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import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Basic
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Yoneda
import Mathlib.Tactic.ApplyFun
#align_import category_theory.limits.concrete_category from "leanprover-community/mathlib"@"c3019c79074b0619edb4b27553a91b2e82242395"
universe t w v u r
open CategoryTheory
namespace CategoryTheory.Limits
attribute [local instance] ConcreteCategory.instFunLike ConcreteCategory.hasCoeToSort
section Colimits
section
variable {C : Type u} [Category.{v} C] [ConcreteCategory.{t} C] {J : Type w} [Category.{r} J]
(F : J ⥤ C) [PreservesColimit F (forget C)]
theorem Concrete.from_union_surjective_of_isColimit {D : Cocone F} (hD : IsColimit D) :
let ff : (Σj : J, F.obj j) → D.pt := fun a => D.ι.app a.1 a.2
Function.Surjective ff := by
intro ff x
let E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
let hE : IsColimit E := isColimitOfPreserves (forget C) hD
obtain ⟨j, y, hy⟩ := Types.jointly_surjective_of_isColimit hE x
exact ⟨⟨j, y⟩, hy⟩
#align category_theory.limits.concrete.from_union_surjective_of_is_colimit CategoryTheory.Limits.Concrete.from_union_surjective_of_isColimit
theorem Concrete.isColimit_exists_rep {D : Cocone F} (hD : IsColimit D) (x : D.pt) :
∃ (j : J) (y : F.obj j), D.ι.app j y = x := by
obtain ⟨a, rfl⟩ := Concrete.from_union_surjective_of_isColimit F hD x
exact ⟨a.1, a.2, rfl⟩
#align category_theory.limits.concrete.is_colimit_exists_rep CategoryTheory.Limits.Concrete.isColimit_exists_rep
theorem Concrete.colimit_exists_rep [HasColimit F] (x : ↑(colimit F)) :
∃ (j : J) (y : F.obj j), colimit.ι F j y = x :=
Concrete.isColimit_exists_rep F (colimit.isColimit _) x
#align category_theory.limits.concrete.colimit_exists_rep CategoryTheory.Limits.Concrete.colimit_exists_rep
| Mathlib/CategoryTheory/Limits/ConcreteCategory.lean | 97 | 106 | theorem Concrete.isColimit_rep_eq_of_exists {D : Cocone F} {i j : J} (x : F.obj i) (y : F.obj j)
(h : ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y) :
D.ι.app i x = D.ι.app j y := by |
let E := (forget C).mapCocone D
obtain ⟨k, f, g, (hfg : (F ⋙ forget C).map f x = F.map g y)⟩ := h
let h1 : (F ⋙ forget C).map f ≫ E.ι.app k = E.ι.app i := E.ι.naturality f
let h2 : (F ⋙ forget C).map g ≫ E.ι.app k = E.ι.app j := E.ι.naturality g
show E.ι.app i x = E.ι.app j y
rw [← h1, types_comp_apply, hfg]
exact congrFun h2 y
| 7 | 1,096.633158 | 2 | 1.6 | 5 | 1,736 |
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Basic
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Yoneda
import Mathlib.Tactic.ApplyFun
#align_import category_theory.limits.concrete_category from "leanprover-community/mathlib"@"c3019c79074b0619edb4b27553a91b2e82242395"
universe t w v u r
open CategoryTheory
namespace CategoryTheory.Limits
attribute [local instance] ConcreteCategory.instFunLike ConcreteCategory.hasCoeToSort
section Colimits
section
variable {C : Type u} [Category.{v} C] [ConcreteCategory.{t} C] {J : Type w} [Category.{r} J]
(F : J ⥤ C) [PreservesColimit F (forget C)]
theorem Concrete.from_union_surjective_of_isColimit {D : Cocone F} (hD : IsColimit D) :
let ff : (Σj : J, F.obj j) → D.pt := fun a => D.ι.app a.1 a.2
Function.Surjective ff := by
intro ff x
let E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
let hE : IsColimit E := isColimitOfPreserves (forget C) hD
obtain ⟨j, y, hy⟩ := Types.jointly_surjective_of_isColimit hE x
exact ⟨⟨j, y⟩, hy⟩
#align category_theory.limits.concrete.from_union_surjective_of_is_colimit CategoryTheory.Limits.Concrete.from_union_surjective_of_isColimit
theorem Concrete.isColimit_exists_rep {D : Cocone F} (hD : IsColimit D) (x : D.pt) :
∃ (j : J) (y : F.obj j), D.ι.app j y = x := by
obtain ⟨a, rfl⟩ := Concrete.from_union_surjective_of_isColimit F hD x
exact ⟨a.1, a.2, rfl⟩
#align category_theory.limits.concrete.is_colimit_exists_rep CategoryTheory.Limits.Concrete.isColimit_exists_rep
theorem Concrete.colimit_exists_rep [HasColimit F] (x : ↑(colimit F)) :
∃ (j : J) (y : F.obj j), colimit.ι F j y = x :=
Concrete.isColimit_exists_rep F (colimit.isColimit _) x
#align category_theory.limits.concrete.colimit_exists_rep CategoryTheory.Limits.Concrete.colimit_exists_rep
theorem Concrete.isColimit_rep_eq_of_exists {D : Cocone F} {i j : J} (x : F.obj i) (y : F.obj j)
(h : ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y) :
D.ι.app i x = D.ι.app j y := by
let E := (forget C).mapCocone D
obtain ⟨k, f, g, (hfg : (F ⋙ forget C).map f x = F.map g y)⟩ := h
let h1 : (F ⋙ forget C).map f ≫ E.ι.app k = E.ι.app i := E.ι.naturality f
let h2 : (F ⋙ forget C).map g ≫ E.ι.app k = E.ι.app j := E.ι.naturality g
show E.ι.app i x = E.ι.app j y
rw [← h1, types_comp_apply, hfg]
exact congrFun h2 y
#align category_theory.limits.concrete.is_colimit_rep_eq_of_exists CategoryTheory.Limits.Concrete.isColimit_rep_eq_of_exists
theorem Concrete.colimit_rep_eq_of_exists [HasColimit F] {i j : J} (x : F.obj i) (y : F.obj j)
(h : ∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y) :
colimit.ι F i x = colimit.ι F j y :=
Concrete.isColimit_rep_eq_of_exists F x y h
#align category_theory.limits.concrete.colimit_rep_eq_of_exists CategoryTheory.Limits.Concrete.colimit_rep_eq_of_exists
end
section FilteredColimits
variable {C : Type u} [Category.{v} C] [ConcreteCategory.{max t w} C] {J : Type w} [Category.{r} J]
(F : J ⥤ C) [PreservesColimit F (forget C)] [IsFiltered J]
| Mathlib/CategoryTheory/Limits/ConcreteCategory.lean | 122 | 127 | theorem Concrete.isColimit_exists_of_rep_eq {D : Cocone F} {i j : J} (hD : IsColimit D)
(x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :
∃ (k : _) (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y := by |
let E := (forget C).mapCocone D
let hE : IsColimit E := isColimitOfPreserves _ hD
exact (Types.FilteredColimit.isColimit_eq_iff (F ⋙ forget C) hE).mp h
| 3 | 20.085537 | 1 | 1.6 | 5 | 1,736 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Tactic.Linarith
#align_import combinatorics.simple_graph.acyclic from "leanprover-community/mathlib"@"b07688016d62f81d14508ff339ea3415558d6353"
universe u v
namespace SimpleGraph
open Walk
variable {V : Type u} (G : SimpleGraph V)
def IsAcyclic : Prop := ∀ ⦃v : V⦄ (c : G.Walk v v), ¬c.IsCycle
#align simple_graph.is_acyclic SimpleGraph.IsAcyclic
@[mk_iff]
structure IsTree : Prop where
protected isConnected : G.Connected
protected IsAcyclic : G.IsAcyclic
#align simple_graph.is_tree SimpleGraph.IsTree
variable {G}
@[simp] lemma isAcyclic_bot : IsAcyclic (⊥ : SimpleGraph V) := fun _a _w hw ↦ hw.ne_bot rfl
| Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 68 | 80 | theorem isAcyclic_iff_forall_adj_isBridge :
G.IsAcyclic ↔ ∀ ⦃v w : V⦄, G.Adj v w → G.IsBridge s(v, w) := by |
simp_rw [isBridge_iff_adj_and_forall_cycle_not_mem]
constructor
· intro ha v w hvw
apply And.intro hvw
intro u p hp
cases ha p hp
· rintro hb v (_ | ⟨ha, p⟩) hp
· exact hp.not_of_nil
· apply (hb ha).2 _ hp
rw [Walk.edges_cons]
apply List.mem_cons_self
| 11 | 59,874.141715 | 2 | 1.6 | 5 | 1,737 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Tactic.Linarith
#align_import combinatorics.simple_graph.acyclic from "leanprover-community/mathlib"@"b07688016d62f81d14508ff339ea3415558d6353"
universe u v
namespace SimpleGraph
open Walk
variable {V : Type u} (G : SimpleGraph V)
def IsAcyclic : Prop := ∀ ⦃v : V⦄ (c : G.Walk v v), ¬c.IsCycle
#align simple_graph.is_acyclic SimpleGraph.IsAcyclic
@[mk_iff]
structure IsTree : Prop where
protected isConnected : G.Connected
protected IsAcyclic : G.IsAcyclic
#align simple_graph.is_tree SimpleGraph.IsTree
variable {G}
@[simp] lemma isAcyclic_bot : IsAcyclic (⊥ : SimpleGraph V) := fun _a _w hw ↦ hw.ne_bot rfl
theorem isAcyclic_iff_forall_adj_isBridge :
G.IsAcyclic ↔ ∀ ⦃v w : V⦄, G.Adj v w → G.IsBridge s(v, w) := by
simp_rw [isBridge_iff_adj_and_forall_cycle_not_mem]
constructor
· intro ha v w hvw
apply And.intro hvw
intro u p hp
cases ha p hp
· rintro hb v (_ | ⟨ha, p⟩) hp
· exact hp.not_of_nil
· apply (hb ha).2 _ hp
rw [Walk.edges_cons]
apply List.mem_cons_self
#align simple_graph.is_acyclic_iff_forall_adj_is_bridge SimpleGraph.isAcyclic_iff_forall_adj_isBridge
| Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 83 | 85 | theorem isAcyclic_iff_forall_edge_isBridge :
G.IsAcyclic ↔ ∀ ⦃e⦄, e ∈ (G.edgeSet) → G.IsBridge e := by |
simp [isAcyclic_iff_forall_adj_isBridge, Sym2.forall]
| 1 | 2.718282 | 0 | 1.6 | 5 | 1,737 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Tactic.Linarith
#align_import combinatorics.simple_graph.acyclic from "leanprover-community/mathlib"@"b07688016d62f81d14508ff339ea3415558d6353"
universe u v
namespace SimpleGraph
open Walk
variable {V : Type u} (G : SimpleGraph V)
def IsAcyclic : Prop := ∀ ⦃v : V⦄ (c : G.Walk v v), ¬c.IsCycle
#align simple_graph.is_acyclic SimpleGraph.IsAcyclic
@[mk_iff]
structure IsTree : Prop where
protected isConnected : G.Connected
protected IsAcyclic : G.IsAcyclic
#align simple_graph.is_tree SimpleGraph.IsTree
variable {G}
@[simp] lemma isAcyclic_bot : IsAcyclic (⊥ : SimpleGraph V) := fun _a _w hw ↦ hw.ne_bot rfl
theorem isAcyclic_iff_forall_adj_isBridge :
G.IsAcyclic ↔ ∀ ⦃v w : V⦄, G.Adj v w → G.IsBridge s(v, w) := by
simp_rw [isBridge_iff_adj_and_forall_cycle_not_mem]
constructor
· intro ha v w hvw
apply And.intro hvw
intro u p hp
cases ha p hp
· rintro hb v (_ | ⟨ha, p⟩) hp
· exact hp.not_of_nil
· apply (hb ha).2 _ hp
rw [Walk.edges_cons]
apply List.mem_cons_self
#align simple_graph.is_acyclic_iff_forall_adj_is_bridge SimpleGraph.isAcyclic_iff_forall_adj_isBridge
theorem isAcyclic_iff_forall_edge_isBridge :
G.IsAcyclic ↔ ∀ ⦃e⦄, e ∈ (G.edgeSet) → G.IsBridge e := by
simp [isAcyclic_iff_forall_adj_isBridge, Sym2.forall]
#align simple_graph.is_acyclic_iff_forall_edge_is_bridge SimpleGraph.isAcyclic_iff_forall_edge_isBridge
| Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 88 | 115 | theorem IsAcyclic.path_unique {G : SimpleGraph V} (h : G.IsAcyclic) {v w : V} (p q : G.Path v w) :
p = q := by |
obtain ⟨p, hp⟩ := p
obtain ⟨q, hq⟩ := q
rw [Subtype.mk.injEq]
induction p with
| nil =>
cases (Walk.isPath_iff_eq_nil _).mp hq
rfl
| cons ph p ih =>
rw [isAcyclic_iff_forall_adj_isBridge] at h
specialize h ph
rw [isBridge_iff_adj_and_forall_walk_mem_edges] at h
replace h := h.2 (q.append p.reverse)
simp only [Walk.edges_append, Walk.edges_reverse, List.mem_append, List.mem_reverse] at h
cases' h with h h
· cases q with
| nil => simp [Walk.isPath_def] at hp
| cons _ q =>
rw [Walk.cons_isPath_iff] at hp hq
simp only [Walk.edges_cons, List.mem_cons, Sym2.eq_iff, true_and] at h
rcases h with (⟨h, rfl⟩ | ⟨rfl, rfl⟩) | h
· cases ih hp.1 q hq.1
rfl
· simp at hq
· exact absurd (Walk.fst_mem_support_of_mem_edges _ h) hq.2
· rw [Walk.cons_isPath_iff] at hp
exact absurd (Walk.fst_mem_support_of_mem_edges _ h) hp.2
| 26 | 195,729,609,428.83878 | 2 | 1.6 | 5 | 1,737 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Tactic.Linarith
#align_import combinatorics.simple_graph.acyclic from "leanprover-community/mathlib"@"b07688016d62f81d14508ff339ea3415558d6353"
universe u v
namespace SimpleGraph
open Walk
variable {V : Type u} (G : SimpleGraph V)
def IsAcyclic : Prop := ∀ ⦃v : V⦄ (c : G.Walk v v), ¬c.IsCycle
#align simple_graph.is_acyclic SimpleGraph.IsAcyclic
@[mk_iff]
structure IsTree : Prop where
protected isConnected : G.Connected
protected IsAcyclic : G.IsAcyclic
#align simple_graph.is_tree SimpleGraph.IsTree
variable {G}
@[simp] lemma isAcyclic_bot : IsAcyclic (⊥ : SimpleGraph V) := fun _a _w hw ↦ hw.ne_bot rfl
theorem isAcyclic_iff_forall_adj_isBridge :
G.IsAcyclic ↔ ∀ ⦃v w : V⦄, G.Adj v w → G.IsBridge s(v, w) := by
simp_rw [isBridge_iff_adj_and_forall_cycle_not_mem]
constructor
· intro ha v w hvw
apply And.intro hvw
intro u p hp
cases ha p hp
· rintro hb v (_ | ⟨ha, p⟩) hp
· exact hp.not_of_nil
· apply (hb ha).2 _ hp
rw [Walk.edges_cons]
apply List.mem_cons_self
#align simple_graph.is_acyclic_iff_forall_adj_is_bridge SimpleGraph.isAcyclic_iff_forall_adj_isBridge
theorem isAcyclic_iff_forall_edge_isBridge :
G.IsAcyclic ↔ ∀ ⦃e⦄, e ∈ (G.edgeSet) → G.IsBridge e := by
simp [isAcyclic_iff_forall_adj_isBridge, Sym2.forall]
#align simple_graph.is_acyclic_iff_forall_edge_is_bridge SimpleGraph.isAcyclic_iff_forall_edge_isBridge
theorem IsAcyclic.path_unique {G : SimpleGraph V} (h : G.IsAcyclic) {v w : V} (p q : G.Path v w) :
p = q := by
obtain ⟨p, hp⟩ := p
obtain ⟨q, hq⟩ := q
rw [Subtype.mk.injEq]
induction p with
| nil =>
cases (Walk.isPath_iff_eq_nil _).mp hq
rfl
| cons ph p ih =>
rw [isAcyclic_iff_forall_adj_isBridge] at h
specialize h ph
rw [isBridge_iff_adj_and_forall_walk_mem_edges] at h
replace h := h.2 (q.append p.reverse)
simp only [Walk.edges_append, Walk.edges_reverse, List.mem_append, List.mem_reverse] at h
cases' h with h h
· cases q with
| nil => simp [Walk.isPath_def] at hp
| cons _ q =>
rw [Walk.cons_isPath_iff] at hp hq
simp only [Walk.edges_cons, List.mem_cons, Sym2.eq_iff, true_and] at h
rcases h with (⟨h, rfl⟩ | ⟨rfl, rfl⟩) | h
· cases ih hp.1 q hq.1
rfl
· simp at hq
· exact absurd (Walk.fst_mem_support_of_mem_edges _ h) hq.2
· rw [Walk.cons_isPath_iff] at hp
exact absurd (Walk.fst_mem_support_of_mem_edges _ h) hp.2
#align simple_graph.is_acyclic.path_unique SimpleGraph.IsAcyclic.path_unique
| Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 118 | 127 | theorem isAcyclic_of_path_unique (h : ∀ (v w : V) (p q : G.Path v w), p = q) : G.IsAcyclic := by |
intro v c hc
simp only [Walk.isCycle_def, Ne] at hc
cases c with
| nil => cases hc.2.1 rfl
| cons ha c' =>
simp only [Walk.cons_isTrail_iff, Walk.support_cons, List.tail_cons, true_and_iff] at hc
specialize h _ _ ⟨c', by simp only [Walk.isPath_def, hc.2]⟩ (Path.singleton ha.symm)
rw [Path.singleton, Subtype.mk.injEq] at h
simp [h] at hc
| 9 | 8,103.083928 | 2 | 1.6 | 5 | 1,737 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Tactic.Linarith
#align_import combinatorics.simple_graph.acyclic from "leanprover-community/mathlib"@"b07688016d62f81d14508ff339ea3415558d6353"
universe u v
namespace SimpleGraph
open Walk
variable {V : Type u} (G : SimpleGraph V)
def IsAcyclic : Prop := ∀ ⦃v : V⦄ (c : G.Walk v v), ¬c.IsCycle
#align simple_graph.is_acyclic SimpleGraph.IsAcyclic
@[mk_iff]
structure IsTree : Prop where
protected isConnected : G.Connected
protected IsAcyclic : G.IsAcyclic
#align simple_graph.is_tree SimpleGraph.IsTree
variable {G}
@[simp] lemma isAcyclic_bot : IsAcyclic (⊥ : SimpleGraph V) := fun _a _w hw ↦ hw.ne_bot rfl
theorem isAcyclic_iff_forall_adj_isBridge :
G.IsAcyclic ↔ ∀ ⦃v w : V⦄, G.Adj v w → G.IsBridge s(v, w) := by
simp_rw [isBridge_iff_adj_and_forall_cycle_not_mem]
constructor
· intro ha v w hvw
apply And.intro hvw
intro u p hp
cases ha p hp
· rintro hb v (_ | ⟨ha, p⟩) hp
· exact hp.not_of_nil
· apply (hb ha).2 _ hp
rw [Walk.edges_cons]
apply List.mem_cons_self
#align simple_graph.is_acyclic_iff_forall_adj_is_bridge SimpleGraph.isAcyclic_iff_forall_adj_isBridge
theorem isAcyclic_iff_forall_edge_isBridge :
G.IsAcyclic ↔ ∀ ⦃e⦄, e ∈ (G.edgeSet) → G.IsBridge e := by
simp [isAcyclic_iff_forall_adj_isBridge, Sym2.forall]
#align simple_graph.is_acyclic_iff_forall_edge_is_bridge SimpleGraph.isAcyclic_iff_forall_edge_isBridge
theorem IsAcyclic.path_unique {G : SimpleGraph V} (h : G.IsAcyclic) {v w : V} (p q : G.Path v w) :
p = q := by
obtain ⟨p, hp⟩ := p
obtain ⟨q, hq⟩ := q
rw [Subtype.mk.injEq]
induction p with
| nil =>
cases (Walk.isPath_iff_eq_nil _).mp hq
rfl
| cons ph p ih =>
rw [isAcyclic_iff_forall_adj_isBridge] at h
specialize h ph
rw [isBridge_iff_adj_and_forall_walk_mem_edges] at h
replace h := h.2 (q.append p.reverse)
simp only [Walk.edges_append, Walk.edges_reverse, List.mem_append, List.mem_reverse] at h
cases' h with h h
· cases q with
| nil => simp [Walk.isPath_def] at hp
| cons _ q =>
rw [Walk.cons_isPath_iff] at hp hq
simp only [Walk.edges_cons, List.mem_cons, Sym2.eq_iff, true_and] at h
rcases h with (⟨h, rfl⟩ | ⟨rfl, rfl⟩) | h
· cases ih hp.1 q hq.1
rfl
· simp at hq
· exact absurd (Walk.fst_mem_support_of_mem_edges _ h) hq.2
· rw [Walk.cons_isPath_iff] at hp
exact absurd (Walk.fst_mem_support_of_mem_edges _ h) hp.2
#align simple_graph.is_acyclic.path_unique SimpleGraph.IsAcyclic.path_unique
theorem isAcyclic_of_path_unique (h : ∀ (v w : V) (p q : G.Path v w), p = q) : G.IsAcyclic := by
intro v c hc
simp only [Walk.isCycle_def, Ne] at hc
cases c with
| nil => cases hc.2.1 rfl
| cons ha c' =>
simp only [Walk.cons_isTrail_iff, Walk.support_cons, List.tail_cons, true_and_iff] at hc
specialize h _ _ ⟨c', by simp only [Walk.isPath_def, hc.2]⟩ (Path.singleton ha.symm)
rw [Path.singleton, Subtype.mk.injEq] at h
simp [h] at hc
#align simple_graph.is_acyclic_of_path_unique SimpleGraph.isAcyclic_of_path_unique
theorem isAcyclic_iff_path_unique : G.IsAcyclic ↔ ∀ ⦃v w : V⦄ (p q : G.Path v w), p = q :=
⟨IsAcyclic.path_unique, isAcyclic_of_path_unique⟩
#align simple_graph.is_acyclic_iff_path_unique SimpleGraph.isAcyclic_iff_path_unique
| Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 134 | 154 | theorem isTree_iff_existsUnique_path :
G.IsTree ↔ Nonempty V ∧ ∀ v w : V, ∃! p : G.Walk v w, p.IsPath := by |
classical
rw [isTree_iff, isAcyclic_iff_path_unique]
constructor
· rintro ⟨hc, hu⟩
refine ⟨hc.nonempty, ?_⟩
intro v w
let q := (hc v w).some.toPath
use q
simp only [true_and_iff, Path.isPath]
intro p hp
specialize hu ⟨p, hp⟩ q
exact Subtype.ext_iff.mp hu
· rintro ⟨hV, h⟩
refine ⟨Connected.mk ?_, ?_⟩
· intro v w
obtain ⟨p, _⟩ := h v w
exact p.reachable
· rintro v w ⟨p, hp⟩ ⟨q, hq⟩
simp only [ExistsUnique.unique (h v w) hp hq]
| 19 | 178,482,300.963187 | 2 | 1.6 | 5 | 1,737 |
import Mathlib.Data.Real.Pi.Bounds
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
-- TODO. Rewrite some of the FLT results on the disciminant using the definitions and results of
-- this file
namespace NumberField
open FiniteDimensional NumberField NumberField.InfinitePlace Matrix
open scoped Classical Real nonZeroDivisors
variable (K : Type*) [Field K] [NumberField K]
noncomputable abbrev discr : ℤ := Algebra.discr ℤ (RingOfIntegers.basis K)
theorem coe_discr : (discr K : ℚ) = Algebra.discr ℚ (integralBasis K) :=
(Algebra.discr_localizationLocalization ℤ _ K (RingOfIntegers.basis K)).symm
| Mathlib/NumberTheory/NumberField/Discriminant.lean | 46 | 48 | theorem discr_ne_zero : discr K ≠ 0 := by |
rw [← (Int.cast_injective (α := ℚ)).ne_iff, coe_discr]
exact Algebra.discr_not_zero_of_basis ℚ (integralBasis K)
| 2 | 7.389056 | 1 | 1.6 | 5 | 1,738 |
import Mathlib.Data.Real.Pi.Bounds
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
-- TODO. Rewrite some of the FLT results on the disciminant using the definitions and results of
-- this file
namespace NumberField
open FiniteDimensional NumberField NumberField.InfinitePlace Matrix
open scoped Classical Real nonZeroDivisors
variable (K : Type*) [Field K] [NumberField K]
noncomputable abbrev discr : ℤ := Algebra.discr ℤ (RingOfIntegers.basis K)
theorem coe_discr : (discr K : ℚ) = Algebra.discr ℚ (integralBasis K) :=
(Algebra.discr_localizationLocalization ℤ _ K (RingOfIntegers.basis K)).symm
theorem discr_ne_zero : discr K ≠ 0 := by
rw [← (Int.cast_injective (α := ℚ)).ne_iff, coe_discr]
exact Algebra.discr_not_zero_of_basis ℚ (integralBasis K)
| Mathlib/NumberTheory/NumberField/Discriminant.lean | 50 | 53 | theorem discr_eq_discr {ι : Type*} [Fintype ι] [DecidableEq ι] (b : Basis ι ℤ (𝓞 K)) :
Algebra.discr ℤ b = discr K := by |
let b₀ := Basis.reindex (RingOfIntegers.basis K) (Basis.indexEquiv (RingOfIntegers.basis K) b)
rw [Algebra.discr_eq_discr (𝓞 K) b b₀, Basis.coe_reindex, Algebra.discr_reindex]
| 2 | 7.389056 | 1 | 1.6 | 5 | 1,738 |
import Mathlib.Data.Real.Pi.Bounds
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
-- TODO. Rewrite some of the FLT results on the disciminant using the definitions and results of
-- this file
namespace NumberField
open FiniteDimensional NumberField NumberField.InfinitePlace Matrix
open scoped Classical Real nonZeroDivisors
variable (K : Type*) [Field K] [NumberField K]
noncomputable abbrev discr : ℤ := Algebra.discr ℤ (RingOfIntegers.basis K)
theorem coe_discr : (discr K : ℚ) = Algebra.discr ℚ (integralBasis K) :=
(Algebra.discr_localizationLocalization ℤ _ K (RingOfIntegers.basis K)).symm
theorem discr_ne_zero : discr K ≠ 0 := by
rw [← (Int.cast_injective (α := ℚ)).ne_iff, coe_discr]
exact Algebra.discr_not_zero_of_basis ℚ (integralBasis K)
theorem discr_eq_discr {ι : Type*} [Fintype ι] [DecidableEq ι] (b : Basis ι ℤ (𝓞 K)) :
Algebra.discr ℤ b = discr K := by
let b₀ := Basis.reindex (RingOfIntegers.basis K) (Basis.indexEquiv (RingOfIntegers.basis K) b)
rw [Algebra.discr_eq_discr (𝓞 K) b b₀, Basis.coe_reindex, Algebra.discr_reindex]
| Mathlib/NumberTheory/NumberField/Discriminant.lean | 55 | 66 | theorem discr_eq_discr_of_algEquiv {L : Type*} [Field L] [NumberField L] (f : K ≃ₐ[ℚ] L) :
discr K = discr L := by |
let f₀ : 𝓞 K ≃ₗ[ℤ] 𝓞 L := (f.restrictScalars ℤ).mapIntegralClosure.toLinearEquiv
rw [← Rat.intCast_inj, coe_discr, Algebra.discr_eq_discr_of_algEquiv (integralBasis K) f,
← discr_eq_discr L ((RingOfIntegers.basis K).map f₀)]
change _ = algebraMap ℤ ℚ _
rw [← Algebra.discr_localizationLocalization ℤ (nonZeroDivisors ℤ) L]
congr
ext
simp only [Function.comp_apply, integralBasis_apply, Basis.localizationLocalization_apply,
Basis.map_apply]
rfl
| 10 | 22,026.465795 | 2 | 1.6 | 5 | 1,738 |
import Mathlib.Data.Real.Pi.Bounds
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
-- TODO. Rewrite some of the FLT results on the disciminant using the definitions and results of
-- this file
namespace NumberField
open FiniteDimensional NumberField NumberField.InfinitePlace Matrix
open scoped Classical Real nonZeroDivisors
variable (K : Type*) [Field K] [NumberField K]
noncomputable abbrev discr : ℤ := Algebra.discr ℤ (RingOfIntegers.basis K)
theorem coe_discr : (discr K : ℚ) = Algebra.discr ℚ (integralBasis K) :=
(Algebra.discr_localizationLocalization ℤ _ K (RingOfIntegers.basis K)).symm
theorem discr_ne_zero : discr K ≠ 0 := by
rw [← (Int.cast_injective (α := ℚ)).ne_iff, coe_discr]
exact Algebra.discr_not_zero_of_basis ℚ (integralBasis K)
theorem discr_eq_discr {ι : Type*} [Fintype ι] [DecidableEq ι] (b : Basis ι ℤ (𝓞 K)) :
Algebra.discr ℤ b = discr K := by
let b₀ := Basis.reindex (RingOfIntegers.basis K) (Basis.indexEquiv (RingOfIntegers.basis K) b)
rw [Algebra.discr_eq_discr (𝓞 K) b b₀, Basis.coe_reindex, Algebra.discr_reindex]
theorem discr_eq_discr_of_algEquiv {L : Type*} [Field L] [NumberField L] (f : K ≃ₐ[ℚ] L) :
discr K = discr L := by
let f₀ : 𝓞 K ≃ₗ[ℤ] 𝓞 L := (f.restrictScalars ℤ).mapIntegralClosure.toLinearEquiv
rw [← Rat.intCast_inj, coe_discr, Algebra.discr_eq_discr_of_algEquiv (integralBasis K) f,
← discr_eq_discr L ((RingOfIntegers.basis K).map f₀)]
change _ = algebraMap ℤ ℚ _
rw [← Algebra.discr_localizationLocalization ℤ (nonZeroDivisors ℤ) L]
congr
ext
simp only [Function.comp_apply, integralBasis_apply, Basis.localizationLocalization_apply,
Basis.map_apply]
rfl
open MeasureTheory MeasureTheory.Measure Zspan NumberField.mixedEmbedding
NumberField.InfinitePlace ENNReal NNReal Complex
| Mathlib/NumberTheory/NumberField/Discriminant.lean | 71 | 103 | theorem _root_.NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis :
volume (fundamentalDomain (latticeBasis K)) =
(2 : ℝ≥0∞)⁻¹ ^ NrComplexPlaces K * sqrt ‖discr K‖₊ := by |
let f : Module.Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
(canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ _)
let e : (index K) ≃ Module.Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
let M := (mixedEmbedding.stdBasis K).toMatrix ((latticeBasis K).reindex e.symm)
let N := Algebra.embeddingsMatrixReindex ℚ ℂ (integralBasis K ∘ f.symm)
RingHom.equivRatAlgHom
suffices M.map Complex.ofReal = (matrixToStdBasis K) *
(Matrix.reindex (indexEquiv K).symm (indexEquiv K).symm N).transpose by
calc volume (fundamentalDomain (latticeBasis K))
_ = ‖((mixedEmbedding.stdBasis K).toMatrix ((latticeBasis K).reindex e.symm)).det‖₊ := by
rw [← fundamentalDomain_reindex _ e.symm, ← norm_toNNReal, measure_fundamentalDomain
((latticeBasis K).reindex e.symm), volume_fundamentalDomain_stdBasis, mul_one]
rfl
_ = ‖(matrixToStdBasis K).det * N.det‖₊ := by
rw [← nnnorm_real, ← ofReal_eq_coe, RingHom.map_det, RingHom.mapMatrix_apply, this,
det_mul, det_transpose, det_reindex_self]
_ = (2 : ℝ≥0∞)⁻¹ ^ Fintype.card {w : InfinitePlace K // IsComplex w} * sqrt ‖N.det ^ 2‖₊ := by
have : ‖Complex.I‖₊ = 1 := by rw [← norm_toNNReal, norm_eq_abs, abs_I, Real.toNNReal_one]
rw [det_matrixToStdBasis, nnnorm_mul, nnnorm_pow, nnnorm_mul, this, mul_one, nnnorm_inv,
coe_mul, ENNReal.coe_pow, ← norm_toNNReal, RCLike.norm_two, Real.toNNReal_ofNat,
coe_inv two_ne_zero, coe_ofNat, nnnorm_pow, NNReal.sqrt_sq]
_ = (2 : ℝ≥0∞)⁻¹ ^ Fintype.card { w // IsComplex w } * NNReal.sqrt ‖discr K‖₊ := by
rw [← Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two, Algebra.discr_reindex,
← coe_discr, map_intCast, ← Complex.nnnorm_int]
ext : 2
dsimp only [M]
rw [Matrix.map_apply, Basis.toMatrix_apply, Basis.coe_reindex, Function.comp_apply,
Equiv.symm_symm, latticeBasis_apply, ← commMap_canonical_eq_mixed, Complex.ofReal_eq_coe,
stdBasis_repr_eq_matrixToStdBasis_mul K _ (fun _ => rfl)]
rfl
| 30 | 10,686,474,581,524.463 | 2 | 1.6 | 5 | 1,738 |
import Mathlib.Data.Real.Pi.Bounds
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
-- TODO. Rewrite some of the FLT results on the disciminant using the definitions and results of
-- this file
namespace NumberField
open FiniteDimensional NumberField NumberField.InfinitePlace Matrix
open scoped Classical Real nonZeroDivisors
variable (K : Type*) [Field K] [NumberField K]
noncomputable abbrev discr : ℤ := Algebra.discr ℤ (RingOfIntegers.basis K)
theorem coe_discr : (discr K : ℚ) = Algebra.discr ℚ (integralBasis K) :=
(Algebra.discr_localizationLocalization ℤ _ K (RingOfIntegers.basis K)).symm
theorem discr_ne_zero : discr K ≠ 0 := by
rw [← (Int.cast_injective (α := ℚ)).ne_iff, coe_discr]
exact Algebra.discr_not_zero_of_basis ℚ (integralBasis K)
theorem discr_eq_discr {ι : Type*} [Fintype ι] [DecidableEq ι] (b : Basis ι ℤ (𝓞 K)) :
Algebra.discr ℤ b = discr K := by
let b₀ := Basis.reindex (RingOfIntegers.basis K) (Basis.indexEquiv (RingOfIntegers.basis K) b)
rw [Algebra.discr_eq_discr (𝓞 K) b b₀, Basis.coe_reindex, Algebra.discr_reindex]
theorem discr_eq_discr_of_algEquiv {L : Type*} [Field L] [NumberField L] (f : K ≃ₐ[ℚ] L) :
discr K = discr L := by
let f₀ : 𝓞 K ≃ₗ[ℤ] 𝓞 L := (f.restrictScalars ℤ).mapIntegralClosure.toLinearEquiv
rw [← Rat.intCast_inj, coe_discr, Algebra.discr_eq_discr_of_algEquiv (integralBasis K) f,
← discr_eq_discr L ((RingOfIntegers.basis K).map f₀)]
change _ = algebraMap ℤ ℚ _
rw [← Algebra.discr_localizationLocalization ℤ (nonZeroDivisors ℤ) L]
congr
ext
simp only [Function.comp_apply, integralBasis_apply, Basis.localizationLocalization_apply,
Basis.map_apply]
rfl
open MeasureTheory MeasureTheory.Measure Zspan NumberField.mixedEmbedding
NumberField.InfinitePlace ENNReal NNReal Complex
theorem _root_.NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis :
volume (fundamentalDomain (latticeBasis K)) =
(2 : ℝ≥0∞)⁻¹ ^ NrComplexPlaces K * sqrt ‖discr K‖₊ := by
let f : Module.Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
(canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ _)
let e : (index K) ≃ Module.Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
let M := (mixedEmbedding.stdBasis K).toMatrix ((latticeBasis K).reindex e.symm)
let N := Algebra.embeddingsMatrixReindex ℚ ℂ (integralBasis K ∘ f.symm)
RingHom.equivRatAlgHom
suffices M.map Complex.ofReal = (matrixToStdBasis K) *
(Matrix.reindex (indexEquiv K).symm (indexEquiv K).symm N).transpose by
calc volume (fundamentalDomain (latticeBasis K))
_ = ‖((mixedEmbedding.stdBasis K).toMatrix ((latticeBasis K).reindex e.symm)).det‖₊ := by
rw [← fundamentalDomain_reindex _ e.symm, ← norm_toNNReal, measure_fundamentalDomain
((latticeBasis K).reindex e.symm), volume_fundamentalDomain_stdBasis, mul_one]
rfl
_ = ‖(matrixToStdBasis K).det * N.det‖₊ := by
rw [← nnnorm_real, ← ofReal_eq_coe, RingHom.map_det, RingHom.mapMatrix_apply, this,
det_mul, det_transpose, det_reindex_self]
_ = (2 : ℝ≥0∞)⁻¹ ^ Fintype.card {w : InfinitePlace K // IsComplex w} * sqrt ‖N.det ^ 2‖₊ := by
have : ‖Complex.I‖₊ = 1 := by rw [← norm_toNNReal, norm_eq_abs, abs_I, Real.toNNReal_one]
rw [det_matrixToStdBasis, nnnorm_mul, nnnorm_pow, nnnorm_mul, this, mul_one, nnnorm_inv,
coe_mul, ENNReal.coe_pow, ← norm_toNNReal, RCLike.norm_two, Real.toNNReal_ofNat,
coe_inv two_ne_zero, coe_ofNat, nnnorm_pow, NNReal.sqrt_sq]
_ = (2 : ℝ≥0∞)⁻¹ ^ Fintype.card { w // IsComplex w } * NNReal.sqrt ‖discr K‖₊ := by
rw [← Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two, Algebra.discr_reindex,
← coe_discr, map_intCast, ← Complex.nnnorm_int]
ext : 2
dsimp only [M]
rw [Matrix.map_apply, Basis.toMatrix_apply, Basis.coe_reindex, Function.comp_apply,
Equiv.symm_symm, latticeBasis_apply, ← commMap_canonical_eq_mixed, Complex.ofReal_eq_coe,
stdBasis_repr_eq_matrixToStdBasis_mul K _ (fun _ => rfl)]
rfl
| Mathlib/NumberTheory/NumberField/Discriminant.lean | 105 | 151 | theorem exists_ne_zero_mem_ideal_of_norm_le_mul_sqrt_discr (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ) :
∃ a ∈ (I : FractionalIdeal (𝓞 K)⁰ K), a ≠ 0 ∧
|Algebra.norm ℚ (a:K)| ≤ FractionalIdeal.absNorm I.1 * (4 / π) ^ NrComplexPlaces K *
(finrank ℚ K).factorial / (finrank ℚ K) ^ (finrank ℚ K) * Real.sqrt |discr K| := by |
-- The smallest possible value for `exists_ne_zero_mem_ideal_of_norm_le`
let B := (minkowskiBound K I * (convexBodySumFactor K)⁻¹).toReal ^ (1 / (finrank ℚ K : ℝ))
have h_le : (minkowskiBound K I) ≤ volume (convexBodySum K B) := by
refine le_of_eq ?_
rw [convexBodySum_volume, ← ENNReal.ofReal_pow (by positivity), ← Real.rpow_natCast,
← Real.rpow_mul toReal_nonneg, div_mul_cancel₀, Real.rpow_one, ofReal_toReal, mul_comm,
mul_assoc, ← coe_mul, inv_mul_cancel (convexBodySumFactor_ne_zero K), ENNReal.coe_one,
mul_one]
· exact mul_ne_top (ne_of_lt (minkowskiBound_lt_top K I)) coe_ne_top
· exact (Nat.cast_ne_zero.mpr (ne_of_gt finrank_pos))
convert exists_ne_zero_mem_ideal_of_norm_le K I h_le
rw [div_pow B, ← Real.rpow_natCast B, ← Real.rpow_mul (by positivity), div_mul_cancel₀ _
(Nat.cast_ne_zero.mpr <| ne_of_gt finrank_pos), Real.rpow_one, mul_comm_div, mul_div_assoc']
congr 1
rw [eq_comm]
calc
_ = FractionalIdeal.absNorm I.1 * (2 : ℝ)⁻¹ ^ NrComplexPlaces K * sqrt ‖discr K‖₊ *
(2 : ℝ) ^ finrank ℚ K * ((2 : ℝ) ^ NrRealPlaces K * (π / 2) ^ NrComplexPlaces K /
(Nat.factorial (finrank ℚ K)))⁻¹ := by
simp_rw [minkowskiBound, convexBodySumFactor,
volume_fundamentalDomain_fractionalIdealLatticeBasis,
volume_fundamentalDomain_latticeBasis, toReal_mul, toReal_pow, toReal_inv, coe_toReal,
toReal_ofNat, mixedEmbedding.finrank, mul_assoc]
rw [ENNReal.toReal_ofReal (Rat.cast_nonneg.mpr (FractionalIdeal.absNorm_nonneg I.1))]
simp_rw [NNReal.coe_inv, NNReal.coe_div, NNReal.coe_mul, NNReal.coe_pow, NNReal.coe_div,
coe_real_pi, NNReal.coe_ofNat, NNReal.coe_natCast]
_ = FractionalIdeal.absNorm I.1 * (2 : ℝ) ^ (finrank ℚ K - NrComplexPlaces K - NrRealPlaces K +
NrComplexPlaces K : ℤ) * Real.sqrt ‖discr K‖ * Nat.factorial (finrank ℚ K) *
π⁻¹ ^ (NrComplexPlaces K) := by
simp_rw [inv_div, div_eq_mul_inv, mul_inv, ← zpow_neg_one, ← zpow_natCast, mul_zpow,
← zpow_mul, neg_one_mul, mul_neg_one, neg_neg, Real.coe_sqrt, coe_nnnorm, sub_eq_add_neg,
zpow_add₀ (two_ne_zero : (2 : ℝ) ≠ 0)]
ring
_ = FractionalIdeal.absNorm I.1 * (2 : ℝ) ^ (2 * NrComplexPlaces K : ℤ) * Real.sqrt ‖discr K‖ *
Nat.factorial (finrank ℚ K) * π⁻¹ ^ (NrComplexPlaces K) := by
congr
rw [← card_add_two_mul_card_eq_rank, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat]
ring
_ = FractionalIdeal.absNorm I.1 * (4 / π) ^ NrComplexPlaces K * (finrank ℚ K).factorial *
Real.sqrt |discr K| := by
rw [Int.norm_eq_abs, zpow_mul, show (2 : ℝ) ^ (2 : ℤ) = 4 by norm_cast, div_pow,
inv_eq_one_div, div_pow, one_pow, zpow_natCast]
ring
| 43 | 4,727,839,468,229,346,000 | 2 | 1.6 | 5 | 1,738 |
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@"431589bce478b2229eba14b14a283250428217db"
noncomputable section
open scoped Classical
open Manifold Topology
open Set Filter TopologicalSpace
variable {H M H' M' X : Type*}
variable [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M]
variable [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M']
variable [TopologicalSpace X]
namespace StructureGroupoid
variable (G : StructureGroupoid H) (G' : StructureGroupoid H')
structure LocalInvariantProp (P : (H → H') → Set H → H → Prop) : Prop where
is_local : ∀ {s x u} {f : H → H'}, IsOpen u → x ∈ u → (P f s x ↔ P f (s ∩ u) x)
right_invariance' : ∀ {s x f} {e : PartialHomeomorph H H},
e ∈ G → x ∈ e.source → P f s x → P (f ∘ e.symm) (e.symm ⁻¹' s) (e x)
congr_of_forall : ∀ {s x} {f g : H → H'}, (∀ y ∈ s, f y = g y) → f x = g x → P f s x → P g s x
left_invariance' : ∀ {s x f} {e' : PartialHomeomorph H' H'},
e' ∈ G' → s ⊆ f ⁻¹' e'.source → f x ∈ e'.source → P f s x → P (e' ∘ f) s x
#align structure_groupoid.local_invariant_prop StructureGroupoid.LocalInvariantProp
variable {G G'} {P : (H → H') → Set H → H → Prop} {s t u : Set H} {x : H}
variable (hG : G.LocalInvariantProp G' P)
namespace LocalInvariantProp
| Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 82 | 85 | theorem congr_set {s t : Set H} {x : H} {f : H → H'} (hu : s =ᶠ[𝓝 x] t) : P f s x ↔ P f t x := by |
obtain ⟨o, host, ho, hxo⟩ := mem_nhds_iff.mp hu.mem_iff
simp_rw [subset_def, mem_setOf, ← and_congr_left_iff, ← mem_inter_iff, ← Set.ext_iff] at host
rw [hG.is_local ho hxo, host, ← hG.is_local ho hxo]
| 3 | 20.085537 | 1 | 1.6 | 5 | 1,739 |
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@"431589bce478b2229eba14b14a283250428217db"
noncomputable section
open scoped Classical
open Manifold Topology
open Set Filter TopologicalSpace
variable {H M H' M' X : Type*}
variable [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M]
variable [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M']
variable [TopologicalSpace X]
namespace StructureGroupoid
variable (G : StructureGroupoid H) (G' : StructureGroupoid H')
structure LocalInvariantProp (P : (H → H') → Set H → H → Prop) : Prop where
is_local : ∀ {s x u} {f : H → H'}, IsOpen u → x ∈ u → (P f s x ↔ P f (s ∩ u) x)
right_invariance' : ∀ {s x f} {e : PartialHomeomorph H H},
e ∈ G → x ∈ e.source → P f s x → P (f ∘ e.symm) (e.symm ⁻¹' s) (e x)
congr_of_forall : ∀ {s x} {f g : H → H'}, (∀ y ∈ s, f y = g y) → f x = g x → P f s x → P g s x
left_invariance' : ∀ {s x f} {e' : PartialHomeomorph H' H'},
e' ∈ G' → s ⊆ f ⁻¹' e'.source → f x ∈ e'.source → P f s x → P (e' ∘ f) s x
#align structure_groupoid.local_invariant_prop StructureGroupoid.LocalInvariantProp
variable {G G'} {P : (H → H') → Set H → H → Prop} {s t u : Set H} {x : H}
variable (hG : G.LocalInvariantProp G' P)
namespace LocalInvariantProp
theorem congr_set {s t : Set H} {x : H} {f : H → H'} (hu : s =ᶠ[𝓝 x] t) : P f s x ↔ P f t x := by
obtain ⟨o, host, ho, hxo⟩ := mem_nhds_iff.mp hu.mem_iff
simp_rw [subset_def, mem_setOf, ← and_congr_left_iff, ← mem_inter_iff, ← Set.ext_iff] at host
rw [hG.is_local ho hxo, host, ← hG.is_local ho hxo]
#align structure_groupoid.local_invariant_prop.congr_set StructureGroupoid.LocalInvariantProp.congr_set
theorem is_local_nhds {s u : Set H} {x : H} {f : H → H'} (hu : u ∈ 𝓝[s] x) :
P f s x ↔ P f (s ∩ u) x :=
hG.congr_set <| mem_nhdsWithin_iff_eventuallyEq.mp hu
#align structure_groupoid.local_invariant_prop.is_local_nhds StructureGroupoid.LocalInvariantProp.is_local_nhds
| Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 93 | 96 | theorem congr_iff_nhdsWithin {s : Set H} {x : H} {f g : H → H'} (h1 : f =ᶠ[𝓝[s] x] g)
(h2 : f x = g x) : P f s x ↔ P g s x := by |
simp_rw [hG.is_local_nhds h1]
exact ⟨hG.congr_of_forall (fun y hy ↦ hy.2) h2, hG.congr_of_forall (fun y hy ↦ hy.2.symm) h2.symm⟩
| 2 | 7.389056 | 1 | 1.6 | 5 | 1,739 |
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@"431589bce478b2229eba14b14a283250428217db"
noncomputable section
open scoped Classical
open Manifold Topology
open Set Filter TopologicalSpace
variable {H M H' M' X : Type*}
variable [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M]
variable [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M']
variable [TopologicalSpace X]
namespace StructureGroupoid
variable (G : StructureGroupoid H) (G' : StructureGroupoid H')
structure LocalInvariantProp (P : (H → H') → Set H → H → Prop) : Prop where
is_local : ∀ {s x u} {f : H → H'}, IsOpen u → x ∈ u → (P f s x ↔ P f (s ∩ u) x)
right_invariance' : ∀ {s x f} {e : PartialHomeomorph H H},
e ∈ G → x ∈ e.source → P f s x → P (f ∘ e.symm) (e.symm ⁻¹' s) (e x)
congr_of_forall : ∀ {s x} {f g : H → H'}, (∀ y ∈ s, f y = g y) → f x = g x → P f s x → P g s x
left_invariance' : ∀ {s x f} {e' : PartialHomeomorph H' H'},
e' ∈ G' → s ⊆ f ⁻¹' e'.source → f x ∈ e'.source → P f s x → P (e' ∘ f) s x
#align structure_groupoid.local_invariant_prop StructureGroupoid.LocalInvariantProp
variable {G G'} {P : (H → H') → Set H → H → Prop} {s t u : Set H} {x : H}
variable (hG : G.LocalInvariantProp G' P)
namespace LocalInvariantProp
theorem congr_set {s t : Set H} {x : H} {f : H → H'} (hu : s =ᶠ[𝓝 x] t) : P f s x ↔ P f t x := by
obtain ⟨o, host, ho, hxo⟩ := mem_nhds_iff.mp hu.mem_iff
simp_rw [subset_def, mem_setOf, ← and_congr_left_iff, ← mem_inter_iff, ← Set.ext_iff] at host
rw [hG.is_local ho hxo, host, ← hG.is_local ho hxo]
#align structure_groupoid.local_invariant_prop.congr_set StructureGroupoid.LocalInvariantProp.congr_set
theorem is_local_nhds {s u : Set H} {x : H} {f : H → H'} (hu : u ∈ 𝓝[s] x) :
P f s x ↔ P f (s ∩ u) x :=
hG.congr_set <| mem_nhdsWithin_iff_eventuallyEq.mp hu
#align structure_groupoid.local_invariant_prop.is_local_nhds StructureGroupoid.LocalInvariantProp.is_local_nhds
theorem congr_iff_nhdsWithin {s : Set H} {x : H} {f g : H → H'} (h1 : f =ᶠ[𝓝[s] x] g)
(h2 : f x = g x) : P f s x ↔ P g s x := by
simp_rw [hG.is_local_nhds h1]
exact ⟨hG.congr_of_forall (fun y hy ↦ hy.2) h2, hG.congr_of_forall (fun y hy ↦ hy.2.symm) h2.symm⟩
#align structure_groupoid.local_invariant_prop.congr_iff_nhds_within StructureGroupoid.LocalInvariantProp.congr_iff_nhdsWithin
theorem congr_nhdsWithin {s : Set H} {x : H} {f g : H → H'} (h1 : f =ᶠ[𝓝[s] x] g) (h2 : f x = g x)
(hP : P f s x) : P g s x :=
(hG.congr_iff_nhdsWithin h1 h2).mp hP
#align structure_groupoid.local_invariant_prop.congr_nhds_within StructureGroupoid.LocalInvariantProp.congr_nhdsWithin
theorem congr_nhdsWithin' {s : Set H} {x : H} {f g : H → H'} (h1 : f =ᶠ[𝓝[s] x] g) (h2 : f x = g x)
(hP : P g s x) : P f s x :=
(hG.congr_iff_nhdsWithin h1 h2).mpr hP
#align structure_groupoid.local_invariant_prop.congr_nhds_within' StructureGroupoid.LocalInvariantProp.congr_nhdsWithin'
theorem congr_iff {s : Set H} {x : H} {f g : H → H'} (h : f =ᶠ[𝓝 x] g) : P f s x ↔ P g s x :=
hG.congr_iff_nhdsWithin (mem_nhdsWithin_of_mem_nhds h) (mem_of_mem_nhds h : _)
#align structure_groupoid.local_invariant_prop.congr_iff StructureGroupoid.LocalInvariantProp.congr_iff
theorem congr {s : Set H} {x : H} {f g : H → H'} (h : f =ᶠ[𝓝 x] g) (hP : P f s x) : P g s x :=
(hG.congr_iff h).mp hP
#align structure_groupoid.local_invariant_prop.congr StructureGroupoid.LocalInvariantProp.congr
theorem congr' {s : Set H} {x : H} {f g : H → H'} (h : f =ᶠ[𝓝 x] g) (hP : P g s x) : P f s x :=
hG.congr h.symm hP
#align structure_groupoid.local_invariant_prop.congr' StructureGroupoid.LocalInvariantProp.congr'
| Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 121 | 136 | theorem left_invariance {s : Set H} {x : H} {f : H → H'} {e' : PartialHomeomorph H' H'}
(he' : e' ∈ G') (hfs : ContinuousWithinAt f s x) (hxe' : f x ∈ e'.source) :
P (e' ∘ f) s x ↔ P f s x := by |
have h2f := hfs.preimage_mem_nhdsWithin (e'.open_source.mem_nhds hxe')
have h3f :=
((e'.continuousAt hxe').comp_continuousWithinAt hfs).preimage_mem_nhdsWithin <|
e'.symm.open_source.mem_nhds <| e'.mapsTo hxe'
constructor
· intro h
rw [hG.is_local_nhds h3f] at h
have h2 := hG.left_invariance' (G'.symm he') inter_subset_right (e'.mapsTo hxe') h
rw [← hG.is_local_nhds h3f] at h2
refine hG.congr_nhdsWithin ?_ (e'.left_inv hxe') h2
exact eventually_of_mem h2f fun x' ↦ e'.left_inv
· simp_rw [hG.is_local_nhds h2f]
exact hG.left_invariance' he' inter_subset_right hxe'
| 13 | 442,413.392009 | 2 | 1.6 | 5 | 1,739 |
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@"431589bce478b2229eba14b14a283250428217db"
noncomputable section
open scoped Classical
open Manifold Topology
open Set Filter TopologicalSpace
variable {H M H' M' X : Type*}
variable [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M]
variable [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M']
variable [TopologicalSpace X]
namespace StructureGroupoid
variable (G : StructureGroupoid H) (G' : StructureGroupoid H')
structure LocalInvariantProp (P : (H → H') → Set H → H → Prop) : Prop where
is_local : ∀ {s x u} {f : H → H'}, IsOpen u → x ∈ u → (P f s x ↔ P f (s ∩ u) x)
right_invariance' : ∀ {s x f} {e : PartialHomeomorph H H},
e ∈ G → x ∈ e.source → P f s x → P (f ∘ e.symm) (e.symm ⁻¹' s) (e x)
congr_of_forall : ∀ {s x} {f g : H → H'}, (∀ y ∈ s, f y = g y) → f x = g x → P f s x → P g s x
left_invariance' : ∀ {s x f} {e' : PartialHomeomorph H' H'},
e' ∈ G' → s ⊆ f ⁻¹' e'.source → f x ∈ e'.source → P f s x → P (e' ∘ f) s x
#align structure_groupoid.local_invariant_prop StructureGroupoid.LocalInvariantProp
variable {G G'} {P : (H → H') → Set H → H → Prop} {s t u : Set H} {x : H}
variable (hG : G.LocalInvariantProp G' P)
section LocalStructomorph
variable (G)
open PartialHomeomorph
def IsLocalStructomorphWithinAt (f : H → H) (s : Set H) (x : H) : Prop :=
x ∈ s → ∃ e : PartialHomeomorph H H, e ∈ G ∧ EqOn f e.toFun (s ∩ e.source) ∧ x ∈ e.source
#align structure_groupoid.is_local_structomorph_within_at StructureGroupoid.IsLocalStructomorphWithinAt
| Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 605 | 643 | theorem isLocalStructomorphWithinAt_localInvariantProp [ClosedUnderRestriction G] :
LocalInvariantProp G G (IsLocalStructomorphWithinAt G) :=
{ is_local := by |
intro s x u f hu hux
constructor
· rintro h hx
rcases h hx.1 with ⟨e, heG, hef, hex⟩
have : s ∩ u ∩ e.source ⊆ s ∩ e.source := by mfld_set_tac
exact ⟨e, heG, hef.mono this, hex⟩
· rintro h hx
rcases h ⟨hx, hux⟩ with ⟨e, heG, hef, hex⟩
refine ⟨e.restr (interior u), ?_, ?_, ?_⟩
· exact closedUnderRestriction' heG isOpen_interior
· have : s ∩ u ∩ e.source = s ∩ (e.source ∩ u) := by mfld_set_tac
simpa only [this, interior_interior, hu.interior_eq, mfld_simps] using hef
· simp only [*, interior_interior, hu.interior_eq, mfld_simps]
right_invariance' := by
intro s x f e' he'G he'x h hx
have hxs : x ∈ s := by simpa only [e'.left_inv he'x, mfld_simps] using hx
rcases h hxs with ⟨e, heG, hef, hex⟩
refine ⟨e'.symm.trans e, G.trans (G.symm he'G) heG, ?_, ?_⟩
· intro y hy
simp only [mfld_simps] at hy
simp only [hef ⟨hy.1, hy.2.2⟩, mfld_simps]
· simp only [hex, he'x, mfld_simps]
congr_of_forall := by
intro s x f g hfgs _ h hx
rcases h hx with ⟨e, heG, hef, hex⟩
refine ⟨e, heG, ?_, hex⟩
intro y hy
rw [← hef hy, hfgs y hy.1]
left_invariance' := by
intro s x f e' he'G _ hfx h hx
rcases h hx with ⟨e, heG, hef, hex⟩
refine ⟨e.trans e', G.trans heG he'G, ?_, ?_⟩
· intro y hy
simp only [mfld_simps] at hy
simp only [hef ⟨hy.1, hy.2.1⟩, mfld_simps]
· simpa only [hex, hef ⟨hx, hex⟩, mfld_simps] using hfx }
| 36 | 4,311,231,547,115,195 | 2 | 1.6 | 5 | 1,739 |
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@"431589bce478b2229eba14b14a283250428217db"
noncomputable section
open scoped Classical
open Manifold Topology
open Set Filter TopologicalSpace
variable {H M H' M' X : Type*}
variable [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M]
variable [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M']
variable [TopologicalSpace X]
namespace StructureGroupoid
variable (G : StructureGroupoid H) (G' : StructureGroupoid H')
structure LocalInvariantProp (P : (H → H') → Set H → H → Prop) : Prop where
is_local : ∀ {s x u} {f : H → H'}, IsOpen u → x ∈ u → (P f s x ↔ P f (s ∩ u) x)
right_invariance' : ∀ {s x f} {e : PartialHomeomorph H H},
e ∈ G → x ∈ e.source → P f s x → P (f ∘ e.symm) (e.symm ⁻¹' s) (e x)
congr_of_forall : ∀ {s x} {f g : H → H'}, (∀ y ∈ s, f y = g y) → f x = g x → P f s x → P g s x
left_invariance' : ∀ {s x f} {e' : PartialHomeomorph H' H'},
e' ∈ G' → s ⊆ f ⁻¹' e'.source → f x ∈ e'.source → P f s x → P (e' ∘ f) s x
#align structure_groupoid.local_invariant_prop StructureGroupoid.LocalInvariantProp
variable {G G'} {P : (H → H') → Set H → H → Prop} {s t u : Set H} {x : H}
variable (hG : G.LocalInvariantProp G' P)
section LocalStructomorph
variable (G)
open PartialHomeomorph
def IsLocalStructomorphWithinAt (f : H → H) (s : Set H) (x : H) : Prop :=
x ∈ s → ∃ e : PartialHomeomorph H H, e ∈ G ∧ EqOn f e.toFun (s ∩ e.source) ∧ x ∈ e.source
#align structure_groupoid.is_local_structomorph_within_at StructureGroupoid.IsLocalStructomorphWithinAt
theorem isLocalStructomorphWithinAt_localInvariantProp [ClosedUnderRestriction G] :
LocalInvariantProp G G (IsLocalStructomorphWithinAt G) :=
{ is_local := by
intro s x u f hu hux
constructor
· rintro h hx
rcases h hx.1 with ⟨e, heG, hef, hex⟩
have : s ∩ u ∩ e.source ⊆ s ∩ e.source := by mfld_set_tac
exact ⟨e, heG, hef.mono this, hex⟩
· rintro h hx
rcases h ⟨hx, hux⟩ with ⟨e, heG, hef, hex⟩
refine ⟨e.restr (interior u), ?_, ?_, ?_⟩
· exact closedUnderRestriction' heG isOpen_interior
· have : s ∩ u ∩ e.source = s ∩ (e.source ∩ u) := by mfld_set_tac
simpa only [this, interior_interior, hu.interior_eq, mfld_simps] using hef
· simp only [*, interior_interior, hu.interior_eq, mfld_simps]
right_invariance' := by
intro s x f e' he'G he'x h hx
have hxs : x ∈ s := by simpa only [e'.left_inv he'x, mfld_simps] using hx
rcases h hxs with ⟨e, heG, hef, hex⟩
refine ⟨e'.symm.trans e, G.trans (G.symm he'G) heG, ?_, ?_⟩
· intro y hy
simp only [mfld_simps] at hy
simp only [hef ⟨hy.1, hy.2.2⟩, mfld_simps]
· simp only [hex, he'x, mfld_simps]
congr_of_forall := by
intro s x f g hfgs _ h hx
rcases h hx with ⟨e, heG, hef, hex⟩
refine ⟨e, heG, ?_, hex⟩
intro y hy
rw [← hef hy, hfgs y hy.1]
left_invariance' := by
intro s x f e' he'G _ hfx h hx
rcases h hx with ⟨e, heG, hef, hex⟩
refine ⟨e.trans e', G.trans heG he'G, ?_, ?_⟩
· intro y hy
simp only [mfld_simps] at hy
simp only [hef ⟨hy.1, hy.2.1⟩, mfld_simps]
· simpa only [hex, hef ⟨hx, hex⟩, mfld_simps] using hfx }
#align structure_groupoid.is_local_structomorph_within_at_local_invariant_prop StructureGroupoid.isLocalStructomorphWithinAt_localInvariantProp
| Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 648 | 666 | theorem _root_.PartialHomeomorph.isLocalStructomorphWithinAt_iff {G : StructureGroupoid H}
[ClosedUnderRestriction G] (f : PartialHomeomorph H H) {s : Set H} {x : H}
(hx : x ∈ f.source ∪ sᶜ) :
G.IsLocalStructomorphWithinAt (⇑f) s x ↔
x ∈ s → ∃ e : PartialHomeomorph H H,
e ∈ G ∧ e.source ⊆ f.source ∧ EqOn f (⇑e) (s ∩ e.source) ∧ x ∈ e.source := by |
constructor
· intro hf h2x
obtain ⟨e, he, hfe, hxe⟩ := hf h2x
refine ⟨e.restr f.source, closedUnderRestriction' he f.open_source, ?_, ?_, hxe, ?_⟩
· simp_rw [PartialHomeomorph.restr_source]
exact inter_subset_right.trans interior_subset
· intro x' hx'
exact hfe ⟨hx'.1, hx'.2.1⟩
· rw [f.open_source.interior_eq]
exact Or.resolve_right hx (not_not.mpr h2x)
· intro hf hx
obtain ⟨e, he, _, hfe, hxe⟩ := hf hx
exact ⟨e, he, hfe, hxe⟩
| 13 | 442,413.392009 | 2 | 1.6 | 5 | 1,739 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Language.{u, v}) [L.Structure M]
open FirstOrder FirstOrder.Language FirstOrder.Language.Structure
variable {α : Type u₁} {β : Type*}
def Definable (s : Set (α → M)) : Prop :=
∃ φ : L[[A]].Formula α, s = setOf φ.Realize
#align set.definable Set.Definable
variable {L} {A} {B : Set M} {s : Set (α → M)}
| Mathlib/ModelTheory/Definability.lean | 52 | 57 | theorem Definable.map_expansion {L' : FirstOrder.Language} [L'.Structure M] (h : A.Definable L s)
(φ : L →ᴸ L') [φ.IsExpansionOn M] : A.Definable L' s := by |
obtain ⟨ψ, rfl⟩ := h
refine ⟨(φ.addConstants A).onFormula ψ, ?_⟩
ext x
simp only [mem_setOf_eq, LHom.realize_onFormula]
| 4 | 54.59815 | 2 | 1.6 | 10 | 1,740 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Language.{u, v}) [L.Structure M]
open FirstOrder FirstOrder.Language FirstOrder.Language.Structure
variable {α : Type u₁} {β : Type*}
def Definable (s : Set (α → M)) : Prop :=
∃ φ : L[[A]].Formula α, s = setOf φ.Realize
#align set.definable Set.Definable
variable {L} {A} {B : Set M} {s : Set (α → M)}
theorem Definable.map_expansion {L' : FirstOrder.Language} [L'.Structure M] (h : A.Definable L s)
(φ : L →ᴸ L') [φ.IsExpansionOn M] : A.Definable L' s := by
obtain ⟨ψ, rfl⟩ := h
refine ⟨(φ.addConstants A).onFormula ψ, ?_⟩
ext x
simp only [mem_setOf_eq, LHom.realize_onFormula]
#align set.definable.map_expansion Set.Definable.map_expansion
| Mathlib/ModelTheory/Definability.lean | 60 | 73 | theorem definable_iff_exists_formula_sum :
A.Definable L s ↔ ∃ φ : L.Formula (A ⊕ α), s = {v | φ.Realize (Sum.elim (↑) v)} := by |
rw [Definable, Equiv.exists_congr_left (BoundedFormula.constantsVarsEquiv)]
refine exists_congr (fun φ => iff_iff_eq.2 (congr_arg (s = ·) ?_))
ext
simp only [Formula.Realize, BoundedFormula.constantsVarsEquiv, constantsOn, mk₂_Relations,
BoundedFormula.mapTermRelEquiv_symm_apply, mem_setOf_eq]
refine BoundedFormula.realize_mapTermRel_id ?_ (fun _ _ _ => rfl)
intros
simp only [Term.constantsVarsEquivLeft_symm_apply, Term.realize_varsToConstants,
coe_con, Term.realize_relabel]
congr
ext a
rcases a with (_ | _) | _ <;> rfl
| 12 | 162,754.791419 | 2 | 1.6 | 10 | 1,740 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Language.{u, v}) [L.Structure M]
open FirstOrder FirstOrder.Language FirstOrder.Language.Structure
variable {α : Type u₁} {β : Type*}
def Definable (s : Set (α → M)) : Prop :=
∃ φ : L[[A]].Formula α, s = setOf φ.Realize
#align set.definable Set.Definable
variable {L} {A} {B : Set M} {s : Set (α → M)}
theorem Definable.map_expansion {L' : FirstOrder.Language} [L'.Structure M] (h : A.Definable L s)
(φ : L →ᴸ L') [φ.IsExpansionOn M] : A.Definable L' s := by
obtain ⟨ψ, rfl⟩ := h
refine ⟨(φ.addConstants A).onFormula ψ, ?_⟩
ext x
simp only [mem_setOf_eq, LHom.realize_onFormula]
#align set.definable.map_expansion Set.Definable.map_expansion
theorem definable_iff_exists_formula_sum :
A.Definable L s ↔ ∃ φ : L.Formula (A ⊕ α), s = {v | φ.Realize (Sum.elim (↑) v)} := by
rw [Definable, Equiv.exists_congr_left (BoundedFormula.constantsVarsEquiv)]
refine exists_congr (fun φ => iff_iff_eq.2 (congr_arg (s = ·) ?_))
ext
simp only [Formula.Realize, BoundedFormula.constantsVarsEquiv, constantsOn, mk₂_Relations,
BoundedFormula.mapTermRelEquiv_symm_apply, mem_setOf_eq]
refine BoundedFormula.realize_mapTermRel_id ?_ (fun _ _ _ => rfl)
intros
simp only [Term.constantsVarsEquivLeft_symm_apply, Term.realize_varsToConstants,
coe_con, Term.realize_relabel]
congr
ext a
rcases a with (_ | _) | _ <;> rfl
| Mathlib/ModelTheory/Definability.lean | 75 | 78 | theorem empty_definable_iff :
(∅ : Set M).Definable L s ↔ ∃ φ : L.Formula α, s = setOf φ.Realize := by |
rw [Definable, Equiv.exists_congr_left (LEquiv.addEmptyConstants L (∅ : Set M)).onFormula]
simp [-constantsOn]
| 2 | 7.389056 | 1 | 1.6 | 10 | 1,740 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Language.{u, v}) [L.Structure M]
open FirstOrder FirstOrder.Language FirstOrder.Language.Structure
variable {α : Type u₁} {β : Type*}
def Definable (s : Set (α → M)) : Prop :=
∃ φ : L[[A]].Formula α, s = setOf φ.Realize
#align set.definable Set.Definable
variable {L} {A} {B : Set M} {s : Set (α → M)}
theorem Definable.map_expansion {L' : FirstOrder.Language} [L'.Structure M] (h : A.Definable L s)
(φ : L →ᴸ L') [φ.IsExpansionOn M] : A.Definable L' s := by
obtain ⟨ψ, rfl⟩ := h
refine ⟨(φ.addConstants A).onFormula ψ, ?_⟩
ext x
simp only [mem_setOf_eq, LHom.realize_onFormula]
#align set.definable.map_expansion Set.Definable.map_expansion
theorem definable_iff_exists_formula_sum :
A.Definable L s ↔ ∃ φ : L.Formula (A ⊕ α), s = {v | φ.Realize (Sum.elim (↑) v)} := by
rw [Definable, Equiv.exists_congr_left (BoundedFormula.constantsVarsEquiv)]
refine exists_congr (fun φ => iff_iff_eq.2 (congr_arg (s = ·) ?_))
ext
simp only [Formula.Realize, BoundedFormula.constantsVarsEquiv, constantsOn, mk₂_Relations,
BoundedFormula.mapTermRelEquiv_symm_apply, mem_setOf_eq]
refine BoundedFormula.realize_mapTermRel_id ?_ (fun _ _ _ => rfl)
intros
simp only [Term.constantsVarsEquivLeft_symm_apply, Term.realize_varsToConstants,
coe_con, Term.realize_relabel]
congr
ext a
rcases a with (_ | _) | _ <;> rfl
theorem empty_definable_iff :
(∅ : Set M).Definable L s ↔ ∃ φ : L.Formula α, s = setOf φ.Realize := by
rw [Definable, Equiv.exists_congr_left (LEquiv.addEmptyConstants L (∅ : Set M)).onFormula]
simp [-constantsOn]
#align set.empty_definable_iff Set.empty_definable_iff
theorem definable_iff_empty_definable_with_params :
A.Definable L s ↔ (∅ : Set M).Definable (L[[A]]) s :=
empty_definable_iff.symm
#align set.definable_iff_empty_definable_with_params Set.definable_iff_empty_definable_with_params
| Mathlib/ModelTheory/Definability.lean | 86 | 88 | theorem Definable.mono (hAs : A.Definable L s) (hAB : A ⊆ B) : B.Definable L s := by |
rw [definable_iff_empty_definable_with_params] at *
exact hAs.map_expansion (L.lhomWithConstantsMap (Set.inclusion hAB))
| 2 | 7.389056 | 1 | 1.6 | 10 | 1,740 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Language.{u, v}) [L.Structure M]
open FirstOrder FirstOrder.Language FirstOrder.Language.Structure
variable {α : Type u₁} {β : Type*}
def Definable (s : Set (α → M)) : Prop :=
∃ φ : L[[A]].Formula α, s = setOf φ.Realize
#align set.definable Set.Definable
variable {L} {A} {B : Set M} {s : Set (α → M)}
theorem Definable.map_expansion {L' : FirstOrder.Language} [L'.Structure M] (h : A.Definable L s)
(φ : L →ᴸ L') [φ.IsExpansionOn M] : A.Definable L' s := by
obtain ⟨ψ, rfl⟩ := h
refine ⟨(φ.addConstants A).onFormula ψ, ?_⟩
ext x
simp only [mem_setOf_eq, LHom.realize_onFormula]
#align set.definable.map_expansion Set.Definable.map_expansion
theorem definable_iff_exists_formula_sum :
A.Definable L s ↔ ∃ φ : L.Formula (A ⊕ α), s = {v | φ.Realize (Sum.elim (↑) v)} := by
rw [Definable, Equiv.exists_congr_left (BoundedFormula.constantsVarsEquiv)]
refine exists_congr (fun φ => iff_iff_eq.2 (congr_arg (s = ·) ?_))
ext
simp only [Formula.Realize, BoundedFormula.constantsVarsEquiv, constantsOn, mk₂_Relations,
BoundedFormula.mapTermRelEquiv_symm_apply, mem_setOf_eq]
refine BoundedFormula.realize_mapTermRel_id ?_ (fun _ _ _ => rfl)
intros
simp only [Term.constantsVarsEquivLeft_symm_apply, Term.realize_varsToConstants,
coe_con, Term.realize_relabel]
congr
ext a
rcases a with (_ | _) | _ <;> rfl
theorem empty_definable_iff :
(∅ : Set M).Definable L s ↔ ∃ φ : L.Formula α, s = setOf φ.Realize := by
rw [Definable, Equiv.exists_congr_left (LEquiv.addEmptyConstants L (∅ : Set M)).onFormula]
simp [-constantsOn]
#align set.empty_definable_iff Set.empty_definable_iff
theorem definable_iff_empty_definable_with_params :
A.Definable L s ↔ (∅ : Set M).Definable (L[[A]]) s :=
empty_definable_iff.symm
#align set.definable_iff_empty_definable_with_params Set.definable_iff_empty_definable_with_params
theorem Definable.mono (hAs : A.Definable L s) (hAB : A ⊆ B) : B.Definable L s := by
rw [definable_iff_empty_definable_with_params] at *
exact hAs.map_expansion (L.lhomWithConstantsMap (Set.inclusion hAB))
#align set.definable.mono Set.Definable.mono
@[simp]
theorem definable_empty : A.Definable L (∅ : Set (α → M)) :=
⟨⊥, by
ext
simp⟩
#align set.definable_empty Set.definable_empty
@[simp]
theorem definable_univ : A.Definable L (univ : Set (α → M)) :=
⟨⊤, by
ext
simp⟩
#align set.definable_univ Set.definable_univ
@[simp]
| Mathlib/ModelTheory/Definability.lean | 106 | 112 | theorem Definable.inter {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∩ g) := by |
rcases hf with ⟨φ, rfl⟩
rcases hg with ⟨θ, rfl⟩
refine ⟨φ ⊓ θ, ?_⟩
ext
simp
| 5 | 148.413159 | 2 | 1.6 | 10 | 1,740 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Language.{u, v}) [L.Structure M]
open FirstOrder FirstOrder.Language FirstOrder.Language.Structure
variable {α : Type u₁} {β : Type*}
def Definable (s : Set (α → M)) : Prop :=
∃ φ : L[[A]].Formula α, s = setOf φ.Realize
#align set.definable Set.Definable
variable {L} {A} {B : Set M} {s : Set (α → M)}
theorem Definable.map_expansion {L' : FirstOrder.Language} [L'.Structure M] (h : A.Definable L s)
(φ : L →ᴸ L') [φ.IsExpansionOn M] : A.Definable L' s := by
obtain ⟨ψ, rfl⟩ := h
refine ⟨(φ.addConstants A).onFormula ψ, ?_⟩
ext x
simp only [mem_setOf_eq, LHom.realize_onFormula]
#align set.definable.map_expansion Set.Definable.map_expansion
theorem definable_iff_exists_formula_sum :
A.Definable L s ↔ ∃ φ : L.Formula (A ⊕ α), s = {v | φ.Realize (Sum.elim (↑) v)} := by
rw [Definable, Equiv.exists_congr_left (BoundedFormula.constantsVarsEquiv)]
refine exists_congr (fun φ => iff_iff_eq.2 (congr_arg (s = ·) ?_))
ext
simp only [Formula.Realize, BoundedFormula.constantsVarsEquiv, constantsOn, mk₂_Relations,
BoundedFormula.mapTermRelEquiv_symm_apply, mem_setOf_eq]
refine BoundedFormula.realize_mapTermRel_id ?_ (fun _ _ _ => rfl)
intros
simp only [Term.constantsVarsEquivLeft_symm_apply, Term.realize_varsToConstants,
coe_con, Term.realize_relabel]
congr
ext a
rcases a with (_ | _) | _ <;> rfl
theorem empty_definable_iff :
(∅ : Set M).Definable L s ↔ ∃ φ : L.Formula α, s = setOf φ.Realize := by
rw [Definable, Equiv.exists_congr_left (LEquiv.addEmptyConstants L (∅ : Set M)).onFormula]
simp [-constantsOn]
#align set.empty_definable_iff Set.empty_definable_iff
theorem definable_iff_empty_definable_with_params :
A.Definable L s ↔ (∅ : Set M).Definable (L[[A]]) s :=
empty_definable_iff.symm
#align set.definable_iff_empty_definable_with_params Set.definable_iff_empty_definable_with_params
theorem Definable.mono (hAs : A.Definable L s) (hAB : A ⊆ B) : B.Definable L s := by
rw [definable_iff_empty_definable_with_params] at *
exact hAs.map_expansion (L.lhomWithConstantsMap (Set.inclusion hAB))
#align set.definable.mono Set.Definable.mono
@[simp]
theorem definable_empty : A.Definable L (∅ : Set (α → M)) :=
⟨⊥, by
ext
simp⟩
#align set.definable_empty Set.definable_empty
@[simp]
theorem definable_univ : A.Definable L (univ : Set (α → M)) :=
⟨⊤, by
ext
simp⟩
#align set.definable_univ Set.definable_univ
@[simp]
theorem Definable.inter {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∩ g) := by
rcases hf with ⟨φ, rfl⟩
rcases hg with ⟨θ, rfl⟩
refine ⟨φ ⊓ θ, ?_⟩
ext
simp
#align set.definable.inter Set.Definable.inter
@[simp]
| Mathlib/ModelTheory/Definability.lean | 116 | 122 | theorem Definable.union {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∪ g) := by |
rcases hf with ⟨φ, hφ⟩
rcases hg with ⟨θ, hθ⟩
refine ⟨φ ⊔ θ, ?_⟩
ext
rw [hφ, hθ, mem_setOf_eq, Formula.realize_sup, mem_union, mem_setOf_eq, mem_setOf_eq]
| 5 | 148.413159 | 2 | 1.6 | 10 | 1,740 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Language.{u, v}) [L.Structure M]
open FirstOrder FirstOrder.Language FirstOrder.Language.Structure
variable {α : Type u₁} {β : Type*}
def Definable (s : Set (α → M)) : Prop :=
∃ φ : L[[A]].Formula α, s = setOf φ.Realize
#align set.definable Set.Definable
variable {L} {A} {B : Set M} {s : Set (α → M)}
theorem Definable.map_expansion {L' : FirstOrder.Language} [L'.Structure M] (h : A.Definable L s)
(φ : L →ᴸ L') [φ.IsExpansionOn M] : A.Definable L' s := by
obtain ⟨ψ, rfl⟩ := h
refine ⟨(φ.addConstants A).onFormula ψ, ?_⟩
ext x
simp only [mem_setOf_eq, LHom.realize_onFormula]
#align set.definable.map_expansion Set.Definable.map_expansion
theorem definable_iff_exists_formula_sum :
A.Definable L s ↔ ∃ φ : L.Formula (A ⊕ α), s = {v | φ.Realize (Sum.elim (↑) v)} := by
rw [Definable, Equiv.exists_congr_left (BoundedFormula.constantsVarsEquiv)]
refine exists_congr (fun φ => iff_iff_eq.2 (congr_arg (s = ·) ?_))
ext
simp only [Formula.Realize, BoundedFormula.constantsVarsEquiv, constantsOn, mk₂_Relations,
BoundedFormula.mapTermRelEquiv_symm_apply, mem_setOf_eq]
refine BoundedFormula.realize_mapTermRel_id ?_ (fun _ _ _ => rfl)
intros
simp only [Term.constantsVarsEquivLeft_symm_apply, Term.realize_varsToConstants,
coe_con, Term.realize_relabel]
congr
ext a
rcases a with (_ | _) | _ <;> rfl
theorem empty_definable_iff :
(∅ : Set M).Definable L s ↔ ∃ φ : L.Formula α, s = setOf φ.Realize := by
rw [Definable, Equiv.exists_congr_left (LEquiv.addEmptyConstants L (∅ : Set M)).onFormula]
simp [-constantsOn]
#align set.empty_definable_iff Set.empty_definable_iff
theorem definable_iff_empty_definable_with_params :
A.Definable L s ↔ (∅ : Set M).Definable (L[[A]]) s :=
empty_definable_iff.symm
#align set.definable_iff_empty_definable_with_params Set.definable_iff_empty_definable_with_params
theorem Definable.mono (hAs : A.Definable L s) (hAB : A ⊆ B) : B.Definable L s := by
rw [definable_iff_empty_definable_with_params] at *
exact hAs.map_expansion (L.lhomWithConstantsMap (Set.inclusion hAB))
#align set.definable.mono Set.Definable.mono
@[simp]
theorem definable_empty : A.Definable L (∅ : Set (α → M)) :=
⟨⊥, by
ext
simp⟩
#align set.definable_empty Set.definable_empty
@[simp]
theorem definable_univ : A.Definable L (univ : Set (α → M)) :=
⟨⊤, by
ext
simp⟩
#align set.definable_univ Set.definable_univ
@[simp]
theorem Definable.inter {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∩ g) := by
rcases hf with ⟨φ, rfl⟩
rcases hg with ⟨θ, rfl⟩
refine ⟨φ ⊓ θ, ?_⟩
ext
simp
#align set.definable.inter Set.Definable.inter
@[simp]
theorem Definable.union {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∪ g) := by
rcases hf with ⟨φ, hφ⟩
rcases hg with ⟨θ, hθ⟩
refine ⟨φ ⊔ θ, ?_⟩
ext
rw [hφ, hθ, mem_setOf_eq, Formula.realize_sup, mem_union, mem_setOf_eq, mem_setOf_eq]
#align set.definable.union Set.Definable.union
| Mathlib/ModelTheory/Definability.lean | 125 | 130 | theorem definable_finset_inf {ι : Type*} {f : ι → Set (α → M)} (hf : ∀ i, A.Definable L (f i))
(s : Finset ι) : A.Definable L (s.inf f) := by |
classical
refine Finset.induction definable_univ (fun i s _ h => ?_) s
rw [Finset.inf_insert]
exact (hf i).inter h
| 4 | 54.59815 | 2 | 1.6 | 10 | 1,740 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Language.{u, v}) [L.Structure M]
open FirstOrder FirstOrder.Language FirstOrder.Language.Structure
variable {α : Type u₁} {β : Type*}
def Definable (s : Set (α → M)) : Prop :=
∃ φ : L[[A]].Formula α, s = setOf φ.Realize
#align set.definable Set.Definable
variable {L} {A} {B : Set M} {s : Set (α → M)}
theorem Definable.map_expansion {L' : FirstOrder.Language} [L'.Structure M] (h : A.Definable L s)
(φ : L →ᴸ L') [φ.IsExpansionOn M] : A.Definable L' s := by
obtain ⟨ψ, rfl⟩ := h
refine ⟨(φ.addConstants A).onFormula ψ, ?_⟩
ext x
simp only [mem_setOf_eq, LHom.realize_onFormula]
#align set.definable.map_expansion Set.Definable.map_expansion
theorem definable_iff_exists_formula_sum :
A.Definable L s ↔ ∃ φ : L.Formula (A ⊕ α), s = {v | φ.Realize (Sum.elim (↑) v)} := by
rw [Definable, Equiv.exists_congr_left (BoundedFormula.constantsVarsEquiv)]
refine exists_congr (fun φ => iff_iff_eq.2 (congr_arg (s = ·) ?_))
ext
simp only [Formula.Realize, BoundedFormula.constantsVarsEquiv, constantsOn, mk₂_Relations,
BoundedFormula.mapTermRelEquiv_symm_apply, mem_setOf_eq]
refine BoundedFormula.realize_mapTermRel_id ?_ (fun _ _ _ => rfl)
intros
simp only [Term.constantsVarsEquivLeft_symm_apply, Term.realize_varsToConstants,
coe_con, Term.realize_relabel]
congr
ext a
rcases a with (_ | _) | _ <;> rfl
theorem empty_definable_iff :
(∅ : Set M).Definable L s ↔ ∃ φ : L.Formula α, s = setOf φ.Realize := by
rw [Definable, Equiv.exists_congr_left (LEquiv.addEmptyConstants L (∅ : Set M)).onFormula]
simp [-constantsOn]
#align set.empty_definable_iff Set.empty_definable_iff
theorem definable_iff_empty_definable_with_params :
A.Definable L s ↔ (∅ : Set M).Definable (L[[A]]) s :=
empty_definable_iff.symm
#align set.definable_iff_empty_definable_with_params Set.definable_iff_empty_definable_with_params
theorem Definable.mono (hAs : A.Definable L s) (hAB : A ⊆ B) : B.Definable L s := by
rw [definable_iff_empty_definable_with_params] at *
exact hAs.map_expansion (L.lhomWithConstantsMap (Set.inclusion hAB))
#align set.definable.mono Set.Definable.mono
@[simp]
theorem definable_empty : A.Definable L (∅ : Set (α → M)) :=
⟨⊥, by
ext
simp⟩
#align set.definable_empty Set.definable_empty
@[simp]
theorem definable_univ : A.Definable L (univ : Set (α → M)) :=
⟨⊤, by
ext
simp⟩
#align set.definable_univ Set.definable_univ
@[simp]
theorem Definable.inter {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∩ g) := by
rcases hf with ⟨φ, rfl⟩
rcases hg with ⟨θ, rfl⟩
refine ⟨φ ⊓ θ, ?_⟩
ext
simp
#align set.definable.inter Set.Definable.inter
@[simp]
theorem Definable.union {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∪ g) := by
rcases hf with ⟨φ, hφ⟩
rcases hg with ⟨θ, hθ⟩
refine ⟨φ ⊔ θ, ?_⟩
ext
rw [hφ, hθ, mem_setOf_eq, Formula.realize_sup, mem_union, mem_setOf_eq, mem_setOf_eq]
#align set.definable.union Set.Definable.union
theorem definable_finset_inf {ι : Type*} {f : ι → Set (α → M)} (hf : ∀ i, A.Definable L (f i))
(s : Finset ι) : A.Definable L (s.inf f) := by
classical
refine Finset.induction definable_univ (fun i s _ h => ?_) s
rw [Finset.inf_insert]
exact (hf i).inter h
#align set.definable_finset_inf Set.definable_finset_inf
| Mathlib/ModelTheory/Definability.lean | 133 | 138 | theorem definable_finset_sup {ι : Type*} {f : ι → Set (α → M)} (hf : ∀ i, A.Definable L (f i))
(s : Finset ι) : A.Definable L (s.sup f) := by |
classical
refine Finset.induction definable_empty (fun i s _ h => ?_) s
rw [Finset.sup_insert]
exact (hf i).union h
| 4 | 54.59815 | 2 | 1.6 | 10 | 1,740 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Language.{u, v}) [L.Structure M]
open FirstOrder FirstOrder.Language FirstOrder.Language.Structure
variable {α : Type u₁} {β : Type*}
def Definable (s : Set (α → M)) : Prop :=
∃ φ : L[[A]].Formula α, s = setOf φ.Realize
#align set.definable Set.Definable
variable {L} {A} {B : Set M} {s : Set (α → M)}
theorem Definable.map_expansion {L' : FirstOrder.Language} [L'.Structure M] (h : A.Definable L s)
(φ : L →ᴸ L') [φ.IsExpansionOn M] : A.Definable L' s := by
obtain ⟨ψ, rfl⟩ := h
refine ⟨(φ.addConstants A).onFormula ψ, ?_⟩
ext x
simp only [mem_setOf_eq, LHom.realize_onFormula]
#align set.definable.map_expansion Set.Definable.map_expansion
theorem definable_iff_exists_formula_sum :
A.Definable L s ↔ ∃ φ : L.Formula (A ⊕ α), s = {v | φ.Realize (Sum.elim (↑) v)} := by
rw [Definable, Equiv.exists_congr_left (BoundedFormula.constantsVarsEquiv)]
refine exists_congr (fun φ => iff_iff_eq.2 (congr_arg (s = ·) ?_))
ext
simp only [Formula.Realize, BoundedFormula.constantsVarsEquiv, constantsOn, mk₂_Relations,
BoundedFormula.mapTermRelEquiv_symm_apply, mem_setOf_eq]
refine BoundedFormula.realize_mapTermRel_id ?_ (fun _ _ _ => rfl)
intros
simp only [Term.constantsVarsEquivLeft_symm_apply, Term.realize_varsToConstants,
coe_con, Term.realize_relabel]
congr
ext a
rcases a with (_ | _) | _ <;> rfl
theorem empty_definable_iff :
(∅ : Set M).Definable L s ↔ ∃ φ : L.Formula α, s = setOf φ.Realize := by
rw [Definable, Equiv.exists_congr_left (LEquiv.addEmptyConstants L (∅ : Set M)).onFormula]
simp [-constantsOn]
#align set.empty_definable_iff Set.empty_definable_iff
theorem definable_iff_empty_definable_with_params :
A.Definable L s ↔ (∅ : Set M).Definable (L[[A]]) s :=
empty_definable_iff.symm
#align set.definable_iff_empty_definable_with_params Set.definable_iff_empty_definable_with_params
theorem Definable.mono (hAs : A.Definable L s) (hAB : A ⊆ B) : B.Definable L s := by
rw [definable_iff_empty_definable_with_params] at *
exact hAs.map_expansion (L.lhomWithConstantsMap (Set.inclusion hAB))
#align set.definable.mono Set.Definable.mono
@[simp]
theorem definable_empty : A.Definable L (∅ : Set (α → M)) :=
⟨⊥, by
ext
simp⟩
#align set.definable_empty Set.definable_empty
@[simp]
theorem definable_univ : A.Definable L (univ : Set (α → M)) :=
⟨⊤, by
ext
simp⟩
#align set.definable_univ Set.definable_univ
@[simp]
theorem Definable.inter {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∩ g) := by
rcases hf with ⟨φ, rfl⟩
rcases hg with ⟨θ, rfl⟩
refine ⟨φ ⊓ θ, ?_⟩
ext
simp
#align set.definable.inter Set.Definable.inter
@[simp]
theorem Definable.union {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∪ g) := by
rcases hf with ⟨φ, hφ⟩
rcases hg with ⟨θ, hθ⟩
refine ⟨φ ⊔ θ, ?_⟩
ext
rw [hφ, hθ, mem_setOf_eq, Formula.realize_sup, mem_union, mem_setOf_eq, mem_setOf_eq]
#align set.definable.union Set.Definable.union
theorem definable_finset_inf {ι : Type*} {f : ι → Set (α → M)} (hf : ∀ i, A.Definable L (f i))
(s : Finset ι) : A.Definable L (s.inf f) := by
classical
refine Finset.induction definable_univ (fun i s _ h => ?_) s
rw [Finset.inf_insert]
exact (hf i).inter h
#align set.definable_finset_inf Set.definable_finset_inf
theorem definable_finset_sup {ι : Type*} {f : ι → Set (α → M)} (hf : ∀ i, A.Definable L (f i))
(s : Finset ι) : A.Definable L (s.sup f) := by
classical
refine Finset.induction definable_empty (fun i s _ h => ?_) s
rw [Finset.sup_insert]
exact (hf i).union h
#align set.definable_finset_sup Set.definable_finset_sup
| Mathlib/ModelTheory/Definability.lean | 141 | 144 | theorem definable_finset_biInter {ι : Type*} {f : ι → Set (α → M)}
(hf : ∀ i, A.Definable L (f i)) (s : Finset ι) : A.Definable L (⋂ i ∈ s, f i) := by |
rw [← Finset.inf_set_eq_iInter]
exact definable_finset_inf hf s
| 2 | 7.389056 | 1 | 1.6 | 10 | 1,740 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Language.{u, v}) [L.Structure M]
open FirstOrder FirstOrder.Language FirstOrder.Language.Structure
variable {α : Type u₁} {β : Type*}
def Definable (s : Set (α → M)) : Prop :=
∃ φ : L[[A]].Formula α, s = setOf φ.Realize
#align set.definable Set.Definable
variable {L} {A} {B : Set M} {s : Set (α → M)}
theorem Definable.map_expansion {L' : FirstOrder.Language} [L'.Structure M] (h : A.Definable L s)
(φ : L →ᴸ L') [φ.IsExpansionOn M] : A.Definable L' s := by
obtain ⟨ψ, rfl⟩ := h
refine ⟨(φ.addConstants A).onFormula ψ, ?_⟩
ext x
simp only [mem_setOf_eq, LHom.realize_onFormula]
#align set.definable.map_expansion Set.Definable.map_expansion
theorem definable_iff_exists_formula_sum :
A.Definable L s ↔ ∃ φ : L.Formula (A ⊕ α), s = {v | φ.Realize (Sum.elim (↑) v)} := by
rw [Definable, Equiv.exists_congr_left (BoundedFormula.constantsVarsEquiv)]
refine exists_congr (fun φ => iff_iff_eq.2 (congr_arg (s = ·) ?_))
ext
simp only [Formula.Realize, BoundedFormula.constantsVarsEquiv, constantsOn, mk₂_Relations,
BoundedFormula.mapTermRelEquiv_symm_apply, mem_setOf_eq]
refine BoundedFormula.realize_mapTermRel_id ?_ (fun _ _ _ => rfl)
intros
simp only [Term.constantsVarsEquivLeft_symm_apply, Term.realize_varsToConstants,
coe_con, Term.realize_relabel]
congr
ext a
rcases a with (_ | _) | _ <;> rfl
theorem empty_definable_iff :
(∅ : Set M).Definable L s ↔ ∃ φ : L.Formula α, s = setOf φ.Realize := by
rw [Definable, Equiv.exists_congr_left (LEquiv.addEmptyConstants L (∅ : Set M)).onFormula]
simp [-constantsOn]
#align set.empty_definable_iff Set.empty_definable_iff
theorem definable_iff_empty_definable_with_params :
A.Definable L s ↔ (∅ : Set M).Definable (L[[A]]) s :=
empty_definable_iff.symm
#align set.definable_iff_empty_definable_with_params Set.definable_iff_empty_definable_with_params
theorem Definable.mono (hAs : A.Definable L s) (hAB : A ⊆ B) : B.Definable L s := by
rw [definable_iff_empty_definable_with_params] at *
exact hAs.map_expansion (L.lhomWithConstantsMap (Set.inclusion hAB))
#align set.definable.mono Set.Definable.mono
@[simp]
theorem definable_empty : A.Definable L (∅ : Set (α → M)) :=
⟨⊥, by
ext
simp⟩
#align set.definable_empty Set.definable_empty
@[simp]
theorem definable_univ : A.Definable L (univ : Set (α → M)) :=
⟨⊤, by
ext
simp⟩
#align set.definable_univ Set.definable_univ
@[simp]
theorem Definable.inter {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∩ g) := by
rcases hf with ⟨φ, rfl⟩
rcases hg with ⟨θ, rfl⟩
refine ⟨φ ⊓ θ, ?_⟩
ext
simp
#align set.definable.inter Set.Definable.inter
@[simp]
theorem Definable.union {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∪ g) := by
rcases hf with ⟨φ, hφ⟩
rcases hg with ⟨θ, hθ⟩
refine ⟨φ ⊔ θ, ?_⟩
ext
rw [hφ, hθ, mem_setOf_eq, Formula.realize_sup, mem_union, mem_setOf_eq, mem_setOf_eq]
#align set.definable.union Set.Definable.union
theorem definable_finset_inf {ι : Type*} {f : ι → Set (α → M)} (hf : ∀ i, A.Definable L (f i))
(s : Finset ι) : A.Definable L (s.inf f) := by
classical
refine Finset.induction definable_univ (fun i s _ h => ?_) s
rw [Finset.inf_insert]
exact (hf i).inter h
#align set.definable_finset_inf Set.definable_finset_inf
theorem definable_finset_sup {ι : Type*} {f : ι → Set (α → M)} (hf : ∀ i, A.Definable L (f i))
(s : Finset ι) : A.Definable L (s.sup f) := by
classical
refine Finset.induction definable_empty (fun i s _ h => ?_) s
rw [Finset.sup_insert]
exact (hf i).union h
#align set.definable_finset_sup Set.definable_finset_sup
theorem definable_finset_biInter {ι : Type*} {f : ι → Set (α → M)}
(hf : ∀ i, A.Definable L (f i)) (s : Finset ι) : A.Definable L (⋂ i ∈ s, f i) := by
rw [← Finset.inf_set_eq_iInter]
exact definable_finset_inf hf s
#align set.definable_finset_bInter Set.definable_finset_biInter
| Mathlib/ModelTheory/Definability.lean | 147 | 150 | theorem definable_finset_biUnion {ι : Type*} {f : ι → Set (α → M)}
(hf : ∀ i, A.Definable L (f i)) (s : Finset ι) : A.Definable L (⋃ i ∈ s, f i) := by |
rw [← Finset.sup_set_eq_biUnion]
exact definable_finset_sup hf s
| 2 | 7.389056 | 1 | 1.6 | 10 | 1,740 |
import Mathlib.Data.List.Basic
open Function
open Nat hiding one_pos
assert_not_exists Set.range
namespace List
universe u v w
variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w} {l₁ l₂ : List α}
section InsertNth
variable {a : α}
@[simp]
theorem insertNth_zero (s : List α) (x : α) : insertNth 0 x s = x :: s :=
rfl
#align list.insert_nth_zero List.insertNth_zero
@[simp]
theorem insertNth_succ_nil (n : ℕ) (a : α) : insertNth (n + 1) a [] = [] :=
rfl
#align list.insert_nth_succ_nil List.insertNth_succ_nil
@[simp]
theorem insertNth_succ_cons (s : List α) (hd x : α) (n : ℕ) :
insertNth (n + 1) x (hd :: s) = hd :: insertNth n x s :=
rfl
#align list.insert_nth_succ_cons List.insertNth_succ_cons
theorem length_insertNth : ∀ n as, n ≤ length as → length (insertNth n a as) = length as + 1
| 0, _, _ => rfl
| _ + 1, [], h => (Nat.not_succ_le_zero _ h).elim
| n + 1, _ :: as, h => congr_arg Nat.succ <| length_insertNth n as (Nat.le_of_succ_le_succ h)
#align list.length_insert_nth List.length_insertNth
| Mathlib/Data/List/InsertNth.lean | 52 | 54 | theorem eraseIdx_insertNth (n : ℕ) (l : List α) : (l.insertNth n a).eraseIdx n = l := by |
rw [eraseIdx_eq_modifyNthTail, insertNth, modifyNthTail_modifyNthTail_same]
exact modifyNthTail_id _ _
| 2 | 7.389056 | 1 | 1.6 | 5 | 1,741 |
import Mathlib.Data.List.Basic
open Function
open Nat hiding one_pos
assert_not_exists Set.range
namespace List
universe u v w
variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w} {l₁ l₂ : List α}
section InsertNth
variable {a : α}
@[simp]
theorem insertNth_zero (s : List α) (x : α) : insertNth 0 x s = x :: s :=
rfl
#align list.insert_nth_zero List.insertNth_zero
@[simp]
theorem insertNth_succ_nil (n : ℕ) (a : α) : insertNth (n + 1) a [] = [] :=
rfl
#align list.insert_nth_succ_nil List.insertNth_succ_nil
@[simp]
theorem insertNth_succ_cons (s : List α) (hd x : α) (n : ℕ) :
insertNth (n + 1) x (hd :: s) = hd :: insertNth n x s :=
rfl
#align list.insert_nth_succ_cons List.insertNth_succ_cons
theorem length_insertNth : ∀ n as, n ≤ length as → length (insertNth n a as) = length as + 1
| 0, _, _ => rfl
| _ + 1, [], h => (Nat.not_succ_le_zero _ h).elim
| n + 1, _ :: as, h => congr_arg Nat.succ <| length_insertNth n as (Nat.le_of_succ_le_succ h)
#align list.length_insert_nth List.length_insertNth
theorem eraseIdx_insertNth (n : ℕ) (l : List α) : (l.insertNth n a).eraseIdx n = l := by
rw [eraseIdx_eq_modifyNthTail, insertNth, modifyNthTail_modifyNthTail_same]
exact modifyNthTail_id _ _
#align list.remove_nth_insert_nth List.eraseIdx_insertNth
@[deprecated (since := "2024-05-04")] alias removeNth_insertNth := eraseIdx_insertNth
theorem insertNth_eraseIdx_of_ge :
∀ n m as,
n < length as → n ≤ m → insertNth m a (as.eraseIdx n) = (as.insertNth (m + 1) a).eraseIdx n
| 0, 0, [], has, _ => (lt_irrefl _ has).elim
| 0, 0, _ :: as, _, _ => by simp [eraseIdx, insertNth]
| 0, m + 1, a :: as, _, _ => rfl
| n + 1, m + 1, a :: as, has, hmn =>
congr_arg (cons a) <|
insertNth_eraseIdx_of_ge n m as (Nat.lt_of_succ_lt_succ has) (Nat.le_of_succ_le_succ hmn)
#align list.insert_nth_remove_nth_of_ge List.insertNth_eraseIdx_of_ge
@[deprecated (since := "2024-05-04")] alias insertNth_removeNth_of_ge := insertNth_eraseIdx_of_ge
theorem insertNth_eraseIdx_of_le :
∀ n m as,
n < length as → m ≤ n → insertNth m a (as.eraseIdx n) = (as.insertNth m a).eraseIdx (n + 1)
| _, 0, _ :: _, _, _ => rfl
| n + 1, m + 1, a :: as, has, hmn =>
congr_arg (cons a) <|
insertNth_eraseIdx_of_le n m as (Nat.lt_of_succ_lt_succ has) (Nat.le_of_succ_le_succ hmn)
#align list.insert_nth_remove_nth_of_le List.insertNth_eraseIdx_of_le
@[deprecated (since := "2024-05-04")] alias insertNth_removeNth_of_le := insertNth_eraseIdx_of_le
theorem insertNth_comm (a b : α) :
∀ (i j : ℕ) (l : List α) (_ : i ≤ j) (_ : j ≤ length l),
(l.insertNth i a).insertNth (j + 1) b = (l.insertNth j b).insertNth i a
| 0, j, l => by simp [insertNth]
| i + 1, 0, l => fun h => (Nat.not_lt_zero _ h).elim
| i + 1, j + 1, [] => by simp
| i + 1, j + 1, c :: l => fun h₀ h₁ => by
simp only [insertNth_succ_cons, cons.injEq, true_and]
exact insertNth_comm a b i j l (Nat.le_of_succ_le_succ h₀) (Nat.le_of_succ_le_succ h₁)
#align list.insert_nth_comm List.insertNth_comm
theorem mem_insertNth {a b : α} :
∀ {n : ℕ} {l : List α} (_ : n ≤ l.length), a ∈ l.insertNth n b ↔ a = b ∨ a ∈ l
| 0, as, _ => by simp
| n + 1, [], h => (Nat.not_succ_le_zero _ h).elim
| n + 1, a' :: as, h => by
rw [List.insertNth_succ_cons, mem_cons, mem_insertNth (Nat.le_of_succ_le_succ h),
← or_assoc, @or_comm (a = a'), or_assoc, mem_cons]
#align list.mem_insert_nth List.mem_insertNth
| Mathlib/Data/List/InsertNth.lean | 103 | 112 | theorem insertNth_of_length_lt (l : List α) (x : α) (n : ℕ) (h : l.length < n) :
insertNth n x l = l := by |
induction' l with hd tl IH generalizing n
· cases n
· simp at h
· simp
· cases n
· simp at h
· simp only [Nat.succ_lt_succ_iff, length] at h
simpa using IH _ h
| 8 | 2,980.957987 | 2 | 1.6 | 5 | 1,741 |
import Mathlib.Data.List.Basic
open Function
open Nat hiding one_pos
assert_not_exists Set.range
namespace List
universe u v w
variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w} {l₁ l₂ : List α}
section InsertNth
variable {a : α}
@[simp]
theorem insertNth_zero (s : List α) (x : α) : insertNth 0 x s = x :: s :=
rfl
#align list.insert_nth_zero List.insertNth_zero
@[simp]
theorem insertNth_succ_nil (n : ℕ) (a : α) : insertNth (n + 1) a [] = [] :=
rfl
#align list.insert_nth_succ_nil List.insertNth_succ_nil
@[simp]
theorem insertNth_succ_cons (s : List α) (hd x : α) (n : ℕ) :
insertNth (n + 1) x (hd :: s) = hd :: insertNth n x s :=
rfl
#align list.insert_nth_succ_cons List.insertNth_succ_cons
theorem length_insertNth : ∀ n as, n ≤ length as → length (insertNth n a as) = length as + 1
| 0, _, _ => rfl
| _ + 1, [], h => (Nat.not_succ_le_zero _ h).elim
| n + 1, _ :: as, h => congr_arg Nat.succ <| length_insertNth n as (Nat.le_of_succ_le_succ h)
#align list.length_insert_nth List.length_insertNth
theorem eraseIdx_insertNth (n : ℕ) (l : List α) : (l.insertNth n a).eraseIdx n = l := by
rw [eraseIdx_eq_modifyNthTail, insertNth, modifyNthTail_modifyNthTail_same]
exact modifyNthTail_id _ _
#align list.remove_nth_insert_nth List.eraseIdx_insertNth
@[deprecated (since := "2024-05-04")] alias removeNth_insertNth := eraseIdx_insertNth
theorem insertNth_eraseIdx_of_ge :
∀ n m as,
n < length as → n ≤ m → insertNth m a (as.eraseIdx n) = (as.insertNth (m + 1) a).eraseIdx n
| 0, 0, [], has, _ => (lt_irrefl _ has).elim
| 0, 0, _ :: as, _, _ => by simp [eraseIdx, insertNth]
| 0, m + 1, a :: as, _, _ => rfl
| n + 1, m + 1, a :: as, has, hmn =>
congr_arg (cons a) <|
insertNth_eraseIdx_of_ge n m as (Nat.lt_of_succ_lt_succ has) (Nat.le_of_succ_le_succ hmn)
#align list.insert_nth_remove_nth_of_ge List.insertNth_eraseIdx_of_ge
@[deprecated (since := "2024-05-04")] alias insertNth_removeNth_of_ge := insertNth_eraseIdx_of_ge
theorem insertNth_eraseIdx_of_le :
∀ n m as,
n < length as → m ≤ n → insertNth m a (as.eraseIdx n) = (as.insertNth m a).eraseIdx (n + 1)
| _, 0, _ :: _, _, _ => rfl
| n + 1, m + 1, a :: as, has, hmn =>
congr_arg (cons a) <|
insertNth_eraseIdx_of_le n m as (Nat.lt_of_succ_lt_succ has) (Nat.le_of_succ_le_succ hmn)
#align list.insert_nth_remove_nth_of_le List.insertNth_eraseIdx_of_le
@[deprecated (since := "2024-05-04")] alias insertNth_removeNth_of_le := insertNth_eraseIdx_of_le
theorem insertNth_comm (a b : α) :
∀ (i j : ℕ) (l : List α) (_ : i ≤ j) (_ : j ≤ length l),
(l.insertNth i a).insertNth (j + 1) b = (l.insertNth j b).insertNth i a
| 0, j, l => by simp [insertNth]
| i + 1, 0, l => fun h => (Nat.not_lt_zero _ h).elim
| i + 1, j + 1, [] => by simp
| i + 1, j + 1, c :: l => fun h₀ h₁ => by
simp only [insertNth_succ_cons, cons.injEq, true_and]
exact insertNth_comm a b i j l (Nat.le_of_succ_le_succ h₀) (Nat.le_of_succ_le_succ h₁)
#align list.insert_nth_comm List.insertNth_comm
theorem mem_insertNth {a b : α} :
∀ {n : ℕ} {l : List α} (_ : n ≤ l.length), a ∈ l.insertNth n b ↔ a = b ∨ a ∈ l
| 0, as, _ => by simp
| n + 1, [], h => (Nat.not_succ_le_zero _ h).elim
| n + 1, a' :: as, h => by
rw [List.insertNth_succ_cons, mem_cons, mem_insertNth (Nat.le_of_succ_le_succ h),
← or_assoc, @or_comm (a = a'), or_assoc, mem_cons]
#align list.mem_insert_nth List.mem_insertNth
theorem insertNth_of_length_lt (l : List α) (x : α) (n : ℕ) (h : l.length < n) :
insertNth n x l = l := by
induction' l with hd tl IH generalizing n
· cases n
· simp at h
· simp
· cases n
· simp at h
· simp only [Nat.succ_lt_succ_iff, length] at h
simpa using IH _ h
#align list.insert_nth_of_length_lt List.insertNth_of_length_lt
@[simp]
| Mathlib/Data/List/InsertNth.lean | 116 | 119 | theorem insertNth_length_self (l : List α) (x : α) : insertNth l.length x l = l ++ [x] := by |
induction' l with hd tl IH
· simp
· simpa using IH
| 3 | 20.085537 | 1 | 1.6 | 5 | 1,741 |
import Mathlib.Data.List.Basic
open Function
open Nat hiding one_pos
assert_not_exists Set.range
namespace List
universe u v w
variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w} {l₁ l₂ : List α}
section InsertNth
variable {a : α}
@[simp]
theorem insertNth_zero (s : List α) (x : α) : insertNth 0 x s = x :: s :=
rfl
#align list.insert_nth_zero List.insertNth_zero
@[simp]
theorem insertNth_succ_nil (n : ℕ) (a : α) : insertNth (n + 1) a [] = [] :=
rfl
#align list.insert_nth_succ_nil List.insertNth_succ_nil
@[simp]
theorem insertNth_succ_cons (s : List α) (hd x : α) (n : ℕ) :
insertNth (n + 1) x (hd :: s) = hd :: insertNth n x s :=
rfl
#align list.insert_nth_succ_cons List.insertNth_succ_cons
theorem length_insertNth : ∀ n as, n ≤ length as → length (insertNth n a as) = length as + 1
| 0, _, _ => rfl
| _ + 1, [], h => (Nat.not_succ_le_zero _ h).elim
| n + 1, _ :: as, h => congr_arg Nat.succ <| length_insertNth n as (Nat.le_of_succ_le_succ h)
#align list.length_insert_nth List.length_insertNth
theorem eraseIdx_insertNth (n : ℕ) (l : List α) : (l.insertNth n a).eraseIdx n = l := by
rw [eraseIdx_eq_modifyNthTail, insertNth, modifyNthTail_modifyNthTail_same]
exact modifyNthTail_id _ _
#align list.remove_nth_insert_nth List.eraseIdx_insertNth
@[deprecated (since := "2024-05-04")] alias removeNth_insertNth := eraseIdx_insertNth
theorem insertNth_eraseIdx_of_ge :
∀ n m as,
n < length as → n ≤ m → insertNth m a (as.eraseIdx n) = (as.insertNth (m + 1) a).eraseIdx n
| 0, 0, [], has, _ => (lt_irrefl _ has).elim
| 0, 0, _ :: as, _, _ => by simp [eraseIdx, insertNth]
| 0, m + 1, a :: as, _, _ => rfl
| n + 1, m + 1, a :: as, has, hmn =>
congr_arg (cons a) <|
insertNth_eraseIdx_of_ge n m as (Nat.lt_of_succ_lt_succ has) (Nat.le_of_succ_le_succ hmn)
#align list.insert_nth_remove_nth_of_ge List.insertNth_eraseIdx_of_ge
@[deprecated (since := "2024-05-04")] alias insertNth_removeNth_of_ge := insertNth_eraseIdx_of_ge
theorem insertNth_eraseIdx_of_le :
∀ n m as,
n < length as → m ≤ n → insertNth m a (as.eraseIdx n) = (as.insertNth m a).eraseIdx (n + 1)
| _, 0, _ :: _, _, _ => rfl
| n + 1, m + 1, a :: as, has, hmn =>
congr_arg (cons a) <|
insertNth_eraseIdx_of_le n m as (Nat.lt_of_succ_lt_succ has) (Nat.le_of_succ_le_succ hmn)
#align list.insert_nth_remove_nth_of_le List.insertNth_eraseIdx_of_le
@[deprecated (since := "2024-05-04")] alias insertNth_removeNth_of_le := insertNth_eraseIdx_of_le
theorem insertNth_comm (a b : α) :
∀ (i j : ℕ) (l : List α) (_ : i ≤ j) (_ : j ≤ length l),
(l.insertNth i a).insertNth (j + 1) b = (l.insertNth j b).insertNth i a
| 0, j, l => by simp [insertNth]
| i + 1, 0, l => fun h => (Nat.not_lt_zero _ h).elim
| i + 1, j + 1, [] => by simp
| i + 1, j + 1, c :: l => fun h₀ h₁ => by
simp only [insertNth_succ_cons, cons.injEq, true_and]
exact insertNth_comm a b i j l (Nat.le_of_succ_le_succ h₀) (Nat.le_of_succ_le_succ h₁)
#align list.insert_nth_comm List.insertNth_comm
theorem mem_insertNth {a b : α} :
∀ {n : ℕ} {l : List α} (_ : n ≤ l.length), a ∈ l.insertNth n b ↔ a = b ∨ a ∈ l
| 0, as, _ => by simp
| n + 1, [], h => (Nat.not_succ_le_zero _ h).elim
| n + 1, a' :: as, h => by
rw [List.insertNth_succ_cons, mem_cons, mem_insertNth (Nat.le_of_succ_le_succ h),
← or_assoc, @or_comm (a = a'), or_assoc, mem_cons]
#align list.mem_insert_nth List.mem_insertNth
theorem insertNth_of_length_lt (l : List α) (x : α) (n : ℕ) (h : l.length < n) :
insertNth n x l = l := by
induction' l with hd tl IH generalizing n
· cases n
· simp at h
· simp
· cases n
· simp at h
· simp only [Nat.succ_lt_succ_iff, length] at h
simpa using IH _ h
#align list.insert_nth_of_length_lt List.insertNth_of_length_lt
@[simp]
theorem insertNth_length_self (l : List α) (x : α) : insertNth l.length x l = l ++ [x] := by
induction' l with hd tl IH
· simp
· simpa using IH
#align list.insert_nth_length_self List.insertNth_length_self
| Mathlib/Data/List/InsertNth.lean | 122 | 127 | theorem length_le_length_insertNth (l : List α) (x : α) (n : ℕ) :
l.length ≤ (insertNth n x l).length := by |
rcases le_or_lt n l.length with hn | hn
· rw [length_insertNth _ _ hn]
exact (Nat.lt_succ_self _).le
· rw [insertNth_of_length_lt _ _ _ hn]
| 4 | 54.59815 | 2 | 1.6 | 5 | 1,741 |
import Mathlib.Data.List.Basic
open Function
open Nat hiding one_pos
assert_not_exists Set.range
namespace List
universe u v w
variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w} {l₁ l₂ : List α}
section InsertNth
variable {a : α}
@[simp]
theorem insertNth_zero (s : List α) (x : α) : insertNth 0 x s = x :: s :=
rfl
#align list.insert_nth_zero List.insertNth_zero
@[simp]
theorem insertNth_succ_nil (n : ℕ) (a : α) : insertNth (n + 1) a [] = [] :=
rfl
#align list.insert_nth_succ_nil List.insertNth_succ_nil
@[simp]
theorem insertNth_succ_cons (s : List α) (hd x : α) (n : ℕ) :
insertNth (n + 1) x (hd :: s) = hd :: insertNth n x s :=
rfl
#align list.insert_nth_succ_cons List.insertNth_succ_cons
theorem length_insertNth : ∀ n as, n ≤ length as → length (insertNth n a as) = length as + 1
| 0, _, _ => rfl
| _ + 1, [], h => (Nat.not_succ_le_zero _ h).elim
| n + 1, _ :: as, h => congr_arg Nat.succ <| length_insertNth n as (Nat.le_of_succ_le_succ h)
#align list.length_insert_nth List.length_insertNth
theorem eraseIdx_insertNth (n : ℕ) (l : List α) : (l.insertNth n a).eraseIdx n = l := by
rw [eraseIdx_eq_modifyNthTail, insertNth, modifyNthTail_modifyNthTail_same]
exact modifyNthTail_id _ _
#align list.remove_nth_insert_nth List.eraseIdx_insertNth
@[deprecated (since := "2024-05-04")] alias removeNth_insertNth := eraseIdx_insertNth
theorem insertNth_eraseIdx_of_ge :
∀ n m as,
n < length as → n ≤ m → insertNth m a (as.eraseIdx n) = (as.insertNth (m + 1) a).eraseIdx n
| 0, 0, [], has, _ => (lt_irrefl _ has).elim
| 0, 0, _ :: as, _, _ => by simp [eraseIdx, insertNth]
| 0, m + 1, a :: as, _, _ => rfl
| n + 1, m + 1, a :: as, has, hmn =>
congr_arg (cons a) <|
insertNth_eraseIdx_of_ge n m as (Nat.lt_of_succ_lt_succ has) (Nat.le_of_succ_le_succ hmn)
#align list.insert_nth_remove_nth_of_ge List.insertNth_eraseIdx_of_ge
@[deprecated (since := "2024-05-04")] alias insertNth_removeNth_of_ge := insertNth_eraseIdx_of_ge
theorem insertNth_eraseIdx_of_le :
∀ n m as,
n < length as → m ≤ n → insertNth m a (as.eraseIdx n) = (as.insertNth m a).eraseIdx (n + 1)
| _, 0, _ :: _, _, _ => rfl
| n + 1, m + 1, a :: as, has, hmn =>
congr_arg (cons a) <|
insertNth_eraseIdx_of_le n m as (Nat.lt_of_succ_lt_succ has) (Nat.le_of_succ_le_succ hmn)
#align list.insert_nth_remove_nth_of_le List.insertNth_eraseIdx_of_le
@[deprecated (since := "2024-05-04")] alias insertNth_removeNth_of_le := insertNth_eraseIdx_of_le
theorem insertNth_comm (a b : α) :
∀ (i j : ℕ) (l : List α) (_ : i ≤ j) (_ : j ≤ length l),
(l.insertNth i a).insertNth (j + 1) b = (l.insertNth j b).insertNth i a
| 0, j, l => by simp [insertNth]
| i + 1, 0, l => fun h => (Nat.not_lt_zero _ h).elim
| i + 1, j + 1, [] => by simp
| i + 1, j + 1, c :: l => fun h₀ h₁ => by
simp only [insertNth_succ_cons, cons.injEq, true_and]
exact insertNth_comm a b i j l (Nat.le_of_succ_le_succ h₀) (Nat.le_of_succ_le_succ h₁)
#align list.insert_nth_comm List.insertNth_comm
theorem mem_insertNth {a b : α} :
∀ {n : ℕ} {l : List α} (_ : n ≤ l.length), a ∈ l.insertNth n b ↔ a = b ∨ a ∈ l
| 0, as, _ => by simp
| n + 1, [], h => (Nat.not_succ_le_zero _ h).elim
| n + 1, a' :: as, h => by
rw [List.insertNth_succ_cons, mem_cons, mem_insertNth (Nat.le_of_succ_le_succ h),
← or_assoc, @or_comm (a = a'), or_assoc, mem_cons]
#align list.mem_insert_nth List.mem_insertNth
theorem insertNth_of_length_lt (l : List α) (x : α) (n : ℕ) (h : l.length < n) :
insertNth n x l = l := by
induction' l with hd tl IH generalizing n
· cases n
· simp at h
· simp
· cases n
· simp at h
· simp only [Nat.succ_lt_succ_iff, length] at h
simpa using IH _ h
#align list.insert_nth_of_length_lt List.insertNth_of_length_lt
@[simp]
theorem insertNth_length_self (l : List α) (x : α) : insertNth l.length x l = l ++ [x] := by
induction' l with hd tl IH
· simp
· simpa using IH
#align list.insert_nth_length_self List.insertNth_length_self
theorem length_le_length_insertNth (l : List α) (x : α) (n : ℕ) :
l.length ≤ (insertNth n x l).length := by
rcases le_or_lt n l.length with hn | hn
· rw [length_insertNth _ _ hn]
exact (Nat.lt_succ_self _).le
· rw [insertNth_of_length_lt _ _ _ hn]
#align list.length_le_length_insert_nth List.length_le_length_insertNth
| Mathlib/Data/List/InsertNth.lean | 130 | 135 | theorem length_insertNth_le_succ (l : List α) (x : α) (n : ℕ) :
(insertNth n x l).length ≤ l.length + 1 := by |
rcases le_or_lt n l.length with hn | hn
· rw [length_insertNth _ _ hn]
· rw [insertNth_of_length_lt _ _ _ hn]
exact (Nat.lt_succ_self _).le
| 4 | 54.59815 | 2 | 1.6 | 5 | 1,741 |
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Basic
import Mathlib.Order.CompleteLatticeIntervals
#align_import order.sup_indep from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
variable {α β ι ι' : Type*}
namespace Finset
section Lattice
variable [Lattice α] [OrderBot α]
def SupIndep (s : Finset ι) (f : ι → α) : Prop :=
∀ ⦃t⦄, t ⊆ s → ∀ ⦃i⦄, i ∈ s → i ∉ t → Disjoint (f i) (t.sup f)
#align finset.sup_indep Finset.SupIndep
variable {s t : Finset ι} {f : ι → α} {i : ι}
instance [DecidableEq ι] [DecidableEq α] : Decidable (SupIndep s f) := by
refine @Finset.decidableForallOfDecidableSubsets _ _ _ (?_)
rintro t -
refine @Finset.decidableDforallFinset _ _ _ (?_)
rintro i -
have : Decidable (Disjoint (f i) (sup t f)) := decidable_of_iff' (_ = ⊥) disjoint_iff
infer_instance
theorem SupIndep.subset (ht : t.SupIndep f) (h : s ⊆ t) : s.SupIndep f := fun _ hu _ hi =>
ht (hu.trans h) (h hi)
#align finset.sup_indep.subset Finset.SupIndep.subset
@[simp]
theorem supIndep_empty (f : ι → α) : (∅ : Finset ι).SupIndep f := fun _ _ a ha =>
(not_mem_empty a ha).elim
#align finset.sup_indep_empty Finset.supIndep_empty
theorem supIndep_singleton (i : ι) (f : ι → α) : ({i} : Finset ι).SupIndep f :=
fun s hs j hji hj => by
rw [eq_empty_of_ssubset_singleton ⟨hs, fun h => hj (h hji)⟩, sup_empty]
exact disjoint_bot_right
#align finset.sup_indep_singleton Finset.supIndep_singleton
theorem SupIndep.pairwiseDisjoint (hs : s.SupIndep f) : (s : Set ι).PairwiseDisjoint f :=
fun _ ha _ hb hab =>
sup_singleton.subst <| hs (singleton_subset_iff.2 hb) ha <| not_mem_singleton.2 hab
#align finset.sup_indep.pairwise_disjoint Finset.SupIndep.pairwiseDisjoint
| Mathlib/Order/SupIndep.lean | 92 | 96 | theorem SupIndep.le_sup_iff (hs : s.SupIndep f) (hts : t ⊆ s) (hi : i ∈ s) (hf : ∀ i, f i ≠ ⊥) :
f i ≤ t.sup f ↔ i ∈ t := by |
refine ⟨fun h => ?_, le_sup⟩
by_contra hit
exact hf i (disjoint_self.1 <| (hs hts hi hit).mono_right h)
| 3 | 20.085537 | 1 | 1.6 | 5 | 1,742 |
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Basic
import Mathlib.Order.CompleteLatticeIntervals
#align_import order.sup_indep from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
variable {α β ι ι' : Type*}
namespace Finset
section Lattice
variable [Lattice α] [OrderBot α]
def SupIndep (s : Finset ι) (f : ι → α) : Prop :=
∀ ⦃t⦄, t ⊆ s → ∀ ⦃i⦄, i ∈ s → i ∉ t → Disjoint (f i) (t.sup f)
#align finset.sup_indep Finset.SupIndep
variable {s t : Finset ι} {f : ι → α} {i : ι}
instance [DecidableEq ι] [DecidableEq α] : Decidable (SupIndep s f) := by
refine @Finset.decidableForallOfDecidableSubsets _ _ _ (?_)
rintro t -
refine @Finset.decidableDforallFinset _ _ _ (?_)
rintro i -
have : Decidable (Disjoint (f i) (sup t f)) := decidable_of_iff' (_ = ⊥) disjoint_iff
infer_instance
theorem SupIndep.subset (ht : t.SupIndep f) (h : s ⊆ t) : s.SupIndep f := fun _ hu _ hi =>
ht (hu.trans h) (h hi)
#align finset.sup_indep.subset Finset.SupIndep.subset
@[simp]
theorem supIndep_empty (f : ι → α) : (∅ : Finset ι).SupIndep f := fun _ _ a ha =>
(not_mem_empty a ha).elim
#align finset.sup_indep_empty Finset.supIndep_empty
theorem supIndep_singleton (i : ι) (f : ι → α) : ({i} : Finset ι).SupIndep f :=
fun s hs j hji hj => by
rw [eq_empty_of_ssubset_singleton ⟨hs, fun h => hj (h hji)⟩, sup_empty]
exact disjoint_bot_right
#align finset.sup_indep_singleton Finset.supIndep_singleton
theorem SupIndep.pairwiseDisjoint (hs : s.SupIndep f) : (s : Set ι).PairwiseDisjoint f :=
fun _ ha _ hb hab =>
sup_singleton.subst <| hs (singleton_subset_iff.2 hb) ha <| not_mem_singleton.2 hab
#align finset.sup_indep.pairwise_disjoint Finset.SupIndep.pairwiseDisjoint
theorem SupIndep.le_sup_iff (hs : s.SupIndep f) (hts : t ⊆ s) (hi : i ∈ s) (hf : ∀ i, f i ≠ ⊥) :
f i ≤ t.sup f ↔ i ∈ t := by
refine ⟨fun h => ?_, le_sup⟩
by_contra hit
exact hf i (disjoint_self.1 <| (hs hts hi hit).mono_right h)
#align finset.sup_indep.le_sup_iff Finset.SupIndep.le_sup_iff
theorem supIndep_iff_disjoint_erase [DecidableEq ι] :
s.SupIndep f ↔ ∀ i ∈ s, Disjoint (f i) ((s.erase i).sup f) :=
⟨fun hs _ hi => hs (erase_subset _ _) hi (not_mem_erase _ _), fun hs _ ht i hi hit =>
(hs i hi).mono_right (sup_mono fun _ hj => mem_erase.2 ⟨ne_of_mem_of_not_mem hj hit, ht hj⟩)⟩
#align finset.sup_indep_iff_disjoint_erase Finset.supIndep_iff_disjoint_erase
| Mathlib/Order/SupIndep.lean | 106 | 117 | theorem SupIndep.image [DecidableEq ι] {s : Finset ι'} {g : ι' → ι} (hs : s.SupIndep (f ∘ g)) :
(s.image g).SupIndep f := by |
intro t ht i hi hit
rw [mem_image] at hi
obtain ⟨i, hi, rfl⟩ := hi
haveI : DecidableEq ι' := Classical.decEq _
suffices hts : t ⊆ (s.erase i).image g by
refine (supIndep_iff_disjoint_erase.1 hs i hi).mono_right ((sup_mono hts).trans ?_)
rw [sup_image]
rintro j hjt
obtain ⟨j, hj, rfl⟩ := mem_image.1 (ht hjt)
exact mem_image_of_mem _ (mem_erase.2 ⟨ne_of_apply_ne g (ne_of_mem_of_not_mem hjt hit), hj⟩)
| 10 | 22,026.465795 | 2 | 1.6 | 5 | 1,742 |
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Basic
import Mathlib.Order.CompleteLatticeIntervals
#align_import order.sup_indep from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
variable {α β ι ι' : Type*}
namespace Finset
section Lattice
variable [Lattice α] [OrderBot α]
def SupIndep (s : Finset ι) (f : ι → α) : Prop :=
∀ ⦃t⦄, t ⊆ s → ∀ ⦃i⦄, i ∈ s → i ∉ t → Disjoint (f i) (t.sup f)
#align finset.sup_indep Finset.SupIndep
variable {s t : Finset ι} {f : ι → α} {i : ι}
instance [DecidableEq ι] [DecidableEq α] : Decidable (SupIndep s f) := by
refine @Finset.decidableForallOfDecidableSubsets _ _ _ (?_)
rintro t -
refine @Finset.decidableDforallFinset _ _ _ (?_)
rintro i -
have : Decidable (Disjoint (f i) (sup t f)) := decidable_of_iff' (_ = ⊥) disjoint_iff
infer_instance
theorem SupIndep.subset (ht : t.SupIndep f) (h : s ⊆ t) : s.SupIndep f := fun _ hu _ hi =>
ht (hu.trans h) (h hi)
#align finset.sup_indep.subset Finset.SupIndep.subset
@[simp]
theorem supIndep_empty (f : ι → α) : (∅ : Finset ι).SupIndep f := fun _ _ a ha =>
(not_mem_empty a ha).elim
#align finset.sup_indep_empty Finset.supIndep_empty
theorem supIndep_singleton (i : ι) (f : ι → α) : ({i} : Finset ι).SupIndep f :=
fun s hs j hji hj => by
rw [eq_empty_of_ssubset_singleton ⟨hs, fun h => hj (h hji)⟩, sup_empty]
exact disjoint_bot_right
#align finset.sup_indep_singleton Finset.supIndep_singleton
theorem SupIndep.pairwiseDisjoint (hs : s.SupIndep f) : (s : Set ι).PairwiseDisjoint f :=
fun _ ha _ hb hab =>
sup_singleton.subst <| hs (singleton_subset_iff.2 hb) ha <| not_mem_singleton.2 hab
#align finset.sup_indep.pairwise_disjoint Finset.SupIndep.pairwiseDisjoint
theorem SupIndep.le_sup_iff (hs : s.SupIndep f) (hts : t ⊆ s) (hi : i ∈ s) (hf : ∀ i, f i ≠ ⊥) :
f i ≤ t.sup f ↔ i ∈ t := by
refine ⟨fun h => ?_, le_sup⟩
by_contra hit
exact hf i (disjoint_self.1 <| (hs hts hi hit).mono_right h)
#align finset.sup_indep.le_sup_iff Finset.SupIndep.le_sup_iff
theorem supIndep_iff_disjoint_erase [DecidableEq ι] :
s.SupIndep f ↔ ∀ i ∈ s, Disjoint (f i) ((s.erase i).sup f) :=
⟨fun hs _ hi => hs (erase_subset _ _) hi (not_mem_erase _ _), fun hs _ ht i hi hit =>
(hs i hi).mono_right (sup_mono fun _ hj => mem_erase.2 ⟨ne_of_mem_of_not_mem hj hit, ht hj⟩)⟩
#align finset.sup_indep_iff_disjoint_erase Finset.supIndep_iff_disjoint_erase
theorem SupIndep.image [DecidableEq ι] {s : Finset ι'} {g : ι' → ι} (hs : s.SupIndep (f ∘ g)) :
(s.image g).SupIndep f := by
intro t ht i hi hit
rw [mem_image] at hi
obtain ⟨i, hi, rfl⟩ := hi
haveI : DecidableEq ι' := Classical.decEq _
suffices hts : t ⊆ (s.erase i).image g by
refine (supIndep_iff_disjoint_erase.1 hs i hi).mono_right ((sup_mono hts).trans ?_)
rw [sup_image]
rintro j hjt
obtain ⟨j, hj, rfl⟩ := mem_image.1 (ht hjt)
exact mem_image_of_mem _ (mem_erase.2 ⟨ne_of_apply_ne g (ne_of_mem_of_not_mem hjt hit), hj⟩)
#align finset.sup_indep.image Finset.SupIndep.image
| Mathlib/Order/SupIndep.lean | 120 | 126 | theorem supIndep_map {s : Finset ι'} {g : ι' ↪ ι} : (s.map g).SupIndep f ↔ s.SupIndep (f ∘ g) := by |
refine ⟨fun hs t ht i hi hit => ?_, fun hs => ?_⟩
· rw [← sup_map]
exact hs (map_subset_map.2 ht) ((mem_map' _).2 hi) (by rwa [mem_map'])
· classical
rw [map_eq_image]
exact hs.image
| 6 | 403.428793 | 2 | 1.6 | 5 | 1,742 |
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Basic
import Mathlib.Order.CompleteLatticeIntervals
#align_import order.sup_indep from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
variable {α β ι ι' : Type*}
namespace Finset
section Lattice
variable [Lattice α] [OrderBot α]
def SupIndep (s : Finset ι) (f : ι → α) : Prop :=
∀ ⦃t⦄, t ⊆ s → ∀ ⦃i⦄, i ∈ s → i ∉ t → Disjoint (f i) (t.sup f)
#align finset.sup_indep Finset.SupIndep
variable {s t : Finset ι} {f : ι → α} {i : ι}
instance [DecidableEq ι] [DecidableEq α] : Decidable (SupIndep s f) := by
refine @Finset.decidableForallOfDecidableSubsets _ _ _ (?_)
rintro t -
refine @Finset.decidableDforallFinset _ _ _ (?_)
rintro i -
have : Decidable (Disjoint (f i) (sup t f)) := decidable_of_iff' (_ = ⊥) disjoint_iff
infer_instance
theorem SupIndep.subset (ht : t.SupIndep f) (h : s ⊆ t) : s.SupIndep f := fun _ hu _ hi =>
ht (hu.trans h) (h hi)
#align finset.sup_indep.subset Finset.SupIndep.subset
@[simp]
theorem supIndep_empty (f : ι → α) : (∅ : Finset ι).SupIndep f := fun _ _ a ha =>
(not_mem_empty a ha).elim
#align finset.sup_indep_empty Finset.supIndep_empty
theorem supIndep_singleton (i : ι) (f : ι → α) : ({i} : Finset ι).SupIndep f :=
fun s hs j hji hj => by
rw [eq_empty_of_ssubset_singleton ⟨hs, fun h => hj (h hji)⟩, sup_empty]
exact disjoint_bot_right
#align finset.sup_indep_singleton Finset.supIndep_singleton
theorem SupIndep.pairwiseDisjoint (hs : s.SupIndep f) : (s : Set ι).PairwiseDisjoint f :=
fun _ ha _ hb hab =>
sup_singleton.subst <| hs (singleton_subset_iff.2 hb) ha <| not_mem_singleton.2 hab
#align finset.sup_indep.pairwise_disjoint Finset.SupIndep.pairwiseDisjoint
theorem SupIndep.le_sup_iff (hs : s.SupIndep f) (hts : t ⊆ s) (hi : i ∈ s) (hf : ∀ i, f i ≠ ⊥) :
f i ≤ t.sup f ↔ i ∈ t := by
refine ⟨fun h => ?_, le_sup⟩
by_contra hit
exact hf i (disjoint_self.1 <| (hs hts hi hit).mono_right h)
#align finset.sup_indep.le_sup_iff Finset.SupIndep.le_sup_iff
theorem supIndep_iff_disjoint_erase [DecidableEq ι] :
s.SupIndep f ↔ ∀ i ∈ s, Disjoint (f i) ((s.erase i).sup f) :=
⟨fun hs _ hi => hs (erase_subset _ _) hi (not_mem_erase _ _), fun hs _ ht i hi hit =>
(hs i hi).mono_right (sup_mono fun _ hj => mem_erase.2 ⟨ne_of_mem_of_not_mem hj hit, ht hj⟩)⟩
#align finset.sup_indep_iff_disjoint_erase Finset.supIndep_iff_disjoint_erase
theorem SupIndep.image [DecidableEq ι] {s : Finset ι'} {g : ι' → ι} (hs : s.SupIndep (f ∘ g)) :
(s.image g).SupIndep f := by
intro t ht i hi hit
rw [mem_image] at hi
obtain ⟨i, hi, rfl⟩ := hi
haveI : DecidableEq ι' := Classical.decEq _
suffices hts : t ⊆ (s.erase i).image g by
refine (supIndep_iff_disjoint_erase.1 hs i hi).mono_right ((sup_mono hts).trans ?_)
rw [sup_image]
rintro j hjt
obtain ⟨j, hj, rfl⟩ := mem_image.1 (ht hjt)
exact mem_image_of_mem _ (mem_erase.2 ⟨ne_of_apply_ne g (ne_of_mem_of_not_mem hjt hit), hj⟩)
#align finset.sup_indep.image Finset.SupIndep.image
theorem supIndep_map {s : Finset ι'} {g : ι' ↪ ι} : (s.map g).SupIndep f ↔ s.SupIndep (f ∘ g) := by
refine ⟨fun hs t ht i hi hit => ?_, fun hs => ?_⟩
· rw [← sup_map]
exact hs (map_subset_map.2 ht) ((mem_map' _).2 hi) (by rwa [mem_map'])
· classical
rw [map_eq_image]
exact hs.image
#align finset.sup_indep_map Finset.supIndep_map
@[simp]
| Mathlib/Order/SupIndep.lean | 130 | 148 | theorem supIndep_pair [DecidableEq ι] {i j : ι} (hij : i ≠ j) :
({i, j} : Finset ι).SupIndep f ↔ Disjoint (f i) (f j) :=
⟨fun h => h.pairwiseDisjoint (by simp) (by simp) hij,
fun h => by
rw [supIndep_iff_disjoint_erase]
intro k hk
rw [Finset.mem_insert, Finset.mem_singleton] at hk
obtain rfl | rfl := hk
· convert h using 1
rw [Finset.erase_insert, Finset.sup_singleton]
simpa using hij
· convert h.symm using 1
have : ({i, k} : Finset ι).erase k = {i} := by |
ext
rw [mem_erase, mem_insert, mem_singleton, mem_singleton, and_or_left, Ne,
not_and_self_iff, or_false_iff, and_iff_right_of_imp]
rintro rfl
exact hij
rw [this, Finset.sup_singleton]⟩
| 6 | 403.428793 | 2 | 1.6 | 5 | 1,742 |
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Basic
import Mathlib.Order.CompleteLatticeIntervals
#align_import order.sup_indep from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
variable {α β ι ι' : Type*}
namespace Finset
section Lattice
variable [Lattice α] [OrderBot α]
def SupIndep (s : Finset ι) (f : ι → α) : Prop :=
∀ ⦃t⦄, t ⊆ s → ∀ ⦃i⦄, i ∈ s → i ∉ t → Disjoint (f i) (t.sup f)
#align finset.sup_indep Finset.SupIndep
variable {s t : Finset ι} {f : ι → α} {i : ι}
instance [DecidableEq ι] [DecidableEq α] : Decidable (SupIndep s f) := by
refine @Finset.decidableForallOfDecidableSubsets _ _ _ (?_)
rintro t -
refine @Finset.decidableDforallFinset _ _ _ (?_)
rintro i -
have : Decidable (Disjoint (f i) (sup t f)) := decidable_of_iff' (_ = ⊥) disjoint_iff
infer_instance
theorem SupIndep.subset (ht : t.SupIndep f) (h : s ⊆ t) : s.SupIndep f := fun _ hu _ hi =>
ht (hu.trans h) (h hi)
#align finset.sup_indep.subset Finset.SupIndep.subset
@[simp]
theorem supIndep_empty (f : ι → α) : (∅ : Finset ι).SupIndep f := fun _ _ a ha =>
(not_mem_empty a ha).elim
#align finset.sup_indep_empty Finset.supIndep_empty
theorem supIndep_singleton (i : ι) (f : ι → α) : ({i} : Finset ι).SupIndep f :=
fun s hs j hji hj => by
rw [eq_empty_of_ssubset_singleton ⟨hs, fun h => hj (h hji)⟩, sup_empty]
exact disjoint_bot_right
#align finset.sup_indep_singleton Finset.supIndep_singleton
theorem SupIndep.pairwiseDisjoint (hs : s.SupIndep f) : (s : Set ι).PairwiseDisjoint f :=
fun _ ha _ hb hab =>
sup_singleton.subst <| hs (singleton_subset_iff.2 hb) ha <| not_mem_singleton.2 hab
#align finset.sup_indep.pairwise_disjoint Finset.SupIndep.pairwiseDisjoint
theorem SupIndep.le_sup_iff (hs : s.SupIndep f) (hts : t ⊆ s) (hi : i ∈ s) (hf : ∀ i, f i ≠ ⊥) :
f i ≤ t.sup f ↔ i ∈ t := by
refine ⟨fun h => ?_, le_sup⟩
by_contra hit
exact hf i (disjoint_self.1 <| (hs hts hi hit).mono_right h)
#align finset.sup_indep.le_sup_iff Finset.SupIndep.le_sup_iff
theorem supIndep_iff_disjoint_erase [DecidableEq ι] :
s.SupIndep f ↔ ∀ i ∈ s, Disjoint (f i) ((s.erase i).sup f) :=
⟨fun hs _ hi => hs (erase_subset _ _) hi (not_mem_erase _ _), fun hs _ ht i hi hit =>
(hs i hi).mono_right (sup_mono fun _ hj => mem_erase.2 ⟨ne_of_mem_of_not_mem hj hit, ht hj⟩)⟩
#align finset.sup_indep_iff_disjoint_erase Finset.supIndep_iff_disjoint_erase
theorem SupIndep.image [DecidableEq ι] {s : Finset ι'} {g : ι' → ι} (hs : s.SupIndep (f ∘ g)) :
(s.image g).SupIndep f := by
intro t ht i hi hit
rw [mem_image] at hi
obtain ⟨i, hi, rfl⟩ := hi
haveI : DecidableEq ι' := Classical.decEq _
suffices hts : t ⊆ (s.erase i).image g by
refine (supIndep_iff_disjoint_erase.1 hs i hi).mono_right ((sup_mono hts).trans ?_)
rw [sup_image]
rintro j hjt
obtain ⟨j, hj, rfl⟩ := mem_image.1 (ht hjt)
exact mem_image_of_mem _ (mem_erase.2 ⟨ne_of_apply_ne g (ne_of_mem_of_not_mem hjt hit), hj⟩)
#align finset.sup_indep.image Finset.SupIndep.image
theorem supIndep_map {s : Finset ι'} {g : ι' ↪ ι} : (s.map g).SupIndep f ↔ s.SupIndep (f ∘ g) := by
refine ⟨fun hs t ht i hi hit => ?_, fun hs => ?_⟩
· rw [← sup_map]
exact hs (map_subset_map.2 ht) ((mem_map' _).2 hi) (by rwa [mem_map'])
· classical
rw [map_eq_image]
exact hs.image
#align finset.sup_indep_map Finset.supIndep_map
@[simp]
theorem supIndep_pair [DecidableEq ι] {i j : ι} (hij : i ≠ j) :
({i, j} : Finset ι).SupIndep f ↔ Disjoint (f i) (f j) :=
⟨fun h => h.pairwiseDisjoint (by simp) (by simp) hij,
fun h => by
rw [supIndep_iff_disjoint_erase]
intro k hk
rw [Finset.mem_insert, Finset.mem_singleton] at hk
obtain rfl | rfl := hk
· convert h using 1
rw [Finset.erase_insert, Finset.sup_singleton]
simpa using hij
· convert h.symm using 1
have : ({i, k} : Finset ι).erase k = {i} := by
ext
rw [mem_erase, mem_insert, mem_singleton, mem_singleton, and_or_left, Ne,
not_and_self_iff, or_false_iff, and_iff_right_of_imp]
rintro rfl
exact hij
rw [this, Finset.sup_singleton]⟩
#align finset.sup_indep_pair Finset.supIndep_pair
| Mathlib/Order/SupIndep.lean | 151 | 154 | theorem supIndep_univ_bool (f : Bool → α) :
(Finset.univ : Finset Bool).SupIndep f ↔ Disjoint (f false) (f true) :=
haveI : true ≠ false := by | simp only [Ne, not_false_iff]
(supIndep_pair this).trans disjoint_comm
| 2 | 7.389056 | 1 | 1.6 | 5 | 1,742 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Function
variable {α β γ δ ε ζ : Type*}
section NeImp
variable {r : α → α → Prop}
theorem IsRefl.reflexive [IsRefl α r] : Reflexive r := fun x ↦ IsRefl.refl x
#align is_refl.reflexive IsRefl.reflexive
| Mathlib/Logic/Relation.lean | 61 | 64 | theorem Reflexive.rel_of_ne_imp (h : Reflexive r) {x y : α} (hr : x ≠ y → r x y) : r x y := by |
by_cases hxy : x = y
· exact hxy ▸ h x
· exact hr hxy
| 3 | 20.085537 | 1 | 1.6 | 15 | 1,743 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Function
variable {α β γ δ ε ζ : Type*}
namespace Relation
section Comp
variable {r : α → β → Prop} {p : β → γ → Prop} {q : γ → δ → Prop}
def Comp (r : α → β → Prop) (p : β → γ → Prop) (a : α) (c : γ) : Prop :=
∃ b, r a b ∧ p b c
#align relation.comp Relation.Comp
@[inherit_doc]
local infixr:80 " ∘r " => Relation.Comp
theorem comp_eq : r ∘r (· = ·) = r :=
funext fun _ ↦ funext fun b ↦ propext <|
Iff.intro (fun ⟨_, h, Eq⟩ ↦ Eq ▸ h) fun h ↦ ⟨b, h, rfl⟩
#align relation.comp_eq Relation.comp_eq
theorem eq_comp : (· = ·) ∘r r = r :=
funext fun a ↦ funext fun _ ↦ propext <|
Iff.intro (fun ⟨_, Eq, h⟩ ↦ Eq.symm ▸ h) fun h ↦ ⟨a, rfl, h⟩
#align relation.eq_comp Relation.eq_comp
| Mathlib/Logic/Relation.lean | 149 | 151 | theorem iff_comp {r : Prop → α → Prop} : (· ↔ ·) ∘r r = r := by |
have : (· ↔ ·) = (· = ·) := by funext a b; exact iff_eq_eq
rw [this, eq_comp]
| 2 | 7.389056 | 1 | 1.6 | 15 | 1,743 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Function
variable {α β γ δ ε ζ : Type*}
namespace Relation
section Comp
variable {r : α → β → Prop} {p : β → γ → Prop} {q : γ → δ → Prop}
def Comp (r : α → β → Prop) (p : β → γ → Prop) (a : α) (c : γ) : Prop :=
∃ b, r a b ∧ p b c
#align relation.comp Relation.Comp
@[inherit_doc]
local infixr:80 " ∘r " => Relation.Comp
theorem comp_eq : r ∘r (· = ·) = r :=
funext fun _ ↦ funext fun b ↦ propext <|
Iff.intro (fun ⟨_, h, Eq⟩ ↦ Eq ▸ h) fun h ↦ ⟨b, h, rfl⟩
#align relation.comp_eq Relation.comp_eq
theorem eq_comp : (· = ·) ∘r r = r :=
funext fun a ↦ funext fun _ ↦ propext <|
Iff.intro (fun ⟨_, Eq, h⟩ ↦ Eq.symm ▸ h) fun h ↦ ⟨a, rfl, h⟩
#align relation.eq_comp Relation.eq_comp
theorem iff_comp {r : Prop → α → Prop} : (· ↔ ·) ∘r r = r := by
have : (· ↔ ·) = (· = ·) := by funext a b; exact iff_eq_eq
rw [this, eq_comp]
#align relation.iff_comp Relation.iff_comp
| Mathlib/Logic/Relation.lean | 154 | 156 | theorem comp_iff {r : α → Prop → Prop} : r ∘r (· ↔ ·) = r := by |
have : (· ↔ ·) = (· = ·) := by funext a b; exact iff_eq_eq
rw [this, comp_eq]
| 2 | 7.389056 | 1 | 1.6 | 15 | 1,743 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Function
variable {α β γ δ ε ζ : Type*}
namespace Relation
section Comp
variable {r : α → β → Prop} {p : β → γ → Prop} {q : γ → δ → Prop}
def Comp (r : α → β → Prop) (p : β → γ → Prop) (a : α) (c : γ) : Prop :=
∃ b, r a b ∧ p b c
#align relation.comp Relation.Comp
@[inherit_doc]
local infixr:80 " ∘r " => Relation.Comp
theorem comp_eq : r ∘r (· = ·) = r :=
funext fun _ ↦ funext fun b ↦ propext <|
Iff.intro (fun ⟨_, h, Eq⟩ ↦ Eq ▸ h) fun h ↦ ⟨b, h, rfl⟩
#align relation.comp_eq Relation.comp_eq
theorem eq_comp : (· = ·) ∘r r = r :=
funext fun a ↦ funext fun _ ↦ propext <|
Iff.intro (fun ⟨_, Eq, h⟩ ↦ Eq.symm ▸ h) fun h ↦ ⟨a, rfl, h⟩
#align relation.eq_comp Relation.eq_comp
theorem iff_comp {r : Prop → α → Prop} : (· ↔ ·) ∘r r = r := by
have : (· ↔ ·) = (· = ·) := by funext a b; exact iff_eq_eq
rw [this, eq_comp]
#align relation.iff_comp Relation.iff_comp
theorem comp_iff {r : α → Prop → Prop} : r ∘r (· ↔ ·) = r := by
have : (· ↔ ·) = (· = ·) := by funext a b; exact iff_eq_eq
rw [this, comp_eq]
#align relation.comp_iff Relation.comp_iff
| Mathlib/Logic/Relation.lean | 159 | 164 | theorem comp_assoc : (r ∘r p) ∘r q = r ∘r p ∘r q := by |
funext a d
apply propext
constructor
· exact fun ⟨c, ⟨b, hab, hbc⟩, hcd⟩ ↦ ⟨b, hab, c, hbc, hcd⟩
· exact fun ⟨b, hab, c, hbc, hcd⟩ ↦ ⟨c, ⟨b, hab, hbc⟩, hcd⟩
| 5 | 148.413159 | 2 | 1.6 | 15 | 1,743 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Function
variable {α β γ δ ε ζ : Type*}
namespace Relation
section Comp
variable {r : α → β → Prop} {p : β → γ → Prop} {q : γ → δ → Prop}
def Comp (r : α → β → Prop) (p : β → γ → Prop) (a : α) (c : γ) : Prop :=
∃ b, r a b ∧ p b c
#align relation.comp Relation.Comp
@[inherit_doc]
local infixr:80 " ∘r " => Relation.Comp
theorem comp_eq : r ∘r (· = ·) = r :=
funext fun _ ↦ funext fun b ↦ propext <|
Iff.intro (fun ⟨_, h, Eq⟩ ↦ Eq ▸ h) fun h ↦ ⟨b, h, rfl⟩
#align relation.comp_eq Relation.comp_eq
theorem eq_comp : (· = ·) ∘r r = r :=
funext fun a ↦ funext fun _ ↦ propext <|
Iff.intro (fun ⟨_, Eq, h⟩ ↦ Eq.symm ▸ h) fun h ↦ ⟨a, rfl, h⟩
#align relation.eq_comp Relation.eq_comp
theorem iff_comp {r : Prop → α → Prop} : (· ↔ ·) ∘r r = r := by
have : (· ↔ ·) = (· = ·) := by funext a b; exact iff_eq_eq
rw [this, eq_comp]
#align relation.iff_comp Relation.iff_comp
theorem comp_iff {r : α → Prop → Prop} : r ∘r (· ↔ ·) = r := by
have : (· ↔ ·) = (· = ·) := by funext a b; exact iff_eq_eq
rw [this, comp_eq]
#align relation.comp_iff Relation.comp_iff
theorem comp_assoc : (r ∘r p) ∘r q = r ∘r p ∘r q := by
funext a d
apply propext
constructor
· exact fun ⟨c, ⟨b, hab, hbc⟩, hcd⟩ ↦ ⟨b, hab, c, hbc, hcd⟩
· exact fun ⟨b, hab, c, hbc, hcd⟩ ↦ ⟨c, ⟨b, hab, hbc⟩, hcd⟩
#align relation.comp_assoc Relation.comp_assoc
| Mathlib/Logic/Relation.lean | 167 | 172 | theorem flip_comp : flip (r ∘r p) = flip p ∘r flip r := by |
funext c a
apply propext
constructor
· exact fun ⟨b, hab, hbc⟩ ↦ ⟨b, hbc, hab⟩
· exact fun ⟨b, hbc, hab⟩ ↦ ⟨b, hab, hbc⟩
| 5 | 148.413159 | 2 | 1.6 | 15 | 1,743 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Function
variable {α β γ δ ε ζ : Type*}
namespace Relation
section Fibration
variable (rα : α → α → Prop) (rβ : β → β → Prop) (f : α → β)
def Fibration :=
∀ ⦃a b⦄, rβ b (f a) → ∃ a', rα a' a ∧ f a' = b
#align relation.fibration Relation.Fibration
variable {rα rβ}
| Mathlib/Logic/Relation.lean | 192 | 196 | theorem _root_.Acc.of_fibration (fib : Fibration rα rβ f) {a} (ha : Acc rα a) : Acc rβ (f a) := by |
induction' ha with a _ ih
refine Acc.intro (f a) fun b hr ↦ ?_
obtain ⟨a', hr', rfl⟩ := fib hr
exact ih a' hr'
| 4 | 54.59815 | 2 | 1.6 | 15 | 1,743 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Function
variable {α β γ δ ε ζ : Type*}
namespace Relation
variable {r : α → α → Prop} {a b c d : α}
@[mk_iff ReflTransGen.cases_tail_iff]
inductive ReflTransGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflTransGen r a a
| tail {b c} : ReflTransGen r a b → r b c → ReflTransGen r a c
#align relation.refl_trans_gen Relation.ReflTransGen
#align relation.refl_trans_gen.cases_tail_iff Relation.ReflTransGen.cases_tail_iff
attribute [refl] ReflTransGen.refl
@[mk_iff]
inductive ReflGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflGen r a a
| single {b} : r a b → ReflGen r a b
#align relation.refl_gen Relation.ReflGen
#align relation.refl_gen_iff Relation.reflGen_iff
@[mk_iff]
inductive TransGen (r : α → α → Prop) (a : α) : α → Prop
| single {b} : r a b → TransGen r a b
| tail {b c} : TransGen r a b → r b c → TransGen r a c
#align relation.trans_gen Relation.TransGen
#align relation.trans_gen_iff Relation.transGen_iff
attribute [refl] ReflGen.refl
namespace ReflTransGen
@[trans]
| Mathlib/Logic/Relation.lean | 296 | 299 | theorem trans (hab : ReflTransGen r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by |
induction hbc with
| refl => assumption
| tail _ hcd hac => exact hac.tail hcd
| 3 | 20.085537 | 1 | 1.6 | 15 | 1,743 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Function
variable {α β γ δ ε ζ : Type*}
namespace Relation
variable {r : α → α → Prop} {a b c d : α}
@[mk_iff ReflTransGen.cases_tail_iff]
inductive ReflTransGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflTransGen r a a
| tail {b c} : ReflTransGen r a b → r b c → ReflTransGen r a c
#align relation.refl_trans_gen Relation.ReflTransGen
#align relation.refl_trans_gen.cases_tail_iff Relation.ReflTransGen.cases_tail_iff
attribute [refl] ReflTransGen.refl
@[mk_iff]
inductive ReflGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflGen r a a
| single {b} : r a b → ReflGen r a b
#align relation.refl_gen Relation.ReflGen
#align relation.refl_gen_iff Relation.reflGen_iff
@[mk_iff]
inductive TransGen (r : α → α → Prop) (a : α) : α → Prop
| single {b} : r a b → TransGen r a b
| tail {b c} : TransGen r a b → r b c → TransGen r a c
#align relation.trans_gen Relation.TransGen
#align relation.trans_gen_iff Relation.transGen_iff
attribute [refl] ReflGen.refl
namespace ReflTransGen
@[trans]
theorem trans (hab : ReflTransGen r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by
induction hbc with
| refl => assumption
| tail _ hcd hac => exact hac.tail hcd
#align relation.refl_trans_gen.trans Relation.ReflTransGen.trans
theorem single (hab : r a b) : ReflTransGen r a b :=
refl.tail hab
#align relation.refl_trans_gen.single Relation.ReflTransGen.single
| Mathlib/Logic/Relation.lean | 306 | 309 | theorem head (hab : r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by |
induction hbc with
| refl => exact refl.tail hab
| tail _ hcd hac => exact hac.tail hcd
| 3 | 20.085537 | 1 | 1.6 | 15 | 1,743 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Function
variable {α β γ δ ε ζ : Type*}
namespace Relation
variable {r : α → α → Prop} {a b c d : α}
@[mk_iff ReflTransGen.cases_tail_iff]
inductive ReflTransGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflTransGen r a a
| tail {b c} : ReflTransGen r a b → r b c → ReflTransGen r a c
#align relation.refl_trans_gen Relation.ReflTransGen
#align relation.refl_trans_gen.cases_tail_iff Relation.ReflTransGen.cases_tail_iff
attribute [refl] ReflTransGen.refl
@[mk_iff]
inductive ReflGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflGen r a a
| single {b} : r a b → ReflGen r a b
#align relation.refl_gen Relation.ReflGen
#align relation.refl_gen_iff Relation.reflGen_iff
@[mk_iff]
inductive TransGen (r : α → α → Prop) (a : α) : α → Prop
| single {b} : r a b → TransGen r a b
| tail {b c} : TransGen r a b → r b c → TransGen r a c
#align relation.trans_gen Relation.TransGen
#align relation.trans_gen_iff Relation.transGen_iff
attribute [refl] ReflGen.refl
namespace ReflTransGen
@[trans]
theorem trans (hab : ReflTransGen r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by
induction hbc with
| refl => assumption
| tail _ hcd hac => exact hac.tail hcd
#align relation.refl_trans_gen.trans Relation.ReflTransGen.trans
theorem single (hab : r a b) : ReflTransGen r a b :=
refl.tail hab
#align relation.refl_trans_gen.single Relation.ReflTransGen.single
theorem head (hab : r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by
induction hbc with
| refl => exact refl.tail hab
| tail _ hcd hac => exact hac.tail hcd
#align relation.refl_trans_gen.head Relation.ReflTransGen.head
| Mathlib/Logic/Relation.lean | 312 | 316 | theorem symmetric (h : Symmetric r) : Symmetric (ReflTransGen r) := by |
intro x y h
induction' h with z w _ b c
· rfl
· apply Relation.ReflTransGen.head (h b) c
| 4 | 54.59815 | 2 | 1.6 | 15 | 1,743 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Function
variable {α β γ δ ε ζ : Type*}
namespace Relation
variable {r : α → α → Prop} {a b c d : α}
@[mk_iff ReflTransGen.cases_tail_iff]
inductive ReflTransGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflTransGen r a a
| tail {b c} : ReflTransGen r a b → r b c → ReflTransGen r a c
#align relation.refl_trans_gen Relation.ReflTransGen
#align relation.refl_trans_gen.cases_tail_iff Relation.ReflTransGen.cases_tail_iff
attribute [refl] ReflTransGen.refl
@[mk_iff]
inductive ReflGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflGen r a a
| single {b} : r a b → ReflGen r a b
#align relation.refl_gen Relation.ReflGen
#align relation.refl_gen_iff Relation.reflGen_iff
@[mk_iff]
inductive TransGen (r : α → α → Prop) (a : α) : α → Prop
| single {b} : r a b → TransGen r a b
| tail {b c} : TransGen r a b → r b c → TransGen r a c
#align relation.trans_gen Relation.TransGen
#align relation.trans_gen_iff Relation.transGen_iff
attribute [refl] ReflGen.refl
namespace ReflTransGen
@[trans]
theorem trans (hab : ReflTransGen r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by
induction hbc with
| refl => assumption
| tail _ hcd hac => exact hac.tail hcd
#align relation.refl_trans_gen.trans Relation.ReflTransGen.trans
theorem single (hab : r a b) : ReflTransGen r a b :=
refl.tail hab
#align relation.refl_trans_gen.single Relation.ReflTransGen.single
theorem head (hab : r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by
induction hbc with
| refl => exact refl.tail hab
| tail _ hcd hac => exact hac.tail hcd
#align relation.refl_trans_gen.head Relation.ReflTransGen.head
theorem symmetric (h : Symmetric r) : Symmetric (ReflTransGen r) := by
intro x y h
induction' h with z w _ b c
· rfl
· apply Relation.ReflTransGen.head (h b) c
#align relation.refl_trans_gen.symmetric Relation.ReflTransGen.symmetric
theorem cases_tail : ReflTransGen r a b → b = a ∨ ∃ c, ReflTransGen r a c ∧ r c b :=
(cases_tail_iff r a b).1
#align relation.refl_trans_gen.cases_tail Relation.ReflTransGen.cases_tail
@[elab_as_elim]
| Mathlib/Logic/Relation.lean | 324 | 332 | theorem head_induction_on {P : ∀ a : α, ReflTransGen r a b → Prop} {a : α} (h : ReflTransGen r a b)
(refl : P b refl)
(head : ∀ {a c} (h' : r a c) (h : ReflTransGen r c b), P c h → P a (h.head h')) : P a h := by |
induction h with
| refl => exact refl
| @tail b c _ hbc ih =>
apply ih
· exact head hbc _ refl
· exact fun h1 h2 ↦ head h1 (h2.tail hbc)
| 6 | 403.428793 | 2 | 1.6 | 15 | 1,743 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Function
variable {α β γ δ ε ζ : Type*}
namespace Relation
variable {r : α → α → Prop} {a b c d : α}
@[mk_iff ReflTransGen.cases_tail_iff]
inductive ReflTransGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflTransGen r a a
| tail {b c} : ReflTransGen r a b → r b c → ReflTransGen r a c
#align relation.refl_trans_gen Relation.ReflTransGen
#align relation.refl_trans_gen.cases_tail_iff Relation.ReflTransGen.cases_tail_iff
attribute [refl] ReflTransGen.refl
@[mk_iff]
inductive ReflGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflGen r a a
| single {b} : r a b → ReflGen r a b
#align relation.refl_gen Relation.ReflGen
#align relation.refl_gen_iff Relation.reflGen_iff
@[mk_iff]
inductive TransGen (r : α → α → Prop) (a : α) : α → Prop
| single {b} : r a b → TransGen r a b
| tail {b c} : TransGen r a b → r b c → TransGen r a c
#align relation.trans_gen Relation.TransGen
#align relation.trans_gen_iff Relation.transGen_iff
attribute [refl] ReflGen.refl
namespace ReflTransGen
@[trans]
theorem trans (hab : ReflTransGen r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by
induction hbc with
| refl => assumption
| tail _ hcd hac => exact hac.tail hcd
#align relation.refl_trans_gen.trans Relation.ReflTransGen.trans
theorem single (hab : r a b) : ReflTransGen r a b :=
refl.tail hab
#align relation.refl_trans_gen.single Relation.ReflTransGen.single
theorem head (hab : r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by
induction hbc with
| refl => exact refl.tail hab
| tail _ hcd hac => exact hac.tail hcd
#align relation.refl_trans_gen.head Relation.ReflTransGen.head
theorem symmetric (h : Symmetric r) : Symmetric (ReflTransGen r) := by
intro x y h
induction' h with z w _ b c
· rfl
· apply Relation.ReflTransGen.head (h b) c
#align relation.refl_trans_gen.symmetric Relation.ReflTransGen.symmetric
theorem cases_tail : ReflTransGen r a b → b = a ∨ ∃ c, ReflTransGen r a c ∧ r c b :=
(cases_tail_iff r a b).1
#align relation.refl_trans_gen.cases_tail Relation.ReflTransGen.cases_tail
@[elab_as_elim]
theorem head_induction_on {P : ∀ a : α, ReflTransGen r a b → Prop} {a : α} (h : ReflTransGen r a b)
(refl : P b refl)
(head : ∀ {a c} (h' : r a c) (h : ReflTransGen r c b), P c h → P a (h.head h')) : P a h := by
induction h with
| refl => exact refl
| @tail b c _ hbc ih =>
apply ih
· exact head hbc _ refl
· exact fun h1 h2 ↦ head h1 (h2.tail hbc)
#align relation.refl_trans_gen.head_induction_on Relation.ReflTransGen.head_induction_on
@[elab_as_elim]
| Mathlib/Logic/Relation.lean | 336 | 342 | theorem trans_induction_on {P : ∀ {a b : α}, ReflTransGen r a b → Prop} {a b : α}
(h : ReflTransGen r a b) (ih₁ : ∀ a, @P a a refl) (ih₂ : ∀ {a b} (h : r a b), P (single h))
(ih₃ : ∀ {a b c} (h₁ : ReflTransGen r a b) (h₂ : ReflTransGen r b c), P h₁ → P h₂ →
P (h₁.trans h₂)) : P h := by |
induction h with
| refl => exact ih₁ a
| tail hab hbc ih => exact ih₃ hab (single hbc) ih (ih₂ hbc)
| 3 | 20.085537 | 1 | 1.6 | 15 | 1,743 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Function
variable {α β γ δ ε ζ : Type*}
namespace Relation
variable {r : α → α → Prop} {a b c d : α}
@[mk_iff ReflTransGen.cases_tail_iff]
inductive ReflTransGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflTransGen r a a
| tail {b c} : ReflTransGen r a b → r b c → ReflTransGen r a c
#align relation.refl_trans_gen Relation.ReflTransGen
#align relation.refl_trans_gen.cases_tail_iff Relation.ReflTransGen.cases_tail_iff
attribute [refl] ReflTransGen.refl
@[mk_iff]
inductive ReflGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflGen r a a
| single {b} : r a b → ReflGen r a b
#align relation.refl_gen Relation.ReflGen
#align relation.refl_gen_iff Relation.reflGen_iff
@[mk_iff]
inductive TransGen (r : α → α → Prop) (a : α) : α → Prop
| single {b} : r a b → TransGen r a b
| tail {b c} : TransGen r a b → r b c → TransGen r a c
#align relation.trans_gen Relation.TransGen
#align relation.trans_gen_iff Relation.transGen_iff
attribute [refl] ReflGen.refl
namespace ReflTransGen
@[trans]
theorem trans (hab : ReflTransGen r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by
induction hbc with
| refl => assumption
| tail _ hcd hac => exact hac.tail hcd
#align relation.refl_trans_gen.trans Relation.ReflTransGen.trans
theorem single (hab : r a b) : ReflTransGen r a b :=
refl.tail hab
#align relation.refl_trans_gen.single Relation.ReflTransGen.single
theorem head (hab : r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by
induction hbc with
| refl => exact refl.tail hab
| tail _ hcd hac => exact hac.tail hcd
#align relation.refl_trans_gen.head Relation.ReflTransGen.head
theorem symmetric (h : Symmetric r) : Symmetric (ReflTransGen r) := by
intro x y h
induction' h with z w _ b c
· rfl
· apply Relation.ReflTransGen.head (h b) c
#align relation.refl_trans_gen.symmetric Relation.ReflTransGen.symmetric
theorem cases_tail : ReflTransGen r a b → b = a ∨ ∃ c, ReflTransGen r a c ∧ r c b :=
(cases_tail_iff r a b).1
#align relation.refl_trans_gen.cases_tail Relation.ReflTransGen.cases_tail
@[elab_as_elim]
theorem head_induction_on {P : ∀ a : α, ReflTransGen r a b → Prop} {a : α} (h : ReflTransGen r a b)
(refl : P b refl)
(head : ∀ {a c} (h' : r a c) (h : ReflTransGen r c b), P c h → P a (h.head h')) : P a h := by
induction h with
| refl => exact refl
| @tail b c _ hbc ih =>
apply ih
· exact head hbc _ refl
· exact fun h1 h2 ↦ head h1 (h2.tail hbc)
#align relation.refl_trans_gen.head_induction_on Relation.ReflTransGen.head_induction_on
@[elab_as_elim]
theorem trans_induction_on {P : ∀ {a b : α}, ReflTransGen r a b → Prop} {a b : α}
(h : ReflTransGen r a b) (ih₁ : ∀ a, @P a a refl) (ih₂ : ∀ {a b} (h : r a b), P (single h))
(ih₃ : ∀ {a b c} (h₁ : ReflTransGen r a b) (h₂ : ReflTransGen r b c), P h₁ → P h₂ →
P (h₁.trans h₂)) : P h := by
induction h with
| refl => exact ih₁ a
| tail hab hbc ih => exact ih₃ hab (single hbc) ih (ih₂ hbc)
#align relation.refl_trans_gen.trans_induction_on Relation.ReflTransGen.trans_induction_on
| Mathlib/Logic/Relation.lean | 345 | 350 | theorem cases_head (h : ReflTransGen r a b) : a = b ∨ ∃ c, r a c ∧ ReflTransGen r c b := by |
induction h using Relation.ReflTransGen.head_induction_on
· left
rfl
· right
exact ⟨_, by assumption, by assumption⟩;
| 5 | 148.413159 | 2 | 1.6 | 15 | 1,743 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Function
variable {α β γ δ ε ζ : Type*}
namespace Relation
variable {r : α → α → Prop} {a b c d : α}
@[mk_iff ReflTransGen.cases_tail_iff]
inductive ReflTransGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflTransGen r a a
| tail {b c} : ReflTransGen r a b → r b c → ReflTransGen r a c
#align relation.refl_trans_gen Relation.ReflTransGen
#align relation.refl_trans_gen.cases_tail_iff Relation.ReflTransGen.cases_tail_iff
attribute [refl] ReflTransGen.refl
@[mk_iff]
inductive ReflGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflGen r a a
| single {b} : r a b → ReflGen r a b
#align relation.refl_gen Relation.ReflGen
#align relation.refl_gen_iff Relation.reflGen_iff
@[mk_iff]
inductive TransGen (r : α → α → Prop) (a : α) : α → Prop
| single {b} : r a b → TransGen r a b
| tail {b c} : TransGen r a b → r b c → TransGen r a c
#align relation.trans_gen Relation.TransGen
#align relation.trans_gen_iff Relation.transGen_iff
attribute [refl] ReflGen.refl
namespace ReflTransGen
@[trans]
theorem trans (hab : ReflTransGen r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by
induction hbc with
| refl => assumption
| tail _ hcd hac => exact hac.tail hcd
#align relation.refl_trans_gen.trans Relation.ReflTransGen.trans
theorem single (hab : r a b) : ReflTransGen r a b :=
refl.tail hab
#align relation.refl_trans_gen.single Relation.ReflTransGen.single
theorem head (hab : r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by
induction hbc with
| refl => exact refl.tail hab
| tail _ hcd hac => exact hac.tail hcd
#align relation.refl_trans_gen.head Relation.ReflTransGen.head
theorem symmetric (h : Symmetric r) : Symmetric (ReflTransGen r) := by
intro x y h
induction' h with z w _ b c
· rfl
· apply Relation.ReflTransGen.head (h b) c
#align relation.refl_trans_gen.symmetric Relation.ReflTransGen.symmetric
theorem cases_tail : ReflTransGen r a b → b = a ∨ ∃ c, ReflTransGen r a c ∧ r c b :=
(cases_tail_iff r a b).1
#align relation.refl_trans_gen.cases_tail Relation.ReflTransGen.cases_tail
@[elab_as_elim]
theorem head_induction_on {P : ∀ a : α, ReflTransGen r a b → Prop} {a : α} (h : ReflTransGen r a b)
(refl : P b refl)
(head : ∀ {a c} (h' : r a c) (h : ReflTransGen r c b), P c h → P a (h.head h')) : P a h := by
induction h with
| refl => exact refl
| @tail b c _ hbc ih =>
apply ih
· exact head hbc _ refl
· exact fun h1 h2 ↦ head h1 (h2.tail hbc)
#align relation.refl_trans_gen.head_induction_on Relation.ReflTransGen.head_induction_on
@[elab_as_elim]
theorem trans_induction_on {P : ∀ {a b : α}, ReflTransGen r a b → Prop} {a b : α}
(h : ReflTransGen r a b) (ih₁ : ∀ a, @P a a refl) (ih₂ : ∀ {a b} (h : r a b), P (single h))
(ih₃ : ∀ {a b c} (h₁ : ReflTransGen r a b) (h₂ : ReflTransGen r b c), P h₁ → P h₂ →
P (h₁.trans h₂)) : P h := by
induction h with
| refl => exact ih₁ a
| tail hab hbc ih => exact ih₃ hab (single hbc) ih (ih₂ hbc)
#align relation.refl_trans_gen.trans_induction_on Relation.ReflTransGen.trans_induction_on
theorem cases_head (h : ReflTransGen r a b) : a = b ∨ ∃ c, r a c ∧ ReflTransGen r c b := by
induction h using Relation.ReflTransGen.head_induction_on
· left
rfl
· right
exact ⟨_, by assumption, by assumption⟩;
#align relation.refl_trans_gen.cases_head Relation.ReflTransGen.cases_head
| Mathlib/Logic/Relation.lean | 353 | 357 | theorem cases_head_iff : ReflTransGen r a b ↔ a = b ∨ ∃ c, r a c ∧ ReflTransGen r c b := by |
use cases_head
rintro (rfl | ⟨c, hac, hcb⟩)
· rfl
· exact head hac hcb
| 4 | 54.59815 | 2 | 1.6 | 15 | 1,743 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Function
variable {α β γ δ ε ζ : Type*}
namespace Relation
variable {r : α → α → Prop} {a b c d : α}
@[mk_iff ReflTransGen.cases_tail_iff]
inductive ReflTransGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflTransGen r a a
| tail {b c} : ReflTransGen r a b → r b c → ReflTransGen r a c
#align relation.refl_trans_gen Relation.ReflTransGen
#align relation.refl_trans_gen.cases_tail_iff Relation.ReflTransGen.cases_tail_iff
attribute [refl] ReflTransGen.refl
@[mk_iff]
inductive ReflGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflGen r a a
| single {b} : r a b → ReflGen r a b
#align relation.refl_gen Relation.ReflGen
#align relation.refl_gen_iff Relation.reflGen_iff
@[mk_iff]
inductive TransGen (r : α → α → Prop) (a : α) : α → Prop
| single {b} : r a b → TransGen r a b
| tail {b c} : TransGen r a b → r b c → TransGen r a c
#align relation.trans_gen Relation.TransGen
#align relation.trans_gen_iff Relation.transGen_iff
attribute [refl] ReflGen.refl
namespace ReflTransGen
@[trans]
theorem trans (hab : ReflTransGen r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by
induction hbc with
| refl => assumption
| tail _ hcd hac => exact hac.tail hcd
#align relation.refl_trans_gen.trans Relation.ReflTransGen.trans
theorem single (hab : r a b) : ReflTransGen r a b :=
refl.tail hab
#align relation.refl_trans_gen.single Relation.ReflTransGen.single
theorem head (hab : r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by
induction hbc with
| refl => exact refl.tail hab
| tail _ hcd hac => exact hac.tail hcd
#align relation.refl_trans_gen.head Relation.ReflTransGen.head
theorem symmetric (h : Symmetric r) : Symmetric (ReflTransGen r) := by
intro x y h
induction' h with z w _ b c
· rfl
· apply Relation.ReflTransGen.head (h b) c
#align relation.refl_trans_gen.symmetric Relation.ReflTransGen.symmetric
theorem cases_tail : ReflTransGen r a b → b = a ∨ ∃ c, ReflTransGen r a c ∧ r c b :=
(cases_tail_iff r a b).1
#align relation.refl_trans_gen.cases_tail Relation.ReflTransGen.cases_tail
@[elab_as_elim]
theorem head_induction_on {P : ∀ a : α, ReflTransGen r a b → Prop} {a : α} (h : ReflTransGen r a b)
(refl : P b refl)
(head : ∀ {a c} (h' : r a c) (h : ReflTransGen r c b), P c h → P a (h.head h')) : P a h := by
induction h with
| refl => exact refl
| @tail b c _ hbc ih =>
apply ih
· exact head hbc _ refl
· exact fun h1 h2 ↦ head h1 (h2.tail hbc)
#align relation.refl_trans_gen.head_induction_on Relation.ReflTransGen.head_induction_on
@[elab_as_elim]
theorem trans_induction_on {P : ∀ {a b : α}, ReflTransGen r a b → Prop} {a b : α}
(h : ReflTransGen r a b) (ih₁ : ∀ a, @P a a refl) (ih₂ : ∀ {a b} (h : r a b), P (single h))
(ih₃ : ∀ {a b c} (h₁ : ReflTransGen r a b) (h₂ : ReflTransGen r b c), P h₁ → P h₂ →
P (h₁.trans h₂)) : P h := by
induction h with
| refl => exact ih₁ a
| tail hab hbc ih => exact ih₃ hab (single hbc) ih (ih₂ hbc)
#align relation.refl_trans_gen.trans_induction_on Relation.ReflTransGen.trans_induction_on
theorem cases_head (h : ReflTransGen r a b) : a = b ∨ ∃ c, r a c ∧ ReflTransGen r c b := by
induction h using Relation.ReflTransGen.head_induction_on
· left
rfl
· right
exact ⟨_, by assumption, by assumption⟩;
#align relation.refl_trans_gen.cases_head Relation.ReflTransGen.cases_head
theorem cases_head_iff : ReflTransGen r a b ↔ a = b ∨ ∃ c, r a c ∧ ReflTransGen r c b := by
use cases_head
rintro (rfl | ⟨c, hac, hcb⟩)
· rfl
· exact head hac hcb
#align relation.refl_trans_gen.cases_head_iff Relation.ReflTransGen.cases_head_iff
| Mathlib/Logic/Relation.lean | 360 | 369 | theorem total_of_right_unique (U : Relator.RightUnique r) (ab : ReflTransGen r a b)
(ac : ReflTransGen r a c) : ReflTransGen r b c ∨ ReflTransGen r c b := by |
induction' ab with b d _ bd IH
· exact Or.inl ac
· rcases IH with (IH | IH)
· rcases cases_head IH with (rfl | ⟨e, be, ec⟩)
· exact Or.inr (single bd)
· cases U bd be
exact Or.inl ec
· exact Or.inr (IH.tail bd)
| 8 | 2,980.957987 | 2 | 1.6 | 15 | 1,743 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Function
variable {α β γ δ ε ζ : Type*}
namespace Relation
variable {r : α → α → Prop} {a b c d : α}
@[mk_iff ReflTransGen.cases_tail_iff]
inductive ReflTransGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflTransGen r a a
| tail {b c} : ReflTransGen r a b → r b c → ReflTransGen r a c
#align relation.refl_trans_gen Relation.ReflTransGen
#align relation.refl_trans_gen.cases_tail_iff Relation.ReflTransGen.cases_tail_iff
attribute [refl] ReflTransGen.refl
@[mk_iff]
inductive ReflGen (r : α → α → Prop) (a : α) : α → Prop
| refl : ReflGen r a a
| single {b} : r a b → ReflGen r a b
#align relation.refl_gen Relation.ReflGen
#align relation.refl_gen_iff Relation.reflGen_iff
@[mk_iff]
inductive TransGen (r : α → α → Prop) (a : α) : α → Prop
| single {b} : r a b → TransGen r a b
| tail {b c} : TransGen r a b → r b c → TransGen r a c
#align relation.trans_gen Relation.TransGen
#align relation.trans_gen_iff Relation.transGen_iff
attribute [refl] ReflGen.refl
| Mathlib/Logic/Relation.lean | 463 | 467 | theorem _root_.Acc.TransGen (h : Acc r a) : Acc (TransGen r) a := by |
induction' h with x _ H
refine Acc.intro x fun y hy ↦ ?_
cases' hy with _ hyx z _ hyz hzx
exacts [H y hyx, (H z hzx).inv hyz]
| 4 | 54.59815 | 2 | 1.6 | 15 | 1,743 |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
open Real Set
open scoped NNReal
| Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 40 | 44 | theorem strictConvexOn_pow {n : ℕ} (hn : 2 ≤ n) : StrictConvexOn ℝ (Ici 0) fun x : ℝ => x ^ n := by |
apply StrictMonoOn.strictConvexOn_of_deriv (convex_Ici _) (continuousOn_pow _)
rw [deriv_pow', interior_Ici]
exact fun x (hx : 0 < x) y _ hxy => mul_lt_mul_of_pos_left
(pow_lt_pow_left hxy hx.le <| Nat.sub_ne_zero_of_lt hn) (by positivity)
| 4 | 54.59815 | 2 | 1.6 | 10 | 1,744 |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
open Real Set
open scoped NNReal
theorem strictConvexOn_pow {n : ℕ} (hn : 2 ≤ n) : StrictConvexOn ℝ (Ici 0) fun x : ℝ => x ^ n := by
apply StrictMonoOn.strictConvexOn_of_deriv (convex_Ici _) (continuousOn_pow _)
rw [deriv_pow', interior_Ici]
exact fun x (hx : 0 < x) y _ hxy => mul_lt_mul_of_pos_left
(pow_lt_pow_left hxy hx.le <| Nat.sub_ne_zero_of_lt hn) (by positivity)
#align strict_convex_on_pow strictConvexOn_pow
| Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 48 | 54 | theorem Even.strictConvexOn_pow {n : ℕ} (hn : Even n) (h : n ≠ 0) :
StrictConvexOn ℝ Set.univ fun x : ℝ => x ^ n := by |
apply StrictMono.strictConvexOn_univ_of_deriv (continuous_pow n)
rw [deriv_pow']
replace h := Nat.pos_of_ne_zero h
exact StrictMono.const_mul (Odd.strictMono_pow <| Nat.Even.sub_odd h hn <| Nat.odd_iff.2 rfl)
(Nat.cast_pos.2 h)
| 5 | 148.413159 | 2 | 1.6 | 10 | 1,744 |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
open Real Set
open scoped NNReal
theorem strictConvexOn_pow {n : ℕ} (hn : 2 ≤ n) : StrictConvexOn ℝ (Ici 0) fun x : ℝ => x ^ n := by
apply StrictMonoOn.strictConvexOn_of_deriv (convex_Ici _) (continuousOn_pow _)
rw [deriv_pow', interior_Ici]
exact fun x (hx : 0 < x) y _ hxy => mul_lt_mul_of_pos_left
(pow_lt_pow_left hxy hx.le <| Nat.sub_ne_zero_of_lt hn) (by positivity)
#align strict_convex_on_pow strictConvexOn_pow
theorem Even.strictConvexOn_pow {n : ℕ} (hn : Even n) (h : n ≠ 0) :
StrictConvexOn ℝ Set.univ fun x : ℝ => x ^ n := by
apply StrictMono.strictConvexOn_univ_of_deriv (continuous_pow n)
rw [deriv_pow']
replace h := Nat.pos_of_ne_zero h
exact StrictMono.const_mul (Odd.strictMono_pow <| Nat.Even.sub_odd h hn <| Nat.odd_iff.2 rfl)
(Nat.cast_pos.2 h)
#align even.strict_convex_on_pow Even.strictConvexOn_pow
| Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 57 | 69 | theorem Finset.prod_nonneg_of_card_nonpos_even {α β : Type*} [LinearOrderedCommRing β] {f : α → β}
[DecidablePred fun x => f x ≤ 0] {s : Finset α} (h0 : Even (s.filter fun x => f x ≤ 0).card) :
0 ≤ ∏ x ∈ s, f x :=
calc
0 ≤ ∏ x ∈ s, (if f x ≤ 0 then (-1 : β) else 1) * f x :=
Finset.prod_nonneg fun x _ => by
split_ifs with hx
· simp [hx]
simp? at hx ⊢ says simp only [not_le, one_mul] at hx ⊢
exact le_of_lt hx
_ = _ := by |
rw [Finset.prod_mul_distrib, Finset.prod_ite, Finset.prod_const_one, mul_one,
Finset.prod_const, neg_one_pow_eq_pow_mod_two, Nat.even_iff.1 h0, pow_zero, one_mul]
| 2 | 7.389056 | 1 | 1.6 | 10 | 1,744 |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
open Real Set
open scoped NNReal
theorem strictConvexOn_pow {n : ℕ} (hn : 2 ≤ n) : StrictConvexOn ℝ (Ici 0) fun x : ℝ => x ^ n := by
apply StrictMonoOn.strictConvexOn_of_deriv (convex_Ici _) (continuousOn_pow _)
rw [deriv_pow', interior_Ici]
exact fun x (hx : 0 < x) y _ hxy => mul_lt_mul_of_pos_left
(pow_lt_pow_left hxy hx.le <| Nat.sub_ne_zero_of_lt hn) (by positivity)
#align strict_convex_on_pow strictConvexOn_pow
theorem Even.strictConvexOn_pow {n : ℕ} (hn : Even n) (h : n ≠ 0) :
StrictConvexOn ℝ Set.univ fun x : ℝ => x ^ n := by
apply StrictMono.strictConvexOn_univ_of_deriv (continuous_pow n)
rw [deriv_pow']
replace h := Nat.pos_of_ne_zero h
exact StrictMono.const_mul (Odd.strictMono_pow <| Nat.Even.sub_odd h hn <| Nat.odd_iff.2 rfl)
(Nat.cast_pos.2 h)
#align even.strict_convex_on_pow Even.strictConvexOn_pow
theorem Finset.prod_nonneg_of_card_nonpos_even {α β : Type*} [LinearOrderedCommRing β] {f : α → β}
[DecidablePred fun x => f x ≤ 0] {s : Finset α} (h0 : Even (s.filter fun x => f x ≤ 0).card) :
0 ≤ ∏ x ∈ s, f x :=
calc
0 ≤ ∏ x ∈ s, (if f x ≤ 0 then (-1 : β) else 1) * f x :=
Finset.prod_nonneg fun x _ => by
split_ifs with hx
· simp [hx]
simp? at hx ⊢ says simp only [not_le, one_mul] at hx ⊢
exact le_of_lt hx
_ = _ := by
rw [Finset.prod_mul_distrib, Finset.prod_ite, Finset.prod_const_one, mul_one,
Finset.prod_const, neg_one_pow_eq_pow_mod_two, Nat.even_iff.1 h0, pow_zero, one_mul]
#align finset.prod_nonneg_of_card_nonpos_even Finset.prod_nonneg_of_card_nonpos_even
| Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 72 | 85 | theorem int_prod_range_nonneg (m : ℤ) (n : ℕ) (hn : Even n) :
0 ≤ ∏ k ∈ Finset.range n, (m - k) := by |
rcases hn with ⟨n, rfl⟩
induction' n with n ihn
· simp
rw [← two_mul] at ihn
rw [← two_mul, mul_add, mul_one, ← one_add_one_eq_two, ← add_assoc,
Finset.prod_range_succ, Finset.prod_range_succ, mul_assoc]
refine mul_nonneg ihn ?_; generalize (1 + 1) * n = k
rcases le_or_lt m k with hmk | hmk
· have : m ≤ k + 1 := hmk.trans (lt_add_one (k : ℤ)).le
convert mul_nonneg_of_nonpos_of_nonpos (sub_nonpos_of_le hmk) _
convert sub_nonpos_of_le this
· exact mul_nonneg (sub_nonneg_of_le hmk.le) (sub_nonneg_of_le hmk)
| 12 | 162,754.791419 | 2 | 1.6 | 10 | 1,744 |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
open Real Set
open scoped NNReal
theorem strictConvexOn_pow {n : ℕ} (hn : 2 ≤ n) : StrictConvexOn ℝ (Ici 0) fun x : ℝ => x ^ n := by
apply StrictMonoOn.strictConvexOn_of_deriv (convex_Ici _) (continuousOn_pow _)
rw [deriv_pow', interior_Ici]
exact fun x (hx : 0 < x) y _ hxy => mul_lt_mul_of_pos_left
(pow_lt_pow_left hxy hx.le <| Nat.sub_ne_zero_of_lt hn) (by positivity)
#align strict_convex_on_pow strictConvexOn_pow
theorem Even.strictConvexOn_pow {n : ℕ} (hn : Even n) (h : n ≠ 0) :
StrictConvexOn ℝ Set.univ fun x : ℝ => x ^ n := by
apply StrictMono.strictConvexOn_univ_of_deriv (continuous_pow n)
rw [deriv_pow']
replace h := Nat.pos_of_ne_zero h
exact StrictMono.const_mul (Odd.strictMono_pow <| Nat.Even.sub_odd h hn <| Nat.odd_iff.2 rfl)
(Nat.cast_pos.2 h)
#align even.strict_convex_on_pow Even.strictConvexOn_pow
theorem Finset.prod_nonneg_of_card_nonpos_even {α β : Type*} [LinearOrderedCommRing β] {f : α → β}
[DecidablePred fun x => f x ≤ 0] {s : Finset α} (h0 : Even (s.filter fun x => f x ≤ 0).card) :
0 ≤ ∏ x ∈ s, f x :=
calc
0 ≤ ∏ x ∈ s, (if f x ≤ 0 then (-1 : β) else 1) * f x :=
Finset.prod_nonneg fun x _ => by
split_ifs with hx
· simp [hx]
simp? at hx ⊢ says simp only [not_le, one_mul] at hx ⊢
exact le_of_lt hx
_ = _ := by
rw [Finset.prod_mul_distrib, Finset.prod_ite, Finset.prod_const_one, mul_one,
Finset.prod_const, neg_one_pow_eq_pow_mod_two, Nat.even_iff.1 h0, pow_zero, one_mul]
#align finset.prod_nonneg_of_card_nonpos_even Finset.prod_nonneg_of_card_nonpos_even
theorem int_prod_range_nonneg (m : ℤ) (n : ℕ) (hn : Even n) :
0 ≤ ∏ k ∈ Finset.range n, (m - k) := by
rcases hn with ⟨n, rfl⟩
induction' n with n ihn
· simp
rw [← two_mul] at ihn
rw [← two_mul, mul_add, mul_one, ← one_add_one_eq_two, ← add_assoc,
Finset.prod_range_succ, Finset.prod_range_succ, mul_assoc]
refine mul_nonneg ihn ?_; generalize (1 + 1) * n = k
rcases le_or_lt m k with hmk | hmk
· have : m ≤ k + 1 := hmk.trans (lt_add_one (k : ℤ)).le
convert mul_nonneg_of_nonpos_of_nonpos (sub_nonpos_of_le hmk) _
convert sub_nonpos_of_le this
· exact mul_nonneg (sub_nonneg_of_le hmk.le) (sub_nonneg_of_le hmk)
#align int_prod_range_nonneg int_prod_range_nonneg
| Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 88 | 94 | theorem int_prod_range_pos {m : ℤ} {n : ℕ} (hn : Even n) (hm : m ∉ Ico (0 : ℤ) n) :
0 < ∏ k ∈ Finset.range n, (m - k) := by |
refine (int_prod_range_nonneg m n hn).lt_of_ne fun h => hm ?_
rw [eq_comm, Finset.prod_eq_zero_iff] at h
obtain ⟨a, ha, h⟩ := h
rw [sub_eq_zero.1 h]
exact ⟨Int.ofNat_zero_le _, Int.ofNat_lt.2 <| Finset.mem_range.1 ha⟩
| 5 | 148.413159 | 2 | 1.6 | 10 | 1,744 |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
open Real Set
open scoped NNReal
theorem strictConvexOn_pow {n : ℕ} (hn : 2 ≤ n) : StrictConvexOn ℝ (Ici 0) fun x : ℝ => x ^ n := by
apply StrictMonoOn.strictConvexOn_of_deriv (convex_Ici _) (continuousOn_pow _)
rw [deriv_pow', interior_Ici]
exact fun x (hx : 0 < x) y _ hxy => mul_lt_mul_of_pos_left
(pow_lt_pow_left hxy hx.le <| Nat.sub_ne_zero_of_lt hn) (by positivity)
#align strict_convex_on_pow strictConvexOn_pow
theorem Even.strictConvexOn_pow {n : ℕ} (hn : Even n) (h : n ≠ 0) :
StrictConvexOn ℝ Set.univ fun x : ℝ => x ^ n := by
apply StrictMono.strictConvexOn_univ_of_deriv (continuous_pow n)
rw [deriv_pow']
replace h := Nat.pos_of_ne_zero h
exact StrictMono.const_mul (Odd.strictMono_pow <| Nat.Even.sub_odd h hn <| Nat.odd_iff.2 rfl)
(Nat.cast_pos.2 h)
#align even.strict_convex_on_pow Even.strictConvexOn_pow
theorem Finset.prod_nonneg_of_card_nonpos_even {α β : Type*} [LinearOrderedCommRing β] {f : α → β}
[DecidablePred fun x => f x ≤ 0] {s : Finset α} (h0 : Even (s.filter fun x => f x ≤ 0).card) :
0 ≤ ∏ x ∈ s, f x :=
calc
0 ≤ ∏ x ∈ s, (if f x ≤ 0 then (-1 : β) else 1) * f x :=
Finset.prod_nonneg fun x _ => by
split_ifs with hx
· simp [hx]
simp? at hx ⊢ says simp only [not_le, one_mul] at hx ⊢
exact le_of_lt hx
_ = _ := by
rw [Finset.prod_mul_distrib, Finset.prod_ite, Finset.prod_const_one, mul_one,
Finset.prod_const, neg_one_pow_eq_pow_mod_two, Nat.even_iff.1 h0, pow_zero, one_mul]
#align finset.prod_nonneg_of_card_nonpos_even Finset.prod_nonneg_of_card_nonpos_even
theorem int_prod_range_nonneg (m : ℤ) (n : ℕ) (hn : Even n) :
0 ≤ ∏ k ∈ Finset.range n, (m - k) := by
rcases hn with ⟨n, rfl⟩
induction' n with n ihn
· simp
rw [← two_mul] at ihn
rw [← two_mul, mul_add, mul_one, ← one_add_one_eq_two, ← add_assoc,
Finset.prod_range_succ, Finset.prod_range_succ, mul_assoc]
refine mul_nonneg ihn ?_; generalize (1 + 1) * n = k
rcases le_or_lt m k with hmk | hmk
· have : m ≤ k + 1 := hmk.trans (lt_add_one (k : ℤ)).le
convert mul_nonneg_of_nonpos_of_nonpos (sub_nonpos_of_le hmk) _
convert sub_nonpos_of_le this
· exact mul_nonneg (sub_nonneg_of_le hmk.le) (sub_nonneg_of_le hmk)
#align int_prod_range_nonneg int_prod_range_nonneg
theorem int_prod_range_pos {m : ℤ} {n : ℕ} (hn : Even n) (hm : m ∉ Ico (0 : ℤ) n) :
0 < ∏ k ∈ Finset.range n, (m - k) := by
refine (int_prod_range_nonneg m n hn).lt_of_ne fun h => hm ?_
rw [eq_comm, Finset.prod_eq_zero_iff] at h
obtain ⟨a, ha, h⟩ := h
rw [sub_eq_zero.1 h]
exact ⟨Int.ofNat_zero_le _, Int.ofNat_lt.2 <| Finset.mem_range.1 ha⟩
#align int_prod_range_pos int_prod_range_pos
| Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 98 | 110 | theorem strictConvexOn_zpow {m : ℤ} (hm₀ : m ≠ 0) (hm₁ : m ≠ 1) :
StrictConvexOn ℝ (Ioi 0) fun x : ℝ => x ^ m := by |
apply strictConvexOn_of_deriv2_pos' (convex_Ioi 0)
· exact (continuousOn_zpow₀ m).mono fun x hx => ne_of_gt hx
intro x hx
rw [mem_Ioi] at hx
rw [iter_deriv_zpow]
refine mul_pos ?_ (zpow_pos_of_pos hx _)
norm_cast
refine int_prod_range_pos (by decide) fun hm => ?_
rw [← Finset.coe_Ico] at hm
norm_cast at hm
fin_cases hm <;> simp_all -- Porting note: `simp_all` was `cc`
| 11 | 59,874.141715 | 2 | 1.6 | 10 | 1,744 |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
open Real Set
open scoped NNReal
theorem strictConvexOn_pow {n : ℕ} (hn : 2 ≤ n) : StrictConvexOn ℝ (Ici 0) fun x : ℝ => x ^ n := by
apply StrictMonoOn.strictConvexOn_of_deriv (convex_Ici _) (continuousOn_pow _)
rw [deriv_pow', interior_Ici]
exact fun x (hx : 0 < x) y _ hxy => mul_lt_mul_of_pos_left
(pow_lt_pow_left hxy hx.le <| Nat.sub_ne_zero_of_lt hn) (by positivity)
#align strict_convex_on_pow strictConvexOn_pow
theorem Even.strictConvexOn_pow {n : ℕ} (hn : Even n) (h : n ≠ 0) :
StrictConvexOn ℝ Set.univ fun x : ℝ => x ^ n := by
apply StrictMono.strictConvexOn_univ_of_deriv (continuous_pow n)
rw [deriv_pow']
replace h := Nat.pos_of_ne_zero h
exact StrictMono.const_mul (Odd.strictMono_pow <| Nat.Even.sub_odd h hn <| Nat.odd_iff.2 rfl)
(Nat.cast_pos.2 h)
#align even.strict_convex_on_pow Even.strictConvexOn_pow
theorem Finset.prod_nonneg_of_card_nonpos_even {α β : Type*} [LinearOrderedCommRing β] {f : α → β}
[DecidablePred fun x => f x ≤ 0] {s : Finset α} (h0 : Even (s.filter fun x => f x ≤ 0).card) :
0 ≤ ∏ x ∈ s, f x :=
calc
0 ≤ ∏ x ∈ s, (if f x ≤ 0 then (-1 : β) else 1) * f x :=
Finset.prod_nonneg fun x _ => by
split_ifs with hx
· simp [hx]
simp? at hx ⊢ says simp only [not_le, one_mul] at hx ⊢
exact le_of_lt hx
_ = _ := by
rw [Finset.prod_mul_distrib, Finset.prod_ite, Finset.prod_const_one, mul_one,
Finset.prod_const, neg_one_pow_eq_pow_mod_two, Nat.even_iff.1 h0, pow_zero, one_mul]
#align finset.prod_nonneg_of_card_nonpos_even Finset.prod_nonneg_of_card_nonpos_even
theorem int_prod_range_nonneg (m : ℤ) (n : ℕ) (hn : Even n) :
0 ≤ ∏ k ∈ Finset.range n, (m - k) := by
rcases hn with ⟨n, rfl⟩
induction' n with n ihn
· simp
rw [← two_mul] at ihn
rw [← two_mul, mul_add, mul_one, ← one_add_one_eq_two, ← add_assoc,
Finset.prod_range_succ, Finset.prod_range_succ, mul_assoc]
refine mul_nonneg ihn ?_; generalize (1 + 1) * n = k
rcases le_or_lt m k with hmk | hmk
· have : m ≤ k + 1 := hmk.trans (lt_add_one (k : ℤ)).le
convert mul_nonneg_of_nonpos_of_nonpos (sub_nonpos_of_le hmk) _
convert sub_nonpos_of_le this
· exact mul_nonneg (sub_nonneg_of_le hmk.le) (sub_nonneg_of_le hmk)
#align int_prod_range_nonneg int_prod_range_nonneg
theorem int_prod_range_pos {m : ℤ} {n : ℕ} (hn : Even n) (hm : m ∉ Ico (0 : ℤ) n) :
0 < ∏ k ∈ Finset.range n, (m - k) := by
refine (int_prod_range_nonneg m n hn).lt_of_ne fun h => hm ?_
rw [eq_comm, Finset.prod_eq_zero_iff] at h
obtain ⟨a, ha, h⟩ := h
rw [sub_eq_zero.1 h]
exact ⟨Int.ofNat_zero_le _, Int.ofNat_lt.2 <| Finset.mem_range.1 ha⟩
#align int_prod_range_pos int_prod_range_pos
theorem strictConvexOn_zpow {m : ℤ} (hm₀ : m ≠ 0) (hm₁ : m ≠ 1) :
StrictConvexOn ℝ (Ioi 0) fun x : ℝ => x ^ m := by
apply strictConvexOn_of_deriv2_pos' (convex_Ioi 0)
· exact (continuousOn_zpow₀ m).mono fun x hx => ne_of_gt hx
intro x hx
rw [mem_Ioi] at hx
rw [iter_deriv_zpow]
refine mul_pos ?_ (zpow_pos_of_pos hx _)
norm_cast
refine int_prod_range_pos (by decide) fun hm => ?_
rw [← Finset.coe_Ico] at hm
norm_cast at hm
fin_cases hm <;> simp_all -- Porting note: `simp_all` was `cc`
#align strict_convex_on_zpow strictConvexOn_zpow
section SqrtMulLog
| Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 115 | 119 | theorem hasDerivAt_sqrt_mul_log {x : ℝ} (hx : x ≠ 0) :
HasDerivAt (fun x => √x * log x) ((2 + log x) / (2 * √x)) x := by |
convert (hasDerivAt_sqrt hx).mul (hasDerivAt_log hx) using 1
rw [add_div, div_mul_cancel_left₀ two_ne_zero, ← div_eq_mul_inv, sqrt_div_self', add_comm,
one_div, one_div, ← div_eq_inv_mul]
| 3 | 20.085537 | 1 | 1.6 | 10 | 1,744 |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
open Real Set
open scoped NNReal
theorem strictConvexOn_pow {n : ℕ} (hn : 2 ≤ n) : StrictConvexOn ℝ (Ici 0) fun x : ℝ => x ^ n := by
apply StrictMonoOn.strictConvexOn_of_deriv (convex_Ici _) (continuousOn_pow _)
rw [deriv_pow', interior_Ici]
exact fun x (hx : 0 < x) y _ hxy => mul_lt_mul_of_pos_left
(pow_lt_pow_left hxy hx.le <| Nat.sub_ne_zero_of_lt hn) (by positivity)
#align strict_convex_on_pow strictConvexOn_pow
theorem Even.strictConvexOn_pow {n : ℕ} (hn : Even n) (h : n ≠ 0) :
StrictConvexOn ℝ Set.univ fun x : ℝ => x ^ n := by
apply StrictMono.strictConvexOn_univ_of_deriv (continuous_pow n)
rw [deriv_pow']
replace h := Nat.pos_of_ne_zero h
exact StrictMono.const_mul (Odd.strictMono_pow <| Nat.Even.sub_odd h hn <| Nat.odd_iff.2 rfl)
(Nat.cast_pos.2 h)
#align even.strict_convex_on_pow Even.strictConvexOn_pow
theorem Finset.prod_nonneg_of_card_nonpos_even {α β : Type*} [LinearOrderedCommRing β] {f : α → β}
[DecidablePred fun x => f x ≤ 0] {s : Finset α} (h0 : Even (s.filter fun x => f x ≤ 0).card) :
0 ≤ ∏ x ∈ s, f x :=
calc
0 ≤ ∏ x ∈ s, (if f x ≤ 0 then (-1 : β) else 1) * f x :=
Finset.prod_nonneg fun x _ => by
split_ifs with hx
· simp [hx]
simp? at hx ⊢ says simp only [not_le, one_mul] at hx ⊢
exact le_of_lt hx
_ = _ := by
rw [Finset.prod_mul_distrib, Finset.prod_ite, Finset.prod_const_one, mul_one,
Finset.prod_const, neg_one_pow_eq_pow_mod_two, Nat.even_iff.1 h0, pow_zero, one_mul]
#align finset.prod_nonneg_of_card_nonpos_even Finset.prod_nonneg_of_card_nonpos_even
theorem int_prod_range_nonneg (m : ℤ) (n : ℕ) (hn : Even n) :
0 ≤ ∏ k ∈ Finset.range n, (m - k) := by
rcases hn with ⟨n, rfl⟩
induction' n with n ihn
· simp
rw [← two_mul] at ihn
rw [← two_mul, mul_add, mul_one, ← one_add_one_eq_two, ← add_assoc,
Finset.prod_range_succ, Finset.prod_range_succ, mul_assoc]
refine mul_nonneg ihn ?_; generalize (1 + 1) * n = k
rcases le_or_lt m k with hmk | hmk
· have : m ≤ k + 1 := hmk.trans (lt_add_one (k : ℤ)).le
convert mul_nonneg_of_nonpos_of_nonpos (sub_nonpos_of_le hmk) _
convert sub_nonpos_of_le this
· exact mul_nonneg (sub_nonneg_of_le hmk.le) (sub_nonneg_of_le hmk)
#align int_prod_range_nonneg int_prod_range_nonneg
theorem int_prod_range_pos {m : ℤ} {n : ℕ} (hn : Even n) (hm : m ∉ Ico (0 : ℤ) n) :
0 < ∏ k ∈ Finset.range n, (m - k) := by
refine (int_prod_range_nonneg m n hn).lt_of_ne fun h => hm ?_
rw [eq_comm, Finset.prod_eq_zero_iff] at h
obtain ⟨a, ha, h⟩ := h
rw [sub_eq_zero.1 h]
exact ⟨Int.ofNat_zero_le _, Int.ofNat_lt.2 <| Finset.mem_range.1 ha⟩
#align int_prod_range_pos int_prod_range_pos
theorem strictConvexOn_zpow {m : ℤ} (hm₀ : m ≠ 0) (hm₁ : m ≠ 1) :
StrictConvexOn ℝ (Ioi 0) fun x : ℝ => x ^ m := by
apply strictConvexOn_of_deriv2_pos' (convex_Ioi 0)
· exact (continuousOn_zpow₀ m).mono fun x hx => ne_of_gt hx
intro x hx
rw [mem_Ioi] at hx
rw [iter_deriv_zpow]
refine mul_pos ?_ (zpow_pos_of_pos hx _)
norm_cast
refine int_prod_range_pos (by decide) fun hm => ?_
rw [← Finset.coe_Ico] at hm
norm_cast at hm
fin_cases hm <;> simp_all -- Porting note: `simp_all` was `cc`
#align strict_convex_on_zpow strictConvexOn_zpow
section SqrtMulLog
theorem hasDerivAt_sqrt_mul_log {x : ℝ} (hx : x ≠ 0) :
HasDerivAt (fun x => √x * log x) ((2 + log x) / (2 * √x)) x := by
convert (hasDerivAt_sqrt hx).mul (hasDerivAt_log hx) using 1
rw [add_div, div_mul_cancel_left₀ two_ne_zero, ← div_eq_mul_inv, sqrt_div_self', add_comm,
one_div, one_div, ← div_eq_inv_mul]
#align has_deriv_at_sqrt_mul_log hasDerivAt_sqrt_mul_log
| Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 122 | 129 | theorem deriv_sqrt_mul_log (x : ℝ) :
deriv (fun x => √x * log x) x = (2 + log x) / (2 * √x) := by |
cases' lt_or_le 0 x with hx hx
· exact (hasDerivAt_sqrt_mul_log hx.ne').deriv
· rw [sqrt_eq_zero_of_nonpos hx, mul_zero, div_zero]
refine HasDerivWithinAt.deriv_eq_zero ?_ (uniqueDiffOn_Iic 0 x hx)
refine (hasDerivWithinAt_const x _ 0).congr_of_mem (fun x hx => ?_) hx
rw [sqrt_eq_zero_of_nonpos hx, zero_mul]
| 6 | 403.428793 | 2 | 1.6 | 10 | 1,744 |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
open Real Set
open scoped NNReal
theorem strictConvexOn_pow {n : ℕ} (hn : 2 ≤ n) : StrictConvexOn ℝ (Ici 0) fun x : ℝ => x ^ n := by
apply StrictMonoOn.strictConvexOn_of_deriv (convex_Ici _) (continuousOn_pow _)
rw [deriv_pow', interior_Ici]
exact fun x (hx : 0 < x) y _ hxy => mul_lt_mul_of_pos_left
(pow_lt_pow_left hxy hx.le <| Nat.sub_ne_zero_of_lt hn) (by positivity)
#align strict_convex_on_pow strictConvexOn_pow
theorem Even.strictConvexOn_pow {n : ℕ} (hn : Even n) (h : n ≠ 0) :
StrictConvexOn ℝ Set.univ fun x : ℝ => x ^ n := by
apply StrictMono.strictConvexOn_univ_of_deriv (continuous_pow n)
rw [deriv_pow']
replace h := Nat.pos_of_ne_zero h
exact StrictMono.const_mul (Odd.strictMono_pow <| Nat.Even.sub_odd h hn <| Nat.odd_iff.2 rfl)
(Nat.cast_pos.2 h)
#align even.strict_convex_on_pow Even.strictConvexOn_pow
theorem Finset.prod_nonneg_of_card_nonpos_even {α β : Type*} [LinearOrderedCommRing β] {f : α → β}
[DecidablePred fun x => f x ≤ 0] {s : Finset α} (h0 : Even (s.filter fun x => f x ≤ 0).card) :
0 ≤ ∏ x ∈ s, f x :=
calc
0 ≤ ∏ x ∈ s, (if f x ≤ 0 then (-1 : β) else 1) * f x :=
Finset.prod_nonneg fun x _ => by
split_ifs with hx
· simp [hx]
simp? at hx ⊢ says simp only [not_le, one_mul] at hx ⊢
exact le_of_lt hx
_ = _ := by
rw [Finset.prod_mul_distrib, Finset.prod_ite, Finset.prod_const_one, mul_one,
Finset.prod_const, neg_one_pow_eq_pow_mod_two, Nat.even_iff.1 h0, pow_zero, one_mul]
#align finset.prod_nonneg_of_card_nonpos_even Finset.prod_nonneg_of_card_nonpos_even
theorem int_prod_range_nonneg (m : ℤ) (n : ℕ) (hn : Even n) :
0 ≤ ∏ k ∈ Finset.range n, (m - k) := by
rcases hn with ⟨n, rfl⟩
induction' n with n ihn
· simp
rw [← two_mul] at ihn
rw [← two_mul, mul_add, mul_one, ← one_add_one_eq_two, ← add_assoc,
Finset.prod_range_succ, Finset.prod_range_succ, mul_assoc]
refine mul_nonneg ihn ?_; generalize (1 + 1) * n = k
rcases le_or_lt m k with hmk | hmk
· have : m ≤ k + 1 := hmk.trans (lt_add_one (k : ℤ)).le
convert mul_nonneg_of_nonpos_of_nonpos (sub_nonpos_of_le hmk) _
convert sub_nonpos_of_le this
· exact mul_nonneg (sub_nonneg_of_le hmk.le) (sub_nonneg_of_le hmk)
#align int_prod_range_nonneg int_prod_range_nonneg
theorem int_prod_range_pos {m : ℤ} {n : ℕ} (hn : Even n) (hm : m ∉ Ico (0 : ℤ) n) :
0 < ∏ k ∈ Finset.range n, (m - k) := by
refine (int_prod_range_nonneg m n hn).lt_of_ne fun h => hm ?_
rw [eq_comm, Finset.prod_eq_zero_iff] at h
obtain ⟨a, ha, h⟩ := h
rw [sub_eq_zero.1 h]
exact ⟨Int.ofNat_zero_le _, Int.ofNat_lt.2 <| Finset.mem_range.1 ha⟩
#align int_prod_range_pos int_prod_range_pos
theorem strictConvexOn_zpow {m : ℤ} (hm₀ : m ≠ 0) (hm₁ : m ≠ 1) :
StrictConvexOn ℝ (Ioi 0) fun x : ℝ => x ^ m := by
apply strictConvexOn_of_deriv2_pos' (convex_Ioi 0)
· exact (continuousOn_zpow₀ m).mono fun x hx => ne_of_gt hx
intro x hx
rw [mem_Ioi] at hx
rw [iter_deriv_zpow]
refine mul_pos ?_ (zpow_pos_of_pos hx _)
norm_cast
refine int_prod_range_pos (by decide) fun hm => ?_
rw [← Finset.coe_Ico] at hm
norm_cast at hm
fin_cases hm <;> simp_all -- Porting note: `simp_all` was `cc`
#align strict_convex_on_zpow strictConvexOn_zpow
open scoped Real
| Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 168 | 171 | theorem strictConcaveOn_sin_Icc : StrictConcaveOn ℝ (Icc 0 π) sin := by |
apply strictConcaveOn_of_deriv2_neg (convex_Icc _ _) continuousOn_sin fun x hx => ?_
rw [interior_Icc] at hx
simp [sin_pos_of_mem_Ioo hx]
| 3 | 20.085537 | 1 | 1.6 | 10 | 1,744 |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
open Real Set
open scoped NNReal
theorem strictConvexOn_pow {n : ℕ} (hn : 2 ≤ n) : StrictConvexOn ℝ (Ici 0) fun x : ℝ => x ^ n := by
apply StrictMonoOn.strictConvexOn_of_deriv (convex_Ici _) (continuousOn_pow _)
rw [deriv_pow', interior_Ici]
exact fun x (hx : 0 < x) y _ hxy => mul_lt_mul_of_pos_left
(pow_lt_pow_left hxy hx.le <| Nat.sub_ne_zero_of_lt hn) (by positivity)
#align strict_convex_on_pow strictConvexOn_pow
theorem Even.strictConvexOn_pow {n : ℕ} (hn : Even n) (h : n ≠ 0) :
StrictConvexOn ℝ Set.univ fun x : ℝ => x ^ n := by
apply StrictMono.strictConvexOn_univ_of_deriv (continuous_pow n)
rw [deriv_pow']
replace h := Nat.pos_of_ne_zero h
exact StrictMono.const_mul (Odd.strictMono_pow <| Nat.Even.sub_odd h hn <| Nat.odd_iff.2 rfl)
(Nat.cast_pos.2 h)
#align even.strict_convex_on_pow Even.strictConvexOn_pow
theorem Finset.prod_nonneg_of_card_nonpos_even {α β : Type*} [LinearOrderedCommRing β] {f : α → β}
[DecidablePred fun x => f x ≤ 0] {s : Finset α} (h0 : Even (s.filter fun x => f x ≤ 0).card) :
0 ≤ ∏ x ∈ s, f x :=
calc
0 ≤ ∏ x ∈ s, (if f x ≤ 0 then (-1 : β) else 1) * f x :=
Finset.prod_nonneg fun x _ => by
split_ifs with hx
· simp [hx]
simp? at hx ⊢ says simp only [not_le, one_mul] at hx ⊢
exact le_of_lt hx
_ = _ := by
rw [Finset.prod_mul_distrib, Finset.prod_ite, Finset.prod_const_one, mul_one,
Finset.prod_const, neg_one_pow_eq_pow_mod_two, Nat.even_iff.1 h0, pow_zero, one_mul]
#align finset.prod_nonneg_of_card_nonpos_even Finset.prod_nonneg_of_card_nonpos_even
theorem int_prod_range_nonneg (m : ℤ) (n : ℕ) (hn : Even n) :
0 ≤ ∏ k ∈ Finset.range n, (m - k) := by
rcases hn with ⟨n, rfl⟩
induction' n with n ihn
· simp
rw [← two_mul] at ihn
rw [← two_mul, mul_add, mul_one, ← one_add_one_eq_two, ← add_assoc,
Finset.prod_range_succ, Finset.prod_range_succ, mul_assoc]
refine mul_nonneg ihn ?_; generalize (1 + 1) * n = k
rcases le_or_lt m k with hmk | hmk
· have : m ≤ k + 1 := hmk.trans (lt_add_one (k : ℤ)).le
convert mul_nonneg_of_nonpos_of_nonpos (sub_nonpos_of_le hmk) _
convert sub_nonpos_of_le this
· exact mul_nonneg (sub_nonneg_of_le hmk.le) (sub_nonneg_of_le hmk)
#align int_prod_range_nonneg int_prod_range_nonneg
theorem int_prod_range_pos {m : ℤ} {n : ℕ} (hn : Even n) (hm : m ∉ Ico (0 : ℤ) n) :
0 < ∏ k ∈ Finset.range n, (m - k) := by
refine (int_prod_range_nonneg m n hn).lt_of_ne fun h => hm ?_
rw [eq_comm, Finset.prod_eq_zero_iff] at h
obtain ⟨a, ha, h⟩ := h
rw [sub_eq_zero.1 h]
exact ⟨Int.ofNat_zero_le _, Int.ofNat_lt.2 <| Finset.mem_range.1 ha⟩
#align int_prod_range_pos int_prod_range_pos
theorem strictConvexOn_zpow {m : ℤ} (hm₀ : m ≠ 0) (hm₁ : m ≠ 1) :
StrictConvexOn ℝ (Ioi 0) fun x : ℝ => x ^ m := by
apply strictConvexOn_of_deriv2_pos' (convex_Ioi 0)
· exact (continuousOn_zpow₀ m).mono fun x hx => ne_of_gt hx
intro x hx
rw [mem_Ioi] at hx
rw [iter_deriv_zpow]
refine mul_pos ?_ (zpow_pos_of_pos hx _)
norm_cast
refine int_prod_range_pos (by decide) fun hm => ?_
rw [← Finset.coe_Ico] at hm
norm_cast at hm
fin_cases hm <;> simp_all -- Porting note: `simp_all` was `cc`
#align strict_convex_on_zpow strictConvexOn_zpow
open scoped Real
theorem strictConcaveOn_sin_Icc : StrictConcaveOn ℝ (Icc 0 π) sin := by
apply strictConcaveOn_of_deriv2_neg (convex_Icc _ _) continuousOn_sin fun x hx => ?_
rw [interior_Icc] at hx
simp [sin_pos_of_mem_Ioo hx]
#align strict_concave_on_sin_Icc strictConcaveOn_sin_Icc
| Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 174 | 177 | theorem strictConcaveOn_cos_Icc : StrictConcaveOn ℝ (Icc (-(π / 2)) (π / 2)) cos := by |
apply strictConcaveOn_of_deriv2_neg (convex_Icc _ _) continuousOn_cos fun x hx => ?_
rw [interior_Icc] at hx
simp [cos_pos_of_mem_Ioo hx]
| 3 | 20.085537 | 1 | 1.6 | 10 | 1,744 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Constructions.Prod.Integral
open Fintype MeasureTheory MeasureTheory.Measure
variable {𝕜 : Type*} [RCLike 𝕜]
namespace MeasureTheory
| Mathlib/MeasureTheory/Integral/Pi.lean | 26 | 41 | theorem Integrable.fin_nat_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
{f : (i : Fin n) → E i → 𝕜} (hf : ∀ i, Integrable (f i)) :
Integrable (fun (x : (i : Fin n) → E i) ↦ ∏ i, f i (x i)) := by |
induction n with
| zero => simp only [Nat.zero_eq, Finset.univ_eq_empty, Finset.prod_empty, volume_pi,
integrable_const_iff, one_ne_zero, pi_empty_univ, ENNReal.one_lt_top, or_true]
| succ n n_ih =>
have := ((measurePreserving_piFinSuccAbove (fun i => (volume : Measure (E i))) 0).symm)
rw [volume_pi, ← this.integrable_comp_emb (MeasurableEquiv.measurableEmbedding _)]
simp_rw [MeasurableEquiv.piFinSuccAbove_symm_apply,
Fin.prod_univ_succ, Fin.insertNth_zero]
simp only [Fin.zero_succAbove, cast_eq, Function.comp_def, Fin.cons_zero, Fin.cons_succ]
have : Integrable (fun (x : (j : Fin n) → E (Fin.succ j)) ↦ ∏ j, f (Fin.succ j) (x j)) :=
n_ih (fun i ↦ hf _)
exact Integrable.prod_mul (hf 0) this
| 12 | 162,754.791419 | 2 | 1.6 | 5 | 1,745 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Constructions.Prod.Integral
open Fintype MeasureTheory MeasureTheory.Measure
variable {𝕜 : Type*} [RCLike 𝕜]
namespace MeasureTheory
theorem Integrable.fin_nat_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
{f : (i : Fin n) → E i → 𝕜} (hf : ∀ i, Integrable (f i)) :
Integrable (fun (x : (i : Fin n) → E i) ↦ ∏ i, f i (x i)) := by
induction n with
| zero => simp only [Nat.zero_eq, Finset.univ_eq_empty, Finset.prod_empty, volume_pi,
integrable_const_iff, one_ne_zero, pi_empty_univ, ENNReal.one_lt_top, or_true]
| succ n n_ih =>
have := ((measurePreserving_piFinSuccAbove (fun i => (volume : Measure (E i))) 0).symm)
rw [volume_pi, ← this.integrable_comp_emb (MeasurableEquiv.measurableEmbedding _)]
simp_rw [MeasurableEquiv.piFinSuccAbove_symm_apply,
Fin.prod_univ_succ, Fin.insertNth_zero]
simp only [Fin.zero_succAbove, cast_eq, Function.comp_def, Fin.cons_zero, Fin.cons_succ]
have : Integrable (fun (x : (j : Fin n) → E (Fin.succ j)) ↦ ∏ j, f (Fin.succ j) (x j)) :=
n_ih (fun i ↦ hf _)
exact Integrable.prod_mul (hf 0) this
| Mathlib/MeasureTheory/Integral/Pi.lean | 45 | 54 | theorem Integrable.fintype_prod_dep {ι : Type*} [Fintype ι] {E : ι → Type*}
{f : (i : ι) → E i → 𝕜} [∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
(hf : ∀ i, Integrable (f i)) :
Integrable (fun (x : (i : ι) → E i) ↦ ∏ i, f i (x i)) := by |
let e := (equivFin ι).symm
simp_rw [← (volume_measurePreserving_piCongrLeft _ e).integrable_comp_emb
(MeasurableEquiv.measurableEmbedding _),
← e.prod_comp, MeasurableEquiv.coe_piCongrLeft, Function.comp_def,
Equiv.piCongrLeft_apply_apply]
exact .fin_nat_prod (fun i ↦ hf _)
| 6 | 403.428793 | 2 | 1.6 | 5 | 1,745 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Constructions.Prod.Integral
open Fintype MeasureTheory MeasureTheory.Measure
variable {𝕜 : Type*} [RCLike 𝕜]
namespace MeasureTheory
theorem Integrable.fin_nat_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
{f : (i : Fin n) → E i → 𝕜} (hf : ∀ i, Integrable (f i)) :
Integrable (fun (x : (i : Fin n) → E i) ↦ ∏ i, f i (x i)) := by
induction n with
| zero => simp only [Nat.zero_eq, Finset.univ_eq_empty, Finset.prod_empty, volume_pi,
integrable_const_iff, one_ne_zero, pi_empty_univ, ENNReal.one_lt_top, or_true]
| succ n n_ih =>
have := ((measurePreserving_piFinSuccAbove (fun i => (volume : Measure (E i))) 0).symm)
rw [volume_pi, ← this.integrable_comp_emb (MeasurableEquiv.measurableEmbedding _)]
simp_rw [MeasurableEquiv.piFinSuccAbove_symm_apply,
Fin.prod_univ_succ, Fin.insertNth_zero]
simp only [Fin.zero_succAbove, cast_eq, Function.comp_def, Fin.cons_zero, Fin.cons_succ]
have : Integrable (fun (x : (j : Fin n) → E (Fin.succ j)) ↦ ∏ j, f (Fin.succ j) (x j)) :=
n_ih (fun i ↦ hf _)
exact Integrable.prod_mul (hf 0) this
theorem Integrable.fintype_prod_dep {ι : Type*} [Fintype ι] {E : ι → Type*}
{f : (i : ι) → E i → 𝕜} [∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
(hf : ∀ i, Integrable (f i)) :
Integrable (fun (x : (i : ι) → E i) ↦ ∏ i, f i (x i)) := by
let e := (equivFin ι).symm
simp_rw [← (volume_measurePreserving_piCongrLeft _ e).integrable_comp_emb
(MeasurableEquiv.measurableEmbedding _),
← e.prod_comp, MeasurableEquiv.coe_piCongrLeft, Function.comp_def,
Equiv.piCongrLeft_apply_apply]
exact .fin_nat_prod (fun i ↦ hf _)
theorem Integrable.fintype_prod {ι : Type*} [Fintype ι] {E : Type*}
{f : ι → E → 𝕜} [MeasureSpace E] [SigmaFinite (volume : Measure E)]
(hf : ∀ i, Integrable (f i)) :
Integrable (fun (x : ι → E) ↦ ∏ i, f i (x i)) :=
Integrable.fintype_prod_dep hf
| Mathlib/MeasureTheory/Integral/Pi.lean | 65 | 84 | theorem integral_fin_nat_prod_eq_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
(f : (i : Fin n) → E i → 𝕜) :
∫ x : (i : Fin n) → E i, ∏ i, f i (x i) = ∏ i, ∫ x, f i x := by |
induction n with
| zero =>
simp only [Nat.zero_eq, volume_pi, Finset.univ_eq_empty, Finset.prod_empty, integral_const,
pi_empty_univ, ENNReal.one_toReal, smul_eq_mul, mul_one, pow_zero, one_smul]
| succ n n_ih =>
calc
_ = ∫ x : E 0 × ((i : Fin n) → E (Fin.succ i)),
f 0 x.1 * ∏ i : Fin n, f (Fin.succ i) (x.2 i) := by
rw [volume_pi, ← ((measurePreserving_piFinSuccAbove
(fun i => (volume : Measure (E i))) 0).symm).integral_comp']
simp_rw [MeasurableEquiv.piFinSuccAbove_symm_apply,
Fin.prod_univ_succ, Fin.insertNth_zero, Fin.cons_succ, volume_eq_prod, volume_pi,
Fin.zero_succAbove, cast_eq, Fin.cons_zero]
_ = (∫ x, f 0 x) * ∏ i : Fin n, ∫ (x : E (Fin.succ i)), f (Fin.succ i) x := by
rw [← n_ih, ← integral_prod_mul, volume_eq_prod]
_ = ∏ i, ∫ x, f i x := by rw [Fin.prod_univ_succ]
| 16 | 8,886,110.520508 | 2 | 1.6 | 5 | 1,745 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Constructions.Prod.Integral
open Fintype MeasureTheory MeasureTheory.Measure
variable {𝕜 : Type*} [RCLike 𝕜]
namespace MeasureTheory
theorem Integrable.fin_nat_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
{f : (i : Fin n) → E i → 𝕜} (hf : ∀ i, Integrable (f i)) :
Integrable (fun (x : (i : Fin n) → E i) ↦ ∏ i, f i (x i)) := by
induction n with
| zero => simp only [Nat.zero_eq, Finset.univ_eq_empty, Finset.prod_empty, volume_pi,
integrable_const_iff, one_ne_zero, pi_empty_univ, ENNReal.one_lt_top, or_true]
| succ n n_ih =>
have := ((measurePreserving_piFinSuccAbove (fun i => (volume : Measure (E i))) 0).symm)
rw [volume_pi, ← this.integrable_comp_emb (MeasurableEquiv.measurableEmbedding _)]
simp_rw [MeasurableEquiv.piFinSuccAbove_symm_apply,
Fin.prod_univ_succ, Fin.insertNth_zero]
simp only [Fin.zero_succAbove, cast_eq, Function.comp_def, Fin.cons_zero, Fin.cons_succ]
have : Integrable (fun (x : (j : Fin n) → E (Fin.succ j)) ↦ ∏ j, f (Fin.succ j) (x j)) :=
n_ih (fun i ↦ hf _)
exact Integrable.prod_mul (hf 0) this
theorem Integrable.fintype_prod_dep {ι : Type*} [Fintype ι] {E : ι → Type*}
{f : (i : ι) → E i → 𝕜} [∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
(hf : ∀ i, Integrable (f i)) :
Integrable (fun (x : (i : ι) → E i) ↦ ∏ i, f i (x i)) := by
let e := (equivFin ι).symm
simp_rw [← (volume_measurePreserving_piCongrLeft _ e).integrable_comp_emb
(MeasurableEquiv.measurableEmbedding _),
← e.prod_comp, MeasurableEquiv.coe_piCongrLeft, Function.comp_def,
Equiv.piCongrLeft_apply_apply]
exact .fin_nat_prod (fun i ↦ hf _)
theorem Integrable.fintype_prod {ι : Type*} [Fintype ι] {E : Type*}
{f : ι → E → 𝕜} [MeasureSpace E] [SigmaFinite (volume : Measure E)]
(hf : ∀ i, Integrable (f i)) :
Integrable (fun (x : ι → E) ↦ ∏ i, f i (x i)) :=
Integrable.fintype_prod_dep hf
theorem integral_fin_nat_prod_eq_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
(f : (i : Fin n) → E i → 𝕜) :
∫ x : (i : Fin n) → E i, ∏ i, f i (x i) = ∏ i, ∫ x, f i x := by
induction n with
| zero =>
simp only [Nat.zero_eq, volume_pi, Finset.univ_eq_empty, Finset.prod_empty, integral_const,
pi_empty_univ, ENNReal.one_toReal, smul_eq_mul, mul_one, pow_zero, one_smul]
| succ n n_ih =>
calc
_ = ∫ x : E 0 × ((i : Fin n) → E (Fin.succ i)),
f 0 x.1 * ∏ i : Fin n, f (Fin.succ i) (x.2 i) := by
rw [volume_pi, ← ((measurePreserving_piFinSuccAbove
(fun i => (volume : Measure (E i))) 0).symm).integral_comp']
simp_rw [MeasurableEquiv.piFinSuccAbove_symm_apply,
Fin.prod_univ_succ, Fin.insertNth_zero, Fin.cons_succ, volume_eq_prod, volume_pi,
Fin.zero_succAbove, cast_eq, Fin.cons_zero]
_ = (∫ x, f 0 x) * ∏ i : Fin n, ∫ (x : E (Fin.succ i)), f (Fin.succ i) x := by
rw [← n_ih, ← integral_prod_mul, volume_eq_prod]
_ = ∏ i, ∫ x, f i x := by rw [Fin.prod_univ_succ]
| Mathlib/MeasureTheory/Integral/Pi.lean | 87 | 93 | theorem integral_fintype_prod_eq_prod (ι : Type*) [Fintype ι] {E : ι → Type*}
(f : (i : ι) → E i → 𝕜) [∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))] :
∫ x : (i : ι) → E i, ∏ i, f i (x i) = ∏ i, ∫ x, f i x := by |
let e := (equivFin ι).symm
rw [← (volume_measurePreserving_piCongrLeft _ e).integral_comp']
simp_rw [← e.prod_comp, MeasurableEquiv.coe_piCongrLeft, Equiv.piCongrLeft_apply_apply,
MeasureTheory.integral_fin_nat_prod_eq_prod]
| 4 | 54.59815 | 2 | 1.6 | 5 | 1,745 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Constructions.Prod.Integral
open Fintype MeasureTheory MeasureTheory.Measure
variable {𝕜 : Type*} [RCLike 𝕜]
namespace MeasureTheory
theorem Integrable.fin_nat_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
{f : (i : Fin n) → E i → 𝕜} (hf : ∀ i, Integrable (f i)) :
Integrable (fun (x : (i : Fin n) → E i) ↦ ∏ i, f i (x i)) := by
induction n with
| zero => simp only [Nat.zero_eq, Finset.univ_eq_empty, Finset.prod_empty, volume_pi,
integrable_const_iff, one_ne_zero, pi_empty_univ, ENNReal.one_lt_top, or_true]
| succ n n_ih =>
have := ((measurePreserving_piFinSuccAbove (fun i => (volume : Measure (E i))) 0).symm)
rw [volume_pi, ← this.integrable_comp_emb (MeasurableEquiv.measurableEmbedding _)]
simp_rw [MeasurableEquiv.piFinSuccAbove_symm_apply,
Fin.prod_univ_succ, Fin.insertNth_zero]
simp only [Fin.zero_succAbove, cast_eq, Function.comp_def, Fin.cons_zero, Fin.cons_succ]
have : Integrable (fun (x : (j : Fin n) → E (Fin.succ j)) ↦ ∏ j, f (Fin.succ j) (x j)) :=
n_ih (fun i ↦ hf _)
exact Integrable.prod_mul (hf 0) this
theorem Integrable.fintype_prod_dep {ι : Type*} [Fintype ι] {E : ι → Type*}
{f : (i : ι) → E i → 𝕜} [∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
(hf : ∀ i, Integrable (f i)) :
Integrable (fun (x : (i : ι) → E i) ↦ ∏ i, f i (x i)) := by
let e := (equivFin ι).symm
simp_rw [← (volume_measurePreserving_piCongrLeft _ e).integrable_comp_emb
(MeasurableEquiv.measurableEmbedding _),
← e.prod_comp, MeasurableEquiv.coe_piCongrLeft, Function.comp_def,
Equiv.piCongrLeft_apply_apply]
exact .fin_nat_prod (fun i ↦ hf _)
theorem Integrable.fintype_prod {ι : Type*} [Fintype ι] {E : Type*}
{f : ι → E → 𝕜} [MeasureSpace E] [SigmaFinite (volume : Measure E)]
(hf : ∀ i, Integrable (f i)) :
Integrable (fun (x : ι → E) ↦ ∏ i, f i (x i)) :=
Integrable.fintype_prod_dep hf
theorem integral_fin_nat_prod_eq_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
(f : (i : Fin n) → E i → 𝕜) :
∫ x : (i : Fin n) → E i, ∏ i, f i (x i) = ∏ i, ∫ x, f i x := by
induction n with
| zero =>
simp only [Nat.zero_eq, volume_pi, Finset.univ_eq_empty, Finset.prod_empty, integral_const,
pi_empty_univ, ENNReal.one_toReal, smul_eq_mul, mul_one, pow_zero, one_smul]
| succ n n_ih =>
calc
_ = ∫ x : E 0 × ((i : Fin n) → E (Fin.succ i)),
f 0 x.1 * ∏ i : Fin n, f (Fin.succ i) (x.2 i) := by
rw [volume_pi, ← ((measurePreserving_piFinSuccAbove
(fun i => (volume : Measure (E i))) 0).symm).integral_comp']
simp_rw [MeasurableEquiv.piFinSuccAbove_symm_apply,
Fin.prod_univ_succ, Fin.insertNth_zero, Fin.cons_succ, volume_eq_prod, volume_pi,
Fin.zero_succAbove, cast_eq, Fin.cons_zero]
_ = (∫ x, f 0 x) * ∏ i : Fin n, ∫ (x : E (Fin.succ i)), f (Fin.succ i) x := by
rw [← n_ih, ← integral_prod_mul, volume_eq_prod]
_ = ∏ i, ∫ x, f i x := by rw [Fin.prod_univ_succ]
theorem integral_fintype_prod_eq_prod (ι : Type*) [Fintype ι] {E : ι → Type*}
(f : (i : ι) → E i → 𝕜) [∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))] :
∫ x : (i : ι) → E i, ∏ i, f i (x i) = ∏ i, ∫ x, f i x := by
let e := (equivFin ι).symm
rw [← (volume_measurePreserving_piCongrLeft _ e).integral_comp']
simp_rw [← e.prod_comp, MeasurableEquiv.coe_piCongrLeft, Equiv.piCongrLeft_apply_apply,
MeasureTheory.integral_fin_nat_prod_eq_prod]
| Mathlib/MeasureTheory/Integral/Pi.lean | 95 | 98 | theorem integral_fintype_prod_eq_pow {E : Type*} (ι : Type*) [Fintype ι] (f : E → 𝕜)
[MeasureSpace E] [SigmaFinite (volume : Measure E)] :
∫ x : ι → E, ∏ i, f (x i) = (∫ x, f x) ^ (card ι) := by |
rw [integral_fintype_prod_eq_prod, Finset.prod_const, card]
| 1 | 2.718282 | 0 | 1.6 | 5 | 1,745 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.ZMod.Parity
#align_import combinatorics.simple_graph.degree_sum from "leanprover-community/mathlib"@"90659cbe25e59ec302e2fb92b00e9732160cc620"
open Finset
namespace SimpleGraph
universe u
variable {V : Type u} (G : SimpleGraph V)
section DegreeSum
variable [Fintype V] [DecidableRel G.Adj]
-- Porting note: Changed to `Fintype (Sym2 V)` to match Combinatorics.SimpleGraph.Basic
variable [Fintype (Sym2 V)]
| Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | 56 | 64 | theorem dart_fst_fiber [DecidableEq V] (v : V) :
(univ.filter fun d : G.Dart => d.fst = v) = univ.image (G.dartOfNeighborSet v) := by |
ext d
simp only [mem_image, true_and_iff, mem_filter, SetCoe.exists, mem_univ, exists_prop_of_true]
constructor
· rintro rfl
exact ⟨_, d.adj, by ext <;> rfl⟩
· rintro ⟨e, he, rfl⟩
rfl
| 7 | 1,096.633158 | 2 | 1.6 | 5 | 1,746 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.ZMod.Parity
#align_import combinatorics.simple_graph.degree_sum from "leanprover-community/mathlib"@"90659cbe25e59ec302e2fb92b00e9732160cc620"
open Finset
namespace SimpleGraph
universe u
variable {V : Type u} (G : SimpleGraph V)
section DegreeSum
variable [Fintype V] [DecidableRel G.Adj]
-- Porting note: Changed to `Fintype (Sym2 V)` to match Combinatorics.SimpleGraph.Basic
variable [Fintype (Sym2 V)]
theorem dart_fst_fiber [DecidableEq V] (v : V) :
(univ.filter fun d : G.Dart => d.fst = v) = univ.image (G.dartOfNeighborSet v) := by
ext d
simp only [mem_image, true_and_iff, mem_filter, SetCoe.exists, mem_univ, exists_prop_of_true]
constructor
· rintro rfl
exact ⟨_, d.adj, by ext <;> rfl⟩
· rintro ⟨e, he, rfl⟩
rfl
#align simple_graph.dart_fst_fiber SimpleGraph.dart_fst_fiber
| Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | 67 | 70 | theorem dart_fst_fiber_card_eq_degree [DecidableEq V] (v : V) :
(univ.filter fun d : G.Dart => d.fst = v).card = G.degree v := by |
simpa only [dart_fst_fiber, Finset.card_univ, card_neighborSet_eq_degree] using
card_image_of_injective univ (G.dartOfNeighborSet_injective v)
| 2 | 7.389056 | 1 | 1.6 | 5 | 1,746 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.ZMod.Parity
#align_import combinatorics.simple_graph.degree_sum from "leanprover-community/mathlib"@"90659cbe25e59ec302e2fb92b00e9732160cc620"
open Finset
namespace SimpleGraph
universe u
variable {V : Type u} (G : SimpleGraph V)
section DegreeSum
variable [Fintype V] [DecidableRel G.Adj]
-- Porting note: Changed to `Fintype (Sym2 V)` to match Combinatorics.SimpleGraph.Basic
variable [Fintype (Sym2 V)]
theorem dart_fst_fiber [DecidableEq V] (v : V) :
(univ.filter fun d : G.Dart => d.fst = v) = univ.image (G.dartOfNeighborSet v) := by
ext d
simp only [mem_image, true_and_iff, mem_filter, SetCoe.exists, mem_univ, exists_prop_of_true]
constructor
· rintro rfl
exact ⟨_, d.adj, by ext <;> rfl⟩
· rintro ⟨e, he, rfl⟩
rfl
#align simple_graph.dart_fst_fiber SimpleGraph.dart_fst_fiber
theorem dart_fst_fiber_card_eq_degree [DecidableEq V] (v : V) :
(univ.filter fun d : G.Dart => d.fst = v).card = G.degree v := by
simpa only [dart_fst_fiber, Finset.card_univ, card_neighborSet_eq_degree] using
card_image_of_injective univ (G.dartOfNeighborSet_injective v)
#align simple_graph.dart_fst_fiber_card_eq_degree SimpleGraph.dart_fst_fiber_card_eq_degree
| Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | 73 | 76 | theorem dart_card_eq_sum_degrees : Fintype.card G.Dart = ∑ v, G.degree v := by |
haveI := Classical.decEq V
simp only [← card_univ, ← dart_fst_fiber_card_eq_degree]
exact card_eq_sum_card_fiberwise (by simp)
| 3 | 20.085537 | 1 | 1.6 | 5 | 1,746 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.ZMod.Parity
#align_import combinatorics.simple_graph.degree_sum from "leanprover-community/mathlib"@"90659cbe25e59ec302e2fb92b00e9732160cc620"
open Finset
namespace SimpleGraph
universe u
variable {V : Type u} (G : SimpleGraph V)
section DegreeSum
variable [Fintype V] [DecidableRel G.Adj]
-- Porting note: Changed to `Fintype (Sym2 V)` to match Combinatorics.SimpleGraph.Basic
variable [Fintype (Sym2 V)]
theorem dart_fst_fiber [DecidableEq V] (v : V) :
(univ.filter fun d : G.Dart => d.fst = v) = univ.image (G.dartOfNeighborSet v) := by
ext d
simp only [mem_image, true_and_iff, mem_filter, SetCoe.exists, mem_univ, exists_prop_of_true]
constructor
· rintro rfl
exact ⟨_, d.adj, by ext <;> rfl⟩
· rintro ⟨e, he, rfl⟩
rfl
#align simple_graph.dart_fst_fiber SimpleGraph.dart_fst_fiber
theorem dart_fst_fiber_card_eq_degree [DecidableEq V] (v : V) :
(univ.filter fun d : G.Dart => d.fst = v).card = G.degree v := by
simpa only [dart_fst_fiber, Finset.card_univ, card_neighborSet_eq_degree] using
card_image_of_injective univ (G.dartOfNeighborSet_injective v)
#align simple_graph.dart_fst_fiber_card_eq_degree SimpleGraph.dart_fst_fiber_card_eq_degree
theorem dart_card_eq_sum_degrees : Fintype.card G.Dart = ∑ v, G.degree v := by
haveI := Classical.decEq V
simp only [← card_univ, ← dart_fst_fiber_card_eq_degree]
exact card_eq_sum_card_fiberwise (by simp)
#align simple_graph.dart_card_eq_sum_degrees SimpleGraph.dart_card_eq_sum_degrees
variable {G}
theorem Dart.edge_fiber [DecidableEq V] (d : G.Dart) :
(univ.filter fun d' : G.Dart => d'.edge = d.edge) = {d, d.symm} :=
Finset.ext fun d' => by simpa using dart_edge_eq_iff d' d
#align simple_graph.dart.edge_fiber SimpleGraph.Dart.edge_fiber
variable (G)
| Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | 88 | 95 | theorem dart_edge_fiber_card [DecidableEq V] (e : Sym2 V) (h : e ∈ G.edgeSet) :
(univ.filter fun d : G.Dart => d.edge = e).card = 2 := by |
refine Sym2.ind (fun v w h => ?_) e h
let d : G.Dart := ⟨(v, w), h⟩
convert congr_arg card d.edge_fiber
rw [card_insert_of_not_mem, card_singleton]
rw [mem_singleton]
exact d.symm_ne.symm
| 6 | 403.428793 | 2 | 1.6 | 5 | 1,746 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.ZMod.Parity
#align_import combinatorics.simple_graph.degree_sum from "leanprover-community/mathlib"@"90659cbe25e59ec302e2fb92b00e9732160cc620"
open Finset
namespace SimpleGraph
universe u
variable {V : Type u} (G : SimpleGraph V)
section DegreeSum
variable [Fintype V] [DecidableRel G.Adj]
-- Porting note: Changed to `Fintype (Sym2 V)` to match Combinatorics.SimpleGraph.Basic
variable [Fintype (Sym2 V)]
theorem dart_fst_fiber [DecidableEq V] (v : V) :
(univ.filter fun d : G.Dart => d.fst = v) = univ.image (G.dartOfNeighborSet v) := by
ext d
simp only [mem_image, true_and_iff, mem_filter, SetCoe.exists, mem_univ, exists_prop_of_true]
constructor
· rintro rfl
exact ⟨_, d.adj, by ext <;> rfl⟩
· rintro ⟨e, he, rfl⟩
rfl
#align simple_graph.dart_fst_fiber SimpleGraph.dart_fst_fiber
theorem dart_fst_fiber_card_eq_degree [DecidableEq V] (v : V) :
(univ.filter fun d : G.Dart => d.fst = v).card = G.degree v := by
simpa only [dart_fst_fiber, Finset.card_univ, card_neighborSet_eq_degree] using
card_image_of_injective univ (G.dartOfNeighborSet_injective v)
#align simple_graph.dart_fst_fiber_card_eq_degree SimpleGraph.dart_fst_fiber_card_eq_degree
theorem dart_card_eq_sum_degrees : Fintype.card G.Dart = ∑ v, G.degree v := by
haveI := Classical.decEq V
simp only [← card_univ, ← dart_fst_fiber_card_eq_degree]
exact card_eq_sum_card_fiberwise (by simp)
#align simple_graph.dart_card_eq_sum_degrees SimpleGraph.dart_card_eq_sum_degrees
variable {G}
theorem Dart.edge_fiber [DecidableEq V] (d : G.Dart) :
(univ.filter fun d' : G.Dart => d'.edge = d.edge) = {d, d.symm} :=
Finset.ext fun d' => by simpa using dart_edge_eq_iff d' d
#align simple_graph.dart.edge_fiber SimpleGraph.Dart.edge_fiber
variable (G)
theorem dart_edge_fiber_card [DecidableEq V] (e : Sym2 V) (h : e ∈ G.edgeSet) :
(univ.filter fun d : G.Dart => d.edge = e).card = 2 := by
refine Sym2.ind (fun v w h => ?_) e h
let d : G.Dart := ⟨(v, w), h⟩
convert congr_arg card d.edge_fiber
rw [card_insert_of_not_mem, card_singleton]
rw [mem_singleton]
exact d.symm_ne.symm
#align simple_graph.dart_edge_fiber_card SimpleGraph.dart_edge_fiber_card
| Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | 98 | 106 | theorem dart_card_eq_twice_card_edges : Fintype.card G.Dart = 2 * G.edgeFinset.card := by |
classical
rw [← card_univ]
rw [@card_eq_sum_card_fiberwise _ _ _ Dart.edge _ G.edgeFinset fun d _h =>
by rw [mem_edgeFinset]; apply Dart.edge_mem]
rw [← mul_comm, sum_const_nat]
intro e h
apply G.dart_edge_fiber_card e
rwa [← mem_edgeFinset]
| 8 | 2,980.957987 | 2 | 1.6 | 5 | 1,746 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u₀ u v v' v'' u₁' w w'
variable {K R : Type u} {V V₁ V₂ V₃ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set
section Module
section DivisionRing
variable [DivisionRing K]
variable [AddCommGroup V] [Module K V]
variable [AddCommGroup V'] [Module K V']
variable [AddCommGroup V₁] [Module K V₁]
theorem Basis.finite_ofVectorSpaceIndex_of_rank_lt_aleph0 (h : Module.rank K V < ℵ₀) :
(Basis.ofVectorSpaceIndex K V).Finite :=
finite_def.2 <| (Basis.ofVectorSpace K V).nonempty_fintype_index_of_rank_lt_aleph0 h
#align basis.finite_of_vector_space_index_of_rank_lt_aleph_0 Basis.finite_ofVectorSpaceIndex_of_rank_lt_aleph0
| Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 59 | 63 | theorem rank_quotient_add_rank_of_divisionRing (p : Submodule K V) :
Module.rank K (V ⧸ p) + Module.rank K p = Module.rank K V := by |
classical
let ⟨f⟩ := quotient_prod_linearEquiv p
exact rank_prod'.symm.trans f.rank_eq
| 3 | 20.085537 | 1 | 1.625 | 8 | 1,748 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u₀ u v v' v'' u₁' w w'
variable {K R : Type u} {V V₁ V₂ V₃ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set
section Module
section DivisionRing
variable [DivisionRing K]
variable [AddCommGroup V] [Module K V]
variable [AddCommGroup V'] [Module K V']
variable [AddCommGroup V₁] [Module K V₁]
theorem Basis.finite_ofVectorSpaceIndex_of_rank_lt_aleph0 (h : Module.rank K V < ℵ₀) :
(Basis.ofVectorSpaceIndex K V).Finite :=
finite_def.2 <| (Basis.ofVectorSpace K V).nonempty_fintype_index_of_rank_lt_aleph0 h
#align basis.finite_of_vector_space_index_of_rank_lt_aleph_0 Basis.finite_ofVectorSpaceIndex_of_rank_lt_aleph0
theorem rank_quotient_add_rank_of_divisionRing (p : Submodule K V) :
Module.rank K (V ⧸ p) + Module.rank K p = Module.rank K V := by
classical
let ⟨f⟩ := quotient_prod_linearEquiv p
exact rank_prod'.symm.trans f.rank_eq
instance DivisionRing.hasRankNullity : HasRankNullity.{u₀} K where
rank_quotient_add_rank := rank_quotient_add_rank_of_divisionRing
exists_set_linearIndependent V _ _ := by
let b := Module.Free.chooseBasis K V
refine ⟨range b, ?_, b.linearIndependent.to_subtype_range⟩
rw [← lift_injective.eq_iff, mk_range_eq_of_injective b.injective,
Module.Free.rank_eq_card_chooseBasisIndex]
section
variable [AddCommGroup V₂] [Module K V₂]
variable [AddCommGroup V₃] [Module K V₃]
open LinearMap
| Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 81 | 108 | theorem rank_add_rank_split (db : V₂ →ₗ[K] V) (eb : V₃ →ₗ[K] V) (cd : V₁ →ₗ[K] V₂)
(ce : V₁ →ₗ[K] V₃) (hde : ⊤ ≤ LinearMap.range db ⊔ LinearMap.range eb) (hgd : ker cd = ⊥)
(eq : db.comp cd = eb.comp ce) (eq₂ : ∀ d e, db d = eb e → ∃ c, cd c = d ∧ ce c = e) :
Module.rank K V + Module.rank K V₁ = Module.rank K V₂ + Module.rank K V₃ := by |
have hf : Surjective (coprod db eb) := by rwa [← range_eq_top, range_coprod, eq_top_iff]
conv =>
rhs
rw [← rank_prod', rank_eq_of_surjective hf]
congr 1
apply LinearEquiv.rank_eq
let L : V₁ →ₗ[K] ker (coprod db eb) := by -- Porting note: this is needed to avoid a timeout
refine LinearMap.codRestrict _ (prod cd (-ce)) ?_
· intro c
simp only [add_eq_zero_iff_eq_neg, LinearMap.prod_apply, mem_ker, Pi.prod, coprod_apply,
neg_neg, map_neg, neg_apply]
exact LinearMap.ext_iff.1 eq c
refine LinearEquiv.ofBijective L ⟨?_, ?_⟩
· rw [← ker_eq_bot, ker_codRestrict, ker_prod, hgd, bot_inf_eq]
· rw [← range_eq_top, eq_top_iff, range_codRestrict, ← map_le_iff_le_comap, Submodule.map_top,
range_subtype]
rintro ⟨d, e⟩
have h := eq₂ d (-e)
simp only [add_eq_zero_iff_eq_neg, LinearMap.prod_apply, mem_ker, SetLike.mem_coe,
Prod.mk.inj_iff, coprod_apply, map_neg, neg_apply, LinearMap.mem_range, Pi.prod] at h ⊢
intro hde
rcases h hde with ⟨c, h₁, h₂⟩
refine ⟨c, h₁, ?_⟩
rw [h₂, _root_.neg_neg]
| 24 | 26,489,122,129.84347 | 2 | 1.625 | 8 | 1,748 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u₀ u v v' v'' u₁' w w'
variable {K R : Type u} {V V₁ V₂ V₃ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set
section Module
section Basis
open FiniteDimensional
variable [DivisionRing K] [AddCommGroup V] [Module K V]
| Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 123 | 166 | theorem linearIndependent_of_top_le_span_of_card_eq_finrank {ι : Type*} [Fintype ι] {b : ι → V}
(spans : ⊤ ≤ span K (Set.range b)) (card_eq : Fintype.card ι = finrank K V) :
LinearIndependent K b :=
linearIndependent_iff'.mpr fun s g dependent i i_mem_s => by
classical
by_contra gx_ne_zero
-- We'll derive a contradiction by showing `b '' (univ \ {i})` of cardinality `n - 1`
-- spans a vector space of dimension `n`.
refine not_le_of_gt (span_lt_top_of_card_lt_finrank
(show (b '' (Set.univ \ {i})).toFinset.card < finrank K V from ?_)) ?_
· calc
(b '' (Set.univ \ {i})).toFinset.card = ((Set.univ \ {i}).toFinset.image b).card := by |
rw [Set.toFinset_card, Fintype.card_ofFinset]
_ ≤ (Set.univ \ {i}).toFinset.card := Finset.card_image_le
_ = (Finset.univ.erase i).card := (congr_arg Finset.card (Finset.ext (by simp [and_comm])))
_ < Finset.univ.card := Finset.card_erase_lt_of_mem (Finset.mem_univ i)
_ = finrank K V := card_eq
-- We already have that `b '' univ` spans the whole space,
-- so we only need to show that the span of `b '' (univ \ {i})` contains each `b j`.
refine spans.trans (span_le.mpr ?_)
rintro _ ⟨j, rfl, rfl⟩
-- The case that `j ≠ i` is easy because `b j ∈ b '' (univ \ {i})`.
by_cases j_eq : j = i
swap
· refine subset_span ⟨j, (Set.mem_diff _).mpr ⟨Set.mem_univ _, ?_⟩, rfl⟩
exact mt Set.mem_singleton_iff.mp j_eq
-- To show `b i ∈ span (b '' (univ \ {i}))`, we use that it's a weighted sum
-- of the other `b j`s.
rw [j_eq, SetLike.mem_coe, show b i = -((g i)⁻¹ • (s.erase i).sum fun j => g j • b j) from _]
· refine neg_mem (smul_mem _ _ (sum_mem fun k hk => ?_))
obtain ⟨k_ne_i, _⟩ := Finset.mem_erase.mp hk
refine smul_mem _ _ (subset_span ⟨k, ?_, rfl⟩)
simp_all only [Set.mem_univ, Set.mem_diff, Set.mem_singleton_iff, and_self, not_false_eq_true]
-- To show `b i` is a weighted sum of the other `b j`s, we'll rewrite this sum
-- to have the form of the assumption `dependent`.
apply eq_neg_of_add_eq_zero_left
calc
(b i + (g i)⁻¹ • (s.erase i).sum fun j => g j • b j) =
(g i)⁻¹ • (g i • b i + (s.erase i).sum fun j => g j • b j) := by
rw [smul_add, ← mul_smul, inv_mul_cancel gx_ne_zero, one_smul]
_ = (g i)⁻¹ • (0 : V) := congr_arg _ ?_
_ = 0 := smul_zero _
-- And then it's just a bit of manipulation with finite sums.
rwa [← Finset.insert_erase i_mem_s, Finset.sum_insert (Finset.not_mem_erase _ _)] at dependent
| 32 | 78,962,960,182,680.7 | 2 | 1.625 | 8 | 1,748 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u₀ u v v' v'' u₁' w w'
variable {K R : Type u} {V V₁ V₂ V₃ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set
section Module
section Basis
open FiniteDimensional
variable [DivisionRing K] [AddCommGroup V] [Module K V]
theorem linearIndependent_of_top_le_span_of_card_eq_finrank {ι : Type*} [Fintype ι] {b : ι → V}
(spans : ⊤ ≤ span K (Set.range b)) (card_eq : Fintype.card ι = finrank K V) :
LinearIndependent K b :=
linearIndependent_iff'.mpr fun s g dependent i i_mem_s => by
classical
by_contra gx_ne_zero
-- We'll derive a contradiction by showing `b '' (univ \ {i})` of cardinality `n - 1`
-- spans a vector space of dimension `n`.
refine not_le_of_gt (span_lt_top_of_card_lt_finrank
(show (b '' (Set.univ \ {i})).toFinset.card < finrank K V from ?_)) ?_
· calc
(b '' (Set.univ \ {i})).toFinset.card = ((Set.univ \ {i}).toFinset.image b).card := by
rw [Set.toFinset_card, Fintype.card_ofFinset]
_ ≤ (Set.univ \ {i}).toFinset.card := Finset.card_image_le
_ = (Finset.univ.erase i).card := (congr_arg Finset.card (Finset.ext (by simp [and_comm])))
_ < Finset.univ.card := Finset.card_erase_lt_of_mem (Finset.mem_univ i)
_ = finrank K V := card_eq
-- We already have that `b '' univ` spans the whole space,
-- so we only need to show that the span of `b '' (univ \ {i})` contains each `b j`.
refine spans.trans (span_le.mpr ?_)
rintro _ ⟨j, rfl, rfl⟩
-- The case that `j ≠ i` is easy because `b j ∈ b '' (univ \ {i})`.
by_cases j_eq : j = i
swap
· refine subset_span ⟨j, (Set.mem_diff _).mpr ⟨Set.mem_univ _, ?_⟩, rfl⟩
exact mt Set.mem_singleton_iff.mp j_eq
-- To show `b i ∈ span (b '' (univ \ {i}))`, we use that it's a weighted sum
-- of the other `b j`s.
rw [j_eq, SetLike.mem_coe, show b i = -((g i)⁻¹ • (s.erase i).sum fun j => g j • b j) from _]
· refine neg_mem (smul_mem _ _ (sum_mem fun k hk => ?_))
obtain ⟨k_ne_i, _⟩ := Finset.mem_erase.mp hk
refine smul_mem _ _ (subset_span ⟨k, ?_, rfl⟩)
simp_all only [Set.mem_univ, Set.mem_diff, Set.mem_singleton_iff, and_self, not_false_eq_true]
-- To show `b i` is a weighted sum of the other `b j`s, we'll rewrite this sum
-- to have the form of the assumption `dependent`.
apply eq_neg_of_add_eq_zero_left
calc
(b i + (g i)⁻¹ • (s.erase i).sum fun j => g j • b j) =
(g i)⁻¹ • (g i • b i + (s.erase i).sum fun j => g j • b j) := by
rw [smul_add, ← mul_smul, inv_mul_cancel gx_ne_zero, one_smul]
_ = (g i)⁻¹ • (0 : V) := congr_arg _ ?_
_ = 0 := smul_zero _
-- And then it's just a bit of manipulation with finite sums.
rwa [← Finset.insert_erase i_mem_s, Finset.sum_insert (Finset.not_mem_erase _ _)] at dependent
#align linear_independent_of_top_le_span_of_card_eq_finrank linearIndependent_of_top_le_span_of_card_eq_finrank
| Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 171 | 193 | theorem linearIndependent_iff_card_eq_finrank_span {ι : Type*} [Fintype ι] {b : ι → V} :
LinearIndependent K b ↔ Fintype.card ι = (Set.range b).finrank K := by |
constructor
· intro h
exact (finrank_span_eq_card h).symm
· intro hc
let f := Submodule.subtype (span K (Set.range b))
let b' : ι → span K (Set.range b) := fun i =>
⟨b i, mem_span.2 fun p hp => hp (Set.mem_range_self _)⟩
have hs : ⊤ ≤ span K (Set.range b') := by
intro x
have h : span K (f '' Set.range b') = map f (span K (Set.range b')) := span_image f
have hf : f '' Set.range b' = Set.range b := by
ext x
simp [f, Set.mem_image, Set.mem_range]
rw [hf] at h
have hx : (x : V) ∈ span K (Set.range b) := x.property
conv at hx =>
arg 2
rw [h]
simpa [f, mem_map] using hx
have hi : LinearMap.ker f = ⊥ := ker_subtype _
convert (linearIndependent_of_top_le_span_of_card_eq_finrank hs hc).map' _ hi
| 21 | 1,318,815,734.483215 | 2 | 1.625 | 8 | 1,748 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u₀ u v v' v'' u₁' w w'
variable {K R : Type u} {V V₁ V₂ V₃ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set
section Module
section Basis
open FiniteDimensional
variable [DivisionRing K] [AddCommGroup V] [Module K V]
theorem linearIndependent_of_top_le_span_of_card_eq_finrank {ι : Type*} [Fintype ι] {b : ι → V}
(spans : ⊤ ≤ span K (Set.range b)) (card_eq : Fintype.card ι = finrank K V) :
LinearIndependent K b :=
linearIndependent_iff'.mpr fun s g dependent i i_mem_s => by
classical
by_contra gx_ne_zero
-- We'll derive a contradiction by showing `b '' (univ \ {i})` of cardinality `n - 1`
-- spans a vector space of dimension `n`.
refine not_le_of_gt (span_lt_top_of_card_lt_finrank
(show (b '' (Set.univ \ {i})).toFinset.card < finrank K V from ?_)) ?_
· calc
(b '' (Set.univ \ {i})).toFinset.card = ((Set.univ \ {i}).toFinset.image b).card := by
rw [Set.toFinset_card, Fintype.card_ofFinset]
_ ≤ (Set.univ \ {i}).toFinset.card := Finset.card_image_le
_ = (Finset.univ.erase i).card := (congr_arg Finset.card (Finset.ext (by simp [and_comm])))
_ < Finset.univ.card := Finset.card_erase_lt_of_mem (Finset.mem_univ i)
_ = finrank K V := card_eq
-- We already have that `b '' univ` spans the whole space,
-- so we only need to show that the span of `b '' (univ \ {i})` contains each `b j`.
refine spans.trans (span_le.mpr ?_)
rintro _ ⟨j, rfl, rfl⟩
-- The case that `j ≠ i` is easy because `b j ∈ b '' (univ \ {i})`.
by_cases j_eq : j = i
swap
· refine subset_span ⟨j, (Set.mem_diff _).mpr ⟨Set.mem_univ _, ?_⟩, rfl⟩
exact mt Set.mem_singleton_iff.mp j_eq
-- To show `b i ∈ span (b '' (univ \ {i}))`, we use that it's a weighted sum
-- of the other `b j`s.
rw [j_eq, SetLike.mem_coe, show b i = -((g i)⁻¹ • (s.erase i).sum fun j => g j • b j) from _]
· refine neg_mem (smul_mem _ _ (sum_mem fun k hk => ?_))
obtain ⟨k_ne_i, _⟩ := Finset.mem_erase.mp hk
refine smul_mem _ _ (subset_span ⟨k, ?_, rfl⟩)
simp_all only [Set.mem_univ, Set.mem_diff, Set.mem_singleton_iff, and_self, not_false_eq_true]
-- To show `b i` is a weighted sum of the other `b j`s, we'll rewrite this sum
-- to have the form of the assumption `dependent`.
apply eq_neg_of_add_eq_zero_left
calc
(b i + (g i)⁻¹ • (s.erase i).sum fun j => g j • b j) =
(g i)⁻¹ • (g i • b i + (s.erase i).sum fun j => g j • b j) := by
rw [smul_add, ← mul_smul, inv_mul_cancel gx_ne_zero, one_smul]
_ = (g i)⁻¹ • (0 : V) := congr_arg _ ?_
_ = 0 := smul_zero _
-- And then it's just a bit of manipulation with finite sums.
rwa [← Finset.insert_erase i_mem_s, Finset.sum_insert (Finset.not_mem_erase _ _)] at dependent
#align linear_independent_of_top_le_span_of_card_eq_finrank linearIndependent_of_top_le_span_of_card_eq_finrank
theorem linearIndependent_iff_card_eq_finrank_span {ι : Type*} [Fintype ι] {b : ι → V} :
LinearIndependent K b ↔ Fintype.card ι = (Set.range b).finrank K := by
constructor
· intro h
exact (finrank_span_eq_card h).symm
· intro hc
let f := Submodule.subtype (span K (Set.range b))
let b' : ι → span K (Set.range b) := fun i =>
⟨b i, mem_span.2 fun p hp => hp (Set.mem_range_self _)⟩
have hs : ⊤ ≤ span K (Set.range b') := by
intro x
have h : span K (f '' Set.range b') = map f (span K (Set.range b')) := span_image f
have hf : f '' Set.range b' = Set.range b := by
ext x
simp [f, Set.mem_image, Set.mem_range]
rw [hf] at h
have hx : (x : V) ∈ span K (Set.range b) := x.property
conv at hx =>
arg 2
rw [h]
simpa [f, mem_map] using hx
have hi : LinearMap.ker f = ⊥ := ker_subtype _
convert (linearIndependent_of_top_le_span_of_card_eq_finrank hs hc).map' _ hi
#align linear_independent_iff_card_eq_finrank_span linearIndependent_iff_card_eq_finrank_span
| Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 196 | 198 | theorem linearIndependent_iff_card_le_finrank_span {ι : Type*} [Fintype ι] {b : ι → V} :
LinearIndependent K b ↔ Fintype.card ι ≤ (Set.range b).finrank K := by |
rw [linearIndependent_iff_card_eq_finrank_span, (finrank_range_le_card _).le_iff_eq]
| 1 | 2.718282 | 0 | 1.625 | 8 | 1,748 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u₀ u v v' v'' u₁' w w'
variable {K R : Type u} {V V₁ V₂ V₃ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set
section Module
section Cardinal
variable (K)
variable [DivisionRing K]
| Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 239 | 283 | theorem max_aleph0_card_le_rank_fun_nat : max ℵ₀ #K ≤ Module.rank K (ℕ → K) := by |
have aleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K) := (rank_finsupp_self K ℕ).symm.trans_le
(Finsupp.lcoeFun.rank_le_of_injective <| by exact DFunLike.coe_injective)
refine max_le aleph0_le ?_
obtain card_K | card_K := le_or_lt #K ℵ₀
· exact card_K.trans aleph0_le
by_contra!
obtain ⟨⟨ιK, bK⟩⟩ := Module.Free.exists_basis (R := K) (M := ℕ → K)
let L := Subfield.closure (Set.range (fun i : ιK × ℕ ↦ bK i.1 i.2))
have hLK : #L < #K := by
refine (Subfield.cardinal_mk_closure_le_max _).trans_lt
(max_lt_iff.mpr ⟨mk_range_le.trans_lt ?_, card_K⟩)
rwa [mk_prod, ← aleph0, lift_uzero, bK.mk_eq_rank'', mul_aleph0_eq aleph0_le]
letI := Module.compHom K (RingHom.op L.subtype)
obtain ⟨⟨ιL, bL⟩⟩ := Module.Free.exists_basis (R := Lᵐᵒᵖ) (M := K)
have card_ιL : ℵ₀ ≤ #ιL := by
contrapose! hLK
haveI := @Fintype.ofFinite _ (lt_aleph0_iff_finite.mp hLK)
rw [bL.repr.toEquiv.cardinal_eq, mk_finsupp_of_fintype,
← MulOpposite.opEquiv.cardinal_eq] at card_K ⊢
apply power_nat_le
contrapose! card_K
exact (power_lt_aleph0 card_K <| nat_lt_aleph0 _).le
obtain ⟨e⟩ := lift_mk_le'.mp (card_ιL.trans_eq (lift_uzero #ιL).symm)
have rep_e := bK.total_repr (bL ∘ e)
rw [Finsupp.total_apply, Finsupp.sum] at rep_e
set c := bK.repr (bL ∘ e)
set s := c.support
let f i (j : s) : L := ⟨bK j i, Subfield.subset_closure ⟨(j, i), rfl⟩⟩
have : ¬LinearIndependent Lᵐᵒᵖ f := fun h ↦ by
have := h.cardinal_lift_le_rank
rw [lift_uzero, (LinearEquiv.piCongrRight fun _ ↦ MulOpposite.opLinearEquiv Lᵐᵒᵖ).rank_eq,
rank_fun'] at this
exact (nat_lt_aleph0 _).not_le this
obtain ⟨t, g, eq0, i, hi, hgi⟩ := not_linearIndependent_iff.mp this
refine hgi (linearIndependent_iff'.mp (bL.linearIndependent.comp e e.injective) t g ?_ i hi)
clear_value c s
simp_rw [← rep_e, Finset.sum_apply, Pi.smul_apply, Finset.smul_sum]
rw [Finset.sum_comm]
refine Finset.sum_eq_zero fun i hi ↦ ?_
replace eq0 := congr_arg L.subtype (congr_fun eq0 ⟨i, hi⟩)
rw [Finset.sum_apply, map_sum] at eq0
have : SMulCommClass Lᵐᵒᵖ K K := ⟨fun _ _ _ ↦ mul_assoc _ _ _⟩
simp_rw [smul_comm _ (c i), ← Finset.smul_sum]
erw [eq0, smul_zero]
| 44 | 12,851,600,114,359,308,000 | 2 | 1.625 | 8 | 1,748 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u₀ u v v' v'' u₁' w w'
variable {K R : Type u} {V V₁ V₂ V₃ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set
section Module
section Cardinal
variable (K)
variable [DivisionRing K]
theorem max_aleph0_card_le_rank_fun_nat : max ℵ₀ #K ≤ Module.rank K (ℕ → K) := by
have aleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K) := (rank_finsupp_self K ℕ).symm.trans_le
(Finsupp.lcoeFun.rank_le_of_injective <| by exact DFunLike.coe_injective)
refine max_le aleph0_le ?_
obtain card_K | card_K := le_or_lt #K ℵ₀
· exact card_K.trans aleph0_le
by_contra!
obtain ⟨⟨ιK, bK⟩⟩ := Module.Free.exists_basis (R := K) (M := ℕ → K)
let L := Subfield.closure (Set.range (fun i : ιK × ℕ ↦ bK i.1 i.2))
have hLK : #L < #K := by
refine (Subfield.cardinal_mk_closure_le_max _).trans_lt
(max_lt_iff.mpr ⟨mk_range_le.trans_lt ?_, card_K⟩)
rwa [mk_prod, ← aleph0, lift_uzero, bK.mk_eq_rank'', mul_aleph0_eq aleph0_le]
letI := Module.compHom K (RingHom.op L.subtype)
obtain ⟨⟨ιL, bL⟩⟩ := Module.Free.exists_basis (R := Lᵐᵒᵖ) (M := K)
have card_ιL : ℵ₀ ≤ #ιL := by
contrapose! hLK
haveI := @Fintype.ofFinite _ (lt_aleph0_iff_finite.mp hLK)
rw [bL.repr.toEquiv.cardinal_eq, mk_finsupp_of_fintype,
← MulOpposite.opEquiv.cardinal_eq] at card_K ⊢
apply power_nat_le
contrapose! card_K
exact (power_lt_aleph0 card_K <| nat_lt_aleph0 _).le
obtain ⟨e⟩ := lift_mk_le'.mp (card_ιL.trans_eq (lift_uzero #ιL).symm)
have rep_e := bK.total_repr (bL ∘ e)
rw [Finsupp.total_apply, Finsupp.sum] at rep_e
set c := bK.repr (bL ∘ e)
set s := c.support
let f i (j : s) : L := ⟨bK j i, Subfield.subset_closure ⟨(j, i), rfl⟩⟩
have : ¬LinearIndependent Lᵐᵒᵖ f := fun h ↦ by
have := h.cardinal_lift_le_rank
rw [lift_uzero, (LinearEquiv.piCongrRight fun _ ↦ MulOpposite.opLinearEquiv Lᵐᵒᵖ).rank_eq,
rank_fun'] at this
exact (nat_lt_aleph0 _).not_le this
obtain ⟨t, g, eq0, i, hi, hgi⟩ := not_linearIndependent_iff.mp this
refine hgi (linearIndependent_iff'.mp (bL.linearIndependent.comp e e.injective) t g ?_ i hi)
clear_value c s
simp_rw [← rep_e, Finset.sum_apply, Pi.smul_apply, Finset.smul_sum]
rw [Finset.sum_comm]
refine Finset.sum_eq_zero fun i hi ↦ ?_
replace eq0 := congr_arg L.subtype (congr_fun eq0 ⟨i, hi⟩)
rw [Finset.sum_apply, map_sum] at eq0
have : SMulCommClass Lᵐᵒᵖ K K := ⟨fun _ _ _ ↦ mul_assoc _ _ _⟩
simp_rw [smul_comm _ (c i), ← Finset.smul_sum]
erw [eq0, smul_zero]
variable {K}
open Function in
| Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 288 | 300 | theorem rank_fun_infinite {ι : Type v} [hι : Infinite ι] : Module.rank K (ι → K) = #(ι → K) := by |
obtain ⟨⟨ιK, bK⟩⟩ := Module.Free.exists_basis (R := K) (M := ι → K)
obtain ⟨e⟩ := lift_mk_le'.mp ((aleph0_le_mk_iff.mpr hι).trans_eq (lift_uzero #ι).symm)
have := LinearMap.lift_rank_le_of_injective _ <|
LinearMap.funLeft_injective_of_surjective K K _ (invFun_surjective e.injective)
rw [lift_umax.{u,v}, lift_id'.{u,v}] at this
have key := (lift_le.{v}.mpr <| max_aleph0_card_le_rank_fun_nat K).trans this
rw [lift_max, lift_aleph0, max_le_iff] at key
haveI : Infinite ιK := by
rw [← aleph0_le_mk_iff, bK.mk_eq_rank'']; exact key.1
rw [bK.repr.toEquiv.cardinal_eq, mk_finsupp_lift_of_infinite,
lift_umax.{u,v}, lift_id'.{u,v}, bK.mk_eq_rank'', eq_comm, max_eq_left]
exact key.2
| 12 | 162,754.791419 | 2 | 1.625 | 8 | 1,748 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u₀ u v v' v'' u₁' w w'
variable {K R : Type u} {V V₁ V₂ V₃ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
open Cardinal Basis Submodule Function Set
section Module
section Cardinal
variable (K)
variable [DivisionRing K]
theorem max_aleph0_card_le_rank_fun_nat : max ℵ₀ #K ≤ Module.rank K (ℕ → K) := by
have aleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K) := (rank_finsupp_self K ℕ).symm.trans_le
(Finsupp.lcoeFun.rank_le_of_injective <| by exact DFunLike.coe_injective)
refine max_le aleph0_le ?_
obtain card_K | card_K := le_or_lt #K ℵ₀
· exact card_K.trans aleph0_le
by_contra!
obtain ⟨⟨ιK, bK⟩⟩ := Module.Free.exists_basis (R := K) (M := ℕ → K)
let L := Subfield.closure (Set.range (fun i : ιK × ℕ ↦ bK i.1 i.2))
have hLK : #L < #K := by
refine (Subfield.cardinal_mk_closure_le_max _).trans_lt
(max_lt_iff.mpr ⟨mk_range_le.trans_lt ?_, card_K⟩)
rwa [mk_prod, ← aleph0, lift_uzero, bK.mk_eq_rank'', mul_aleph0_eq aleph0_le]
letI := Module.compHom K (RingHom.op L.subtype)
obtain ⟨⟨ιL, bL⟩⟩ := Module.Free.exists_basis (R := Lᵐᵒᵖ) (M := K)
have card_ιL : ℵ₀ ≤ #ιL := by
contrapose! hLK
haveI := @Fintype.ofFinite _ (lt_aleph0_iff_finite.mp hLK)
rw [bL.repr.toEquiv.cardinal_eq, mk_finsupp_of_fintype,
← MulOpposite.opEquiv.cardinal_eq] at card_K ⊢
apply power_nat_le
contrapose! card_K
exact (power_lt_aleph0 card_K <| nat_lt_aleph0 _).le
obtain ⟨e⟩ := lift_mk_le'.mp (card_ιL.trans_eq (lift_uzero #ιL).symm)
have rep_e := bK.total_repr (bL ∘ e)
rw [Finsupp.total_apply, Finsupp.sum] at rep_e
set c := bK.repr (bL ∘ e)
set s := c.support
let f i (j : s) : L := ⟨bK j i, Subfield.subset_closure ⟨(j, i), rfl⟩⟩
have : ¬LinearIndependent Lᵐᵒᵖ f := fun h ↦ by
have := h.cardinal_lift_le_rank
rw [lift_uzero, (LinearEquiv.piCongrRight fun _ ↦ MulOpposite.opLinearEquiv Lᵐᵒᵖ).rank_eq,
rank_fun'] at this
exact (nat_lt_aleph0 _).not_le this
obtain ⟨t, g, eq0, i, hi, hgi⟩ := not_linearIndependent_iff.mp this
refine hgi (linearIndependent_iff'.mp (bL.linearIndependent.comp e e.injective) t g ?_ i hi)
clear_value c s
simp_rw [← rep_e, Finset.sum_apply, Pi.smul_apply, Finset.smul_sum]
rw [Finset.sum_comm]
refine Finset.sum_eq_zero fun i hi ↦ ?_
replace eq0 := congr_arg L.subtype (congr_fun eq0 ⟨i, hi⟩)
rw [Finset.sum_apply, map_sum] at eq0
have : SMulCommClass Lᵐᵒᵖ K K := ⟨fun _ _ _ ↦ mul_assoc _ _ _⟩
simp_rw [smul_comm _ (c i), ← Finset.smul_sum]
erw [eq0, smul_zero]
variable {K}
open Function in
theorem rank_fun_infinite {ι : Type v} [hι : Infinite ι] : Module.rank K (ι → K) = #(ι → K) := by
obtain ⟨⟨ιK, bK⟩⟩ := Module.Free.exists_basis (R := K) (M := ι → K)
obtain ⟨e⟩ := lift_mk_le'.mp ((aleph0_le_mk_iff.mpr hι).trans_eq (lift_uzero #ι).symm)
have := LinearMap.lift_rank_le_of_injective _ <|
LinearMap.funLeft_injective_of_surjective K K _ (invFun_surjective e.injective)
rw [lift_umax.{u,v}, lift_id'.{u,v}] at this
have key := (lift_le.{v}.mpr <| max_aleph0_card_le_rank_fun_nat K).trans this
rw [lift_max, lift_aleph0, max_le_iff] at key
haveI : Infinite ιK := by
rw [← aleph0_le_mk_iff, bK.mk_eq_rank'']; exact key.1
rw [bK.repr.toEquiv.cardinal_eq, mk_finsupp_lift_of_infinite,
lift_umax.{u,v}, lift_id'.{u,v}, bK.mk_eq_rank'', eq_comm, max_eq_left]
exact key.2
| Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 304 | 311 | theorem rank_dual_eq_card_dual_of_aleph0_le_rank' {V : Type*} [AddCommGroup V] [Module K V]
(h : ℵ₀ ≤ Module.rank K V) : Module.rank Kᵐᵒᵖ (V →ₗ[K] K) = #(V →ₗ[K] K) := by |
obtain ⟨⟨ι, b⟩⟩ := Module.Free.exists_basis (R := K) (M := V)
rw [← b.mk_eq_rank'', aleph0_le_mk_iff] at h
have e := (b.constr Kᵐᵒᵖ (M' := K)).symm.trans
(LinearEquiv.piCongrRight fun _ ↦ MulOpposite.opLinearEquiv Kᵐᵒᵖ)
rw [e.rank_eq, e.toEquiv.cardinal_eq]
apply rank_fun_infinite
| 6 | 403.428793 | 2 | 1.625 | 8 | 1,748 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Qify
#align_import group_theory.commuting_probability from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
open scoped Classical
open Fintype
variable (M : Type*) [Mul M]
def commProb : ℚ :=
Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2
#align comm_prob commProb
theorem commProb_def :
commProb M = Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2 :=
rfl
#align comm_prob_def commProb_def
| Mathlib/GroupTheory/CommutingProbability.lean | 47 | 52 | theorem commProb_prod (M' : Type*) [Mul M'] : commProb (M × M') = commProb M * commProb M' := by |
simp_rw [commProb_def, div_mul_div_comm, Nat.card_prod, Nat.cast_mul, mul_pow, ← Nat.cast_mul,
← Nat.card_prod, Commute, SemiconjBy, Prod.ext_iff]
congr 2
exact Nat.card_congr ⟨fun x => ⟨⟨⟨x.1.1.1, x.1.2.1⟩, x.2.1⟩, ⟨⟨x.1.1.2, x.1.2.2⟩, x.2.2⟩⟩,
fun x => ⟨⟨⟨x.1.1.1, x.2.1.1⟩, ⟨x.1.1.2, x.2.1.2⟩⟩, ⟨x.1.2, x.2.2⟩⟩, fun x => rfl, fun x => rfl⟩
| 5 | 148.413159 | 2 | 1.625 | 8 | 1,749 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Qify
#align_import group_theory.commuting_probability from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
open scoped Classical
open Fintype
variable (M : Type*) [Mul M]
def commProb : ℚ :=
Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2
#align comm_prob commProb
theorem commProb_def :
commProb M = Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2 :=
rfl
#align comm_prob_def commProb_def
theorem commProb_prod (M' : Type*) [Mul M'] : commProb (M × M') = commProb M * commProb M' := by
simp_rw [commProb_def, div_mul_div_comm, Nat.card_prod, Nat.cast_mul, mul_pow, ← Nat.cast_mul,
← Nat.card_prod, Commute, SemiconjBy, Prod.ext_iff]
congr 2
exact Nat.card_congr ⟨fun x => ⟨⟨⟨x.1.1.1, x.1.2.1⟩, x.2.1⟩, ⟨⟨x.1.1.2, x.1.2.2⟩, x.2.2⟩⟩,
fun x => ⟨⟨⟨x.1.1.1, x.2.1.1⟩, ⟨x.1.1.2, x.2.1.2⟩⟩, ⟨x.1.2, x.2.2⟩⟩, fun x => rfl, fun x => rfl⟩
| Mathlib/GroupTheory/CommutingProbability.lean | 54 | 60 | theorem commProb_pi {α : Type*} (i : α → Type*) [Fintype α] [∀ a, Mul (i a)] :
commProb (∀ a, i a) = ∏ a, commProb (i a) := by |
simp_rw [commProb_def, Finset.prod_div_distrib, Finset.prod_pow, ← Nat.cast_prod,
← Nat.card_pi, Commute, SemiconjBy, Function.funext_iff]
congr 2
exact Nat.card_congr ⟨fun x a => ⟨⟨x.1.1 a, x.1.2 a⟩, x.2 a⟩, fun x => ⟨⟨fun a => (x a).1.1,
fun a => (x a).1.2⟩, fun a => (x a).2⟩, fun x => rfl, fun x => rfl⟩
| 5 | 148.413159 | 2 | 1.625 | 8 | 1,749 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Qify
#align_import group_theory.commuting_probability from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
open scoped Classical
open Fintype
variable (M : Type*) [Mul M]
def commProb : ℚ :=
Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2
#align comm_prob commProb
theorem commProb_def :
commProb M = Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2 :=
rfl
#align comm_prob_def commProb_def
theorem commProb_prod (M' : Type*) [Mul M'] : commProb (M × M') = commProb M * commProb M' := by
simp_rw [commProb_def, div_mul_div_comm, Nat.card_prod, Nat.cast_mul, mul_pow, ← Nat.cast_mul,
← Nat.card_prod, Commute, SemiconjBy, Prod.ext_iff]
congr 2
exact Nat.card_congr ⟨fun x => ⟨⟨⟨x.1.1.1, x.1.2.1⟩, x.2.1⟩, ⟨⟨x.1.1.2, x.1.2.2⟩, x.2.2⟩⟩,
fun x => ⟨⟨⟨x.1.1.1, x.2.1.1⟩, ⟨x.1.1.2, x.2.1.2⟩⟩, ⟨x.1.2, x.2.2⟩⟩, fun x => rfl, fun x => rfl⟩
theorem commProb_pi {α : Type*} (i : α → Type*) [Fintype α] [∀ a, Mul (i a)] :
commProb (∀ a, i a) = ∏ a, commProb (i a) := by
simp_rw [commProb_def, Finset.prod_div_distrib, Finset.prod_pow, ← Nat.cast_prod,
← Nat.card_pi, Commute, SemiconjBy, Function.funext_iff]
congr 2
exact Nat.card_congr ⟨fun x a => ⟨⟨x.1.1 a, x.1.2 a⟩, x.2 a⟩, fun x => ⟨⟨fun a => (x a).1.1,
fun a => (x a).1.2⟩, fun a => (x a).2⟩, fun x => rfl, fun x => rfl⟩
| Mathlib/GroupTheory/CommutingProbability.lean | 62 | 64 | theorem commProb_function {α β : Type*} [Fintype α] [Mul β] :
commProb (α → β) = (commProb β) ^ Fintype.card α := by |
rw [commProb_pi, Finset.prod_const, Finset.card_univ]
| 1 | 2.718282 | 0 | 1.625 | 8 | 1,749 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Qify
#align_import group_theory.commuting_probability from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
open scoped Classical
open Fintype
variable (M : Type*) [Mul M]
def commProb : ℚ :=
Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2
#align comm_prob commProb
theorem commProb_def :
commProb M = Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2 :=
rfl
#align comm_prob_def commProb_def
theorem commProb_prod (M' : Type*) [Mul M'] : commProb (M × M') = commProb M * commProb M' := by
simp_rw [commProb_def, div_mul_div_comm, Nat.card_prod, Nat.cast_mul, mul_pow, ← Nat.cast_mul,
← Nat.card_prod, Commute, SemiconjBy, Prod.ext_iff]
congr 2
exact Nat.card_congr ⟨fun x => ⟨⟨⟨x.1.1.1, x.1.2.1⟩, x.2.1⟩, ⟨⟨x.1.1.2, x.1.2.2⟩, x.2.2⟩⟩,
fun x => ⟨⟨⟨x.1.1.1, x.2.1.1⟩, ⟨x.1.1.2, x.2.1.2⟩⟩, ⟨x.1.2, x.2.2⟩⟩, fun x => rfl, fun x => rfl⟩
theorem commProb_pi {α : Type*} (i : α → Type*) [Fintype α] [∀ a, Mul (i a)] :
commProb (∀ a, i a) = ∏ a, commProb (i a) := by
simp_rw [commProb_def, Finset.prod_div_distrib, Finset.prod_pow, ← Nat.cast_prod,
← Nat.card_pi, Commute, SemiconjBy, Function.funext_iff]
congr 2
exact Nat.card_congr ⟨fun x a => ⟨⟨x.1.1 a, x.1.2 a⟩, x.2 a⟩, fun x => ⟨⟨fun a => (x a).1.1,
fun a => (x a).1.2⟩, fun a => (x a).2⟩, fun x => rfl, fun x => rfl⟩
theorem commProb_function {α β : Type*} [Fintype α] [Mul β] :
commProb (α → β) = (commProb β) ^ Fintype.card α := by
rw [commProb_pi, Finset.prod_const, Finset.card_univ]
@[simp]
theorem commProb_eq_zero_of_infinite [Infinite M] : commProb M = 0 :=
div_eq_zero_iff.2 (Or.inl (Nat.cast_eq_zero.2 Nat.card_eq_zero_of_infinite))
variable [Finite M]
theorem commProb_pos [h : Nonempty M] : 0 < commProb M :=
h.elim fun x ↦
div_pos (Nat.cast_pos.mpr (Finite.card_pos_iff.mpr ⟨⟨(x, x), rfl⟩⟩))
(pow_pos (Nat.cast_pos.mpr Finite.card_pos) 2)
#align comm_prob_pos commProb_pos
| Mathlib/GroupTheory/CommutingProbability.lean | 78 | 81 | theorem commProb_le_one : commProb M ≤ 1 := by |
refine div_le_one_of_le ?_ (sq_nonneg (Nat.card M : ℚ))
rw [← Nat.cast_pow, Nat.cast_le, sq, ← Nat.card_prod]
apply Finite.card_subtype_le
| 3 | 20.085537 | 1 | 1.625 | 8 | 1,749 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Qify
#align_import group_theory.commuting_probability from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
open scoped Classical
open Fintype
variable (M : Type*) [Mul M]
def commProb : ℚ :=
Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2
#align comm_prob commProb
theorem commProb_def :
commProb M = Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2 :=
rfl
#align comm_prob_def commProb_def
theorem commProb_prod (M' : Type*) [Mul M'] : commProb (M × M') = commProb M * commProb M' := by
simp_rw [commProb_def, div_mul_div_comm, Nat.card_prod, Nat.cast_mul, mul_pow, ← Nat.cast_mul,
← Nat.card_prod, Commute, SemiconjBy, Prod.ext_iff]
congr 2
exact Nat.card_congr ⟨fun x => ⟨⟨⟨x.1.1.1, x.1.2.1⟩, x.2.1⟩, ⟨⟨x.1.1.2, x.1.2.2⟩, x.2.2⟩⟩,
fun x => ⟨⟨⟨x.1.1.1, x.2.1.1⟩, ⟨x.1.1.2, x.2.1.2⟩⟩, ⟨x.1.2, x.2.2⟩⟩, fun x => rfl, fun x => rfl⟩
theorem commProb_pi {α : Type*} (i : α → Type*) [Fintype α] [∀ a, Mul (i a)] :
commProb (∀ a, i a) = ∏ a, commProb (i a) := by
simp_rw [commProb_def, Finset.prod_div_distrib, Finset.prod_pow, ← Nat.cast_prod,
← Nat.card_pi, Commute, SemiconjBy, Function.funext_iff]
congr 2
exact Nat.card_congr ⟨fun x a => ⟨⟨x.1.1 a, x.1.2 a⟩, x.2 a⟩, fun x => ⟨⟨fun a => (x a).1.1,
fun a => (x a).1.2⟩, fun a => (x a).2⟩, fun x => rfl, fun x => rfl⟩
theorem commProb_function {α β : Type*} [Fintype α] [Mul β] :
commProb (α → β) = (commProb β) ^ Fintype.card α := by
rw [commProb_pi, Finset.prod_const, Finset.card_univ]
@[simp]
theorem commProb_eq_zero_of_infinite [Infinite M] : commProb M = 0 :=
div_eq_zero_iff.2 (Or.inl (Nat.cast_eq_zero.2 Nat.card_eq_zero_of_infinite))
variable [Finite M]
theorem commProb_pos [h : Nonempty M] : 0 < commProb M :=
h.elim fun x ↦
div_pos (Nat.cast_pos.mpr (Finite.card_pos_iff.mpr ⟨⟨(x, x), rfl⟩⟩))
(pow_pos (Nat.cast_pos.mpr Finite.card_pos) 2)
#align comm_prob_pos commProb_pos
theorem commProb_le_one : commProb M ≤ 1 := by
refine div_le_one_of_le ?_ (sq_nonneg (Nat.card M : ℚ))
rw [← Nat.cast_pow, Nat.cast_le, sq, ← Nat.card_prod]
apply Finite.card_subtype_le
#align comm_prob_le_one commProb_le_one
variable {M}
| Mathlib/GroupTheory/CommutingProbability.lean | 86 | 93 | theorem commProb_eq_one_iff [h : Nonempty M] :
commProb M = 1 ↔ Commutative ((· * ·) : M → M → M) := by |
haveI := Fintype.ofFinite M
rw [commProb, ← Set.coe_setOf, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card]
rw [div_eq_one_iff_eq, ← Nat.cast_pow, Nat.cast_inj, sq, ← card_prod,
set_fintype_card_eq_univ_iff, Set.eq_univ_iff_forall]
· exact ⟨fun h x y ↦ h (x, y), fun h x ↦ h x.1 x.2⟩
· exact pow_ne_zero 2 (Nat.cast_ne_zero.mpr card_ne_zero)
| 6 | 403.428793 | 2 | 1.625 | 8 | 1,749 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Qify
#align_import group_theory.commuting_probability from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
open scoped Classical
open Fintype
variable (M : Type*) [Mul M]
def commProb : ℚ :=
Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2
#align comm_prob commProb
theorem commProb_def :
commProb M = Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2 :=
rfl
#align comm_prob_def commProb_def
theorem commProb_prod (M' : Type*) [Mul M'] : commProb (M × M') = commProb M * commProb M' := by
simp_rw [commProb_def, div_mul_div_comm, Nat.card_prod, Nat.cast_mul, mul_pow, ← Nat.cast_mul,
← Nat.card_prod, Commute, SemiconjBy, Prod.ext_iff]
congr 2
exact Nat.card_congr ⟨fun x => ⟨⟨⟨x.1.1.1, x.1.2.1⟩, x.2.1⟩, ⟨⟨x.1.1.2, x.1.2.2⟩, x.2.2⟩⟩,
fun x => ⟨⟨⟨x.1.1.1, x.2.1.1⟩, ⟨x.1.1.2, x.2.1.2⟩⟩, ⟨x.1.2, x.2.2⟩⟩, fun x => rfl, fun x => rfl⟩
theorem commProb_pi {α : Type*} (i : α → Type*) [Fintype α] [∀ a, Mul (i a)] :
commProb (∀ a, i a) = ∏ a, commProb (i a) := by
simp_rw [commProb_def, Finset.prod_div_distrib, Finset.prod_pow, ← Nat.cast_prod,
← Nat.card_pi, Commute, SemiconjBy, Function.funext_iff]
congr 2
exact Nat.card_congr ⟨fun x a => ⟨⟨x.1.1 a, x.1.2 a⟩, x.2 a⟩, fun x => ⟨⟨fun a => (x a).1.1,
fun a => (x a).1.2⟩, fun a => (x a).2⟩, fun x => rfl, fun x => rfl⟩
theorem commProb_function {α β : Type*} [Fintype α] [Mul β] :
commProb (α → β) = (commProb β) ^ Fintype.card α := by
rw [commProb_pi, Finset.prod_const, Finset.card_univ]
@[simp]
theorem commProb_eq_zero_of_infinite [Infinite M] : commProb M = 0 :=
div_eq_zero_iff.2 (Or.inl (Nat.cast_eq_zero.2 Nat.card_eq_zero_of_infinite))
variable [Finite M]
theorem commProb_pos [h : Nonempty M] : 0 < commProb M :=
h.elim fun x ↦
div_pos (Nat.cast_pos.mpr (Finite.card_pos_iff.mpr ⟨⟨(x, x), rfl⟩⟩))
(pow_pos (Nat.cast_pos.mpr Finite.card_pos) 2)
#align comm_prob_pos commProb_pos
theorem commProb_le_one : commProb M ≤ 1 := by
refine div_le_one_of_le ?_ (sq_nonneg (Nat.card M : ℚ))
rw [← Nat.cast_pow, Nat.cast_le, sq, ← Nat.card_prod]
apply Finite.card_subtype_le
#align comm_prob_le_one commProb_le_one
variable {M}
theorem commProb_eq_one_iff [h : Nonempty M] :
commProb M = 1 ↔ Commutative ((· * ·) : M → M → M) := by
haveI := Fintype.ofFinite M
rw [commProb, ← Set.coe_setOf, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card]
rw [div_eq_one_iff_eq, ← Nat.cast_pow, Nat.cast_inj, sq, ← card_prod,
set_fintype_card_eq_univ_iff, Set.eq_univ_iff_forall]
· exact ⟨fun h x y ↦ h (x, y), fun h x ↦ h x.1 x.2⟩
· exact pow_ne_zero 2 (Nat.cast_ne_zero.mpr card_ne_zero)
#align comm_prob_eq_one_iff commProb_eq_one_iff
variable (G : Type*) [Group G]
| Mathlib/GroupTheory/CommutingProbability.lean | 98 | 102 | theorem commProb_def' : commProb G = Nat.card (ConjClasses G) / Nat.card G := by |
rw [commProb, card_comm_eq_card_conjClasses_mul_card, Nat.cast_mul, sq]
by_cases h : (Nat.card G : ℚ) = 0
· rw [h, zero_mul, div_zero, div_zero]
· exact mul_div_mul_right _ _ h
| 4 | 54.59815 | 2 | 1.625 | 8 | 1,749 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Qify
#align_import group_theory.commuting_probability from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
open scoped Classical
open Fintype
variable (M : Type*) [Mul M]
def commProb : ℚ :=
Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2
#align comm_prob commProb
theorem commProb_def :
commProb M = Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2 :=
rfl
#align comm_prob_def commProb_def
theorem commProb_prod (M' : Type*) [Mul M'] : commProb (M × M') = commProb M * commProb M' := by
simp_rw [commProb_def, div_mul_div_comm, Nat.card_prod, Nat.cast_mul, mul_pow, ← Nat.cast_mul,
← Nat.card_prod, Commute, SemiconjBy, Prod.ext_iff]
congr 2
exact Nat.card_congr ⟨fun x => ⟨⟨⟨x.1.1.1, x.1.2.1⟩, x.2.1⟩, ⟨⟨x.1.1.2, x.1.2.2⟩, x.2.2⟩⟩,
fun x => ⟨⟨⟨x.1.1.1, x.2.1.1⟩, ⟨x.1.1.2, x.2.1.2⟩⟩, ⟨x.1.2, x.2.2⟩⟩, fun x => rfl, fun x => rfl⟩
theorem commProb_pi {α : Type*} (i : α → Type*) [Fintype α] [∀ a, Mul (i a)] :
commProb (∀ a, i a) = ∏ a, commProb (i a) := by
simp_rw [commProb_def, Finset.prod_div_distrib, Finset.prod_pow, ← Nat.cast_prod,
← Nat.card_pi, Commute, SemiconjBy, Function.funext_iff]
congr 2
exact Nat.card_congr ⟨fun x a => ⟨⟨x.1.1 a, x.1.2 a⟩, x.2 a⟩, fun x => ⟨⟨fun a => (x a).1.1,
fun a => (x a).1.2⟩, fun a => (x a).2⟩, fun x => rfl, fun x => rfl⟩
theorem commProb_function {α β : Type*} [Fintype α] [Mul β] :
commProb (α → β) = (commProb β) ^ Fintype.card α := by
rw [commProb_pi, Finset.prod_const, Finset.card_univ]
@[simp]
theorem commProb_eq_zero_of_infinite [Infinite M] : commProb M = 0 :=
div_eq_zero_iff.2 (Or.inl (Nat.cast_eq_zero.2 Nat.card_eq_zero_of_infinite))
variable [Finite M]
theorem commProb_pos [h : Nonempty M] : 0 < commProb M :=
h.elim fun x ↦
div_pos (Nat.cast_pos.mpr (Finite.card_pos_iff.mpr ⟨⟨(x, x), rfl⟩⟩))
(pow_pos (Nat.cast_pos.mpr Finite.card_pos) 2)
#align comm_prob_pos commProb_pos
theorem commProb_le_one : commProb M ≤ 1 := by
refine div_le_one_of_le ?_ (sq_nonneg (Nat.card M : ℚ))
rw [← Nat.cast_pow, Nat.cast_le, sq, ← Nat.card_prod]
apply Finite.card_subtype_le
#align comm_prob_le_one commProb_le_one
variable {M}
theorem commProb_eq_one_iff [h : Nonempty M] :
commProb M = 1 ↔ Commutative ((· * ·) : M → M → M) := by
haveI := Fintype.ofFinite M
rw [commProb, ← Set.coe_setOf, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card]
rw [div_eq_one_iff_eq, ← Nat.cast_pow, Nat.cast_inj, sq, ← card_prod,
set_fintype_card_eq_univ_iff, Set.eq_univ_iff_forall]
· exact ⟨fun h x y ↦ h (x, y), fun h x ↦ h x.1 x.2⟩
· exact pow_ne_zero 2 (Nat.cast_ne_zero.mpr card_ne_zero)
#align comm_prob_eq_one_iff commProb_eq_one_iff
variable (G : Type*) [Group G]
theorem commProb_def' : commProb G = Nat.card (ConjClasses G) / Nat.card G := by
rw [commProb, card_comm_eq_card_conjClasses_mul_card, Nat.cast_mul, sq]
by_cases h : (Nat.card G : ℚ) = 0
· rw [h, zero_mul, div_zero, div_zero]
· exact mul_div_mul_right _ _ h
#align comm_prob_def' commProb_def'
variable {G}
variable [Finite G] (H : Subgroup G)
| Mathlib/GroupTheory/CommutingProbability.lean | 108 | 116 | theorem Subgroup.commProb_subgroup_le : commProb H ≤ commProb G * (H.index : ℚ) ^ 2 := by |
/- After rewriting with `commProb_def`, we reduce to showing that `G` has at least as many
commuting pairs as `H`. -/
rw [commProb_def, commProb_def, div_le_iff, mul_assoc, ← mul_pow, ← Nat.cast_mul,
mul_comm H.index, H.card_mul_index, div_mul_cancel₀, Nat.cast_le]
· refine Finite.card_le_of_injective (fun p ↦ ⟨⟨p.1.1, p.1.2⟩, Subtype.ext_iff.mp p.2⟩) ?_
exact fun p q h ↦ by simpa only [Subtype.ext_iff, Prod.ext_iff] using h
· exact pow_ne_zero 2 (Nat.cast_ne_zero.mpr Finite.card_pos.ne')
· exact pow_pos (Nat.cast_pos.mpr Finite.card_pos) 2
| 8 | 2,980.957987 | 2 | 1.625 | 8 | 1,749 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Qify
#align_import group_theory.commuting_probability from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
open scoped Classical
open Fintype
variable (M : Type*) [Mul M]
def commProb : ℚ :=
Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2
#align comm_prob commProb
theorem commProb_def :
commProb M = Nat.card { p : M × M // Commute p.1 p.2 } / (Nat.card M : ℚ) ^ 2 :=
rfl
#align comm_prob_def commProb_def
theorem commProb_prod (M' : Type*) [Mul M'] : commProb (M × M') = commProb M * commProb M' := by
simp_rw [commProb_def, div_mul_div_comm, Nat.card_prod, Nat.cast_mul, mul_pow, ← Nat.cast_mul,
← Nat.card_prod, Commute, SemiconjBy, Prod.ext_iff]
congr 2
exact Nat.card_congr ⟨fun x => ⟨⟨⟨x.1.1.1, x.1.2.1⟩, x.2.1⟩, ⟨⟨x.1.1.2, x.1.2.2⟩, x.2.2⟩⟩,
fun x => ⟨⟨⟨x.1.1.1, x.2.1.1⟩, ⟨x.1.1.2, x.2.1.2⟩⟩, ⟨x.1.2, x.2.2⟩⟩, fun x => rfl, fun x => rfl⟩
theorem commProb_pi {α : Type*} (i : α → Type*) [Fintype α] [∀ a, Mul (i a)] :
commProb (∀ a, i a) = ∏ a, commProb (i a) := by
simp_rw [commProb_def, Finset.prod_div_distrib, Finset.prod_pow, ← Nat.cast_prod,
← Nat.card_pi, Commute, SemiconjBy, Function.funext_iff]
congr 2
exact Nat.card_congr ⟨fun x a => ⟨⟨x.1.1 a, x.1.2 a⟩, x.2 a⟩, fun x => ⟨⟨fun a => (x a).1.1,
fun a => (x a).1.2⟩, fun a => (x a).2⟩, fun x => rfl, fun x => rfl⟩
theorem commProb_function {α β : Type*} [Fintype α] [Mul β] :
commProb (α → β) = (commProb β) ^ Fintype.card α := by
rw [commProb_pi, Finset.prod_const, Finset.card_univ]
@[simp]
theorem commProb_eq_zero_of_infinite [Infinite M] : commProb M = 0 :=
div_eq_zero_iff.2 (Or.inl (Nat.cast_eq_zero.2 Nat.card_eq_zero_of_infinite))
variable [Finite M]
theorem commProb_pos [h : Nonempty M] : 0 < commProb M :=
h.elim fun x ↦
div_pos (Nat.cast_pos.mpr (Finite.card_pos_iff.mpr ⟨⟨(x, x), rfl⟩⟩))
(pow_pos (Nat.cast_pos.mpr Finite.card_pos) 2)
#align comm_prob_pos commProb_pos
theorem commProb_le_one : commProb M ≤ 1 := by
refine div_le_one_of_le ?_ (sq_nonneg (Nat.card M : ℚ))
rw [← Nat.cast_pow, Nat.cast_le, sq, ← Nat.card_prod]
apply Finite.card_subtype_le
#align comm_prob_le_one commProb_le_one
variable {M}
theorem commProb_eq_one_iff [h : Nonempty M] :
commProb M = 1 ↔ Commutative ((· * ·) : M → M → M) := by
haveI := Fintype.ofFinite M
rw [commProb, ← Set.coe_setOf, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card]
rw [div_eq_one_iff_eq, ← Nat.cast_pow, Nat.cast_inj, sq, ← card_prod,
set_fintype_card_eq_univ_iff, Set.eq_univ_iff_forall]
· exact ⟨fun h x y ↦ h (x, y), fun h x ↦ h x.1 x.2⟩
· exact pow_ne_zero 2 (Nat.cast_ne_zero.mpr card_ne_zero)
#align comm_prob_eq_one_iff commProb_eq_one_iff
variable (G : Type*) [Group G]
theorem commProb_def' : commProb G = Nat.card (ConjClasses G) / Nat.card G := by
rw [commProb, card_comm_eq_card_conjClasses_mul_card, Nat.cast_mul, sq]
by_cases h : (Nat.card G : ℚ) = 0
· rw [h, zero_mul, div_zero, div_zero]
· exact mul_div_mul_right _ _ h
#align comm_prob_def' commProb_def'
variable {G}
variable [Finite G] (H : Subgroup G)
theorem Subgroup.commProb_subgroup_le : commProb H ≤ commProb G * (H.index : ℚ) ^ 2 := by
rw [commProb_def, commProb_def, div_le_iff, mul_assoc, ← mul_pow, ← Nat.cast_mul,
mul_comm H.index, H.card_mul_index, div_mul_cancel₀, Nat.cast_le]
· refine Finite.card_le_of_injective (fun p ↦ ⟨⟨p.1.1, p.1.2⟩, Subtype.ext_iff.mp p.2⟩) ?_
exact fun p q h ↦ by simpa only [Subtype.ext_iff, Prod.ext_iff] using h
· exact pow_ne_zero 2 (Nat.cast_ne_zero.mpr Finite.card_pos.ne')
· exact pow_pos (Nat.cast_pos.mpr Finite.card_pos) 2
#align subgroup.comm_prob_subgroup_le Subgroup.commProb_subgroup_le
| Mathlib/GroupTheory/CommutingProbability.lean | 119 | 128 | theorem Subgroup.commProb_quotient_le [H.Normal] : commProb (G ⧸ H) ≤ commProb G * Nat.card H := by |
/- After rewriting with `commProb_def'`, we reduce to showing that `G` has at least as many
conjugacy classes as `G ⧸ H`. -/
rw [commProb_def', commProb_def', div_le_iff, mul_assoc, ← Nat.cast_mul, ← Subgroup.index,
H.card_mul_index, div_mul_cancel₀, Nat.cast_le]
· apply Finite.card_le_of_surjective
show Function.Surjective (ConjClasses.map (QuotientGroup.mk' H))
exact ConjClasses.map_surjective Quotient.surjective_Quotient_mk''
· exact Nat.cast_ne_zero.mpr Finite.card_pos.ne'
· exact Nat.cast_pos.mpr Finite.card_pos
| 9 | 8,103.083928 | 2 | 1.625 | 8 | 1,749 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe353f425855fcf0cedf9ea0fe8a4"
noncomputable section
open scoped NNReal ENNReal Function
variable {α : Type*} {E : α → Type*} {p q : ℝ≥0∞} [∀ i, NormedAddCommGroup (E i)]
def Memℓp (f : ∀ i, E i) (p : ℝ≥0∞) : Prop :=
if p = 0 then Set.Finite { i | f i ≠ 0 }
else if p = ∞ then BddAbove (Set.range fun i => ‖f i‖)
else Summable fun i => ‖f i‖ ^ p.toReal
#align mem_ℓp Memℓp
| Mathlib/Analysis/NormedSpace/lpSpace.lean | 81 | 83 | theorem memℓp_zero_iff {f : ∀ i, E i} : Memℓp f 0 ↔ Set.Finite { i | f i ≠ 0 } := by |
dsimp [Memℓp]
rw [if_pos rfl]
| 2 | 7.389056 | 1 | 1.625 | 8 | 1,750 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe353f425855fcf0cedf9ea0fe8a4"
noncomputable section
open scoped NNReal ENNReal Function
variable {α : Type*} {E : α → Type*} {p q : ℝ≥0∞} [∀ i, NormedAddCommGroup (E i)]
def Memℓp (f : ∀ i, E i) (p : ℝ≥0∞) : Prop :=
if p = 0 then Set.Finite { i | f i ≠ 0 }
else if p = ∞ then BddAbove (Set.range fun i => ‖f i‖)
else Summable fun i => ‖f i‖ ^ p.toReal
#align mem_ℓp Memℓp
theorem memℓp_zero_iff {f : ∀ i, E i} : Memℓp f 0 ↔ Set.Finite { i | f i ≠ 0 } := by
dsimp [Memℓp]
rw [if_pos rfl]
#align mem_ℓp_zero_iff memℓp_zero_iff
theorem memℓp_zero {f : ∀ i, E i} (hf : Set.Finite { i | f i ≠ 0 }) : Memℓp f 0 :=
memℓp_zero_iff.2 hf
#align mem_ℓp_zero memℓp_zero
| Mathlib/Analysis/NormedSpace/lpSpace.lean | 90 | 92 | theorem memℓp_infty_iff {f : ∀ i, E i} : Memℓp f ∞ ↔ BddAbove (Set.range fun i => ‖f i‖) := by |
dsimp [Memℓp]
rw [if_neg ENNReal.top_ne_zero, if_pos rfl]
| 2 | 7.389056 | 1 | 1.625 | 8 | 1,750 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe353f425855fcf0cedf9ea0fe8a4"
noncomputable section
open scoped NNReal ENNReal Function
variable {α : Type*} {E : α → Type*} {p q : ℝ≥0∞} [∀ i, NormedAddCommGroup (E i)]
def Memℓp (f : ∀ i, E i) (p : ℝ≥0∞) : Prop :=
if p = 0 then Set.Finite { i | f i ≠ 0 }
else if p = ∞ then BddAbove (Set.range fun i => ‖f i‖)
else Summable fun i => ‖f i‖ ^ p.toReal
#align mem_ℓp Memℓp
theorem memℓp_zero_iff {f : ∀ i, E i} : Memℓp f 0 ↔ Set.Finite { i | f i ≠ 0 } := by
dsimp [Memℓp]
rw [if_pos rfl]
#align mem_ℓp_zero_iff memℓp_zero_iff
theorem memℓp_zero {f : ∀ i, E i} (hf : Set.Finite { i | f i ≠ 0 }) : Memℓp f 0 :=
memℓp_zero_iff.2 hf
#align mem_ℓp_zero memℓp_zero
theorem memℓp_infty_iff {f : ∀ i, E i} : Memℓp f ∞ ↔ BddAbove (Set.range fun i => ‖f i‖) := by
dsimp [Memℓp]
rw [if_neg ENNReal.top_ne_zero, if_pos rfl]
#align mem_ℓp_infty_iff memℓp_infty_iff
theorem memℓp_infty {f : ∀ i, E i} (hf : BddAbove (Set.range fun i => ‖f i‖)) : Memℓp f ∞ :=
memℓp_infty_iff.2 hf
#align mem_ℓp_infty memℓp_infty
| Mathlib/Analysis/NormedSpace/lpSpace.lean | 99 | 103 | theorem memℓp_gen_iff (hp : 0 < p.toReal) {f : ∀ i, E i} :
Memℓp f p ↔ Summable fun i => ‖f i‖ ^ p.toReal := by |
rw [ENNReal.toReal_pos_iff] at hp
dsimp [Memℓp]
rw [if_neg hp.1.ne', if_neg hp.2.ne]
| 3 | 20.085537 | 1 | 1.625 | 8 | 1,750 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe353f425855fcf0cedf9ea0fe8a4"
noncomputable section
open scoped NNReal ENNReal Function
variable {α : Type*} {E : α → Type*} {p q : ℝ≥0∞} [∀ i, NormedAddCommGroup (E i)]
def Memℓp (f : ∀ i, E i) (p : ℝ≥0∞) : Prop :=
if p = 0 then Set.Finite { i | f i ≠ 0 }
else if p = ∞ then BddAbove (Set.range fun i => ‖f i‖)
else Summable fun i => ‖f i‖ ^ p.toReal
#align mem_ℓp Memℓp
theorem memℓp_zero_iff {f : ∀ i, E i} : Memℓp f 0 ↔ Set.Finite { i | f i ≠ 0 } := by
dsimp [Memℓp]
rw [if_pos rfl]
#align mem_ℓp_zero_iff memℓp_zero_iff
theorem memℓp_zero {f : ∀ i, E i} (hf : Set.Finite { i | f i ≠ 0 }) : Memℓp f 0 :=
memℓp_zero_iff.2 hf
#align mem_ℓp_zero memℓp_zero
theorem memℓp_infty_iff {f : ∀ i, E i} : Memℓp f ∞ ↔ BddAbove (Set.range fun i => ‖f i‖) := by
dsimp [Memℓp]
rw [if_neg ENNReal.top_ne_zero, if_pos rfl]
#align mem_ℓp_infty_iff memℓp_infty_iff
theorem memℓp_infty {f : ∀ i, E i} (hf : BddAbove (Set.range fun i => ‖f i‖)) : Memℓp f ∞ :=
memℓp_infty_iff.2 hf
#align mem_ℓp_infty memℓp_infty
theorem memℓp_gen_iff (hp : 0 < p.toReal) {f : ∀ i, E i} :
Memℓp f p ↔ Summable fun i => ‖f i‖ ^ p.toReal := by
rw [ENNReal.toReal_pos_iff] at hp
dsimp [Memℓp]
rw [if_neg hp.1.ne', if_neg hp.2.ne]
#align mem_ℓp_gen_iff memℓp_gen_iff
| Mathlib/Analysis/NormedSpace/lpSpace.lean | 106 | 114 | theorem memℓp_gen {f : ∀ i, E i} (hf : Summable fun i => ‖f i‖ ^ p.toReal) : Memℓp f p := by |
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf
exact (Set.Finite.of_summable_const (by norm_num) H).subset (Set.subset_univ _)
· apply memℓp_infty
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf
simpa using ((Set.Finite.of_summable_const (by norm_num) H).image fun i => ‖f i‖).bddAbove
exact (memℓp_gen_iff hp).2 hf
| 8 | 2,980.957987 | 2 | 1.625 | 8 | 1,750 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe353f425855fcf0cedf9ea0fe8a4"
noncomputable section
open scoped NNReal ENNReal Function
variable {α : Type*} {E : α → Type*} {p q : ℝ≥0∞} [∀ i, NormedAddCommGroup (E i)]
def Memℓp (f : ∀ i, E i) (p : ℝ≥0∞) : Prop :=
if p = 0 then Set.Finite { i | f i ≠ 0 }
else if p = ∞ then BddAbove (Set.range fun i => ‖f i‖)
else Summable fun i => ‖f i‖ ^ p.toReal
#align mem_ℓp Memℓp
theorem memℓp_zero_iff {f : ∀ i, E i} : Memℓp f 0 ↔ Set.Finite { i | f i ≠ 0 } := by
dsimp [Memℓp]
rw [if_pos rfl]
#align mem_ℓp_zero_iff memℓp_zero_iff
theorem memℓp_zero {f : ∀ i, E i} (hf : Set.Finite { i | f i ≠ 0 }) : Memℓp f 0 :=
memℓp_zero_iff.2 hf
#align mem_ℓp_zero memℓp_zero
theorem memℓp_infty_iff {f : ∀ i, E i} : Memℓp f ∞ ↔ BddAbove (Set.range fun i => ‖f i‖) := by
dsimp [Memℓp]
rw [if_neg ENNReal.top_ne_zero, if_pos rfl]
#align mem_ℓp_infty_iff memℓp_infty_iff
theorem memℓp_infty {f : ∀ i, E i} (hf : BddAbove (Set.range fun i => ‖f i‖)) : Memℓp f ∞ :=
memℓp_infty_iff.2 hf
#align mem_ℓp_infty memℓp_infty
theorem memℓp_gen_iff (hp : 0 < p.toReal) {f : ∀ i, E i} :
Memℓp f p ↔ Summable fun i => ‖f i‖ ^ p.toReal := by
rw [ENNReal.toReal_pos_iff] at hp
dsimp [Memℓp]
rw [if_neg hp.1.ne', if_neg hp.2.ne]
#align mem_ℓp_gen_iff memℓp_gen_iff
theorem memℓp_gen {f : ∀ i, E i} (hf : Summable fun i => ‖f i‖ ^ p.toReal) : Memℓp f p := by
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf
exact (Set.Finite.of_summable_const (by norm_num) H).subset (Set.subset_univ _)
· apply memℓp_infty
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf
simpa using ((Set.Finite.of_summable_const (by norm_num) H).image fun i => ‖f i‖).bddAbove
exact (memℓp_gen_iff hp).2 hf
#align mem_ℓp_gen memℓp_gen
| Mathlib/Analysis/NormedSpace/lpSpace.lean | 117 | 127 | theorem memℓp_gen' {C : ℝ} {f : ∀ i, E i} (hf : ∀ s : Finset α, ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C) :
Memℓp f p := by |
apply memℓp_gen
use ⨆ s : Finset α, ∑ i ∈ s, ‖f i‖ ^ p.toReal
apply hasSum_of_isLUB_of_nonneg
· intro b
exact Real.rpow_nonneg (norm_nonneg _) _
apply isLUB_ciSup
use C
rintro - ⟨s, rfl⟩
exact hf s
| 9 | 8,103.083928 | 2 | 1.625 | 8 | 1,750 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe353f425855fcf0cedf9ea0fe8a4"
noncomputable section
open scoped NNReal ENNReal Function
variable {α : Type*} {E : α → Type*} {p q : ℝ≥0∞} [∀ i, NormedAddCommGroup (E i)]
def Memℓp (f : ∀ i, E i) (p : ℝ≥0∞) : Prop :=
if p = 0 then Set.Finite { i | f i ≠ 0 }
else if p = ∞ then BddAbove (Set.range fun i => ‖f i‖)
else Summable fun i => ‖f i‖ ^ p.toReal
#align mem_ℓp Memℓp
theorem memℓp_zero_iff {f : ∀ i, E i} : Memℓp f 0 ↔ Set.Finite { i | f i ≠ 0 } := by
dsimp [Memℓp]
rw [if_pos rfl]
#align mem_ℓp_zero_iff memℓp_zero_iff
theorem memℓp_zero {f : ∀ i, E i} (hf : Set.Finite { i | f i ≠ 0 }) : Memℓp f 0 :=
memℓp_zero_iff.2 hf
#align mem_ℓp_zero memℓp_zero
theorem memℓp_infty_iff {f : ∀ i, E i} : Memℓp f ∞ ↔ BddAbove (Set.range fun i => ‖f i‖) := by
dsimp [Memℓp]
rw [if_neg ENNReal.top_ne_zero, if_pos rfl]
#align mem_ℓp_infty_iff memℓp_infty_iff
theorem memℓp_infty {f : ∀ i, E i} (hf : BddAbove (Set.range fun i => ‖f i‖)) : Memℓp f ∞ :=
memℓp_infty_iff.2 hf
#align mem_ℓp_infty memℓp_infty
theorem memℓp_gen_iff (hp : 0 < p.toReal) {f : ∀ i, E i} :
Memℓp f p ↔ Summable fun i => ‖f i‖ ^ p.toReal := by
rw [ENNReal.toReal_pos_iff] at hp
dsimp [Memℓp]
rw [if_neg hp.1.ne', if_neg hp.2.ne]
#align mem_ℓp_gen_iff memℓp_gen_iff
theorem memℓp_gen {f : ∀ i, E i} (hf : Summable fun i => ‖f i‖ ^ p.toReal) : Memℓp f p := by
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf
exact (Set.Finite.of_summable_const (by norm_num) H).subset (Set.subset_univ _)
· apply memℓp_infty
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf
simpa using ((Set.Finite.of_summable_const (by norm_num) H).image fun i => ‖f i‖).bddAbove
exact (memℓp_gen_iff hp).2 hf
#align mem_ℓp_gen memℓp_gen
theorem memℓp_gen' {C : ℝ} {f : ∀ i, E i} (hf : ∀ s : Finset α, ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C) :
Memℓp f p := by
apply memℓp_gen
use ⨆ s : Finset α, ∑ i ∈ s, ‖f i‖ ^ p.toReal
apply hasSum_of_isLUB_of_nonneg
· intro b
exact Real.rpow_nonneg (norm_nonneg _) _
apply isLUB_ciSup
use C
rintro - ⟨s, rfl⟩
exact hf s
#align mem_ℓp_gen' memℓp_gen'
| Mathlib/Analysis/NormedSpace/lpSpace.lean | 130 | 138 | theorem zero_memℓp : Memℓp (0 : ∀ i, E i) p := by |
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
simp
· apply memℓp_infty
simp only [norm_zero, Pi.zero_apply]
exact bddAbove_singleton.mono Set.range_const_subset
· apply memℓp_gen
simp [Real.zero_rpow hp.ne', summable_zero]
| 8 | 2,980.957987 | 2 | 1.625 | 8 | 1,750 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe353f425855fcf0cedf9ea0fe8a4"
noncomputable section
open scoped NNReal ENNReal Function
variable {α : Type*} {E : α → Type*} {p q : ℝ≥0∞} [∀ i, NormedAddCommGroup (E i)]
def Memℓp (f : ∀ i, E i) (p : ℝ≥0∞) : Prop :=
if p = 0 then Set.Finite { i | f i ≠ 0 }
else if p = ∞ then BddAbove (Set.range fun i => ‖f i‖)
else Summable fun i => ‖f i‖ ^ p.toReal
#align mem_ℓp Memℓp
theorem memℓp_zero_iff {f : ∀ i, E i} : Memℓp f 0 ↔ Set.Finite { i | f i ≠ 0 } := by
dsimp [Memℓp]
rw [if_pos rfl]
#align mem_ℓp_zero_iff memℓp_zero_iff
theorem memℓp_zero {f : ∀ i, E i} (hf : Set.Finite { i | f i ≠ 0 }) : Memℓp f 0 :=
memℓp_zero_iff.2 hf
#align mem_ℓp_zero memℓp_zero
theorem memℓp_infty_iff {f : ∀ i, E i} : Memℓp f ∞ ↔ BddAbove (Set.range fun i => ‖f i‖) := by
dsimp [Memℓp]
rw [if_neg ENNReal.top_ne_zero, if_pos rfl]
#align mem_ℓp_infty_iff memℓp_infty_iff
theorem memℓp_infty {f : ∀ i, E i} (hf : BddAbove (Set.range fun i => ‖f i‖)) : Memℓp f ∞ :=
memℓp_infty_iff.2 hf
#align mem_ℓp_infty memℓp_infty
theorem memℓp_gen_iff (hp : 0 < p.toReal) {f : ∀ i, E i} :
Memℓp f p ↔ Summable fun i => ‖f i‖ ^ p.toReal := by
rw [ENNReal.toReal_pos_iff] at hp
dsimp [Memℓp]
rw [if_neg hp.1.ne', if_neg hp.2.ne]
#align mem_ℓp_gen_iff memℓp_gen_iff
theorem memℓp_gen {f : ∀ i, E i} (hf : Summable fun i => ‖f i‖ ^ p.toReal) : Memℓp f p := by
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf
exact (Set.Finite.of_summable_const (by norm_num) H).subset (Set.subset_univ _)
· apply memℓp_infty
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf
simpa using ((Set.Finite.of_summable_const (by norm_num) H).image fun i => ‖f i‖).bddAbove
exact (memℓp_gen_iff hp).2 hf
#align mem_ℓp_gen memℓp_gen
theorem memℓp_gen' {C : ℝ} {f : ∀ i, E i} (hf : ∀ s : Finset α, ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C) :
Memℓp f p := by
apply memℓp_gen
use ⨆ s : Finset α, ∑ i ∈ s, ‖f i‖ ^ p.toReal
apply hasSum_of_isLUB_of_nonneg
· intro b
exact Real.rpow_nonneg (norm_nonneg _) _
apply isLUB_ciSup
use C
rintro - ⟨s, rfl⟩
exact hf s
#align mem_ℓp_gen' memℓp_gen'
theorem zero_memℓp : Memℓp (0 : ∀ i, E i) p := by
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
simp
· apply memℓp_infty
simp only [norm_zero, Pi.zero_apply]
exact bddAbove_singleton.mono Set.range_const_subset
· apply memℓp_gen
simp [Real.zero_rpow hp.ne', summable_zero]
#align zero_mem_ℓp zero_memℓp
theorem zero_mem_ℓp' : Memℓp (fun i : α => (0 : E i)) p :=
zero_memℓp
#align zero_mem_ℓp' zero_mem_ℓp'
namespace Memℓp
theorem finite_dsupport {f : ∀ i, E i} (hf : Memℓp f 0) : Set.Finite { i | f i ≠ 0 } :=
memℓp_zero_iff.1 hf
#align mem_ℓp.finite_dsupport Memℓp.finite_dsupport
theorem bddAbove {f : ∀ i, E i} (hf : Memℓp f ∞) : BddAbove (Set.range fun i => ‖f i‖) :=
memℓp_infty_iff.1 hf
#align mem_ℓp.bdd_above Memℓp.bddAbove
theorem summable (hp : 0 < p.toReal) {f : ∀ i, E i} (hf : Memℓp f p) :
Summable fun i => ‖f i‖ ^ p.toReal :=
(memℓp_gen_iff hp).1 hf
#align mem_ℓp.summable Memℓp.summable
| Mathlib/Analysis/NormedSpace/lpSpace.lean | 160 | 167 | theorem neg {f : ∀ i, E i} (hf : Memℓp f p) : Memℓp (-f) p := by |
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
simp [hf.finite_dsupport]
· apply memℓp_infty
simpa using hf.bddAbove
· apply memℓp_gen
simpa using hf.summable hp
| 7 | 1,096.633158 | 2 | 1.625 | 8 | 1,750 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe353f425855fcf0cedf9ea0fe8a4"
noncomputable section
open scoped NNReal ENNReal Function
variable {α : Type*} {E : α → Type*} {p q : ℝ≥0∞} [∀ i, NormedAddCommGroup (E i)]
def Memℓp (f : ∀ i, E i) (p : ℝ≥0∞) : Prop :=
if p = 0 then Set.Finite { i | f i ≠ 0 }
else if p = ∞ then BddAbove (Set.range fun i => ‖f i‖)
else Summable fun i => ‖f i‖ ^ p.toReal
#align mem_ℓp Memℓp
theorem memℓp_zero_iff {f : ∀ i, E i} : Memℓp f 0 ↔ Set.Finite { i | f i ≠ 0 } := by
dsimp [Memℓp]
rw [if_pos rfl]
#align mem_ℓp_zero_iff memℓp_zero_iff
theorem memℓp_zero {f : ∀ i, E i} (hf : Set.Finite { i | f i ≠ 0 }) : Memℓp f 0 :=
memℓp_zero_iff.2 hf
#align mem_ℓp_zero memℓp_zero
theorem memℓp_infty_iff {f : ∀ i, E i} : Memℓp f ∞ ↔ BddAbove (Set.range fun i => ‖f i‖) := by
dsimp [Memℓp]
rw [if_neg ENNReal.top_ne_zero, if_pos rfl]
#align mem_ℓp_infty_iff memℓp_infty_iff
theorem memℓp_infty {f : ∀ i, E i} (hf : BddAbove (Set.range fun i => ‖f i‖)) : Memℓp f ∞ :=
memℓp_infty_iff.2 hf
#align mem_ℓp_infty memℓp_infty
theorem memℓp_gen_iff (hp : 0 < p.toReal) {f : ∀ i, E i} :
Memℓp f p ↔ Summable fun i => ‖f i‖ ^ p.toReal := by
rw [ENNReal.toReal_pos_iff] at hp
dsimp [Memℓp]
rw [if_neg hp.1.ne', if_neg hp.2.ne]
#align mem_ℓp_gen_iff memℓp_gen_iff
theorem memℓp_gen {f : ∀ i, E i} (hf : Summable fun i => ‖f i‖ ^ p.toReal) : Memℓp f p := by
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf
exact (Set.Finite.of_summable_const (by norm_num) H).subset (Set.subset_univ _)
· apply memℓp_infty
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf
simpa using ((Set.Finite.of_summable_const (by norm_num) H).image fun i => ‖f i‖).bddAbove
exact (memℓp_gen_iff hp).2 hf
#align mem_ℓp_gen memℓp_gen
theorem memℓp_gen' {C : ℝ} {f : ∀ i, E i} (hf : ∀ s : Finset α, ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C) :
Memℓp f p := by
apply memℓp_gen
use ⨆ s : Finset α, ∑ i ∈ s, ‖f i‖ ^ p.toReal
apply hasSum_of_isLUB_of_nonneg
· intro b
exact Real.rpow_nonneg (norm_nonneg _) _
apply isLUB_ciSup
use C
rintro - ⟨s, rfl⟩
exact hf s
#align mem_ℓp_gen' memℓp_gen'
theorem zero_memℓp : Memℓp (0 : ∀ i, E i) p := by
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
simp
· apply memℓp_infty
simp only [norm_zero, Pi.zero_apply]
exact bddAbove_singleton.mono Set.range_const_subset
· apply memℓp_gen
simp [Real.zero_rpow hp.ne', summable_zero]
#align zero_mem_ℓp zero_memℓp
theorem zero_mem_ℓp' : Memℓp (fun i : α => (0 : E i)) p :=
zero_memℓp
#align zero_mem_ℓp' zero_mem_ℓp'
namespace Memℓp
theorem finite_dsupport {f : ∀ i, E i} (hf : Memℓp f 0) : Set.Finite { i | f i ≠ 0 } :=
memℓp_zero_iff.1 hf
#align mem_ℓp.finite_dsupport Memℓp.finite_dsupport
theorem bddAbove {f : ∀ i, E i} (hf : Memℓp f ∞) : BddAbove (Set.range fun i => ‖f i‖) :=
memℓp_infty_iff.1 hf
#align mem_ℓp.bdd_above Memℓp.bddAbove
theorem summable (hp : 0 < p.toReal) {f : ∀ i, E i} (hf : Memℓp f p) :
Summable fun i => ‖f i‖ ^ p.toReal :=
(memℓp_gen_iff hp).1 hf
#align mem_ℓp.summable Memℓp.summable
theorem neg {f : ∀ i, E i} (hf : Memℓp f p) : Memℓp (-f) p := by
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
simp [hf.finite_dsupport]
· apply memℓp_infty
simpa using hf.bddAbove
· apply memℓp_gen
simpa using hf.summable hp
#align mem_ℓp.neg Memℓp.neg
@[simp]
theorem neg_iff {f : ∀ i, E i} : Memℓp (-f) p ↔ Memℓp f p :=
⟨fun h => neg_neg f ▸ h.neg, Memℓp.neg⟩
#align mem_ℓp.neg_iff Memℓp.neg_iff
| Mathlib/Analysis/NormedSpace/lpSpace.lean | 175 | 211 | theorem of_exponent_ge {p q : ℝ≥0∞} {f : ∀ i, E i} (hfq : Memℓp f q) (hpq : q ≤ p) : Memℓp f p := by |
rcases ENNReal.trichotomy₂ hpq with
(⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ | ⟨rfl, hp⟩ | ⟨rfl, rfl⟩ | ⟨hq, rfl⟩ | ⟨hq, _, hpq'⟩)
· exact hfq
· apply memℓp_infty
obtain ⟨C, hC⟩ := (hfq.finite_dsupport.image fun i => ‖f i‖).bddAbove
use max 0 C
rintro x ⟨i, rfl⟩
by_cases hi : f i = 0
· simp [hi]
· exact (hC ⟨i, hi, rfl⟩).trans (le_max_right _ _)
· apply memℓp_gen
have : ∀ i ∉ hfq.finite_dsupport.toFinset, ‖f i‖ ^ p.toReal = 0 := by
intro i hi
have : f i = 0 := by simpa using hi
simp [this, Real.zero_rpow hp.ne']
exact summable_of_ne_finset_zero this
· exact hfq
· apply memℓp_infty
obtain ⟨A, hA⟩ := (hfq.summable hq).tendsto_cofinite_zero.bddAbove_range_of_cofinite
use A ^ q.toReal⁻¹
rintro x ⟨i, rfl⟩
have : 0 ≤ ‖f i‖ ^ q.toReal := by positivity
simpa [← Real.rpow_mul, mul_inv_cancel hq.ne'] using
Real.rpow_le_rpow this (hA ⟨i, rfl⟩) (inv_nonneg.mpr hq.le)
· apply memℓp_gen
have hf' := hfq.summable hq
refine .of_norm_bounded_eventually _ hf' (@Set.Finite.subset _ { i | 1 ≤ ‖f i‖ } ?_ _ ?_)
· have H : { x : α | 1 ≤ ‖f x‖ ^ q.toReal }.Finite := by
simpa using eventually_lt_of_tendsto_lt (by norm_num) hf'.tendsto_cofinite_zero
exact H.subset fun i hi => Real.one_le_rpow hi hq.le
· show ∀ i, ¬|‖f i‖ ^ p.toReal| ≤ ‖f i‖ ^ q.toReal → 1 ≤ ‖f i‖
intro i hi
have : 0 ≤ ‖f i‖ ^ p.toReal := Real.rpow_nonneg (norm_nonneg _) p.toReal
simp only [abs_of_nonneg, this] at hi
contrapose! hi
exact Real.rpow_le_rpow_of_exponent_ge' (norm_nonneg _) hi.le hq.le hpq'
| 36 | 4,311,231,547,115,195 | 2 | 1.625 | 8 | 1,750 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
section Module
variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V]
noncomputable def Basis.ofRankEqZero [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) : Basis ι K V :=
haveI : Subsingleton V := by
obtain ⟨_, b⟩ := Module.Free.exists_basis (R := K) (M := V)
haveI := mk_eq_zero_iff.1 (hV ▸ b.mk_eq_rank'')
exact b.repr.toEquiv.subsingleton
Basis.empty _
#align basis.of_rank_eq_zero Basis.ofRankEqZero
@[simp]
theorem Basis.ofRankEqZero_apply [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) (i : ι) : Basis.ofRankEqZero hV i = 0 := rfl
#align basis.of_rank_eq_zero_apply Basis.ofRankEqZero_apply
| Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 46 | 60 | theorem le_rank_iff_exists_linearIndependent [Module.Free K V] {c : Cardinal} :
c ≤ Module.rank K V ↔ ∃ s : Set V, #s = c ∧ LinearIndependent K ((↑) : s → V) := by |
haveI := nontrivial_of_invariantBasisNumber K
constructor
· intro h
obtain ⟨κ, t'⟩ := Module.Free.exists_basis (R := K) (M := V)
let t := t'.reindexRange
have : LinearIndependent K ((↑) : Set.range t' → V) := by
convert t.linearIndependent
ext; exact (Basis.reindexRange_apply _ _).symm
rw [← t.mk_eq_rank'', le_mk_iff_exists_subset] at h
rcases h with ⟨s, hst, hsc⟩
exact ⟨s, hsc, this.mono hst⟩
· rintro ⟨s, rfl, si⟩
exact si.cardinal_le_rank
| 13 | 442,413.392009 | 2 | 1.636364 | 11 | 1,751 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
section Module
variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V]
noncomputable def Basis.ofRankEqZero [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) : Basis ι K V :=
haveI : Subsingleton V := by
obtain ⟨_, b⟩ := Module.Free.exists_basis (R := K) (M := V)
haveI := mk_eq_zero_iff.1 (hV ▸ b.mk_eq_rank'')
exact b.repr.toEquiv.subsingleton
Basis.empty _
#align basis.of_rank_eq_zero Basis.ofRankEqZero
@[simp]
theorem Basis.ofRankEqZero_apply [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) (i : ι) : Basis.ofRankEqZero hV i = 0 := rfl
#align basis.of_rank_eq_zero_apply Basis.ofRankEqZero_apply
theorem le_rank_iff_exists_linearIndependent [Module.Free K V] {c : Cardinal} :
c ≤ Module.rank K V ↔ ∃ s : Set V, #s = c ∧ LinearIndependent K ((↑) : s → V) := by
haveI := nontrivial_of_invariantBasisNumber K
constructor
· intro h
obtain ⟨κ, t'⟩ := Module.Free.exists_basis (R := K) (M := V)
let t := t'.reindexRange
have : LinearIndependent K ((↑) : Set.range t' → V) := by
convert t.linearIndependent
ext; exact (Basis.reindexRange_apply _ _).symm
rw [← t.mk_eq_rank'', le_mk_iff_exists_subset] at h
rcases h with ⟨s, hst, hsc⟩
exact ⟨s, hsc, this.mono hst⟩
· rintro ⟨s, rfl, si⟩
exact si.cardinal_le_rank
#align le_rank_iff_exists_linear_independent le_rank_iff_exists_linearIndependent
| Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 63 | 71 | theorem le_rank_iff_exists_linearIndependent_finset
[Module.Free K V] {n : ℕ} : ↑n ≤ Module.rank K V ↔
∃ s : Finset V, s.card = n ∧ LinearIndependent K ((↑) : ↥(s : Set V) → V) := by |
simp only [le_rank_iff_exists_linearIndependent, mk_set_eq_nat_iff_finset]
constructor
· rintro ⟨s, ⟨t, rfl, rfl⟩, si⟩
exact ⟨t, rfl, si⟩
· rintro ⟨s, rfl, si⟩
exact ⟨s, ⟨s, rfl, rfl⟩, si⟩
| 6 | 403.428793 | 2 | 1.636364 | 11 | 1,751 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
section Module
variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V]
noncomputable def Basis.ofRankEqZero [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) : Basis ι K V :=
haveI : Subsingleton V := by
obtain ⟨_, b⟩ := Module.Free.exists_basis (R := K) (M := V)
haveI := mk_eq_zero_iff.1 (hV ▸ b.mk_eq_rank'')
exact b.repr.toEquiv.subsingleton
Basis.empty _
#align basis.of_rank_eq_zero Basis.ofRankEqZero
@[simp]
theorem Basis.ofRankEqZero_apply [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) (i : ι) : Basis.ofRankEqZero hV i = 0 := rfl
#align basis.of_rank_eq_zero_apply Basis.ofRankEqZero_apply
theorem le_rank_iff_exists_linearIndependent [Module.Free K V] {c : Cardinal} :
c ≤ Module.rank K V ↔ ∃ s : Set V, #s = c ∧ LinearIndependent K ((↑) : s → V) := by
haveI := nontrivial_of_invariantBasisNumber K
constructor
· intro h
obtain ⟨κ, t'⟩ := Module.Free.exists_basis (R := K) (M := V)
let t := t'.reindexRange
have : LinearIndependent K ((↑) : Set.range t' → V) := by
convert t.linearIndependent
ext; exact (Basis.reindexRange_apply _ _).symm
rw [← t.mk_eq_rank'', le_mk_iff_exists_subset] at h
rcases h with ⟨s, hst, hsc⟩
exact ⟨s, hsc, this.mono hst⟩
· rintro ⟨s, rfl, si⟩
exact si.cardinal_le_rank
#align le_rank_iff_exists_linear_independent le_rank_iff_exists_linearIndependent
theorem le_rank_iff_exists_linearIndependent_finset
[Module.Free K V] {n : ℕ} : ↑n ≤ Module.rank K V ↔
∃ s : Finset V, s.card = n ∧ LinearIndependent K ((↑) : ↥(s : Set V) → V) := by
simp only [le_rank_iff_exists_linearIndependent, mk_set_eq_nat_iff_finset]
constructor
· rintro ⟨s, ⟨t, rfl, rfl⟩, si⟩
exact ⟨t, rfl, si⟩
· rintro ⟨s, rfl, si⟩
exact ⟨s, ⟨s, rfl, rfl⟩, si⟩
#align le_rank_iff_exists_linear_independent_finset le_rank_iff_exists_linearIndependent_finset
| Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 76 | 100 | theorem rank_le_one_iff [Module.Free K V] :
Module.rank K V ≤ 1 ↔ ∃ v₀ : V, ∀ v, ∃ r : K, r • v₀ = v := by |
obtain ⟨κ, b⟩ := Module.Free.exists_basis (R := K) (M := V)
constructor
· intro hd
rw [← b.mk_eq_rank'', le_one_iff_subsingleton] at hd
rcases isEmpty_or_nonempty κ with hb | ⟨⟨i⟩⟩
· use 0
have h' : ∀ v : V, v = 0 := by
simpa [range_eq_empty, Submodule.eq_bot_iff] using b.span_eq.symm
intro v
simp [h' v]
· use b i
have h' : (K ∙ b i) = ⊤ :=
(subsingleton_range b).eq_singleton_of_mem (mem_range_self i) ▸ b.span_eq
intro v
have hv : v ∈ (⊤ : Submodule K V) := mem_top
rwa [← h', mem_span_singleton] at hv
· rintro ⟨v₀, hv₀⟩
have h : (K ∙ v₀) = ⊤ := by
ext
simp [mem_span_singleton, hv₀]
rw [← rank_top, ← h]
refine (rank_span_le _).trans_eq ?_
simp
| 23 | 9,744,803,446.248903 | 2 | 1.636364 | 11 | 1,751 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
section Module
variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V]
noncomputable def Basis.ofRankEqZero [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) : Basis ι K V :=
haveI : Subsingleton V := by
obtain ⟨_, b⟩ := Module.Free.exists_basis (R := K) (M := V)
haveI := mk_eq_zero_iff.1 (hV ▸ b.mk_eq_rank'')
exact b.repr.toEquiv.subsingleton
Basis.empty _
#align basis.of_rank_eq_zero Basis.ofRankEqZero
@[simp]
theorem Basis.ofRankEqZero_apply [Module.Free K V] {ι : Type*} [IsEmpty ι]
(hV : Module.rank K V = 0) (i : ι) : Basis.ofRankEqZero hV i = 0 := rfl
#align basis.of_rank_eq_zero_apply Basis.ofRankEqZero_apply
theorem le_rank_iff_exists_linearIndependent [Module.Free K V] {c : Cardinal} :
c ≤ Module.rank K V ↔ ∃ s : Set V, #s = c ∧ LinearIndependent K ((↑) : s → V) := by
haveI := nontrivial_of_invariantBasisNumber K
constructor
· intro h
obtain ⟨κ, t'⟩ := Module.Free.exists_basis (R := K) (M := V)
let t := t'.reindexRange
have : LinearIndependent K ((↑) : Set.range t' → V) := by
convert t.linearIndependent
ext; exact (Basis.reindexRange_apply _ _).symm
rw [← t.mk_eq_rank'', le_mk_iff_exists_subset] at h
rcases h with ⟨s, hst, hsc⟩
exact ⟨s, hsc, this.mono hst⟩
· rintro ⟨s, rfl, si⟩
exact si.cardinal_le_rank
#align le_rank_iff_exists_linear_independent le_rank_iff_exists_linearIndependent
theorem le_rank_iff_exists_linearIndependent_finset
[Module.Free K V] {n : ℕ} : ↑n ≤ Module.rank K V ↔
∃ s : Finset V, s.card = n ∧ LinearIndependent K ((↑) : ↥(s : Set V) → V) := by
simp only [le_rank_iff_exists_linearIndependent, mk_set_eq_nat_iff_finset]
constructor
· rintro ⟨s, ⟨t, rfl, rfl⟩, si⟩
exact ⟨t, rfl, si⟩
· rintro ⟨s, rfl, si⟩
exact ⟨s, ⟨s, rfl, rfl⟩, si⟩
#align le_rank_iff_exists_linear_independent_finset le_rank_iff_exists_linearIndependent_finset
theorem rank_le_one_iff [Module.Free K V] :
Module.rank K V ≤ 1 ↔ ∃ v₀ : V, ∀ v, ∃ r : K, r • v₀ = v := by
obtain ⟨κ, b⟩ := Module.Free.exists_basis (R := K) (M := V)
constructor
· intro hd
rw [← b.mk_eq_rank'', le_one_iff_subsingleton] at hd
rcases isEmpty_or_nonempty κ with hb | ⟨⟨i⟩⟩
· use 0
have h' : ∀ v : V, v = 0 := by
simpa [range_eq_empty, Submodule.eq_bot_iff] using b.span_eq.symm
intro v
simp [h' v]
· use b i
have h' : (K ∙ b i) = ⊤ :=
(subsingleton_range b).eq_singleton_of_mem (mem_range_self i) ▸ b.span_eq
intro v
have hv : v ∈ (⊤ : Submodule K V) := mem_top
rwa [← h', mem_span_singleton] at hv
· rintro ⟨v₀, hv₀⟩
have h : (K ∙ v₀) = ⊤ := by
ext
simp [mem_span_singleton, hv₀]
rw [← rank_top, ← h]
refine (rank_span_le _).trans_eq ?_
simp
#align rank_le_one_iff rank_le_one_iff
| Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 105 | 119 | theorem rank_eq_one_iff [Module.Free K V] :
Module.rank K V = 1 ↔ ∃ v₀ : V, v₀ ≠ 0 ∧ ∀ v, ∃ r : K, r • v₀ = v := by |
haveI := nontrivial_of_invariantBasisNumber K
refine ⟨fun h ↦ ?_, fun ⟨v₀, h, hv⟩ ↦ (rank_le_one_iff.2 ⟨v₀, hv⟩).antisymm ?_⟩
· obtain ⟨v₀, hv⟩ := rank_le_one_iff.1 h.le
refine ⟨v₀, fun hzero ↦ ?_, hv⟩
simp_rw [hzero, smul_zero, exists_const] at hv
haveI : Subsingleton V := .intro fun _ _ ↦ by simp_rw [← hv]
exact one_ne_zero (h ▸ rank_subsingleton' K V)
· by_contra H
rw [not_le, lt_one_iff_zero] at H
obtain ⟨κ, b⟩ := Module.Free.exists_basis (R := K) (M := V)
haveI := mk_eq_zero_iff.1 (H ▸ b.mk_eq_rank'')
haveI := b.repr.toEquiv.subsingleton
exact h (Subsingleton.elim _ _)
| 13 | 442,413.392009 | 2 | 1.636364 | 11 | 1,751 |
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