state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
H H' : Subgraph G
x✝¹ x✝ : V
h : Adj H x✝¹ x✝
⊢ x✝ ∈ H.verts ∧ x✝¹ ∈ H.verts ∧ Adj H' x✝¹ x✝ ↔ Adj H' x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp [H.edge_vert h, H.edge_vert h.symm] | lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) :
Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by
ext
· simp [and_comm]
· simp_rw [coeSubgraph_adj, restrict_adj]
simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff]
intro h
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1034_0.BlhiAiIDADcXv8t | lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) :
Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
a b : V
⊢ (G'.Adj \ Sym2.ToRel s) a b → (G'.Adj \ Sym2.ToRel s) b a | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp [G'.adj_comm, Sym2.eq_swap] | /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges
from its edge set, if present.
See also: `SimpleGraph.deleteEdges`. -/
def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where
verts := G'.verts
Adj := G'.Adj \ Sym2.ToRel s
adj_sub h' := G'.adj_sub h'.1
... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1046_0.BlhiAiIDADcXv8t | /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges
from its edge set, if present.
See also: `SimpleGraph.deleteEdges`. -/
def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where
verts | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ s s' : Set (Sym2 V)
⊢ deleteEdges (deleteEdges G' s) s' = deleteEdges G' (s ∪ s') | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | ext | @[simp]
theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) :
(G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1072_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) :
(G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ s s' : Set (Sym2 V)
x✝ : V
⊢ x✝ ∈ (deleteEdges (deleteEdges G' s) s').verts ↔ x✝ ∈ (deleteEdges G' (s ∪ s')).verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp [and_assoc, not_or] | @[simp]
theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) :
(G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1072_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) :
(G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ s s' : Set (Sym2 V)
x✝¹ x✝ : V
⊢ Adj (deleteEdges (deleteEdges G' s) s') x✝¹ x✝ ↔ Adj (deleteEdges G' (s ∪ s')) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp [and_assoc, not_or] | @[simp]
theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) :
(G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1072_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) :
(G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
⊢ deleteEdges G' ∅ = G' | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | ext | @[simp]
theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1078_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
x✝ : V
⊢ x✝ ∈ (deleteEdges G' ∅).verts ↔ x✝ ∈ G'.verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp | @[simp]
theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1078_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
x✝¹ x✝ : V
⊢ Adj (deleteEdges G' ∅) x✝¹ x✝ ↔ Adj G' x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp | @[simp]
theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1078_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
⊢ SimpleGraph.deleteEdges (Subgraph.spanningCoe G') s = Subgraph.spanningCoe (deleteEdges G' s) | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | ext | @[simp]
theorem deleteEdges_spanningCoe_eq :
G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1083_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteEdges_spanningCoe_eq :
G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
x✝¹ x✝ : V
⊢ SimpleGraph.Adj (SimpleGraph.deleteEdges (Subgraph.spanningCoe G') s) x✝¹ x✝ ↔
SimpleGraph.Adj (Subgraph.spanningCoe (deleteEdges G' s)) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp | @[simp]
theorem deleteEdges_spanningCoe_eq :
G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by
ext
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1083_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteEdges_spanningCoe_eq :
G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ : Set (Sym2 V)
s : Set (Sym2 ↑G'.verts)
⊢ SimpleGraph.deleteEdges (Subgraph.coe G') s = Subgraph.coe (deleteEdges G' (Sym2.map Subtype.val '' s)) | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | ext ⟨v, hv⟩ ⟨w, hw⟩ | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.mk.h.mk.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ : Set (Sym2 V)
s : Set (Sym2 ↑G'.verts)
v : V
hv : v ∈ G'.verts
w : V
hw : w ∈ G'.verts
⊢ SimpleGraph.Adj (SimpleGraph.deleteEdges (Subgraph.coe G') s) { val := v, property := hv }
{ val := w, property := hw } ↔
Simpl... | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff] | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by
ext ⟨v, hv⟩ ⟨w, hw⟩
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.mk.h.mk.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ : Set (Sym2 V)
s : Set (Sym2 ↑G'.verts)
v : V
hv : v ∈ G'.verts
w : V
hw : w ∈ G'.verts
⊢ Adj G' v w →
(⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s ↔ ∀ x ∈ s, ¬Sym2.map Subtype.val x = ⟦(v, w)⟧) | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | intro | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.mk.h.mk.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ : Set (Sym2 V)
s : Set (Sym2 ↑G'.verts)
v : V
hv : v ∈ G'.verts
w : V
hw : w ∈ G'.verts
a✝ : Adj G' v w
⊢ ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s ↔ ∀ x ∈ s, ¬Sym2.map Subtype.val x = ⟦(v, w)⟧ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | constructor | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
intro
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.mk.h.mk.a.mp
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ : Set (Sym2 V)
s : Set (Sym2 ↑G'.verts)
v : V
hv : v ∈ G'.verts
w : V
hw : w ∈ G'.verts
a✝ : Adj G' v w
⊢ ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s → ∀ x ∈ s, ¬Sym2.map Subtype.val x = ⟦(v, w)⟧ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | intro hs | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
intro
constructor
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.mk.h.mk.a.mp
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ : Set (Sym2 V)
s : Set (Sym2 ↑G'.verts)
v : V
hv : v ∈ G'.verts
w : V
hw : w ∈ G'.verts
a✝ : Adj G' v w
hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s
⊢ ∀ x ∈ s, ¬Sym2.map Subtype.val x = ⟦(v, w)⟧ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | refine' Sym2.ind _ | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
intro
constructor
· intro hs
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.mk.h.mk.a.mp
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ : Set (Sym2 V)
s : Set (Sym2 ↑G'.verts)
v : V
hv : v ∈ G'.verts
w : V
hw : w ∈ G'.verts
a✝ : Adj G' v w
hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s
⊢ ∀ (x y : ↑G'.verts), ⟦(x, y)⟧ ∈ s → ¬Sym2.map ... | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | rintro ⟨v', hv'⟩ ⟨w', hw'⟩ | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
intro
constructor
· intro hs
r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.mk.h.mk.a.mp.mk.mk
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ : Set (Sym2 V)
s : Set (Sym2 ↑G'.verts)
v : V
hv : v ∈ G'.verts
w : V
hw : w ∈ G'.verts
a✝ : Adj G' v w
hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s
v' : V
hv' : v' ∈ G'.verts
w' : V
hw' : w'... | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp only [Sym2.map_pair_eq, Quotient.eq] | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
intro
constructor
· intro hs
r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.mk.h.mk.a.mp.mk.mk
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ : Set (Sym2 V)
s : Set (Sym2 ↑G'.verts)
v : V
hv : v ∈ G'.verts
w : V
hw : w ∈ G'.verts
a✝ : Adj G' v w
hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s
v' : V
hv' : v' ∈ G'.verts
w' : V
hw' : w'... | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | contrapose! | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
intro
constructor
· intro hs
r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.mk.h.mk.a.mp.mk.mk
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ : Set (Sym2 V)
s : Set (Sym2 ↑G'.verts)
v : V
hv : v ∈ G'.verts
w : V
hw : w ∈ G'.verts
a✝ : Adj G' v w
hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s
v' : V
hv' : v' ∈ G'.verts
w' : V
hw' : w'... | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | rintro (_ | _) | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
intro
constructor
· intro hs
r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.mk.h.mk.a.mp.mk.mk.refl
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ : Set (Sym2 V)
s : Set (Sym2 ↑G'.verts)
v : V
hv : v ∈ G'.verts
w : V
hw : w ∈ G'.verts
a✝ : Adj G' v w
hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s
hv' : v ∈ G'.verts
hw' : w ∈ G'.verts... | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simpa only [Sym2.eq_swap] | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
intro
constructor
· intro hs
r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.mk.h.mk.a.mp.mk.mk.swap
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ : Set (Sym2 V)
s : Set (Sym2 ↑G'.verts)
v : V
hv : v ∈ G'.verts
w : V
hw : w ∈ G'.verts
a✝ : Adj G' v w
hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s
hw' : v ∈ G'.verts
hv' : w ∈ G'.verts... | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simpa only [Sym2.eq_swap] | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
intro
constructor
· intro hs
r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.mk.h.mk.a.mpr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ : Set (Sym2 V)
s : Set (Sym2 ↑G'.verts)
v : V
hv : v ∈ G'.verts
w : V
hw : w ∈ G'.verts
a✝ : Adj G' v w
⊢ (∀ x ∈ s, ¬Sym2.map Subtype.val x = ⟦(v, w)⟧) → ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | intro h' hs | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
intro
constructor
· intro hs
r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.mk.h.mk.a.mpr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ : Set (Sym2 V)
s : Set (Sym2 ↑G'.verts)
v : V
hv : v ∈ G'.verts
w : V
hw : w ∈ G'.verts
a✝ : Adj G' v w
h' : ∀ x ∈ s, ¬Sym2.map Subtype.val x = ⟦(v, w)⟧
hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∈ ... | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | exact h' _ hs rfl | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
intro
constructor
· intro hs
r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ s : Set (Sym2 V)
⊢ Subgraph.coe (deleteEdges G' s) = SimpleGraph.deleteEdges (Subgraph.coe G') (Sym2.map Subtype.val ⁻¹' s) | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | ext ⟨v, hv⟩ ⟨w, hw⟩ | theorem coe_deleteEdges_eq (s : Set (Sym2 V)) :
(G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1107_0.BlhiAiIDADcXv8t | theorem coe_deleteEdges_eq (s : Set (Sym2 V)) :
(G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.mk.h.mk.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ s : Set (Sym2 V)
v : V
hv : v ∈ (deleteEdges G' s).verts
w : V
hw : w ∈ (deleteEdges G' s).verts
⊢ SimpleGraph.Adj (Subgraph.coe (deleteEdges G' s)) { val := v, property := hv } { val := w, property := hw } ↔
SimpleGraph.Ad... | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp | theorem coe_deleteEdges_eq (s : Set (Sym2 V)) :
(G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by
ext ⟨v, hv⟩ ⟨w, hw⟩
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1107_0.BlhiAiIDADcXv8t | theorem coe_deleteEdges_eq (s : Set (Sym2 V)) :
(G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
⊢ deleteEdges G' s ≤ G' | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | constructor | theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1113_0.BlhiAiIDADcXv8t | theorem deleteEdges_le : G'.deleteEdges s ≤ G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case left
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
⊢ (deleteEdges G' s).verts ⊆ G'.verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) [subset_rfl] | theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by
constructor <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1113_0.BlhiAiIDADcXv8t | theorem deleteEdges_le : G'.deleteEdges s ≤ G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
⊢ ∀ ⦃v w : V⦄, Adj (deleteEdges G' s) v w → Adj G' v w | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) [subset_rfl] | theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by
constructor <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1113_0.BlhiAiIDADcXv8t | theorem deleteEdges_le : G'.deleteEdges s ≤ G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ s s' : Set (Sym2 V)
h : s ⊆ s'
⊢ deleteEdges G' s' ≤ deleteEdges G' s | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | constructor | theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') :
G'.deleteEdges s' ≤ G'.deleteEdges s := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1117_0.BlhiAiIDADcXv8t | theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') :
G'.deleteEdges s' ≤ G'.deleteEdges s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case left
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ s s' : Set (Sym2 V)
h : s ⊆ s'
⊢ (deleteEdges G' s').verts ⊆ (deleteEdges G' s).verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj,
true_and_iff, and_imp, subset_rfl] | theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') :
G'.deleteEdges s' ≤ G'.deleteEdges s := by
constructor <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1117_0.BlhiAiIDADcXv8t | theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') :
G'.deleteEdges s' ≤ G'.deleteEdges s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ s s' : Set (Sym2 V)
h : s ⊆ s'
⊢ ∀ ⦃v w : V⦄, Adj (deleteEdges G' s') v w → Adj (deleteEdges G' s) v w | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj,
true_and_iff, and_imp, subset_rfl] | theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') :
G'.deleteEdges s' ≤ G'.deleteEdges s := by
constructor <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1117_0.BlhiAiIDADcXv8t | theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') :
G'.deleteEdges s' ≤ G'.deleteEdges s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ s s' : Set (Sym2 V)
h : s ⊆ s'
⊢ ∀ ⦃v w : V⦄, Adj G' v w → ⟦(v, w)⟧ ∉ s' → ⟦(v, w)⟧ ∉ s | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | exact fun _ _ _ hs' hs ↦ hs' (h hs) | theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') :
G'.deleteEdges s' ≤ G'.deleteEdges s := by
constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj,
true_and_iff, and_imp, subset_rfl]
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1117_0.BlhiAiIDADcXv8t | theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') :
G'.deleteEdges s' ≤ G'.deleteEdges s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
⊢ deleteEdges G' (edgeSet G' ∩ s) = deleteEdges G' s | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | ext | @[simp]
theorem deleteEdges_inter_edgeSet_left_eq :
G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1124_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteEdges_inter_edgeSet_left_eq :
G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
x✝ : V
⊢ x✝ ∈ (deleteEdges G' (edgeSet G' ∩ s)).verts ↔ x✝ ∈ (deleteEdges G' s).verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) [imp_false] | @[simp]
theorem deleteEdges_inter_edgeSet_left_eq :
G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1124_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteEdges_inter_edgeSet_left_eq :
G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
x✝¹ x✝ : V
⊢ Adj (deleteEdges G' (edgeSet G' ∩ s)) x✝¹ x✝ ↔ Adj (deleteEdges G' s) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) [imp_false] | @[simp]
theorem deleteEdges_inter_edgeSet_left_eq :
G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1124_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteEdges_inter_edgeSet_left_eq :
G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
⊢ deleteEdges G' (s ∩ edgeSet G') = deleteEdges G' s | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | ext | @[simp]
theorem deleteEdges_inter_edgeSet_right_eq :
G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1130_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteEdges_inter_edgeSet_right_eq :
G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
x✝ : V
⊢ x✝ ∈ (deleteEdges G' (s ∩ edgeSet G')).verts ↔ x✝ ∈ (deleteEdges G' s).verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) [imp_false] | @[simp]
theorem deleteEdges_inter_edgeSet_right_eq :
G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1130_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteEdges_inter_edgeSet_right_eq :
G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
x✝¹ x✝ : V
⊢ Adj (deleteEdges G' (s ∩ edgeSet G')) x✝¹ x✝ ↔ Adj (deleteEdges G' s) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) [imp_false] | @[simp]
theorem deleteEdges_inter_edgeSet_right_eq :
G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1130_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteEdges_inter_edgeSet_right_eq :
G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
⊢ Subgraph.coe (deleteEdges G' s) ≤ Subgraph.coe G' | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | intro v w | theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1136_0.BlhiAiIDADcXv8t | theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set (Sym2 V)
v w : ↑(deleteEdges G' s).verts
⊢ SimpleGraph.Adj (Subgraph.coe (deleteEdges G' s)) v w → SimpleGraph.Adj (Subgraph.coe G') v w | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) | theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by
intro v w
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1136_0.BlhiAiIDADcXv8t | theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
s : Set V
⊢ SimpleGraph.induce s G = Subgraph.coe (induce ⊤ s) | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | ext | theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) :
G.induce s = ((⊤ : G.Subgraph).induce s).coe := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1166_0.BlhiAiIDADcXv8t | theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) :
G.induce s = ((⊤ : G.Subgraph).induce s).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
s : Set V
x✝¹ x✝ : ↑s
⊢ SimpleGraph.Adj (SimpleGraph.induce s G) x✝¹ x✝ ↔ SimpleGraph.Adj (Subgraph.coe (induce ⊤ s)) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp | theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) :
G.induce s = ((⊤ : G.Subgraph).induce s).coe := by
ext
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1166_0.BlhiAiIDADcXv8t | theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) :
G.induce s = ((⊤ : G.Subgraph).induce s).coe | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
hg : G' ≤ G''
hs : s ⊆ s'
⊢ induce G' s ≤ induce G'' s' | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | constructor | theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1176_0.BlhiAiIDADcXv8t | theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case left
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
hg : G' ≤ G''
hs : s ⊆ s'
⊢ (induce G' s).verts ⊆ (induce G'' s').verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp [hs] | theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by
constructor
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.1176_0.BlhiAiIDADcXv8t | theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
hg : G' ≤ G''
hs : s ⊆ s'
⊢ ∀ ⦃v w : V⦄, Adj (induce G' s) v w → Adj (induce G'' s') v w | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] | theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by
constructor
· simp [hs]
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.1176_0.BlhiAiIDADcXv8t | theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
hg : G' ≤ G''
hs : s ⊆ s'
⊢ ∀ ⦃v w : V⦄, v ∈ s → w ∈ s → Adj G' v w → v ∈ s' ∧ w ∈ s' ∧ Adj G'' v w | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | intro v w hv hw ha | theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by
constructor
· simp [hs]
· simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp]
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1176_0.BlhiAiIDADcXv8t | theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
hg : G' ≤ G''
hs : s ⊆ s'
v w : V
hv : v ∈ s
hw : w ∈ s
ha : Adj G' v w
⊢ v ∈ s' ∧ w ∈ s' ∧ Adj G'' v w | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | exact ⟨hs hv, hs hw, hg.2 ha⟩ | theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by
constructor
· simp [hs]
· simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp]
intro v w hv hw ha
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1176_0.BlhiAiIDADcXv8t | theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
⊢ induce G' ∅ = ⊥ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | ext | @[simp]
theorem induce_empty : G'.induce ∅ = ⊥ := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1194_0.BlhiAiIDADcXv8t | @[simp]
theorem induce_empty : G'.induce ∅ = ⊥ | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
x✝ : V
⊢ x✝ ∈ (induce G' ∅).verts ↔ x✝ ∈ ⊥.verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp | @[simp]
theorem induce_empty : G'.induce ∅ = ⊥ := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1194_0.BlhiAiIDADcXv8t | @[simp]
theorem induce_empty : G'.induce ∅ = ⊥ | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
x✝¹ x✝ : V
⊢ Adj (induce G' ∅) x✝¹ x✝ ↔ Adj ⊥ x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp | @[simp]
theorem induce_empty : G'.induce ∅ = ⊥ := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1194_0.BlhiAiIDADcXv8t | @[simp]
theorem induce_empty : G'.induce ∅ = ⊥ | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
⊢ induce G' G'.verts = G' | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | ext | @[simp]
theorem induce_self_verts : G'.induce G'.verts = G' := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1199_0.BlhiAiIDADcXv8t | @[simp]
theorem induce_self_verts : G'.induce G'.verts = G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
x✝ : V
⊢ x✝ ∈ (induce G' G'.verts).verts ↔ x✝ ∈ G'.verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp | @[simp]
theorem induce_self_verts : G'.induce G'.verts = G' := by
ext
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.1199_0.BlhiAiIDADcXv8t | @[simp]
theorem induce_self_verts : G'.induce G'.verts = G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
x✝¹ x✝ : V
⊢ Adj (induce G' G'.verts) x✝¹ x✝ ↔ Adj G' x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | constructor | @[simp]
theorem induce_self_verts : G'.induce G'.verts = G' := by
ext
· simp
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.1199_0.BlhiAiIDADcXv8t | @[simp]
theorem induce_self_verts : G'.induce G'.verts = G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mp
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
x✝¹ x✝ : V
⊢ Adj (induce G' G'.verts) x✝¹ x✝ → Adj G' x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] | @[simp]
theorem induce_self_verts : G'.induce G'.verts = G' := by
ext
· simp
· constructor <;>
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1199_0.BlhiAiIDADcXv8t | @[simp]
theorem induce_self_verts : G'.induce G'.verts = G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
x✝¹ x✝ : V
⊢ Adj G' x✝¹ x✝ → Adj (induce G' G'.verts) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] | @[simp]
theorem induce_self_verts : G'.induce G'.verts = G' := by
ext
· simp
· constructor <;>
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1199_0.BlhiAiIDADcXv8t | @[simp]
theorem induce_self_verts : G'.induce G'.verts = G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
x✝¹ x✝ : V
⊢ Adj G' x✝¹ x✝ → x✝¹ ∈ G'.verts ∧ x✝ ∈ G'.verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ | @[simp]
theorem induce_self_verts : G'.induce G'.verts = G' := by
ext
· simp
· constructor <;>
simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff]
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1199_0.BlhiAiIDADcXv8t | @[simp]
theorem induce_self_verts : G'.induce G'.verts = G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
⊢ induce G' s ⊔ induce G' s' ≤ induce G' (s ∪ s') | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | constructor | lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1212_0.BlhiAiIDADcXv8t | lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case left
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
⊢ (induce G' s ⊔ induce G' s').verts ⊆ (induce G' (s ∪ s')).verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp only [verts_sup, induce_verts, Set.Subset.rfl] | lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by
constructor
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.1212_0.BlhiAiIDADcXv8t | lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
⊢ ∀ ⦃v w : V⦄, Adj (induce G' s ⊔ induce G' s') v w → Adj (induce G' (s ∪ s')) v w | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp only [sup_adj, induce_adj, Set.mem_union] | lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by
constructor
· simp only [verts_sup, induce_verts, Set.Subset.rfl]
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.1212_0.BlhiAiIDADcXv8t | lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
⊢ ∀ ⦃v w : V⦄,
v ∈ s ∧ w ∈ s ∧ Adj G' v w ∨ v ∈ s' ∧ w ∈ s' ∧ Adj G' v w → (v ∈ s ∨ v ∈ s') ∧ (w ∈ s ∨ w ∈ s') ∧ Adj G' v w | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | rintro v w (h | h) | lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by
constructor
· simp only [verts_sup, induce_verts, Set.Subset.rfl]
· simp only [sup_adj, induce_adj, Set.mem_union]
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1212_0.BlhiAiIDADcXv8t | lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right.inl
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v w : V
h : v ∈ s ∧ w ∈ s ∧ Adj G' v w
⊢ (v ∈ s ∨ v ∈ s') ∧ (w ∈ s ∨ w ∈ s') ∧ Adj G' v w | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp [h] | lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by
constructor
· simp only [verts_sup, induce_verts, Set.Subset.rfl]
· simp only [sup_adj, induce_adj, Set.mem_union]
rintro v w (h | h) <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1212_0.BlhiAiIDADcXv8t | lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right.inr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v w : V
h : v ∈ s' ∧ w ∈ s' ∧ Adj G' v w
⊢ (v ∈ s ∨ v ∈ s') ∧ (w ∈ s ∨ w ∈ s') ∧ Adj G' v w | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp [h] | lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by
constructor
· simp only [verts_sup, induce_verts, Set.Subset.rfl]
· simp only [sup_adj, induce_adj, Set.mem_union]
rintro v w (h | h) <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1212_0.BlhiAiIDADcXv8t | lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
⊢ induce G' s ≤ induce G' (s ∪ s') | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | exact (sup_le_iff.mp le_induce_union).1 | lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1218_0.BlhiAiIDADcXv8t | lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
⊢ induce G' s' ≤ induce G' (s ∪ s') | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | exact (sup_le_iff.mp le_induce_union).2 | lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1221_0.BlhiAiIDADcXv8t | lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v : V
⊢ SimpleGraph.singletonSubgraph G v = induce ⊤ {v} | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | ext | theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} :=
by | Mathlib.Combinatorics.SimpleGraph.Subgraph.1224_0.BlhiAiIDADcXv8t | theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v x✝ : V
⊢ x✝ ∈ (SimpleGraph.singletonSubgraph G v).verts ↔ x✝ ∈ (induce ⊤ {v}).verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] | theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} :=
by ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1224_0.BlhiAiIDADcXv8t | theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v x✝¹ x✝ : V
⊢ Adj (SimpleGraph.singletonSubgraph G v) x✝¹ x✝ ↔ Adj (induce ⊤ {v}) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] | theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} :=
by ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1224_0.BlhiAiIDADcXv8t | theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v w : V
hvw : SimpleGraph.Adj G v w
⊢ subgraphOfAdj G hvw = induce ⊤ {v, w} | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | ext | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v w : V
hvw : SimpleGraph.Adj G v w
x✝ : V
⊢ x✝ ∈ (subgraphOfAdj G hvw).verts ↔ x✝ ∈ (induce ⊤ {v, w}).verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v w : V
hvw : SimpleGraph.Adj G v w
x✝¹ x✝ : V
⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ ↔ Adj (induce ⊤ {v, w}) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | constructor | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mp
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v w : V
hvw : SimpleGraph.Adj G v w
x✝¹ x✝ : V
⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ → Adj (induce ⊤ {v, w}) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | intro h | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mp
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v w : V
hvw : SimpleGraph.Adj G v w
x✝¹ x✝ : V
h : Adj (subgraphOfAdj G hvw) x✝¹ x✝
⊢ Adj (induce ⊤ {v, w}) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mp
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v w : V
hvw : SimpleGraph.Adj G v w
x✝¹ x✝ : V
h : v = x✝¹ ∧ w = x✝ ∨ v = x✝ ∧ w = x✝¹
⊢ Adj (induce ⊤ {v, w}) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mp.inl.intro
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v w : V
hvw : SimpleGraph.Adj G v w
⊢ Adj (induce ⊤ {v, w}) v w | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp [hvw, hvw.symm] | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mp.inr.intro
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v w : V
hvw : SimpleGraph.Adj G v w
⊢ Adj (induce ⊤ {v, w}) w v | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp [hvw, hvw.symm] | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v w : V
hvw : SimpleGraph.Adj G v w
x✝¹ x✝ : V
⊢ Adj (induce ⊤ {v, w}) x✝¹ x✝ → Adj (subgraphOfAdj G hvw) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | intro h | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm]
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v w : V
hvw : SimpleGraph.Adj G v w
x✝¹ x✝ : V
h : Adj (induce ⊤ {v, w}) x✝¹ x✝
⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm]
· intro h
... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v w : V
hvw : SimpleGraph.Adj G v w
x✝¹ x✝ : V
h : (x✝¹ = v ∨ x✝¹ = w) ∧ (x✝ = v ∨ x✝ = w) ∧ SimpleGraph.Adj G x✝¹ x✝
⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | obtain ⟨rfl | rfl, rfl | rfl, ha⟩ := h | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm]
· intro h
... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr.intro.inl.intro.inl
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
w x✝ : V
hvw : SimpleGraph.Adj G x✝ w
ha : SimpleGraph.Adj G x✝ x✝
⊢ Adj (subgraphOfAdj G hvw) x✝ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | first |exact (ha.ne rfl).elim|simp | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm]
· intro h
... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr.intro.inl.intro.inl
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
w x✝ : V
hvw : SimpleGraph.Adj G x✝ w
ha : SimpleGraph.Adj G x✝ x✝
⊢ Adj (subgraphOfAdj G hvw) x✝ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | exact (ha.ne rfl).elim | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm]
· intro h
... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr.intro.inl.intro.inr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
x✝¹ x✝ : V
ha hvw : SimpleGraph.Adj G x✝¹ x✝
⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | first |exact (ha.ne rfl).elim|simp | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm]
· intro h
... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr.intro.inl.intro.inr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
x✝¹ x✝ : V
ha hvw : SimpleGraph.Adj G x✝¹ x✝
⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | exact (ha.ne rfl).elim | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm]
· intro h
... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr.intro.inl.intro.inr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
x✝¹ x✝ : V
ha hvw : SimpleGraph.Adj G x✝¹ x✝
⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm]
· intro h
... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr.intro.inr.intro.inl
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
x✝¹ x✝ : V
ha : SimpleGraph.Adj G x✝¹ x✝
hvw : SimpleGraph.Adj G x✝ x✝¹
⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | first |exact (ha.ne rfl).elim|simp | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm]
· intro h
... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr.intro.inr.intro.inl
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
x✝¹ x✝ : V
ha : SimpleGraph.Adj G x✝¹ x✝
hvw : SimpleGraph.Adj G x✝ x✝¹
⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | exact (ha.ne rfl).elim | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm]
· intro h
... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr.intro.inr.intro.inl
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
x✝¹ x✝ : V
ha : SimpleGraph.Adj G x✝¹ x✝
hvw : SimpleGraph.Adj G x✝ x✝¹
⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm]
· intro h
... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr.intro.inr.intro.inr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v x✝ : V
hvw : SimpleGraph.Adj G v x✝
ha : SimpleGraph.Adj G x✝ x✝
⊢ Adj (subgraphOfAdj G hvw) x✝ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | first |exact (ha.ne rfl).elim|simp | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm]
· intro h
... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr.intro.inr.intro.inr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : Subgraph G
s s' : Set V
v x✝ : V
hvw : SimpleGraph.Adj G v x✝
ha : SimpleGraph.Adj G x✝ x✝
⊢ Adj (subgraphOfAdj G hvw) x✝ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | exact (ha.ne rfl).elim | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm]
· intro h
... | Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set V
u v : V
⊢ Adj (deleteVerts G' s) u v ↔ u ∈ G'.verts ∧ u ∉ s ∧ v ∈ G'.verts ∧ v ∉ s ∧ Adj G' u v | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp [and_assoc] | theorem deleteVerts_adj {u v : V} :
(G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1258_0.BlhiAiIDADcXv8t | theorem deleteVerts_adj {u v : V} :
(G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ s s' : Set V
⊢ deleteVerts (deleteVerts G' s) s' = deleteVerts G' (s ∪ s') | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | ext | @[simp]
theorem deleteVerts_deleteVerts (s s' : Set V) :
(G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1263_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteVerts_deleteVerts (s s' : Set V) :
(G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ s s' : Set V
x✝ : V
⊢ x✝ ∈ (deleteVerts (deleteVerts G' s) s').verts ↔ x✝ ∈ (deleteVerts G' (s ∪ s')).verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) [not_or, and_assoc] | @[simp]
theorem deleteVerts_deleteVerts (s s' : Set V) :
(G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1263_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteVerts_deleteVerts (s s' : Set V) :
(G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s✝ s s' : Set V
x✝¹ x✝ : V
⊢ Adj (deleteVerts (deleteVerts G' s) s') x✝¹ x✝ ↔ Adj (deleteVerts G' (s ∪ s')) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) [not_or, and_assoc] | @[simp]
theorem deleteVerts_deleteVerts (s s' : Set V) :
(G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1263_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteVerts_deleteVerts (s s' : Set V) :
(G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set V
⊢ deleteVerts G' ∅ = G' | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp [deleteVerts] | @[simp]
theorem deleteVerts_empty : G'.deleteVerts ∅ = G' := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1269_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteVerts_empty : G'.deleteVerts ∅ = G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set V
⊢ deleteVerts G' s ≤ G' | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | constructor | theorem deleteVerts_le : G'.deleteVerts s ≤ G' := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1274_0.BlhiAiIDADcXv8t | theorem deleteVerts_le : G'.deleteVerts s ≤ G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case left
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set V
⊢ (deleteVerts G' s).verts ⊆ G'.verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp [Set.diff_subset] | theorem deleteVerts_le : G'.deleteVerts s ≤ G' := by
constructor <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1274_0.BlhiAiIDADcXv8t | theorem deleteVerts_le : G'.deleteVerts s ≤ G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set V
⊢ ∀ ⦃v w : V⦄, Adj (deleteVerts G' s) v w → Adj G' v w | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp [Set.diff_subset] | theorem deleteVerts_le : G'.deleteVerts s ≤ G' := by
constructor <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1274_0.BlhiAiIDADcXv8t | theorem deleteVerts_le : G'.deleteVerts s ≤ G' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set V
⊢ deleteVerts G' (G'.verts ∩ s) = deleteVerts G' s | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | ext | @[simp]
theorem deleteVerts_inter_verts_left_eq : G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1289_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteVerts_inter_verts_left_eq : G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set V
x✝ : V
⊢ x✝ ∈ (deleteVerts G' (G'.verts ∩ s)).verts ↔ x✝ ∈ (deleteVerts G' s).verts | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) [imp_false] | @[simp]
theorem deleteVerts_inter_verts_left_eq : G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1289_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteVerts_inter_verts_left_eq : G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
s : Set V
x✝¹ x✝ : V
⊢ Adj (deleteVerts G' (G'.verts ∩ s)) x✝¹ x✝ ↔ Adj (deleteVerts G' s) x✝¹ x✝ | /-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6... | simp (config := { contextual := true }) [imp_false] | @[simp]
theorem deleteVerts_inter_verts_left_eq : G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.1289_0.BlhiAiIDADcXv8t | @[simp]
theorem deleteVerts_inter_verts_left_eq : G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s | Mathlib_Combinatorics_SimpleGraph_Subgraph |
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