Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 70 | 72 | theorem inv_inv : inv (inv r) = r := by |
ext x y
rfl
| 1,062 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 86 | 88 | theorem codom_inv : r.inv.codom = r.dom := by |
ext x
rfl
| 1,062 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 91 | 93 | theorem dom_inv : r.inv.dom = r.codom := by |
ext x
rfl
| 1,062 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 104 | 108 | theorem comp_assoc {δ : Type*} (r : Rel α β) (s : Rel β γ) (t : Rel γ δ) :
(r • s) • t = r • (s • t) := by |
unfold comp; ext (x w); constructor
· rintro ⟨z, ⟨y, rxy, syz⟩, tzw⟩; exact ⟨y, rxy, z, syz, tzw⟩
· rintro ⟨y, rxy, z, syz, tzw⟩; exact ⟨z, ⟨y, rxy, syz⟩, tzw⟩
| 1,062 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 112 | 115 | theorem comp_right_id (r : Rel α β) : r • @Eq β = r := by |
unfold comp
ext y
simp
| 1,062 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 119 | 122 | theorem comp_left_id (r : Rel α β) : @Eq α • r = r := by |
unfold comp
ext x
simp
| 1,062 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 126 | 128 | theorem comp_right_bot (r : Rel α β) : r • (⊥ : Rel β γ) = ⊥ := by |
ext x y
simp [comp, Bot.bot]
| 1,062 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 131 | 133 | theorem comp_left_bot (r : Rel α β) : (⊥ : Rel γ α) • r = ⊥ := by |
ext x y
simp [comp, Bot.bot]
| 1,062 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 136 | 138 | theorem comp_right_top (r : Rel α β) : r • (⊤ : Rel β γ) = fun x _ ↦ x ∈ r.dom := by |
ext x z
simp [comp, Top.top, dom]
| 1,062 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 141 | 143 | theorem comp_left_top (r : Rel α β) : (⊤ : Rel γ α) • r = fun _ y ↦ y ∈ r.codom := by |
ext x z
simp [comp, Top.top, codom]
| 1,062 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 145 | 147 | theorem inv_id : inv (@Eq α) = @Eq α := by |
ext x y
constructor <;> apply Eq.symm
| 1,062 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 150 | 152 | theorem inv_comp (r : Rel α β) (s : Rel β γ) : inv (r • s) = inv s • inv r := by |
ext x z
simp [comp, inv, flip, and_comm]
| 1,062 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 156 | 158 | theorem inv_bot : (⊥ : Rel α β).inv = (⊥ : Rel β α) := by |
#adaptation_note /-- nightly-2024-03-16: simp was `simp [Bot.bot, inv, flip]` -/
simp [Bot.bot, inv, Function.flip_def]
| 1,062 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 375 | 380 | theorem graph_injective : Injective (graph : (α → β) → Rel α β) := by |
intro _ g h
ext x
have h2 := congr_fun₂ h x (g x)
simp only [graph_def, eq_iff_iff, iff_true] at h2
exact h2
| 1,062 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 384 | 384 | theorem graph_id : graph id = @Eq α := by | simp (config := { unfoldPartialApp := true }) [graph]
| 1,062 |
import Mathlib.Data.Part
import Mathlib.Data.Rel
#align_import data.pfun from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Function
def PFun (α β : Type*) :=
α → Part β
#align pfun PFun
infixr:25 " →. " => PFun
namespace PFun
variable {α β γ δ ε ι : Type*}
instance inhab... | Mathlib/Data/PFun.lean | 80 | 80 | theorem mem_dom (f : α →. β) (x : α) : x ∈ Dom f ↔ ∃ y, y ∈ f x := by | simp [Dom, Part.dom_iff_mem]
| 1,063 |
import Mathlib.Data.Part
import Mathlib.Data.Rel
#align_import data.pfun from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Function
def PFun (α β : Type*) :=
α → Part β
#align pfun PFun
infixr:25 " →. " => PFun
namespace PFun
variable {α β γ δ ε ι : Type*}
instance inhab... | Mathlib/Data/PFun.lean | 180 | 181 | theorem mem_restrict {f : α →. β} {s : Set α} (h : s ⊆ f.Dom) (a : α) (b : β) :
b ∈ f.restrict h a ↔ a ∈ s ∧ b ∈ f a := by | simp [restrict]
| 1,063 |
import Mathlib.Data.Part
import Mathlib.Data.Rel
#align_import data.pfun from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Function
def PFun (α β : Type*) :=
α → Part β
#align pfun PFun
infixr:25 " →. " => PFun
namespace PFun
variable {α β γ δ ε ι : Type*}
instance inhab... | Mathlib/Data/PFun.lean | 189 | 190 | theorem mem_res (f : α → β) (s : Set α) (a : α) (b : β) : b ∈ res f s a ↔ a ∈ s ∧ f a = b := by |
simp [res, @eq_comm _ b]
| 1,063 |
import Mathlib.Order.Filter.Basic
import Mathlib.Data.PFun
#align_import order.filter.partial from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
universe u v w
namespace Filter
variable {α : Type u} {β : Type v} {γ : Type w}
open Filter
def rmap (r : Rel α β) (l : Filter α) : F... | Mathlib/Order/Filter/Partial.lean | 130 | 136 | theorem rtendsto_iff_le_rcomap (r : Rel α β) (l₁ : Filter α) (l₂ : Filter β) :
RTendsto r l₁ l₂ ↔ l₁ ≤ l₂.rcomap r := by |
rw [rtendsto_def]
simp_rw [← l₂.mem_sets]
simp [Filter.le_def, rcomap, Rel.mem_image]; constructor
· exact fun h s t tl₂ => mem_of_superset (h t tl₂)
· exact fun h t tl₂ => h _ t tl₂ Set.Subset.rfl
| 1,064 |
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Basic
import Mathlib.Tactic.ApplyFun
#align_import computability.turing_machine from "leanprover-commu... | Mathlib/Computability/TuringMachine.lean | 85 | 88 | theorem BlankExtends.trans {Γ} [Inhabited Γ] {l₁ l₂ l₃ : List Γ} :
BlankExtends l₁ l₂ → BlankExtends l₂ l₃ → BlankExtends l₁ l₃ := by |
rintro ⟨i, rfl⟩ ⟨j, rfl⟩
exact ⟨i + j, by simp [List.replicate_add]⟩
| 1,065 |
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Basic
import Mathlib.Tactic.ApplyFun
#align_import computability.turing_machine from "leanprover-commu... | Mathlib/Computability/TuringMachine.lean | 91 | 95 | theorem BlankExtends.below_of_le {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} :
BlankExtends l l₁ → BlankExtends l l₂ → l₁.length ≤ l₂.length → BlankExtends l₁ l₂ := by |
rintro ⟨i, rfl⟩ ⟨j, rfl⟩ h; use j - i
simp only [List.length_append, Nat.add_le_add_iff_left, List.length_replicate] at h
simp only [← List.replicate_add, Nat.add_sub_cancel' h, List.append_assoc]
| 1,065 |
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Basic
import Mathlib.Tactic.ApplyFun
#align_import computability.turing_machine from "leanprover-commu... | Mathlib/Computability/TuringMachine.lean | 106 | 113 | theorem BlankExtends.above_of_le {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} :
BlankExtends l₁ l → BlankExtends l₂ l → l₁.length ≤ l₂.length → BlankExtends l₁ l₂ := by |
rintro ⟨i, rfl⟩ ⟨j, e⟩ h; use i - j
refine List.append_cancel_right (e.symm.trans ?_)
rw [List.append_assoc, ← List.replicate_add, Nat.sub_add_cancel]
apply_fun List.length at e
simp only [List.length_append, List.length_replicate] at e
rwa [← Nat.add_le_add_iff_left, e, Nat.add_le_add_iff_right]
| 1,065 |
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Basic
import Mathlib.Tactic.ApplyFun
#align_import computability.turing_machine from "leanprover-commu... | Mathlib/Computability/TuringMachine.lean | 133 | 143 | theorem BlankRel.trans {Γ} [Inhabited Γ] {l₁ l₂ l₃ : List Γ} :
BlankRel l₁ l₂ → BlankRel l₂ l₃ → BlankRel l₁ l₃ := by |
rintro (h₁ | h₁) (h₂ | h₂)
· exact Or.inl (h₁.trans h₂)
· rcases le_total l₁.length l₃.length with h | h
· exact Or.inl (h₁.above_of_le h₂ h)
· exact Or.inr (h₂.above_of_le h₁ h)
· rcases le_total l₁.length l₃.length with h | h
· exact Or.inl (h₁.below_of_le h₂ h)
· exact Or.inr (h₂.below_of_le... | 1,065 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 140 | 140 | theorem zero'_eval : zero'.eval = fun v => pure (0 :: v) := by | simp [eval]
| 1,066 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 143 | 143 | theorem succ_eval : succ.eval = fun v => pure [v.headI.succ] := by | simp [eval]
| 1,066 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 146 | 146 | theorem tail_eval : tail.eval = fun v => pure v.tail := by | simp [eval]
| 1,066 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 149 | 152 | theorem cons_eval (f fs) : (cons f fs).eval = fun v => do {
let n ← Code.eval f v
let ns ← Code.eval fs v
pure (n.headI :: ns) } := by | simp [eval]
| 1,066 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 155 | 155 | theorem comp_eval (f g) : (comp f g).eval = fun v => g.eval v >>= f.eval := by | simp [eval]
| 1,066 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 158 | 160 | theorem case_eval (f g) :
(case f g).eval = fun v => v.headI.rec (f.eval v.tail) fun y _ => g.eval (y::v.tail) := by |
simp [eval]
| 1,066 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 163 | 166 | theorem fix_eval (f) : (fix f).eval =
PFun.fix fun v => (f.eval v).map fun v =>
if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tail := by |
simp [eval]
| 1,066 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 174 | 174 | theorem nil_eval (v) : nil.eval v = pure [] := by | simp [nil]
| 1,066 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 183 | 183 | theorem id_eval (v) : id.eval v = pure v := by | simp [id]
| 1,066 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 192 | 192 | theorem head_eval (v) : head.eval v = pure [v.headI] := by | simp [head]
| 1,066 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 201 | 201 | theorem zero_eval (v) : zero.eval v = pure [0] := by | simp [zero]
| 1,066 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 211 | 212 | theorem pred_eval (v) : pred.eval v = pure [v.headI.pred] := by |
simp [pred]; cases v.headI <;> simp
| 1,066 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 264 | 282 | theorem exists_code.comp {m n} {f : Vector ℕ n →. ℕ} {g : Fin n → Vector ℕ m →. ℕ}
(hf : ∃ c : Code, ∀ v : Vector ℕ n, c.eval v.1 = pure <$> f v)
(hg : ∀ i, ∃ c : Code, ∀ v : Vector ℕ m, c.eval v.1 = pure <$> g i v) :
∃ c : Code, ∀ v : Vector ℕ m, c.eval v.1 = pure <$> ((Vector.mOfFn fun i => g i v) >>= f) ... |
rsuffices ⟨cg, hg⟩ :
∃ c : Code, ∀ v : Vector ℕ m, c.eval v.1 = Subtype.val <$> Vector.mOfFn fun i => g i v
· obtain ⟨cf, hf⟩ := hf
exact
⟨cf.comp cg, fun v => by
simp [hg, hf, map_bind, seq_bind_eq, Function.comp]
rfl⟩
clear hf f; induction' n with n IH
· exact ⟨nil, fun v => by ... | 1,066 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 550 | 554 | theorem Cont.then_eval {k k' : Cont} {v} : (k.then k').eval v = k.eval v >>= k'.eval := by |
induction' k with _ _ _ _ _ _ _ _ _ k_ih _ _ k_ih generalizing v <;>
simp only [Cont.eval, Cont.then, bind_assoc, pure_bind, *]
· simp only [← k_ih]
· split_ifs <;> [rfl; simp only [← k_ih, bind_assoc]]
| 1,066 |
import Mathlib.Data.Finset.Option
import Mathlib.Data.PFun
import Mathlib.Data.Part
#align_import data.finset.pimage from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α β : Type*}
namespace Part
def toFinset (o : Part α) [Decidable o.Dom] : Finset α :=
o.toOption.toFins... | Mathlib/Data/Finset/PImage.lean | 34 | 35 | theorem mem_toFinset {o : Part α} [Decidable o.Dom] {x : α} : x ∈ o.toFinset ↔ x ∈ o := by |
simp [toFinset]
| 1,067 |
import Mathlib.Data.Finset.Option
import Mathlib.Data.PFun
import Mathlib.Data.Part
#align_import data.finset.pimage from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α β : Type*}
namespace Part
def toFinset (o : Part α) [Decidable o.Dom] : Finset α :=
o.toOption.toFins... | Mathlib/Data/Finset/PImage.lean | 39 | 40 | theorem toFinset_none [Decidable (none : Part α).Dom] : none.toFinset = (∅ : Finset α) := by |
simp [toFinset]
| 1,067 |
import Mathlib.Data.Finset.Option
import Mathlib.Data.PFun
import Mathlib.Data.Part
#align_import data.finset.pimage from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α β : Type*}
namespace Part
def toFinset (o : Part α) [Decidable o.Dom] : Finset α :=
o.toOption.toFins... | Mathlib/Data/Finset/PImage.lean | 44 | 45 | theorem toFinset_some {a : α} [Decidable (some a).Dom] : (some a).toFinset = {a} := by |
simp [toFinset]
| 1,067 |
import Mathlib.CategoryTheory.Iso
import Mathlib.CategoryTheory.EssentialImage
import Mathlib.CategoryTheory.Types
import Mathlib.CategoryTheory.Opposites
import Mathlib.Data.Rel
#align_import category_theory.category.Rel from "leanprover-community/mathlib"@"afad8e438d03f9d89da2914aa06cb4964ba87a18"
namespace Cate... | Mathlib/CategoryTheory/Category/RelCat.lean | 62 | 63 | theorem rel_id_apply₂ (X : RelCat) (x y : X) : (𝟙 X) x y ↔ x = y := by |
rw [RelCat.Hom.rel_id]
| 1,068 |
import Mathlib.CategoryTheory.Iso
import Mathlib.CategoryTheory.EssentialImage
import Mathlib.CategoryTheory.Types
import Mathlib.CategoryTheory.Opposites
import Mathlib.Data.Rel
#align_import category_theory.category.Rel from "leanprover-community/mathlib"@"afad8e438d03f9d89da2914aa06cb4964ba87a18"
namespace Cate... | Mathlib/CategoryTheory/Category/RelCat.lean | 65 | 66 | theorem rel_comp_apply₂ {X Y Z : RelCat} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) (z : Z) :
(f ≫ g) x z ↔ ∃ y, f x y ∧ g y z := by | rfl
| 1,068 |
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Data.Rel
#align_import combinatorics.hall.basic from "leanprover-community/mathlib"@"8195826f5c428fc283510bc67303dd4472d78498"
open Finset CategoryTheory
universe u v
def hallMatchingsOn {ι : Type u} {α : Typ... | Mathlib/Combinatorics/Hall/Basic.lean | 77 | 86 | theorem hallMatchingsOn.nonempty {ι : Type u} {α : Type v} [DecidableEq α] (t : ι → Finset α)
(h : ∀ s : Finset ι, s.card ≤ (s.biUnion t).card) (ι' : Finset ι) :
Nonempty (hallMatchingsOn t ι') := by |
classical
refine ⟨Classical.indefiniteDescription _ ?_⟩
apply (all_card_le_biUnion_card_iff_existsInjective' fun i : ι' => t i).mp
intro s'
convert h (s'.image (↑)) using 1
· simp only [card_image_of_injective s' Subtype.coe_injective]
· rw [image_biUnion]
| 1,069 |
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Data.Rel
#align_import combinatorics.hall.basic from "leanprover-community/mathlib"@"8195826f5c428fc283510bc67303dd4472d78498"
open Finset CategoryTheory
universe u v
def hallMatchingsOn {ι : Type u} {α : Typ... | Mathlib/Combinatorics/Hall/Basic.lean | 123 | 163 | theorem Finset.all_card_le_biUnion_card_iff_exists_injective {ι : Type u} {α : Type v}
[DecidableEq α] (t : ι → Finset α) :
(∀ s : Finset ι, s.card ≤ (s.biUnion t).card) ↔
∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x := by |
constructor
· intro h
-- Set up the functor
haveI : ∀ ι' : (Finset ι)ᵒᵖ, Nonempty ((hallMatchingsFunctor t).obj ι') := fun ι' =>
hallMatchingsOn.nonempty t h ι'.unop
classical
haveI : ∀ ι' : (Finset ι)ᵒᵖ, Finite ((hallMatchingsFunctor t).obj ι') := by
intro ι'
rw [hallMatchi... | 1,069 |
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Rel
import Mathlib.Data.Set.Finite
import Mathlib.Data.Sym.Sym2
#align_import combinatorics.simple_graph.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
-- Porting note: using `aesop` for automation
-- Porting n... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 188 | 190 | theorem ne_of_adj (h : G.Adj a b) : a ≠ b := by |
rintro rfl
exact G.irrefl h
| 1,070 |
import Mathlib.Combinatorics.SimpleGraph.Basic
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
structure Dart extends V × V where
adj : G.Adj fst snd
deriving DecidableEq
#align simple_graph.dart SimpleGraph.Dart
initialize_simps_projections Dart (+toProd, -fst, -snd)
attribute [simp] Dart.a... | Mathlib/Combinatorics/SimpleGraph/Dart.lean | 33 | 34 | theorem Dart.ext_iff (d₁ d₂ : G.Dart) : d₁ = d₂ ↔ d₁.toProd = d₂.toProd := by |
cases d₁; cases d₂; simp
| 1,071 |
import Mathlib.Combinatorics.SimpleGraph.Basic
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
structure Dart extends V × V where
adj : G.Adj fst snd
deriving DecidableEq
#align simple_graph.dart SimpleGraph.Dart
initialize_simps_projections Dart (+toProd, -fst, -snd)
attribute [simp] Dart.a... | Mathlib/Combinatorics/SimpleGraph/Dart.lean | 107 | 109 | theorem dart_edge_eq_iff : ∀ d₁ d₂ : G.Dart, d₁.edge = d₂.edge ↔ d₁ = d₂ ∨ d₁ = d₂.symm := by |
rintro ⟨p, hp⟩ ⟨q, hq⟩
simp
| 1,071 |
import Mathlib.Combinatorics.SimpleGraph.Basic
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
structure Dart extends V × V where
adj : G.Adj fst snd
deriving DecidableEq
#align simple_graph.dart SimpleGraph.Dart
initialize_simps_projections Dart (+toProd, -fst, -snd)
attribute [simp] Dart.a... | Mathlib/Combinatorics/SimpleGraph/Dart.lean | 112 | 115 | theorem dart_edge_eq_mk'_iff :
∀ {d : G.Dart} {p : V × V}, d.edge = Sym2.mk p ↔ d.toProd = p ∨ d.toProd = p.swap := by |
rintro ⟨p, h⟩
apply Sym2.mk_eq_mk_iff
| 1,071 |
import Mathlib.Combinatorics.SimpleGraph.Basic
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
structure Dart extends V × V where
adj : G.Adj fst snd
deriving DecidableEq
#align simple_graph.dart SimpleGraph.Dart
initialize_simps_projections Dart (+toProd, -fst, -snd)
attribute [simp] Dart.a... | Mathlib/Combinatorics/SimpleGraph/Dart.lean | 118 | 123 | theorem dart_edge_eq_mk'_iff' :
∀ {d : G.Dart} {u v : V},
d.edge = s(u, v) ↔ d.fst = u ∧ d.snd = v ∨ d.fst = v ∧ d.snd = u := by |
rintro ⟨⟨a, b⟩, h⟩ u v
rw [dart_edge_eq_mk'_iff]
simp
| 1,071 |
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
nam... | 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
| 1,072 |
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
nam... | 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)
| 1,072 |
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
nam... | 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)
| 1,072 |
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
nam... | 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
| 1,072 |
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
nam... | 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]
| 1,072 |
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Data.FunLike.Fintype
open Function
namespace SimpleGraph
variable {V W X : Type*} (G : SimpleGraph V) (G' : SimpleGraph W) {u v : V}
protected def map (f : V ↪ W) (G : SimpleGraph V) : SimpleGraph W where
Adj := Relation.Map G.Adj f f
symm a b... | Mathlib/Combinatorics/SimpleGraph/Maps.lean | 76 | 78 | theorem map_monotone (f : V ↪ W) : Monotone (SimpleGraph.map f) := by |
rintro G G' h _ _ ⟨u, v, ha, rfl, rfl⟩
exact ⟨_, _, h ha, rfl, rfl⟩
| 1,073 |
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Data.FunLike.Fintype
open Function
namespace SimpleGraph
variable {V W X : Type*} (G : SimpleGraph V) (G' : SimpleGraph W) {u v : V}
protected def map (f : V ↪ W) (G : SimpleGraph V) : SimpleGraph W where
Adj := Relation.Map G.Adj f f
symm a b... | Mathlib/Combinatorics/SimpleGraph/Maps.lean | 123 | 125 | theorem comap_monotone (f : V ↪ W) : Monotone (SimpleGraph.comap f) := by |
intro G G' h _ _ ha
exact h ha
| 1,073 |
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Data.FunLike.Fintype
open Function
namespace SimpleGraph
variable {V W X : Type*} (G : SimpleGraph V) (G' : SimpleGraph W) {u v : V}
protected def map (f : V ↪ W) (G : SimpleGraph V) : SimpleGraph W where
Adj := Relation.Map G.Adj f f
symm a b... | Mathlib/Combinatorics/SimpleGraph/Maps.lean | 129 | 131 | theorem comap_map_eq (f : V ↪ W) (G : SimpleGraph V) : (G.map f).comap f = G := by |
ext
simp
| 1,073 |
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Data.FunLike.Fintype
open Function
namespace SimpleGraph
variable {V W X : Type*} (G : SimpleGraph V) (G' : SimpleGraph W) {u v : V}
protected def map (f : V ↪ W) (G : SimpleGraph V) : SimpleGraph W where
Adj := Relation.Map G.Adj f f
symm a b... | Mathlib/Combinatorics/SimpleGraph/Maps.lean | 154 | 155 | theorem map_comap_le (f : V ↪ W) (G : SimpleGraph W) : (G.comap f).map f ≤ G := by |
rw [map_le_iff_le_comap]
| 1,073 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b"
universe u v
namespace SimpleGraph
@[ext]
structure Subgraph {V : Type u} (G : SimpleGra... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 177 | 178 | theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by |
simp [Subgraph.spanningCoe]
| 1,074 |
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.finsubgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b"
open Set CategoryTheory
universe u v
variable {V : Type u} {W : Type v} {G : Simple... | Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean | 93 | 95 | theorem singletonFinsubgraph_le_adj_left {u v : V} {e : G.Adj u v} :
singletonFinsubgraph u ≤ finsubgraphOfAdj e := by |
simp [singletonFinsubgraph, finsubgraphOfAdj]
| 1,075 |
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.finsubgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b"
open Set CategoryTheory
universe u v
variable {V : Type u} {W : Type v} {G : Simple... | Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean | 98 | 100 | theorem singletonFinsubgraph_le_adj_right {u v : V} {e : G.Adj u v} :
singletonFinsubgraph v ≤ finsubgraphOfAdj e := by |
simp [singletonFinsubgraph, finsubgraphOfAdj]
| 1,075 |
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.finsubgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b"
open Set CategoryTheory
universe u v
variable {V : Type u} {W : Type v} {G : Simple... | Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean | 119 | 153 | theorem nonempty_hom_of_forall_finite_subgraph_hom [Finite W]
(h : ∀ G' : G.Subgraph, G'.verts.Finite → G'.coe →g F) : Nonempty (G →g F) := by |
-- Obtain a `Fintype` instance for `W`.
cases nonempty_fintype W
-- Establish the required interface instances.
haveI : ∀ G' : G.Finsubgraphᵒᵖ, Nonempty ((finsubgraphHomFunctor G F).obj G') := fun G' =>
⟨h G'.unop G'.unop.property⟩
haveI : ∀ G' : G.Finsubgraphᵒᵖ, Fintype ((finsubgraphHomFunctor G F).obj ... | 1,075 |
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 133 | 137 | theorem copy_copy {u v u' v' u'' v''} (p : G.Walk u v)
(hu : u = u') (hv : v = v') (hu' : u' = u'') (hv' : v' = v'') :
(p.copy hu hv).copy hu' hv' = p.copy (hu.trans hu') (hv.trans hv') := by |
subst_vars
rfl
| 1,076 |
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 141 | 143 | theorem copy_nil {u u'} (hu : u = u') : (Walk.nil : G.Walk u u).copy hu hu = Walk.nil := by |
subst_vars
rfl
| 1,076 |
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 146 | 149 | theorem copy_cons {u v w u' w'} (h : G.Adj u v) (p : G.Walk v w) (hu : u = u') (hw : w = w') :
(Walk.cons h p).copy hu hw = Walk.cons (hu ▸ h) (p.copy rfl hw) := by |
subst_vars
rfl
| 1,076 |
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 153 | 156 | theorem cons_copy {u v w v' w'} (h : G.Adj u v) (p : G.Walk v' w') (hv : v' = v) (hw : w' = w) :
Walk.cons h (p.copy hv hw) = (Walk.cons (hv ▸ h) p).copy rfl hw := by |
subst_vars
rfl
| 1,076 |
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 208 | 208 | theorem getVert_zero {u v} (w : G.Walk u v) : w.getVert 0 = u := by | cases w <;> rfl
| 1,076 |
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 211 | 218 | theorem getVert_of_length_le {u v} (w : G.Walk u v) {i : ℕ} (hi : w.length ≤ i) :
w.getVert i = v := by |
induction w generalizing i with
| nil => rfl
| cons _ _ ih =>
cases i
· cases hi
· exact ih (Nat.succ_le_succ_iff.1 hi)
| 1,076 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
namespace SimpleGraph
universe u v
variable {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'}
namespace Subgraph
protected structure Preconnected (H : G.Subgraph) : Prop where
protected coe : H.coe.Preconnected
instance {H : G.Subgraph}... | Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean | 64 | 69 | theorem singletonSubgraph_connected {v : V} : (G.singletonSubgraph v).Connected := by |
refine ⟨⟨?_⟩⟩
rintro ⟨a, ha⟩ ⟨b, hb⟩
simp only [singletonSubgraph_verts, Set.mem_singleton_iff] at ha hb
subst_vars
rfl
| 1,077 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
namespace SimpleGraph
universe u v
variable {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'}
namespace Subgraph
protected structure Preconnected (H : G.Subgraph) : Prop where
protected coe : H.coe.Preconnected
instance {H : G.Subgraph}... | Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean | 73 | 78 | theorem subgraphOfAdj_connected {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).Connected := by |
refine ⟨⟨?_⟩⟩
rintro ⟨a, ha⟩ ⟨b, hb⟩
simp only [subgraphOfAdj_verts, Set.mem_insert_iff, Set.mem_singleton_iff] at ha hb
obtain rfl | rfl := ha <;> obtain rfl | rfl := hb <;>
first | rfl | (apply Adj.reachable; simp)
| 1,077 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 64 | 65 | theorem apply_diag_ne [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i : V) :
¬A i i = 1 := by | simp [h.apply_diag i]
| 1,078 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 69 | 70 | theorem apply_ne_one_iff [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i j : V) :
¬A i j = 1 ↔ A i j = 0 := by | obtain h | h := h.zero_or_one i j <;> simp [h]
| 1,078 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 74 | 75 | theorem apply_ne_zero_iff [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i j : V) :
¬A i j = 0 ↔ A i j = 1 := by | rw [← apply_ne_one_iff h, Classical.not_not]
| 1,078 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 105 | 105 | theorem compl_apply_diag [Zero α] [One α] (i : V) : A.compl i i = 0 := by | simp [compl]
| 1,078 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 109 | 111 | theorem compl_apply [Zero α] [One α] (i j : V) : A.compl i j = 0 ∨ A.compl i j = 1 := by |
unfold compl
split_ifs <;> simp
| 1,078 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 115 | 117 | theorem isSymm_compl [Zero α] [One α] (h : A.IsSymm) : A.compl.IsSymm := by |
ext
simp [compl, h.apply, eq_comm]
| 1,078 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 121 | 122 | theorem isAdjMatrix_compl [Zero α] [One α] (h : A.IsSymm) : IsAdjMatrix A.compl :=
{ symm := by | simp [h] }
| 1,078 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
#align_import combinatorics.simple_graph.prod from "leanprover-community/mathlib"@"2985fa3c31a27274aed06c433510bc14b73d6488"
variable {α β γ : Type*}
namespace SimpleGraph
-- Porting note: pruned variables to keep things out of local contexts, which
-- can im... | Mathlib/Combinatorics/SimpleGraph/Prod.lean | 59 | 60 | theorem boxProd_adj_left : (G □ H).Adj (a₁, b) (a₂, b) ↔ G.Adj a₁ a₂ := by |
simp only [boxProd_adj, and_true, SimpleGraph.irrefl, false_and, or_false]
| 1,079 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
#align_import combinatorics.simple_graph.prod from "leanprover-community/mathlib"@"2985fa3c31a27274aed06c433510bc14b73d6488"
variable {α β γ : Type*}
namespace SimpleGraph
-- Porting note: pruned variables to keep things out of local contexts, which
-- can im... | Mathlib/Combinatorics/SimpleGraph/Prod.lean | 65 | 66 | theorem boxProd_adj_right : (G □ H).Adj (a, b₁) (a, b₂) ↔ H.Adj b₁ b₂ := by |
simp only [boxProd_adj, SimpleGraph.irrefl, false_and, and_true, false_or]
| 1,079 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
#align_import combinatorics.simple_graph.prod from "leanprover-community/mathlib"@"2985fa3c31a27274aed06c433510bc14b73d6488"
variable {α β γ : Type*}
namespace SimpleGraph
-- Porting note: pruned variables to keep things out of local contexts, which
-- can im... | Mathlib/Combinatorics/SimpleGraph/Prod.lean | 69 | 73 | theorem boxProd_neighborSet (x : α × β) :
(G □ H).neighborSet x = G.neighborSet x.1 ×ˢ {x.2} ∪ {x.1} ×ˢ H.neighborSet x.2 := by |
ext ⟨a', b'⟩
simp only [mem_neighborSet, Set.mem_union, boxProd_adj, Set.mem_prod, Set.mem_singleton_iff]
simp only [eq_comm, and_comm]
| 1,079 |
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Finite.Set
#align_import combinatorics.simple_graph.ends.defs from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
universe u
variable {V : Type u} (G : SimpleGraph V... | Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean | 44 | 49 | theorem ComponentCompl.supp_injective :
Function.Injective (ComponentCompl.supp : G.ComponentCompl K → Set V) := by |
refine ConnectedComponent.ind₂ ?_
rintro ⟨v, hv⟩ ⟨w, hw⟩ h
simp only [Set.ext_iff, ConnectedComponent.eq, Set.mem_setOf_eq, ComponentCompl.supp] at h ⊢
exact ((h v).mp ⟨hv, Reachable.refl _⟩).choose_spec
| 1,080 |
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Finite.Set
#align_import combinatorics.simple_graph.ends.defs from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
universe u
variable {V : Type u} (G : SimpleGraph V... | Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean | 71 | 75 | theorem componentComplMk_eq_of_adj (G : SimpleGraph V) {v w : V} (vK : v ∉ K) (wK : w ∉ K)
(a : G.Adj v w) : G.componentComplMk vK = G.componentComplMk wK := by |
rw [ConnectedComponent.eq]
apply Adj.reachable
exact a
| 1,080 |
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 IsAcy... | 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
| 1,081 |
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 IsAcy... | 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]
| 1,081 |
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 IsAcy... | 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.a... | 1,081 |
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 IsAcy... | 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.singlet... | 1,081 |
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 IsAcy... | 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⟩
r... | 1,081 |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 63 | 67 | theorem IsMatching.toEdge_eq_of_adj {M : Subgraph G} (h : M.IsMatching) {v w : V} (hv : v ∈ M.verts)
(hvw : M.Adj v w) : h.toEdge ⟨v, hv⟩ = ⟨s(v, w), hvw⟩ := by |
simp only [IsMatching.toEdge, Subtype.mk_eq_mk]
congr
exact ((h (M.edge_vert hvw)).choose_spec.2 w hvw).symm
| 1,082 |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 70 | 74 | theorem IsMatching.toEdge.surjective {M : Subgraph G} (h : M.IsMatching) :
Function.Surjective h.toEdge := by |
rintro ⟨e, he⟩
refine Sym2.ind (fun x y he => ?_) e he
exact ⟨⟨x, M.edge_vert he⟩, h.toEdge_eq_of_adj _ he⟩
| 1,082 |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 77 | 80 | theorem IsMatching.toEdge_eq_toEdge_of_adj {M : Subgraph G} {v w : V} (h : M.IsMatching)
(hv : v ∈ M.verts) (hw : w ∈ M.verts) (ha : M.Adj v w) :
h.toEdge ⟨v, hv⟩ = h.toEdge ⟨w, hw⟩ := by |
rw [h.toEdge_eq_of_adj hv ha, h.toEdge_eq_of_adj hw (M.symm ha), Subtype.mk_eq_mk, Sym2.eq_swap]
| 1,082 |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 90 | 93 | theorem IsMatching.support_eq_verts {M : Subgraph G} (h : M.IsMatching) : M.support = M.verts := by |
refine M.support_subset_verts.antisymm fun v hv => ?_
obtain ⟨w, hvw, -⟩ := h hv
exact ⟨_, hvw⟩
| 1,082 |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 96 | 98 | theorem isMatching_iff_forall_degree {M : Subgraph G} [∀ v : V, Fintype (M.neighborSet v)] :
M.IsMatching ↔ ∀ v : V, v ∈ M.verts → M.degree v = 1 := by |
simp only [degree_eq_one_iff_unique_adj, IsMatching]
| 1,082 |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 101 | 111 | theorem IsMatching.even_card {M : Subgraph G} [Fintype M.verts] (h : M.IsMatching) :
Even M.verts.toFinset.card := by |
classical
rw [isMatching_iff_forall_degree] at h
use M.coe.edgeFinset.card
rw [← two_mul, ← M.coe.sum_degrees_eq_twice_card_edges]
-- Porting note: `SimpleGraph.Subgraph.coe_degree` does not trigger because it uses
-- instance arguments instead of implicit arguments for the first `Fintype` argument.
-- U... | 1,082 |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 114 | 119 | theorem isPerfectMatching_iff : M.IsPerfectMatching ↔ ∀ v, ∃! w, M.Adj v w := by |
refine ⟨?_, fun hm => ⟨fun v _ => hm v, fun v => ?_⟩⟩
· rintro ⟨hm, hs⟩ v
exact hm (hs v)
· obtain ⟨w, hw, -⟩ := hm v
exact M.edge_vert hw
| 1,082 |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 122 | 124 | theorem isPerfectMatching_iff_forall_degree {M : Subgraph G} [∀ v, Fintype (M.neighborSet v)] :
M.IsPerfectMatching ↔ ∀ v, M.degree v = 1 := by |
simp [degree_eq_one_iff_unique_adj, isPerfectMatching_iff]
| 1,082 |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 127 | 130 | theorem IsPerfectMatching.even_card {M : Subgraph G} [Fintype V] (h : M.IsPerfectMatching) :
Even (Fintype.card V) := by |
classical
simpa only [h.2.card_verts] using IsMatching.even_card h.1
| 1,082 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
namespace Iso
variable {G} {W : Type*} {G' : SimpleGraph W} (f : G ≃g G')
| Mathlib/Combinatorics/SimpleGraph/Operations.lean | 35 | 39 | theorem card_edgeFinset_eq [Fintype G.edgeSet] [Fintype G'.edgeSet] :
G.edgeFinset.card = G'.edgeFinset.card := by |
apply Finset.card_eq_of_equiv
simp only [Set.mem_toFinset]
exact f.mapEdgeSet
| 1,083 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 76 | 80 | theorem edgeSet_replaceVertex_of_not_adj (hn : ¬G.Adj s t) : (G.replaceVertex s t).edgeSet =
G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s) := by |
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
| 1,083 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 82 | 86 | theorem edgeSet_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeSet =
(G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s)) \ {s(t, t)} := by |
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
| 1,083 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 92 | 96 | theorem edgeFinset_replaceVertex_of_not_adj (hn : ¬G.Adj s t) : (G.replaceVertex s t).edgeFinset =
G.edgeFinset \ G.incidenceFinset t ∪ (G.neighborFinset s).image (s(·, t)) := by |
simp only [incidenceFinset, neighborFinset, ← Set.toFinset_diff, ← Set.toFinset_image,
← Set.toFinset_union]
exact Set.toFinset_congr (G.edgeSet_replaceVertex_of_not_adj hn)
| 1,083 |
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