chars int64 40 24k | content_type stringclasses 9
values | id stringlengths 13 64 | source stringclasses 5
values | subset stringclasses 31
values | text stringlengths 40 24k |
|---|---|---|---|---|---|
2,857 | lean_file_chunk | 1c29ac69414cb82bd8c5e0d8de0174cfaeaca958b0af159e54eda1cab5cc9b42 | lean_source_file | pkg_mathlib_source_file_chunks | _ = (∫ x : α, f x ∂μ) + 2 * δ := by
simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf]; ring
_ = (∫ x : α, f x ∂μ) + ε := by congr 1; simp [field, δ]
case aelt =>
show ∀ᵐ x : α ∂μ, g x < ⊤
filter_upwards [gp_lt_top] with ?_ hx
simp only [g, sub_eq_add_neg, Ne, (EReal.add_lt... |
23,988 | lean_file_chunk | b8d0c62a659270e953ac051a8e632dabb9bf373da3e8e4ec719bac85689459c4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
module
public import Mathlib.MeasureTheory.Integral.SetToL1
/-!
# Bochner integral
The Bochner integral extends the... |
1,332 | lean_file_chunk | 2bfc4319915dc2d25670feba48f4913ede91c259f5f76c30e41792583dd97ed6 | lean_source_file | pkg_mathlib_source_file_chunks | norm_Integral_le_one
theorem norm_integral_le (f : α →₁[μ] E) : ‖integral f‖ ≤ ‖f‖ :=
calc
‖integral f‖ = ‖integralCLM f‖ := by simp only [integral]
_ ≤ ‖integralCLM (α := α) (μ := μ)‖ * ‖f‖ := le_opNorm _ _
_ ≤ 1 * ‖f‖ := mul_le_mul_of_nonneg_right norm_Integral_le_one <| norm_nonneg _
_ = ‖f‖ := on... |
7,188 | lean_file_chunk | abd0d89b393f99fcef0e1bd898b41e12731c1519847525d634eb9fcbc5c69530 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
module
public import Mathlib.Algebra.BigOperators.Group.Finset.Defs
public import Mathlib.Data.Finset.NatAntidiagonal
public import Mathlib.Data.Nat.Choose.C... |
8,754 | lean_file_chunk | bebec4c7042aa19141473514b953ec08d30c37ef33dced76c4250817264b9038 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Algebra.BigOperators.Ring.Finset
public import Mathlib.Algebra.Order.BigOperators.Group.Finset
public import Mathlib.Algebra.Order.Ring.Nat
... |
8,632 | lean_file_chunk | 420d760dd7df35597d5b807a3846988ab3ef1c73cbf67a988c4b989949e10306 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Algebra.BigOperators.Pi
public import Mathlib.Algebra.BigOperators.Ring.Finset
public import Mathlib.Algebra.Module.BigOperators
/-!
# Incl... |
22,349 | lean_file_chunk | 60eb661a7d524cddd6b4f2ed8c428556328277a8454365808165fb67241c7e44 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
module
public import Mathlib.Combinatorics.Enumerative.Catalan
import Batteries.Data.List.Count
import Mathlib.Tactic.Positivity.Finset
/-!
# Dyck words
A Dyck word is a ... |
23,972 | lean_file_chunk | 005976cc9fa520d0b1ce8fcbff209f531319a2a7896830caec4cc5ecfe09da0d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Algebra.BigOperators.Fin
public import Mathlib.Algebra.Order.BigOperators.Group.Finset
public import Mathlib.Data.Finset.Sort
/-!... |
14,959 | lean_file_chunk | 7106822077e1528228fc57a36302ff6a7c110741ed6db7f7edad11efb0664041 | lean_source_file | pkg_mathlib_source_file_chunks | (zero : motive 0 (ones 0))
(single_append : ∀ k n c, motive n c →
motive (k + 1 + n) (append (single (k + 1) k.succ_pos) c)) :
motive n c :=
match n, c with
| _, ⟨blocks, blocks_pos, rfl⟩ =>
match blocks with
| [] => zero
| 0 :: _ => by simp at blocks_pos
| (k + 1) :: l =>
single... |
4,367 | lean_file_chunk | 9930281596205039c59e259c46a1c8d00b7a8ae2c0e3f780a9717afcf980e864 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Weijie Jiang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Weijie Jiang
-/
module
public import Mathlib.Algebra.BigOperators.Group.Finset.Defs
public import Mathlib.Algebra.Group.Even
public import Mathlib.Order.Interval.Finset.Nat
import Math... |
7,594 | lean_file_chunk | c066316b1fcec62cd0b2753c868018ac0402f86dd65ae1053327cf8de6ea80f1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Antoine Chambert-Loir & María-Inés de Frutos—Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María-Inés de Frutos—Fernández, Yu Shao, Beibei Xiong, Weijie Jiang
-/
module
public import Mathlib.Data.Nat.Choose.Mult... |
133 | lean_file_chunk | 6a32a54cdf96ee8af5e812104ba82e00a18a846e4fbc58a771bd325217a9ac29 | lean_source_file | pkg_mathlib_source_file_chunks | module -- shake: keep-all
public import Mathlib.Combinatorics.Enumerative.Partition.Basic
deprecated_module (since := "2025-11-15") |
23,984 | lean_file_chunk | 98039fbd26d252910912149e31027f77fb009e456899564f15ae3b8c3ec02256 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Alex J. Best, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
module
public import Mathlib.Algebra.Algebra.Defs
public import Mathlib.Algebra.BigOperators.Ring.Finset
public import Mathlib.Algebra.Module... |
2,443 | lean_file_chunk | cc60bde300bbfb7839dd1bfc228358471e2847784a0b031d8a3e1a0011318762 | lean_source_file | pkg_mathlib_source_file_chunks | /-- The Cartesian product of two incidence algebras. -/
protected def prod : IncidenceAlgebra 𝕜 (α × β) where
toFun x y := f x.1 y.1 * g x.2 y.2
eq_zero_of_not_le' x y hxy := by
rw [Prod.le_def, not_and_or] at hxy
obtain hxy | hxy := hxy <;> simp [apply_eq_zero_of_not_le hxy]
lemma prod_mk (a₁ a₂ : α) (b₁... |
5,965 | lean_file_chunk | a55885de697d0561fdb3ce7da1fb7753b6a13967a22fe92462ddd978294fa0a5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Beibei Xiong. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Beibei Xiong, Yu Shao, Weijie Jiang, Zhengfeng Yang
-/
module
public import Mathlib.Data.Nat.Factorial.Basic
public import Mathlib.Data.Nat.Choose.Basic
public import Mathlib.Tactic.Nor... |
8,395 | lean_file_chunk | 4c76b6e5072bbe5d8612c7228a7591fcb0fd5e48d3f1849c9c4664f605abcc64 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Martin Dvorak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Dvorak
-/
module
public import Mathlib.Algebra.BigOperators.Fin
public import Mathlib.Algebra.Order.BigOperators.Group.Multiset
public import Mathlib.Data.Fin.VecNotation
public... |
22,673 | lean_file_chunk | 777059a5945f2d1bfd1e2c7b5c3da2e2619c7c2f5e69864bb1e0463eb2a16b38 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson, Jun Kwon
-/
module
public import Mathlib.Data.Set.Basic
public import Mathlib.Data.Sym.Sym2
/-!
# Multigraphs
A multigraph is a set of vertices and a set of edges,
toget... |
7,470 | lean_file_chunk | 1ecdf0b150606d50e7655a7a0eec090b433e0e2c797d927907939a0d25544ba2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Jun Kwon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson, Jun Kwon
-/
module
public import Mathlib.Combinatorics.Graph.Subgraph
/-!
# Deletion of edges and vertices
This file defines the deletion of edges and vertices from a graph.... |
15,917 | lean_file_chunk | 4723e430ebdc9989b010947c0dd0ee9cf27179b76965da06fc6ec659ecfff129 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson, Jun Kwon
-/
module
public import Mathlib.Combinatorics.Graph.Basic
public import Mathlib.Tactic.TFAE
/-!
# Subgraphs of multigraphs
This file develops the basic theory o... |
11,693 | lean_file_chunk | 009ca40c5c5e49c59abf9627ec1a2cfd1ba9d69671f57734ff80843c929a6c0c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
module
public import Mathlib.Combinatorics.Quiver.Cast
public import Mathlib.Combinatorics.Quiver.Symmetric
public import Mathlib... |
6,295 | lean_file_chunk | f9ec07a195e4339da04dc74cf59275f2fb1115c7786df062de57d61af72d2936 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Mario Carneiro and Emily Riehl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Emily Riehl
-/
module
public import Mathlib.Data.Set.Function
public import Mathlib.CategoryTheory.Category.Cat
/-!
# Reflexive Quivers
This module d... |
5,064 | lean_file_chunk | 3cd7c696e1a57f2368372dc0411559797e92837d42e6670ae7b0accf6e250981 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
module
public import Mathlib.Combinatorics.Quiver.Path
public import Mathlib.Combinatorics.Quiver.Subquiver
public import Mathlib.Order.WellFounded
/-!
# Arborescences
A q... |
4,649 | lean_file_chunk | c481958cffb1e95e659271a98cebcab9ab43e8debafe0d702e098d9e2751f0fc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Kim Morrison
-/
module
public import Mathlib.Data.Opposite
public import Mathlib.Tactic.ToDual
/-!
# Quivers
This module defines quivers. A quiver on a type `V` of vertices ... |
12,344 | lean_file_chunk | d9d8b3e6cafe5508582181cc49fd11e4fe2a9398b7f7c3777aa57f8636d1bb5b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 David Wärn,. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Kim Morrison, Matteo Cipollina
-/
module
public import Mathlib.Combinatorics.Quiver.Prefunctor
public import Mathlib.Logic.Lemmas
public import Batteries.Data.List.Basic
/-!... |
5,771 | lean_file_chunk | 82ff53a4a462078a63fdedb56f1727eba6d038e6564170c621db45630ccea88b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Runtian Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Runtian Zhou
-/
module
public import Mathlib.Algebra.Group.Action.Defs
public import Mathlib.Algebra.Group.Basic
public import Mathlib.Combinatorics.Quiver.Covering
public import Mathli... |
4,270 | lean_file_chunk | 34d8cdb33e468dbc408ed46d02fea31d74584e329e4f64656ddf1e4ab0571d0d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Kim Morrison
-/
module
public import Mathlib.Combinatorics.Quiver.Basic
/-!
# Morphisms of quivers
-/
@[expose] public section
universe v₁ v₂ u u₁ u₂
/-- A morphism of qui... |
4,271 | lean_file_chunk | 70f9d407f4a55e825a0fcb5eba5f443ba06a76a5c6b8130c6ff05c22ac3c2f2e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
module
public import Mathlib.Combinatorics.Quiver.Cast
public import Mathlib.Combinatorics.Quiver.Symmetric
/-!
# Single-object quiver
Single object quiver with ... |
2,366 | lean_file_chunk | 9c334b1ea99c1ddff4a7c7a8ae252101eff0a340aa301c36f3f7a875f152c703 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
module
public import Mathlib.Order.Notation
public import Mathlib.Combinatorics.Quiver.Basic
/-!
## Wide subquivers
A wide subquiver `H` of a quiver `H` consists of a subs... |
7,384 | lean_file_chunk | 818eb4f1584a86dd4fe1aedc6b8bd0977a73027d052e2b7de9b31b788af37ebb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Antoine Labelle, Rémi Bottinelli
-/
module
public import Mathlib.Combinatorics.Quiver.Path
public import Mathlib.Combinatorics.Quiver.Push
/-!
## Symmetric quivers and arrow ... |
4,297 | lean_file_chunk | f12c819c53922f0fcfe06f9d2690e40a0f402c1f59a1f867a86aa27e557d8248 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
module
public import Mathlib.Combinatorics.Quiver.Basic
public import Mathlib.Combinatorics.Quiver.Path
/-!
# Rewriting arrows ... |
7,320 | lean_file_chunk | a2dd8b0efe79ceb630756b4068fb52dad5331d2431ea2cb57c6be91d0f68a80d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Matteo Cipollina
-/
module
public import Mathlib.Combinatorics.Quiver.Subquiver
public import Mathlib.Combinatorics.Quiver.Path
public import Mathlib.Combinatorics.Quiver.Symm... |
2,409 | lean_file_chunk | 6afc7bc1919db88e984e20136ab41fe2e962e78ccf86c8a45e391c4daff3aa45 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli
-/
module
public import Mathlib.Combinatorics.Quiver.Prefunctor
/-!
# Pushing a quiver structure along a map
Given a map `σ : V → W` and a `Quiver` instance on `V`... |
12,868 | lean_file_chunk | 3a4666801e6ee44f93976af15ddaa20cae2e655216c788835f438612b5057492 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
module
public import Mathlib.Combinatorics.Matroid.Map
public import Mathlib.Logic.Embedding.Set
/-!
# Sums of matroids
The *sum* `M` of a collection `M₁, M₂, ..` of m... |
513 | lean_file_chunk | d08a43edf45e7dfabdf9e3e0348565d9669a40ddb0df96bf3cdabce5c5550850 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
module
public import Mathlib.Init
public import Aesop
/-!
# Matroid Rule Set
This module defines the `Matroid` Aesop rule set which is used by the
`aesop_mat` tactic. ... |
23,945 | lean_file_chunk | 191f6c8e69b65da664f43987fce4b34d0c4c36c48d4c42ebcf10866a23a89a06 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
module
public import Mathlib.Combinatorics.Matroid.Closure
/-!
# Matroid IsCircuits
A 'Circuit' of a matroid `M` is a minimal set `C` that is dependent in `M`.
A matro... |
12,784 | lean_file_chunk | 1b1c223a464fd9cbed901a3024901dd9c1d6176220af09cf2c2bbeddd74d1c7c | lean_source_file | pkg_mathlib_source_file_chunks | as an axiom, it is usually stated with the extra assumption that `e ∈ C₁ ∩ C₂`. -/
lemma IsCircuit.elimination (hC₁ : M.IsCircuit C₁) (hC₂ : M.IsCircuit C₂) (h : C₁ ≠ C₂) (e : α) :
∃ C ⊆ (C₁ ∪ C₂) \ {e}, M.IsCircuit C := by
have hnss : ¬ (C₁ ⊆ C₂) := fun hss ↦ h <| hC₁.eq_of_subset_isCircuit hC₂ hss
obtain ⟨f, ... |
23,980 | lean_file_chunk | ca455d6e3bf37f18ff53a7a232a93c35d5dc9f05c32b1b0de24586a53433cb9d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
module
public import Mathlib.Combinatorics.Matroid.Map
public import Mathlib.Order.Closure
public import Mathlib.Order.CompleteLatticeIntervals
/-!
# Matroid Closure
A... |
23,979 | lean_file_chunk | 9ff232c66ab5248f4fe8738164dd36f374220148d7ec3630f76d69ab9b94ae0f | lean_source_file | pkg_mathlib_source_file_chunks | rw [← singleton_union, ← diff_subset_iff, diff_singleton_eq_self hfX] at hIs
exact hIs.trans diff_subset
exact heEI.2 (hIs _ hX' heX)
lemma closure_iInter_eq_iInter_closure_of_iUnion_indep [hι : Nonempty ι] (Is : ι → Set α)
(h : M.Indep (⋃ i, Is i)) : M.closure (⋂ i, Is i) = (⋂ i, M.closure (Is i)) := by
c... |
2,607 | lean_file_chunk | 5e3f77262b9fae5c33c547623814625d716f3129735e2c019446b2bf7fd3dfbd | lean_source_file | pkg_mathlib_source_file_chunks | -- Use a choice of basis and extensionality to change the goal to a statement about independence.
obtain ⟨I, hI⟩ := (M.comap f).exists_isBasis' X
obtain ⟨hI', hIinj, -⟩ := comap_isBasis'_iff.1 hI
simp_rw [← hI.closure_eq_closure, ← hI'.closure_eq_closure, Set.ext_iff,
hI.indep.mem_closure_iff', comap_ground_e... |
23,926 | lean_file_chunk | 28abec24d3e86786ea688b09e692042fa12521208ba804463642e3398d4b15d1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
module
public import Mathlib.Combinatorics.Matroid.Constructions
public import Mathlib.Data.Set.Notation
/-!
# Maps between matroids
This file defines maps and comaps,... |
8,734 | lean_file_chunk | d7ac8b33cf5cc8760a3af677ca9d2f74603295ba30706c2333d278b7583dc0d7 | lean_source_file | pkg_mathlib_source_file_chunks | @[simp] lemma mapSetEquiv_indep_iff (M : Matroid α) {E : Set β} (e : M.E ≃ E) {I : Set β} :
(M.mapSetEquiv e).Indep I ↔ M.Indep ↑(e.symm '' (E ↓∩ I)) ∧ I ⊆ E := Iff.rfl
@[simp] lemma mapSetEquiv.ground (M : Matroid α) {E : Set β} (e : M.E ≃ E) :
(M.mapSetEquiv e).E = E := rfl
end mapSetEquiv
section mapEmbedd... |
23,981 | lean_file_chunk | 3e24fccec2277d886f4e459981e70f3cd80c07abc64b5a528370e26be9cd1901 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
module
public import Mathlib.Combinatorics.Matroid.Init
public import Mathlib.Data.Finite.Prod
public import Mathlib.Data.Set.Card
public import Mathlib.Data.Set.Finite.... |
23,966 | lean_file_chunk | 24511c0c1404fbb3bf521342087c82e129d37213e1113908e7c7c67c3558f375 | lean_source_file | pkg_mathlib_source_file_chunks | rw [Dep, and_iff_left hX]
apply em
theorem Indep.not_dep (hI : M.Indep I) : ¬ M.Dep I :=
fun h ↦ h.1 hI
theorem Dep.not_indep (hD : M.Dep D) : ¬ M.Indep D :=
hD.1
theorem dep_of_not_indep (hD : ¬ M.Indep D) (hDE : D ⊆ M.E := by aesop_mat) : M.Dep D :=
⟨hD, hDE⟩
theorem indep_of_not_dep (hI : ¬ M.Dep I) (hIE... |
2,064 | lean_file_chunk | 3b3563e3536a87183c4dbaf4533d21fdfeadafde4ab90dcd4f49383ff6ac4e8b | lean_source_file | pkg_mathlib_source_file_chunks | (fun _ he ↦ hIX.mem_of_insert_indep he.2 (hJ.subset (insert_subset he.1 hIJ)))
theorem IsBasis'.inter_eq_of_subset_indep (hI : M.IsBasis' I X) (hIJ : I ⊆ J) (hJ : M.Indep J) :
J ∩ X = I := by
rw [← hI.isBasis_inter_ground.inter_eq_of_subset_indep hIJ hJ, inter_comm X, ← inter_assoc,
inter_eq_self_of_subset_l... |
10,222 | lean_file_chunk | b1f3bb01f05baa0f9da04d39d2d0ff28e0630a9f79ab363b16ba38dfce25e7ea | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
module
public import Mathlib.Combinatorics.Matroid.Minor.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, nam... |
23,991 | lean_file_chunk | 527a29d4c1fa6590ba0297a172dbaa6b58711027f2d9cb55d072217067dcde43 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
module
public import Mathlib.Combinatorics.Matroid.Circuit
public import Mathlib.Tactic.TFAE
/-!
# Matroid loops and coloops
## Loops
A 'loop' of a matroid `M` is an e... |
13,325 | lean_file_chunk | 88f5b3e16e7f8ad47b914b3ac7bb4e8849cb09adf37e729c99789ac9c2d29e01 | lean_source_file | pkg_mathlib_source_file_chunks | lemma isColoop_iff_forall_mem_closure_iff_mem' :
M.IsColoop e ↔ (∀ X, X ⊆ M.E → (e ∈ M.closure X ↔ e ∈ X)) ∧ e ∈ M.E := by
refine ⟨fun h ↦ ⟨fun X _ ↦ isColoop_iff_forall_mem_closure_iff_mem.1 h X, h.mem_ground⟩,
fun ⟨h, he⟩ ↦ isColoop_iff_forall_mem_closure_iff_mem.2 fun X ↦ ?_⟩
rw [← closure_inter_ground, ... |
23,976 | lean_file_chunk | 5c79720a296caea9907c8ba6ae9a8140298fc2b9a4b480279a37c591243acbef | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
module
public import Mathlib.Combinatorics.Matroid.Basic
public import Mathlib.Data.Set.Finite.Lattice
public import Mathlib.Order.Interval.Finset.Nat
/-!
# Matroid Ind... |
3,766 | lean_file_chunk | 83cc32d5b7f572de030421c927f10d671fed2e1023affdefbe102a968cc0a87c | lean_source_file | pkg_mathlib_source_file_chunks | IndepMatroid.mk (E := E) (Indep := Indep)
(indep_empty := by obtain ⟨M, -, rfl⟩ := hex; exact M.empty_indep)
(indep_subset := by obtain ⟨M, -, rfl⟩ := hex; exact fun I J hJ hIJ ↦ hJ.subset hIJ)
(indep_aug := by obtain ⟨M, -, rfl⟩ := hex; exact Indep.exists_insert_of_not_maximal M)
(indep_maximal := by obtain ⟨M... |
11,962 | lean_file_chunk | 1801d4f0287c2acd94fa3803f1040ba5b0483159ea1e2465567000dc51dddfc4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
module
public import Mathlib.Combinatorics.Matroid.IndepAxioms
/-!
# Matroid Duality
For a matroid `M` on ground set `E`, the collection of complements of the bases of... |
23,969 | lean_file_chunk | 1321fa7f320bd986ba1ec429b15b2074923b06027d9ea089b5a346fd3930de5d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Walk.Decomp
public import Mathlib.Combinatorics.SimpleGraph.Walk.Maps
public import Mathlib.Combinatorics.SimpleGrap... |
11,333 | lean_file_chunk | e35b3bdfbbb636ac46da9ca4a2294da655508d8508dbf54ec32b84d4d5df016d | lean_source_file | pkg_mathlib_source_file_chunks | /-- Taking a strict initial segment of a path removes the end vertex from the support. -/
lemma endpoint_notMem_support_takeUntil {p : G.Walk u v} (hp : p.IsPath) (hw : w ∈ p.support)
(h : v ≠ w) : v ∉ (p.takeUntil w hw).support := by
intro hv
rw [Walk.mem_support_iff_exists_getVert] at hv
obtain ⟨n, ⟨hn, hnl... |
23,928 | lean_file_chunk | c645e9315eb9f573cdf592abe22c8d9fd46ccb15649d51db8cccca6699b23d6d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Maps
public import Mathlib.Data.Finset.Max
... |
1,820 | lean_file_chunk | ab5f6fe799bdc4310fdccb361d37c48bf4980f0c737b79ad3c5db95839932b05 | lean_source_file | pkg_mathlib_source_file_chunks | ((G.induce s).neighborFinset v).map (.subtype (· ∈ s)) = G.neighborFinset v ∩ s.toFinset := by
ext; simp
theorem map_neighborFinset_induce_of_neighborSet_subset {v : s} (h : G.neighborSet v ⊆ s) :
((G.induce s).neighborFinset v).map (.subtype (· ∈ s)) = G.neighborFinset v := by
classical
rwa [← Set.toFinset_... |
3,341 | lean_file_chunk | 7298793f3c47c51297827c7fe542ca2f5ae53a2bd07188ee835cdec793a76817 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Otte
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Clique
public import Mathlib.Combinatorics.SimpleGraph.Connectivity.Represents
public import Mathlib.Combinatorics.SimpleG... |
6,057 | lean_file_chunk | 50e2df2bf2ffec03da4ccc81495cfe696e0856aef89dc9fd42f1997a2ffa1a94 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Basic
public import Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring
public import Mathlib.Combinatorics.... |
7,593 | lean_file_chunk | 182d36312ff82502ad7786ff2b9c6514485b99a9cd90888b819383769a3ed920 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Joanna Choules. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joanna Choules
-/
module
public import Mathlib.CategoryTheory.CofilteredSystem
public import Mathlib.Combinatorics.SimpleGraph.Subgraph
/-!
# Homomorphisms from finite subgraphs
Thi... |
22,101 | lean_file_chunk | 97fcd916c846349b97760eec75e7ed72efe61f52d4af61cd1db70d16561650e1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 John Talbot and Lian Bremner Tattersall. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: John Talbot, Lian Bremner Tattersall
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring
public import Mathlib.Combinatorics.Sim... |
15,781 | lean_file_chunk | 63da353eded5f888074f4403079038713e2f83762f0c211bc3cccf47e2d88f98 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Walk.Counting
public import Mathlib.LinearAlgebra.Matrix.Symmetric
pu... |
8,255 | lean_file_chunk | a4df951936f728f072af3bc59133931c1993570b59d396e0451a7bc7f97d6c76 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Vlad Tsyrklevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vlad Tsyrklevich
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Clique
public import Mathlib.Data.ENat.Lattice
public import Mathlib.Data.Set.Card
import Mathlib.Data.Fi... |
6,499 | lean_file_chunk | 3c7c015b4c6e033b0b96d8423652864776f65a256b1b970f7c56ceaedccc5a97 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Iván Renison, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison, Bhavik Mehta
-/
module
public import Mathlib.Algebra.Group.Fin.Basic
public import Mathlib.Combinatorics.SimpleGraph.Hasse
public import Mathlib.Algebra.Grou... |
3,992 | lean_file_chunk | c90bbfeb3e5805dd2fde11cbb0c478e14f1e7b5abdb58605510ab9f3d59d6b32 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Basic
public import Mathlib.Data.Fintype.Sigma
/-!
# Darts in graphs
A `Dart` or half-edge or bond in a graph is a... |
14,992 | lean_file_chunk | 1769894c8b07229892683c27eac9ab44fad814379601e7a731d464f1333d4373 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Vincent Beffara, Rida Hamadani
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
public import Mathlib.Data.ENat.Lattice
/-!
# Graph metric
... |
23,965 | lean_file_chunk | 9123e3ae319190f2f0dd77eb06018925d872daf043ed54a1958a9837419b487f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Bipartite
public import Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph
public import Mathlib.Combinatorics.... |
6,867 | lean_file_chunk | 80e12ead1701e0ec2a82803dd0378915940d1bea8b69eb4e6cd8269121e3b670 | lean_source_file | pkg_mathlib_source_file_chunks | obtain ⟨u, adj_vu, hu⟩ := degree_eq_one_iff_existsUnique_adj.mp hdeg
refine (connected_iff _).mpr ⟨?_, u, by aesop⟩
/- There exists a walk between any two vertices w and x in G.induce {v}ᶜ
via the unique vertex u adjacent to vertex v. -/
intro w x
obtain ⟨pwu, hpwu⟩ := hconn.exists_isPath w u
obtain ⟨pux, h... |
3,215 | lean_file_chunk | c4af93d1c127ffcc5674504eec11c84a1ddef09c3a0775cac7a5c69cbe54099f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Snir Broshi
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Copy
/-!
# LineGraph
## Main definitions
* `SimpleGraph.lineGraph` is the line graph of a simple g... |
23,996 | lean_file_chunk | c4d1c74a8d5b9124f845184991314a6b3e66ea28c22db63d61e4a352c8d92176 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Dart
public import Mathlib.Data.FunLike.Fintype
public import Mathlib.Logic.Embedding.Set
/-!
# Ma... |
1,085 | lean_file_chunk | b720d03b0230d2ad9ac748a87f2d67ce3454d88cf92b096a7b1dfa9cd8ea06cf | lean_source_file | pkg_mathlib_source_file_chunks | f.trans f'
@[simp]
theorem coe_comp (f' : G' ≃g G'') (f : G ≃g G') : ⇑(f'.comp f) = f' ∘ f :=
rfl
end Iso
/-- The graph induced on `Set.univ` is isomorphic to the original graph. -/
@[simps!]
def induceUnivIso (G : SimpleGraph V) : G.induce Set.univ ≃g G where
toEquiv := Equiv.Set.univ V
map_rel_iff' := by sim... |
3,265 | lean_file_chunk | 10dd0d695aac1e8ed2f00fcf84b0e80dfabc746392223d2ff95ae8bcffedf6bc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Acyclic
public import Mathlib.Data.ENat.Lattice
/-!
# Girth of a simple graph
This file defines the girth and th... |
23,962 | lean_file_chunk | eff2b1b63214e02d5359dee5283a8f22e29fa5d5342ba6c5ad725ea879165cf8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Alena Gusakov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Arthur Paulino, Kyle Miller, Pim Otte
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Clique
public import Mathlib.Combinatorics.SimpleGraph.Connectivity.Finit... |
2,838 | lean_file_chunk | be678b45ec33fc75332da15f7a55fd068e1d48ffea73a5faf445e61815adbcd0 | lean_source_file | pkg_mathlib_source_file_chunks | lemma IsAlternating.sup_edge {u x : V} (halt : G.IsAlternating G') (hnadj : ¬G'.Adj u x)
(hu' : ∀ u', u' ≠ u → G.Adj x u' → G'.Adj x u')
(hx' : ∀ x', x' ≠ x → G.Adj x' u → G'.Adj x' u) : (G ⊔ edge u x).IsAlternating G' := by
by_cases hadj : G.Adj u x
· rwa [sup_edge_of_adj G hadj]
intro v w w' hww' hvw hv... |
531 | lean_file_chunk | d4dd4d4fa67836aab830c42b8c742580422253ecf1eccbb3e2dfe1d2151d0274 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Jannis Limperg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jannis Limperg
-/
module
public import Mathlib.Init
public import Aesop
/-!
# SimpleGraph Rule Set
This module defines the `SimpleGraph` Aesop rule set which is used by the
`aesop_g... |
11,333 | lean_file_chunk | df437a55321f922598326917246fbeb53980b39b0085673938d8449f60fb216f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 George Peter Banyard, Yaël Dillies, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: George Peter Banyard, Yaël Dillies, Kyle Miller
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Metric
public import Mathlib.Combinatorics.S... |
10,482 | lean_file_chunk | bda1249961ee8f45ccd7f9932be18c085634682abc83a66c81d27a06c586b1f5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
module
public import Mathlib.Algebra.Ring.Defs
public import Mathlib.Combinatorics.... |
4,944 | lean_file_chunk | fcad2a073514fe7088dcbb6e5db46212dd9d584dc101105da0c5fa245abc3554 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph
public import Mathlib.Combinatorics.SimpleGraph.Copy
public import Mathlib.Combinatorics.Sim... |
17,550 | lean_file_chunk | aa1c13c0e276029b6866635881877145be2e3b2cc60e0f6b634a9bf5fa7f6ab4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Otte
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Matching
public import Mathlib.Combinatorics.SimpleGraph.Metric
public import Mathlib.Combinatorics.SimpleGraph.Operations... |
22,370 | lean_file_chunk | 437362007264f0fac51a918e369f9deabc0ee6ec0ffa7073894803a2d3a86eff | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 John Talbot and Lian Bremner Tattersall. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: John Talbot, Lian Bremner Tattersall
-/
module
public import Mathlib.Algebra.BigOperators.Ring.Finset
public import Mathlib.Algebra.Order.BigOperators.Group.F... |
15,723 | lean_file_chunk | 7232c62b3c58e0e12fe1c083e92ab860a4e818191ba6df91969fca882d6c3674 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
module
public import Mathlib.Algebra.Order.Field.Basic
public import Mathlib.Combinatorics.SimpleGraph.Basic
public import Mathlib.Data.Rat.C... |
23,943 | lean_file_chunk | 683b4257de03b42a797e5b9ab7a59b9d93f6f920634e9f7056b1607fbd07a45f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Init
public import Mathlib.D... |
9,055 | lean_file_chunk | 263d0e57df53f267fe61d29a012d2e2b00b2b17e70065f5b6db2830e4517fae2 | lean_source_file | pkg_mathlib_source_file_chunks | conv_rhs => rw [← Set.diff_union_inter s Sym2.diagSet]
rw [← disjoint_edgeSet, edgeSet_fromEdgeSet]
grind [edgeSet_subset_compl_diagSet]
@[simp] lemma fromEdgeSet_disjoint : Disjoint (fromEdgeSet s) G ↔ Disjoint s G.edgeSet := by
rw [disjoint_comm, disjoint_fromEdgeSet, disjoint_comm]
instance [DecidableEq V] [... |
6,784 | lean_file_chunk | 3e22738b76d89df705137a453245a0d373d4f29329b54ba24a97b25a24859421 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Gabriel Moise. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Moise, Yaël Dillies, Kyle Miller
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Finite
public import Mathlib.Data.Finset.Sym
public import Mathlib.Data.Matrix.Mul
/... |
8,769 | lean_file_chunk | 346e010e08542eeab14547c52d0dc40bbead94e3afc3d0ba51b5c5d9ed451cfc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Finite
public import Mathlib.Combinatorics.SimpleGraph.Maps
public import Mathlib.Combinatorics.SimpleGraph.Subgraph
... |
5,751 | lean_file_chunk | 27b0a3b360cd4d5e30f5234c43c529a765eadc939fa2da74561334f195df9c46 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
module
public import Mathlib.Algebra.Ring.Parity
public import Mathlib.Combinatorics.SimpleGraph.Paths
/-!
# Trails and Eulerian trails
This module contains additional ... |
5,718 | lean_file_chunk | 933b1a29180b2c422b966e6947ba1f7061620cee37cf32cb1014611f5c1bbed8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Vlad Tsyrklevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vlad Tsyrklevich
-/
module
public import Mathlib.Combinatorics.Hall.Basic
public import Mathlib.Combinatorics.SimpleGraph.Bipartite
public import Mathlib.Combinatorics.SimpleGraph.M... |
7,033 | lean_file_chunk | 5e7520fc5202c58e659726852ea2ebfe7fd17b3c8145dd5d616136373b324338 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Olivia Röhrig
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Maps
public import Mathlib.Combinatorics.SimpleGraph.Finite
public import Mathlib.Data.Fintype.BigO... |
9,470 | lean_file_chunk | fc92a7ae4da67e852f7fa265a064e64d6740eec887ecd7473f13d26496873136 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Alena Gusakov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Jeremy Tan
-/
module
public import Mathlib.Combinatorics.Enumerative.DoubleCounting
public import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
/-!
# Strongly regular gra... |
16,359 | lean_file_chunk | d85a1f200f5fec5a36860188cb17d99dc118a41e438b7fad60741282835c4c0a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Rida Hamadani. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rida Hamadani
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Metric
/-!
# Diameter of a simple graph
This module defines the eccentricity of vertices, the diameter, and th... |
11,726 | lean_file_chunk | af7dfd235710ddc7454d481334a9428408f1db7205fdd4834761a77a13bd225a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Bhavik Mehta, Rishi Mehta, Linus Sommer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Rishi Mehta, Linus Sommer, Yue Sun
-/
module
public import Mathlib.Algebra.GroupWithZero.Nat
public import Mathlib.Algebra.Order.Group.Nat
publi... |
306 | lean_file_chunk | 6c1d028926e4a4b24fe4e7f3b287cc8b9b495be8ed3665520b194856b6d6d617 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
module -- shake: keep-all
public import Mathlib.Combinatorics.SimpleGraph.Init
public import Mathlib.Data.Sym.Sym2
deprecated_module (since := "2025-11-12") |
4,840 | lean_file_chunk | 35590d08c0f001d2d7830b312c40a9dce98e41ea1ed0e2f98985378a4bdbeb8e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Edward van de Meent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Edward van de Meent
-/
module
public import Mathlib.Algebra.Group.Pointwise.Set.Basic
public import Mathlib.Combinatorics.SimpleGraph.Basic
/-!
### Definition of Cayley graphs
... |
23,938 | lean_file_chunk | f4c09b731afa7b47a5819b2767c0dfe983d36241dc8882aa66c23e206567dc13 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yaël Dillies, Mitchell Horner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Mitchell Horner
-/
module
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Combinatorics.SimpleGraph.Subgraph
/-!
# Containment of gra... |
2,274 | lean_file_chunk | 92895747c1045eaf03a9c635c4d936e9a24cbabba238cb055527a54e2104211f | lean_source_file | pkg_mathlib_source_file_chunks | let e' : Sym2 G'.verts := Sym2.map (Copy.isoSubgraphMap (.ofLE _ _ _) _).symm e
have he' : e' ∈ G'.coe.edgeSet := (Iso.map_mem_edgeSet_iff _).2 he₀
rw [Subgraph.edgeSet_coe] at he'
have := Subgraph.edgeSet_subset _ he'
simp only [edgeSet_sdiff, edgeSet_fromEdgeSet, edgeSet_sdiff_sdiff_isDiag, Set.mem_diff,
... |
6,293 | lean_file_chunk | 0d0909d7b9b87bfc088f981432fd6f023ae437f4184ee882a2bdf2af6bfc4a68 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
module
public import Mathlib.Algebra.BigOperators.Ring.Finset
public import Mathlib.Combinatorics.SimpleGraph.Dart
public import Mathlib.Combinatorics.SimpleGraph.Finite
p... |
5,400 | lean_file_chunk | 3c71be74e01620e1ae2a5e9220612bfa27182d5d034fbd4fa6b461db44eaf012 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
module
public import Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring
/-!
# Graph partitions
This module provides an interface for dealing wi... |
23,951 | lean_file_chunk | 5dfe8b742d6cfe92d8c7d7693372090564e6d88e001650f58bbb04bc276742a3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
module
public import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
public import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple... |
23,947 | lean_file_chunk | 4356ab4f23e302a67890d47b0db655c09e1d323ec1380de7354f46bfcc5095c1 | lean_source_file | pkg_mathlib_source_file_chunks | @[simp] lemma comap_equiv_top {H : SimpleGraph W} (f : G →g H) : Subgraph.comap f ⊤ = ⊤ := by
ext <;> simp +contextual [f.map_adj]
theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) :
H.map f ≤ H' ↔ H ≤ H'.comap f := by
refine ⟨fun h ↦ ⟨fun v hv ↦ ?_, fun v w hvw... |
754 | lean_file_chunk | a435a9bbec35c1fdc09e8e24101a52609cbb14bb14c58fb3bdee2a45677feb97 | lean_source_file | pkg_mathlib_source_file_chunks | DecidableRel ((⊤ : G.Subgraph).deleteVerts u).coe.Adj :=
fun x y =>
if h : G.Adj x y
then
.isTrue <| SimpleGraph.Subgraph.Adj.coe <| Subgraph.deleteVerts_adj.mpr
⟨by trivial, x.2.2, by trivial, y.2.2, h⟩
else
.isFalse <| fun hadj ↦ h <| Subgraph.coe_adj_sub _ _ _ hadj
set_option backw... |
9,310 | lean_file_chunk | d75de68d6c79580a668b04b1e8d20bc9f36670b1ffae5d865674627f9c3b7543 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Adrian Wüthrich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adrian Wüthrich
-/
module
public import Mathlib.Analysis.Matrix.Order
public import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
public import Mathlib.Combinatorics.SimpleGraph.Connec... |
23,993 | lean_file_chunk | 9e4c4e85b4e6d14bd71448b2ae0cac336b53faf37eebe75e51181c5bb11f8f01 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Mitchell Horner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Horner
-/
module
public import Mathlib.Algebra.Notation.Indicator
public import Mathlib.Combinatorics.Enumerative.DoubleCounting
public import Mathlib.Combinatorics.SimpleGr... |
3,882 | lean_file_chunk | eb310563718f174cfa298070f5c6370c827a9cfaf342f834d7dddfeab7780b20 | lean_source_file | pkg_mathlib_source_file_chunks | | .inl _, .inl _ | .inr _, .inr _ => by simp at hadj
/-- The bipartite double cover of `G` has twice the number of edges as `G`. -/
theorem card_edgeFinset_bipartiteDoubleCover [Fintype V] [DecidableRel G.Adj] :
#G.bipartiteDoubleCover.edgeFinset = 2 * #G.edgeFinset := by
rw [two_mul_card_edgeFinset, eq_comm]
... |
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