SetRel.id",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel.id",
"SetRel",
"congrArg",
"SetRel.inv",
"Membership.mem",
"id",
"Prod.mk",
"Iff",
"SetRel.mem_id... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 143,
"column": 50
} | {
"line": 143,
"column": 55
} | {
"line": 145,
"column": 0
} | [
{
"pp": "α : Type u_1\n⊢ SetRel.id.inv = SetRel.id",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel.id",
"SetRel",
"congrArg",
"SetRel.inv",
"Membership.mem",
"id",
"Prod.mk",
"Iff",
"SetRel.mem_id... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 143,
"column": 50
} | {
"line": 143,
"column": 55
} | {
"line": 145,
"column": 0
} | [
{
"pp": "α : Type u_1\n⊢ SetRel.id.inv = SetRel.id",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel.id",
"SetRel",
"congrArg",
"SetRel.inv",
"Membership.mem",
"id",
"Prod.mk",
"Iff",
"SetRel.mem_id... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 157,
"column": 36
} | {
"line": 157,
"column": 41
} | {
"line": 159,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : SetRel α β\nS : SetRel β γ\nt : SetRel γ δ\n⊢ R ○ S ○ t = R ○ (S ○ t)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Membership.mem",
"Exist... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 157,
"column": 36
} | {
"line": 157,
"column": 41
} | {
"line": 159,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : SetRel α β\nS : SetRel β γ\nt : SetRel γ δ\n⊢ R ○ S ○ t = R ○ (S ○ t)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Membership.mem",
"Exist... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 157,
"column": 36
} | {
"line": 157,
"column": 41
} | {
"line": 159,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : SetRel α β\nS : SetRel β γ\nt : SetRel γ δ\n⊢ R ○ S ○ t = R ○ (S ○ t)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Membership.mem",
"Exist... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 159,
"column": 59
} | {
"line": 159,
"column": 64
} | {
"line": 160,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ R ○ SetRel.id = R",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel.id",
"SetRel",
"congrArg",
"Membership.mem",
"Exists",
"SetRel.mem_comp._simp_1",
"Prod.mk",
"Set... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 159,
"column": 59
} | {
"line": 159,
"column": 64
} | {
"line": 160,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ R ○ SetRel.id = R",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel.id",
"SetRel",
"congrArg",
"Membership.mem",
"Exists",
"SetRel.mem_comp._simp_1",
"Prod.mk",
"Set... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 159,
"column": 59
} | {
"line": 159,
"column": 64
} | {
"line": 160,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ R ○ SetRel.id = R",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel.id",
"SetRel",
"congrArg",
"Membership.mem",
"Exists",
"SetRel.mem_comp._simp_1",
"Prod.mk",
"Set... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 160,
"column": 59
} | {
"line": 160,
"column": 64
} | {
"line": 162,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ SetRel.id ○ R = R",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel.id",
"SetRel",
"congrArg",
"Membership.mem",
"Exists",
"SetRel.mem_comp._simp_1",
"exists_eq_left'._sim... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 160,
"column": 59
} | {
"line": 160,
"column": 64
} | {
"line": 162,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ SetRel.id ○ R = R",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel.id",
"SetRel",
"congrArg",
"Membership.mem",
"Exists",
"SetRel.mem_comp._simp_1",
"exists_eq_left'._sim... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 160,
"column": 59
} | {
"line": 160,
"column": 64
} | {
"line": 162,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ SetRel.id ○ R = R",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel.id",
"SetRel",
"congrArg",
"Membership.mem",
"Exists",
"SetRel.mem_comp._simp_1",
"exists_eq_left'._sim... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 162,
"column": 93
} | {
"line": 162,
"column": 98
} | {
"line": 164,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\nS : SetRel β γ\n⊢ (R ○ S).inv = S.inv ○ R.inv",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"SetRel.inv",
"Membership.mem",
"Exists",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 162,
"column": 93
} | {
"line": 162,
"column": 98
} | {
"line": 164,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\nS : SetRel β γ\n⊢ (R ○ S).inv = S.inv ○ R.inv",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"SetRel.inv",
"Membership.mem",
"Exists",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 162,
"column": 93
} | {
"line": 162,
"column": 98
} | {
"line": 164,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\nS : SetRel β γ\n⊢ (R ○ S).inv = S.inv ○ R.inv",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"SetRel.inv",
"Membership.mem",
"Exists",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 164,
"column": 75
} | {
"line": 164,
"column": 80
} | {
"line": 165,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\n⊢ R ○ ∅ = ∅",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"False",
"SetRel",
"iff_false",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"Membership.mem"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 164,
"column": 75
} | {
"line": 164,
"column": 80
} | {
"line": 165,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\n⊢ R ○ ∅ = ∅",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"False",
"SetRel",
"iff_false",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"Membership.mem"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 164,
"column": 75
} | {
"line": 164,
"column": 80
} | {
"line": 165,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\n⊢ R ○ ∅ = ∅",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"False",
"SetRel",
"iff_false",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"Membership.mem"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 165,
"column": 75
} | {
"line": 165,
"column": 80
} | {
"line": 167,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nS : SetRel β γ\n⊢ ∅ ○ S = ∅",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"False",
"SetRel",
"iff_false",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"false_and",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 165,
"column": 75
} | {
"line": 165,
"column": 80
} | {
"line": 167,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nS : SetRel β γ\n⊢ ∅ ○ S = ∅",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"False",
"SetRel",
"iff_false",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"false_and",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 165,
"column": 75
} | {
"line": 165,
"column": 80
} | {
"line": 167,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nS : SetRel β γ\n⊢ ∅ ○ S = ∅",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"False",
"SetRel",
"iff_false",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"false_and",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 7
} | {
"line": 171,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\n⊢ R ○ Set.univ = {(a, _c) | a ∈ R.dom}",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"and_true",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 7
} | {
"line": 171,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\n⊢ R ○ Set.univ = {(a, _c) | a ∈ R.dom}",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"and_true",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 7
} | {
"line": 171,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\n⊢ R ○ Set.univ = {(a, _c) | a ∈ R.dom}",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"and_true",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 7
} | {
"line": 175,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nS : SetRel β γ\n⊢ Set.univ ○ S = {(_b, c) | c ∈ S.cod}",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel.cod",
"SetRel",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 7
} | {
"line": 175,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nS : SetRel β γ\n⊢ Set.univ ○ S = {(_b, c) | c ∈ S.cod}",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel.cod",
"SetRel",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 7
} | {
"line": 175,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nS : SetRel β γ\n⊢ Set.univ ○ S = {(_b, c) | c ∈ S.cod}",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel.cod",
"SetRel",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 175,
"column": 92
} | {
"line": 175,
"column": 97
} | {
"line": 176,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_5\nR : SetRel α β\nS : ι → SetRel β γ\n⊢ R ○ ⋃ i, S i = ⋃ i, R ○ S i",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Membe... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 175,
"column": 92
} | {
"line": 175,
"column": 97
} | {
"line": 176,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_5\nR : SetRel α β\nS : ι → SetRel β γ\n⊢ R ○ ⋃ i, S i = ⋃ i, R ○ S i",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Membe... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 175,
"column": 92
} | {
"line": 175,
"column": 97
} | {
"line": 176,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_5\nR : SetRel α β\nS : ι → SetRel β γ\n⊢ R ○ ⋃ i, S i = ⋃ i, R ○ S i",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Membe... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Exact.Basic | {
"line": 177,
"column": 6
} | {
"line": 177,
"column": 57
} | {
"line": 178,
"column": 4
} | [
{
"pp": "case mpr.mp\nM₁ : Type u_8\nM₂ : Type u_9\nM₃ : Type u_10\nN₁ : Type u_11\nN₂ : Type u_12\nN₃ : Type u_13\ninst✝⁵ : CommMonoid M₁\ninst✝⁴ : CommMonoid M₂\ninst✝³ : CommMonoid M₃\ninst✝² : CommMonoid N₁\ninst✝¹ : CommMonoid N₂\ninst✝ : CommMonoid N₃\nf : M₁ →* M₂\ng : M₂ →* M₃\nf' : N₁ →* N₂\ng' : N₂ →*... | [] | exact ⟨x₁, h₂.1 (by simpa only [comm₁₂] using hy₁)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Rel | {
"line": 176,
"column": 94
} | {
"line": 176,
"column": 99
} | {
"line": 177,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_5\nR : ι → SetRel α β\nS : SetRel β γ\n⊢ (⋃ i, R i) ○ S = ⋃ i, R i ○ S",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Mem... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 176,
"column": 94
} | {
"line": 176,
"column": 99
} | {
"line": 177,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_5\nR : ι → SetRel α β\nS : SetRel β γ\n⊢ (⋃ i, R i) ○ S = ⋃ i, R i ○ S",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Mem... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 176,
"column": 94
} | {
"line": 176,
"column": 99
} | {
"line": 177,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_5\nR : ι → SetRel α β\nS : SetRel β γ\n⊢ (⋃ i, R i) ○ S = ⋃ i, R i ○ S",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Mem... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 177,
"column": 92
} | {
"line": 177,
"column": 97
} | {
"line": 178,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\n𝒮 : Set (SetRel β γ)\n⊢ R ○ ⋃₀ 𝒮 = ⋃ S ∈ 𝒮, R ○ S",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Set.sUnion",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 177,
"column": 92
} | {
"line": 177,
"column": 97
} | {
"line": 178,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\n𝒮 : Set (SetRel β γ)\n⊢ R ○ ⋃₀ 𝒮 = ⋃ S ∈ 𝒮, R ○ S",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Set.sUnion",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 177,
"column": 92
} | {
"line": 177,
"column": 97
} | {
"line": 178,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\n𝒮 : Set (SetRel β γ)\n⊢ R ○ ⋃₀ 𝒮 = ⋃ S ∈ 𝒮, R ○ S",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Set.sUnion",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 178,
"column": 92
} | {
"line": 178,
"column": 97
} | {
"line": 180,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nℛ : Set (SetRel α β)\nS : SetRel β γ\n⊢ ⋃₀ ℛ ○ S = ⋃ R ∈ ℛ, R ○ S",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Set.sUnion",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 178,
"column": 92
} | {
"line": 178,
"column": 97
} | {
"line": 180,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nℛ : Set (SetRel α β)\nS : SetRel β γ\n⊢ ⋃₀ ℛ ○ S = ⋃ R ∈ ℛ, R ○ S",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Set.sUnion",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 178,
"column": 92
} | {
"line": 178,
"column": 97
} | {
"line": 180,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nℛ : Set (SetRel α β)\nS : SetRel β γ\n⊢ ⋃₀ ℛ ○ S = ⋃ R ∈ ℛ, R ○ S",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Set.sUnion",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 197,
"column": 37
} | {
"line": 197,
"column": 42
} | {
"line": 199,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nt₁ t₂ : Set β\nht : (t₁ ∩ t₂).Nonempty\ns : Set α\nu : Set γ\n⊢ s ×ˢ t₁ ○ t₂ ×ˢ u = s ×ˢ u",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Set.ext",
"Eq.mpr",
"SetRel",
"and_true",
"SPr... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 197,
"column": 37
} | {
"line": 197,
"column": 42
} | {
"line": 199,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nt₁ t₂ : Set β\nht : (t₁ ∩ t₂).Nonempty\ns : Set α\nu : Set γ\n⊢ s ×ˢ t₁ ○ t₂ ×ˢ u = s ×ˢ u",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Set.ext",
"Eq.mpr",
"SetRel",
"and_true",
"SPr... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 197,
"column": 37
} | {
"line": 197,
"column": 42
} | {
"line": 199,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nt₁ t₂ : Set β\nht : (t₁ ∩ t₂).Nonempty\ns : Set α\nu : Set γ\n⊢ s ×ˢ t₁ ○ t₂ ×ˢ u = s ×ˢ u",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Set.ext",
"Eq.mpr",
"SetRel",
"and_true",
"SPr... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 241,
"column": 54
} | {
"line": 241,
"column": 59
} | {
"line": 242,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ R.image ∅ = ∅",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"SetRel",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"false_and",
"Membership.mem",
"Exists",
"P... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 241,
"column": 54
} | {
"line": 241,
"column": 59
} | {
"line": 242,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ R.image ∅ = ∅",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"SetRel",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"false_and",
"Membership.mem",
"Exists",
"P... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 241,
"column": 54
} | {
"line": 241,
"column": 59
} | {
"line": 242,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ R.image ∅ = ∅",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"SetRel",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"false_and",
"Membership.mem",
"Exists",
"P... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 242,
"column": 60
} | {
"line": 242,
"column": 65
} | {
"line": 244,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ R.preimage ∅ = ∅",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"SetRel",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"false_and",
"Membership.mem",
"Exists",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 242,
"column": 60
} | {
"line": 242,
"column": 65
} | {
"line": 244,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ R.preimage ∅ = ∅",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"SetRel",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"false_and",
"Membership.mem",
"Exists",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 242,
"column": 60
} | {
"line": 242,
"column": 65
} | {
"line": 244,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ R.preimage ∅ = ∅",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"SetRel",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"false_and",
"Membership.mem",
"Exists",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 244,
"column": 61
} | {
"line": 244,
"column": 66
} | {
"line": 245,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ R.image Set.univ = R.cod",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
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"SetRel.cod",
"SetRel",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"Membership.mem",
"Exists",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 244,
"column": 61
} | {
"line": 244,
"column": 66
} | {
"line": 245,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ R.image Set.univ = R.cod",
"ppTerm": "?m.7",
"assigned": true,
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"SetRel.cod",
"SetRel",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"Membership.mem",
"Exists",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 244,
"column": 61
} | {
"line": 244,
"column": 66
} | {
"line": 245,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ R.image Set.univ = R.cod",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
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"SetRel.cod",
"SetRel",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"Membership.mem",
"Exists",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 245,
"column": 67
} | {
"line": 245,
"column": 72
} | {
"line": 247,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ R.preimage Set.univ = R.dom",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"Membership.mem",
"Exists",
"Prod.mk",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 245,
"column": 67
} | {
"line": 245,
"column": 72
} | {
"line": 247,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ R.preimage Set.univ = R.dom",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"Membership.mem",
"Exists",
"Prod.mk",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 245,
"column": 67
} | {
"line": 245,
"column": 72
} | {
"line": 247,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\n⊢ R.preimage Set.univ = R.dom",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"Membership.mem",
"Exists",
"Prod.mk",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 255,
"column": 70
} | {
"line": 255,
"column": 75
} | {
"line": 257,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\ns₁ s₂ : Set α\n⊢ R.image (s₁ ∪ s₂) = R.image s₁ ∪ R.image s₂",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"true_or",
"Membership.mem",
"Exists",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 255,
"column": 70
} | {
"line": 255,
"column": 75
} | {
"line": 257,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\ns₁ s₂ : Set α\n⊢ R.image (s₁ ∪ s₂) = R.image s₁ ∪ R.image s₂",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"true_or",
"Membership.mem",
"Exists",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 255,
"column": 70
} | {
"line": 255,
"column": 75
} | {
"line": 257,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\ns₁ s₂ : Set α\n⊢ R.image (s₁ ∪ s₂) = R.image s₁ ∪ R.image s₂",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"true_or",
"Membership.mem",
"Exists",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 258,
"column": 83
} | {
"line": 258,
"column": 88
} | {
"line": 260,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\nR : SetRel α β\ns : ι → Set α\n⊢ R.image (⋃ i, s i) = ⋃ i, R.image (s i)",
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"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Membership.m... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 258,
"column": 83
} | {
"line": 258,
"column": 88
} | {
"line": 260,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\nR : SetRel α β\ns : ι → Set α\n⊢ R.image (⋃ i, s i) = ⋃ i, R.image (s i)",
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"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Membership.m... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 258,
"column": 83
} | {
"line": 258,
"column": 88
} | {
"line": 260,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\nR : SetRel α β\ns : ι → Set α\n⊢ R.image (⋃ i, s i) = ⋃ i, R.image (s i)",
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"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Membership.m... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 261,
"column": 81
} | {
"line": 261,
"column": 86
} | {
"line": 263,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\nS : Set (Set α)\n⊢ R.image (⋃₀ S) = ⋃ s ∈ S, R.image s",
"ppTerm": "?m.17",
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"Set.ext",
"Eq.mpr",
"SetRel",
"Iff.of_eq",
"congrArg",
"Set.mem_iUnion._simp_1",
"Set.sUnio... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 261,
"column": 81
} | {
"line": 261,
"column": 86
} | {
"line": 263,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\nS : Set (Set α)\n⊢ R.image (⋃₀ S) = ⋃ s ∈ S, R.image s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"Iff.of_eq",
"congrArg",
"Set.mem_iUnion._simp_1",
"Set.sUnio... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 261,
"column": 81
} | {
"line": 261,
"column": 86
} | {
"line": 263,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\nS : Set (Set α)\n⊢ R.image (⋃₀ S) = ⋃ s ∈ S, R.image s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"Iff.of_eq",
"congrArg",
"Set.mem_iUnion._simp_1",
"Set.sUnio... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 264,
"column": 82
} | {
"line": 264,
"column": 87
} | {
"line": 266,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\nt₁ t₂ : Set β\n⊢ R.preimage (t₁ ∪ t₂) = R.preimage t₁ ∪ R.preimage t₂",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"true_or",
"Membership.mem",
"Exis... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 264,
"column": 82
} | {
"line": 264,
"column": 87
} | {
"line": 266,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\nt₁ t₂ : Set β\n⊢ R.preimage (t₁ ∪ t₂) = R.preimage t₁ ∪ R.preimage t₂",
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"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"true_or",
"Membership.mem",
"Exis... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 264,
"column": 82
} | {
"line": 264,
"column": 87
} | {
"line": 266,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\nt₁ t₂ : Set β\n⊢ R.preimage (t₁ ∪ t₂) = R.preimage t₁ ∪ R.preimage t₂",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"true_or",
"Membership.mem",
"Exis... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 267,
"column": 92
} | {
"line": 267,
"column": 97
} | {
"line": 269,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\nR : SetRel α β\nt : ι → Set β\n⊢ R.preimage (⋃ i, t i) = ⋃ i, R.preimage (t i)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Member... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 267,
"column": 92
} | {
"line": 267,
"column": 97
} | {
"line": 269,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\nR : SetRel α β\nt : ι → Set β\n⊢ R.preimage (⋃ i, t i) = ⋃ i, R.preimage (t i)",
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"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Member... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 267,
"column": 92
} | {
"line": 267,
"column": 97
} | {
"line": 269,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\nR : SetRel α β\nt : ι → Set β\n⊢ R.preimage (⋃ i, t i) = ⋃ i, R.preimage (t i)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Set.mem_iUnion._simp_1",
"Member... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Exact.Basic | {
"line": 268,
"column": 6
} | {
"line": 268,
"column": 16
} | {
"line": 268,
"column": 17
} | [
{
"pp": "R : Type u_8\nN : Type u_10\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nQ : Submodule R N\n⊢ Exact ⇑Q.subtype ⇑Q.mkQ",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Function.Exact",
"Eq.mpr",
"Submodule",
"Submodule.Quotient.instZeroQuot... | [
"R : Type u_8\nN : Type u_10\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nQ : Submodule R N\n⊢ Q.mkQ.ker = Q.subtype.range"
] | exact_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Rel | {
"line": 270,
"column": 90
} | {
"line": 270,
"column": 95
} | {
"line": 272,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\nT : Set (Set β)\n⊢ R.preimage (⋃₀ T) = ⋃ t ∈ T, R.preimage t",
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"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"Iff.of_eq",
"congrArg",
"Set.mem_iUnion._simp_1",
"Set... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 270,
"column": 90
} | {
"line": 270,
"column": 95
} | {
"line": 272,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\nT : Set (Set β)\n⊢ R.preimage (⋃₀ T) = ⋃ t ∈ T, R.preimage t",
"ppTerm": "?m.17",
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"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"Iff.of_eq",
"congrArg",
"Set.mem_iUnion._simp_1",
"Set... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 270,
"column": 90
} | {
"line": 270,
"column": 95
} | {
"line": 272,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α β\nT : Set (Set β)\n⊢ R.preimage (⋃₀ T) = ⋃ t ∈ T, R.preimage t",
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"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"Iff.of_eq",
"congrArg",
"Set.mem_iUnion._simp_1",
"Set... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 273,
"column": 47
} | {
"line": 273,
"column": 52
} | {
"line": 275,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\n⊢ SetRel.id.image s = s",
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"SetRel",
"congrArg",
"Membership.mem",
"Exists",
"Prod.mk",
"iff_self",
"funext",
"And",
"Iff"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 273,
"column": 47
} | {
"line": 273,
"column": 52
} | {
"line": 275,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\n⊢ SetRel.id.image s = s",
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"usedConstants": [
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"SetRel",
"congrArg",
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"Exists",
"Prod.mk",
"iff_self",
"funext",
"And",
"Iff"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 273,
"column": 47
} | {
"line": 273,
"column": 52
} | {
"line": 275,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\n⊢ SetRel.id.image s = s",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
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"SetRel.id",
"SetRel",
"congrArg",
"Membership.mem",
"Exists",
"Prod.mk",
"iff_self",
"funext",
"And",
"Iff"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 276,
"column": 53
} | {
"line": 276,
"column": 58
} | {
"line": 278,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\n⊢ SetRel.id.preimage s = s",
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"usedConstants": [
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"SetRel",
"congrArg",
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"Exists",
"Prod.mk",
"iff_self",
"exists_eq_right'._simp_1",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 276,
"column": 53
} | {
"line": 276,
"column": 58
} | {
"line": 278,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\n⊢ SetRel.id.preimage s = s",
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"Exists",
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"iff_self",
"exists_eq_right'._simp_1",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 276,
"column": 53
} | {
"line": 276,
"column": 58
} | {
"line": 278,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\n⊢ SetRel.id.preimage s = s",
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"iff_self",
"exists_eq_right'._simp_1",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 279,
"column": 63
} | {
"line": 279,
"column": 68
} | {
"line": 281,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\nS : SetRel β γ\ns : Set α\n⊢ (R ○ S).image s = S.image (R.image s)",
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"assigned": true,
"usedConstants": [
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"Eq.mpr",
"SetRel",
"congrArg",
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"Exists",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 279,
"column": 63
} | {
"line": 279,
"column": 68
} | {
"line": 281,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\nS : SetRel β γ\ns : Set α\n⊢ (R ○ S).image s = S.image (R.image s)",
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"assigned": true,
"usedConstants": [
"Set.ext",
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"congrArg",
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"Exists",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rel | {
"line": 279,
"column": 63
} | {
"line": 279,
"column": 68
} | {
"line": 281,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\nS : SetRel β γ\ns : Set α\n⊢ (R ○ S).image s = S.image (R.image s)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Membership.mem",
"Exists",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 282,
"column": 75
} | {
"line": 282,
"column": 80
} | {
"line": 284,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\nS : SetRel β γ\nu : Set γ\n⊢ (R ○ S).preimage u = R.preimage (S.preimage u)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Membership.mem",
"E... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Rel | {
"line": 282,
"column": 75
} | {
"line": 282,
"column": 80
} | {
"line": 284,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nR : SetRel α β\nS : SetRel β γ\nu : Set γ\n⊢ (R ○ S).preimage u = R.preimage (S.preimage u)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetRel",
"congrArg",
"Membership.mem",
"E... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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