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 |
|---|---|---|---|---|---|---|
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G₁ G₂ : Subgraph G
a b : V
G' : Subgraph G
v : V
inst✝¹ : Fintype ↑(neighborSet G' v)
inst✝ : Fintype ↑(SimpleGraph.neighborSet (Subgraph.spanningCoe G') v)
⊢ SimpleGraph.degree (Subgraph.spanningCoe G') v = degree G' 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... | rw [← card_neighborSet_eq_degree, Subgraph.degree] | @[simp]
theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)]
[Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.820_0.BlhiAiIDADcXv8t | @[simp]
theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)]
[Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G₁ G₂ : Subgraph G
a b : V
G' : Subgraph G
v : V
inst✝¹ : Fintype ↑(neighborSet G' v)
inst✝ : Fintype ↑(SimpleGraph.neighborSet (Subgraph.spanningCoe G') v)
⊢ Fintype.card ↑(SimpleGraph.neighborSet (Subgraph.spanningCoe G') v) = Fintype.card ↑(neighborSet G' 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... | congr! | @[simp]
theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)]
[Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by
rw [← card_neighborSet_eq_degree, Subgraph.degree]
| Mathlib.Combinatorics.SimpleGraph.Subgraph.820_0.BlhiAiIDADcXv8t | @[simp]
theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)]
[Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G₁ G₂ : Subgraph G
a b : V
G' : Subgraph G
v : V
inst✝ : Fintype ↑(neighborSet G' v)
⊢ degree G' v = 1 ↔ ∃! 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... | rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] | theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] :
G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.827_0.BlhiAiIDADcXv8t | theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] :
G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G₁ G₂ : Subgraph G
a b : V
G' : Subgraph G
v : V
inst✝ : Fintype ↑(neighborSet G' v)
⊢ (∃! a, a ∈ Set.toFinset (neighborSet G' 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 only [Set.mem_toFinset, mem_neighborSet] | theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] :
G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by
rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem]
| Mathlib.Combinatorics.SimpleGraph.Subgraph.827_0.BlhiAiIDADcXv8t | theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] :
G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v : V
H : Subgraph G
⊢ SimpleGraph.singletonSubgraph G v ≤ H ↔ v ∈ H.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... | refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ | @[simp]
theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) :
G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.846_0.BlhiAiIDADcXv8t | @[simp]
theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) :
G.singletonSubgraph v ≤ H ↔ v ∈ H.verts | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v : V
H : Subgraph G
⊢ v ∈ H.verts → SimpleGraph.singletonSubgraph G v ≤ H | /-
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 | @[simp]
theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) :
G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by
refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩
| Mathlib.Combinatorics.SimpleGraph.Subgraph.846_0.BlhiAiIDADcXv8t | @[simp]
theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) :
G.singletonSubgraph v ≤ H ↔ v ∈ H.verts | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v : V
H : Subgraph G
h : v ∈ H.verts
⊢ SimpleGraph.singletonSubgraph G v ≤ H | /-
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 singletonSubgraph_le_iff (v : V) (H : G.Subgraph) :
G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by
refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩
intro h
| Mathlib.Combinatorics.SimpleGraph.Subgraph.846_0.BlhiAiIDADcXv8t | @[simp]
theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) :
G.singletonSubgraph v ≤ H ↔ v ∈ H.verts | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case left
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v : V
H : Subgraph G
h : v ∈ H.verts
⊢ (SimpleGraph.singletonSubgraph G v).verts ⊆ H.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... | rwa [singletonSubgraph_verts, Set.singleton_subset_iff] | @[simp]
theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) :
G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by
refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩
intro h
constructor
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.846_0.BlhiAiIDADcXv8t | @[simp]
theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) :
G.singletonSubgraph v ≤ H ↔ v ∈ H.verts | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v : V
H : Subgraph G
h : v ∈ H.verts
⊢ ∀ ⦃v_1 w : V⦄, Subgraph.Adj (SimpleGraph.singletonSubgraph G v) v_1 w → Subgraph.Adj H v_1 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 fun _ _ ↦ False.elim | @[simp]
theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) :
G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by
refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩
intro h
constructor
· rwa [singletonSubgraph_verts, Set.singleton_subset_iff]
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.846_0.BlhiAiIDADcXv8t | @[simp]
theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) :
G.singletonSubgraph v ≤ H ↔ v ∈ H.verts | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v : V
⊢ Subgraph.map f (SimpleGraph.singletonSubgraph G v) = SimpleGraph.singletonSubgraph G' (f 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 | @[simp]
theorem map_singletonSubgraph (f : G →g G') {v : V} :
Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.856_0.BlhiAiIDADcXv8t | @[simp]
theorem map_singletonSubgraph (f : G →g G') {v : V} :
Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v : V
x✝ : W
⊢ x✝ ∈ (Subgraph.map f (SimpleGraph.singletonSubgraph G v)).verts ↔ x✝ ∈ (SimpleGraph.singletonSubgraph G' (f 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 only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply,
exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts,
singletonSubgraph_verts, Set.image_singleton] | @[simp]
theorem map_singletonSubgraph (f : G →g G') {v : V} :
Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.856_0.BlhiAiIDADcXv8t | @[simp]
theorem map_singletonSubgraph (f : G →g G') {v : V} :
Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v : V
x✝¹ x✝ : W
⊢ Subgraph.Adj (Subgraph.map f (SimpleGraph.singletonSubgraph G v)) x✝¹ x✝ ↔
Subgraph.Adj (SimpleGraph.singletonSubgraph G' (f 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 only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply,
exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts,
singletonSubgraph_verts, Set.image_singleton] | @[simp]
theorem map_singletonSubgraph (f : G →g G') {v : V} :
Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.856_0.BlhiAiIDADcXv8t | @[simp]
theorem map_singletonSubgraph (f : G →g G') {v : V} :
Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v : V
x✝¹ x✝ : W
⊢ ⊥ → ∃ x, f x = x✝¹ ∧ ∃ x, f 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 False.elim | @[simp]
theorem map_singletonSubgraph (f : G →g G') {v : V} :
Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by
ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply,
exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts,
sin... | Mathlib.Combinatorics.SimpleGraph.Subgraph.856_0.BlhiAiIDADcXv8t | @[simp]
theorem map_singletonSubgraph (f : G →g G') {v : V} :
Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
H : Subgraph G
v : V
⊢ H = SimpleGraph.singletonSubgraph G v ↔ H.verts = {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... | refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.875_0.BlhiAiIDADcXv8t | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
H : Subgraph G
v : V
h : H = SimpleGraph.singletonSubgraph G v
⊢ H.verts = {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... | rw [h, singletonSubgraph_verts] | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} := by
refine' ⟨fun h ↦ by | Mathlib.Combinatorics.SimpleGraph.Subgraph.875_0.BlhiAiIDADcXv8t | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
H : Subgraph G
v : V
h : H.verts = {v}
⊢ H = SimpleGraph.singletonSubgraph G 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 eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} := by
refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩
| Mathlib.Combinatorics.SimpleGraph.Subgraph.875_0.BlhiAiIDADcXv8t | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
H : Subgraph G
v : V
h : H.verts = {v}
x✝ : V
⊢ x✝ ∈ H.verts ↔ x✝ ∈ (SimpleGraph.singletonSubgraph G 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... | rw [h, singletonSubgraph_verts] | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} := by
refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩
ext
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.875_0.BlhiAiIDADcXv8t | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
H : Subgraph G
v : V
h : H.verts = {v}
x✝¹ x✝ : V
⊢ Subgraph.Adj H x✝¹ x✝ ↔ Subgraph.Adj (SimpleGraph.singletonSubgraph G 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 only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} := by
refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩
ext
· rw [h, singletonSubgraph_verts]
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.875_0.BlhiAiIDADcXv8t | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
H : Subgraph G
v : V
h : H.verts = {v}
x✝¹ x✝ : V
⊢ ¬Subgraph.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... | intro ha | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} := by
refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩
ext
· rw [h, singletonSubgraph_verts]
· simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff]
| Mathlib.Combinatorics.SimpleGraph.Subgraph.875_0.BlhiAiIDADcXv8t | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
H : Subgraph G
v : V
h : H.verts = {v}
x✝¹ x✝ : V
ha : Subgraph.Adj H x✝¹ x✝
⊢ False | /-
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... | have ha1 := ha.fst_mem | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} := by
refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩
ext
· rw [h, singletonSubgraph_verts]
· simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff]
i... | Mathlib.Combinatorics.SimpleGraph.Subgraph.875_0.BlhiAiIDADcXv8t | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
H : Subgraph G
v : V
h : H.verts = {v}
x✝¹ x✝ : V
ha : Subgraph.Adj H x✝¹ x✝
ha1 : x✝¹ ∈ H.verts
⊢ False | /-
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... | have ha2 := ha.snd_mem | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} := by
refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩
ext
· rw [h, singletonSubgraph_verts]
· simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff]
i... | Mathlib.Combinatorics.SimpleGraph.Subgraph.875_0.BlhiAiIDADcXv8t | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
H : Subgraph G
v : V
h : H.verts = {v}
x✝¹ x✝ : V
ha : Subgraph.Adj H x✝¹ x✝
ha1 : x✝¹ ∈ H.verts
ha2 : x✝ ∈ H.verts
⊢ False | /-
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... | rw [h, Set.mem_singleton_iff] at ha1 ha2 | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} := by
refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩
ext
· rw [h, singletonSubgraph_verts]
· simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff]
i... | Mathlib.Combinatorics.SimpleGraph.Subgraph.875_0.BlhiAiIDADcXv8t | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
H : Subgraph G
v : V
h : H.verts = {v}
x✝¹ x✝ : V
ha : Subgraph.Adj H x✝¹ x✝
ha1 : x✝¹ = v
ha2 : x✝ = v
⊢ False | /-
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... | subst_vars | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} := by
refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩
ext
· rw [h, singletonSubgraph_verts]
· simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff]
i... | Mathlib.Combinatorics.SimpleGraph.Subgraph.875_0.BlhiAiIDADcXv8t | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
H : Subgraph G
x✝ : V
ha : Subgraph.Adj H x✝ x✝
h : H.verts = {x✝}
⊢ False | /-
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 | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} := by
refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩
ext
· rw [h, singletonSubgraph_verts]
· simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff]
i... | Mathlib.Combinatorics.SimpleGraph.Subgraph.875_0.BlhiAiIDADcXv8t | theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
H = G.singletonSubgraph v ↔ H.verts = {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
hvw : Adj G v w
⊢ v ∈ (subgraphOfAdj G hvw).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 | instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) :
Nonempty (G.subgraphOfAdj hvw).verts :=
⟨⟨v, by | Mathlib.Combinatorics.SimpleGraph.Subgraph.889_0.BlhiAiIDADcXv8t | instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) :
Nonempty (G.subgraphOfAdj hvw).verts | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
hvw : Adj G v w
⊢ Subgraph.edgeSet (subgraphOfAdj G hvw) = {⟦(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 e | @[simp]
theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.894_0.BlhiAiIDADcXv8t | @[simp]
theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
hvw : Adj G v w
e : Sym2 V
⊢ e ∈ Subgraph.edgeSet (subgraphOfAdj G hvw) ↔ e ∈ {⟦(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' e.ind _ | @[simp]
theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by
ext e
| Mathlib.Combinatorics.SimpleGraph.Subgraph.894_0.BlhiAiIDADcXv8t | @[simp]
theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
hvw : Adj G v w
e : Sym2 V
⊢ ∀ (x y : V), ⟦(x, y)⟧ ∈ Subgraph.edgeSet (subgraphOfAdj G hvw) ↔ ⟦(x, y)⟧ ∈ {⟦(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 [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff,
forall₂_true_iff] | @[simp]
theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by
ext e
refine' e.ind _
| Mathlib.Combinatorics.SimpleGraph.Subgraph.894_0.BlhiAiIDADcXv8t | @[simp]
theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
H : Subgraph G
h : Subgraph.Adj H v w
⊢ subgraphOfAdj G (_ : Adj G v w) ≤ H | /-
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 | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.903_0.BlhiAiIDADcXv8t | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case left
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
H : Subgraph G
h : Subgraph.Adj H v w
⊢ (subgraphOfAdj G (_ : Adj G v w)).verts ⊆ H.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... | intro x | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H := by
constructor
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.903_0.BlhiAiIDADcXv8t | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case left
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
H : Subgraph G
h : Subgraph.Adj H v w
x : V
⊢ x ∈ (subgraphOfAdj G (_ : Adj G v w)).verts → x ∈ H.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... | rintro (rfl | rfl) | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H := by
constructor
· intro x
| Mathlib.Combinatorics.SimpleGraph.Subgraph.903_0.BlhiAiIDADcXv8t | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case left.inl
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
w : V
H : Subgraph G
x : V
h : Subgraph.Adj H x w
⊢ x ∈ H.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 [H.edge_vert h, H.edge_vert h.symm] | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H := by
constructor
· intro x
rintro (rfl | rfl) <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.903_0.BlhiAiIDADcXv8t | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case left.inr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v : V
H : Subgraph G
x : V
h : Subgraph.Adj H v x
⊢ x ∈ H.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 [H.edge_vert h, H.edge_vert h.symm] | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H := by
constructor
· intro x
rintro (rfl | rfl) <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.903_0.BlhiAiIDADcXv8t | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
H : Subgraph G
h : Subgraph.Adj H v w
⊢ ∀ ⦃v_1 w_1 : V⦄, Subgraph.Adj (subgraphOfAdj G (_ : Adj G v w)) v_1 w_1 → Subgraph.Adj H v_1 w_1 | /-
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] | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H := by
constructor
· intro x
rintro (rfl | rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm]
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.903_0.BlhiAiIDADcXv8t | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
H : Subgraph G
h : Subgraph.Adj H v w
⊢ ∀ ⦃v_1 w_1 : V⦄, v = v_1 ∧ w = w_1 ∨ v = w_1 ∧ w = v_1 → Subgraph.Adj H v_1 w_1 | /-
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 _ _ (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩) | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H := by
constructor
· intro x
rintro (rfl | rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm]
· simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff]
| Mathlib.Combinatorics.SimpleGraph.Subgraph.903_0.BlhiAiIDADcXv8t | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right.inl.intro
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
H : Subgraph G
h : Subgraph.Adj H v w
⊢ Subgraph.Adj H 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, h.symm] | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H := by
constructor
· intro x
rintro (rfl | rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm]
· simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff]
rintro _ _ (⟨rfl, rf... | Mathlib.Combinatorics.SimpleGraph.Subgraph.903_0.BlhiAiIDADcXv8t | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right.inr.intro
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
H : Subgraph G
h : Subgraph.Adj H v w
⊢ Subgraph.Adj H 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 [h, h.symm] | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H := by
constructor
· intro x
rintro (rfl | rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm]
· simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff]
rintro _ _ (⟨rfl, rf... | Mathlib.Combinatorics.SimpleGraph.Subgraph.903_0.BlhiAiIDADcXv8t | set_option autoImplicit true in
lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :
G.subgraphOfAdj (H.adj_sub h) ≤ H | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
hvw : Adj G v w
⊢ subgraphOfAdj G (_ : Adj G w v) = subgraphOfAdj G hvw | /-
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_symm {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.912_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
hvw : Adj G v w
x✝ : V
⊢ x✝ ∈ (subgraphOfAdj G (_ : Adj G w v)).verts ↔ x✝ ∈ (subgraphOfAdj G hvw).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 [or_comm, and_comm] | theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.912_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
hvw : Adj G v w
x✝¹ x✝ : V
⊢ Subgraph.Adj (subgraphOfAdj G (_ : Adj G w v)) x✝¹ x✝ ↔ Subgraph.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 [or_comm, and_comm] | theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by
ext <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.912_0.BlhiAiIDADcXv8t | theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
⊢ Subgraph.map f (subgraphOfAdj G hvw) = subgraphOfAdj G' (_ : Adj G' (f v) (f 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 | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
x✝ : W
⊢ x✝ ∈ (Subgraph.map f (subgraphOfAdj G hvw)).verts ↔ x✝ ∈ (subgraphOfAdj G' (_ : Adj G' (f v) (f 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 only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff] | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
x✝ : W
⊢ (∃ x, (x = v ∨ x = w) ∧ f x = x✝) ↔ x✝ = f v ∨ x✝ = f 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 | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
| Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h.mp
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
x✝ : W
⊢ (∃ x, (x = v ∨ x = w) ∧ f x = x✝) → x✝ = f v ∨ x✝ = f 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 ⟨u, rfl | rfl, rfl⟩ | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h.mp.intro.intro.inl
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
w u : V
hvw : Adj G u w
⊢ f u = f u ∨ f u = f 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 | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h.mp.intro.intro.inr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v u : V
hvw : Adj G v u
⊢ f u = f v ∨ f u = f u | /-
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 map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h.mpr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
x✝ : W
⊢ x✝ = f v ∨ x✝ = f w → ∃ x, (x = v ∨ x = w) ∧ f 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... | rintro (rfl | rfl) | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h.mpr.inl
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
⊢ ∃ x, (x = v ∨ x = w) ∧ f x = f 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... | use v | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
⊢ (v = v ∨ v = w) ∧ f v = f 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 | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h.mpr.inr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
⊢ ∃ x, (x = v ∨ x = w) ∧ f x = f 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... | use w | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
⊢ (w = v ∨ w = w) ∧ f w = f 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 | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
x✝¹ x✝ : W
⊢ Subgraph.Adj (Subgraph.map f (subgraphOfAdj G hvw)) x✝¹ x✝ ↔
Subgraph.Adj (subgraphOfAdj G' (_ : Adj G' (f v) (f 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 [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
x✝¹ x✝ : W
⊢ (∃ a b, (v = a ∧ w = b ∨ v = b ∧ w = a) ∧ f a = x✝¹ ∧ f b = x✝) ↔ f v = x✝¹ ∧ f w = x✝ ∨ f v = x✝ ∧ f w = x✝¹ | /-
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 map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mp
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
x✝¹ x✝ : W
⊢ (∃ a b, (v = a ∧ w = b ∨ v = b ∧ w = a) ∧ f a = x✝¹ ∧ f b = x✝) → f v = x✝¹ ∧ f w = x✝ ∨ f v = x✝ ∧ f w = x✝¹ | /-
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 ⟨a, b, ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩, rfl, rfl⟩ | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mp.intro.intro.intro.inl.intro.intro
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
⊢ f v = f v ∧ f w = f w ∨ f v = f w ∧ f w = f 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 | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mp.intro.intro.intro.inr.intro.intro
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
⊢ f v = f w ∧ f w = f v ∨ f v = f v ∧ f w = f 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 | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
x✝¹ x✝ : W
⊢ f v = x✝¹ ∧ f w = x✝ ∨ f v = x✝ ∧ f w = x✝¹ → ∃ a b, (v = a ∧ w = b ∨ v = b ∧ w = a) ∧ f a = x✝¹ ∧ f b = 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... | rintro (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩) | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr.inl.intro
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
⊢ ∃ a b, (v = a ∧ w = b ∨ v = b ∧ w = a) ∧ f a = f v ∧ f b = f 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... | use v, w | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
⊢ (v = v ∧ w = w ∨ v = w ∧ w = v) ∧ f v = f v ∧ f w = f 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 | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case Adj.h.h.a.mpr.inr.intro
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
⊢ ∃ a b, (v = a ∧ w = b ∨ v = b ∧ w = a) ∧ f a = f w ∧ f b = f 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... | use w, v | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
v w : V
hvw : Adj G v w
⊢ (v = w ∧ w = v ∨ v = v ∧ w = w) ∧ f w = f w ∧ f v = f 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 | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by
ext
· simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,
Set.mem_singleton_iff]
constructor
· rintro ⟨u, r... | Mathlib.Combinatorics.SimpleGraph.Subgraph.917_0.BlhiAiIDADcXv8t | @[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
hvw : Adj G v w
⊢ Subgraph.neighborSet (subgraphOfAdj G hvw) 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 u | @[simp]
theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet v = {w} := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.945_0.BlhiAiIDADcXv8t | @[simp]
theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet v = {w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
hvw : Adj G v w
u : V
⊢ u ∈ Subgraph.neighborSet (subgraphOfAdj G hvw) v ↔ u ∈ {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... | suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this | @[simp]
theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet v = {w} := by
ext u
| Mathlib.Combinatorics.SimpleGraph.Subgraph.945_0.BlhiAiIDADcXv8t | @[simp]
theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet v = {w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
hvw : Adj G v w
u : V
this : w = u ↔ u = w
⊢ u ∈ Subgraph.neighborSet (subgraphOfAdj G hvw) v ↔ u ∈ {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... | simpa [hvw.ne.symm] using this | @[simp]
theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet v = {w} := by
ext u
suffices w = u ↔ u = w by | Mathlib.Combinatorics.SimpleGraph.Subgraph.945_0.BlhiAiIDADcXv8t | @[simp]
theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet v = {w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
hvw : Adj G v w
u : V
⊢ w = u ↔ u = 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... | rw [eq_comm] | @[simp]
theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet v = {w} := by
ext u
suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this
| Mathlib.Combinatorics.SimpleGraph.Subgraph.945_0.BlhiAiIDADcXv8t | @[simp]
theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet v = {w} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
hvw : Adj G v w
⊢ Subgraph.neighborSet (subgraphOfAdj G hvw) 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... | rw [subgraphOfAdj_symm hvw.symm] | @[simp]
theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet w = {v} := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.953_0.BlhiAiIDADcXv8t | @[simp]
theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet w = {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
hvw : Adj G v w
⊢ Subgraph.neighborSet (subgraphOfAdj G (_ : Adj G w v)) 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... | exact neighborSet_fst_subgraphOfAdj hvw.symm | @[simp]
theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet w = {v} := by
rw [subgraphOfAdj_symm hvw.symm]
| Mathlib.Combinatorics.SimpleGraph.Subgraph.953_0.BlhiAiIDADcXv8t | @[simp]
theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet w = {v} | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
u v w : V
hvw : Adj G v w
hv : u ≠ v
hw : u ≠ w
⊢ Subgraph.neighborSet (subgraphOfAdj G hvw) u = ∅ | /-
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 neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v)
(hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.960_0.BlhiAiIDADcXv8t | @[simp]
theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v)
(hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
u v w : V
hvw : Adj G v w
hv : u ≠ v
hw : u ≠ w
x✝ : V
⊢ x✝ ∈ Subgraph.neighborSet (subgraphOfAdj G hvw) u ↔ 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 [hv.symm, hw.symm] | @[simp]
theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v)
(hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by
ext
| Mathlib.Combinatorics.SimpleGraph.Subgraph.960_0.BlhiAiIDADcXv8t | @[simp]
theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v)
(hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
inst✝ : DecidableEq V
u v w : V
hvw : Adj G v w
⊢ Subgraph.neighborSet (subgraphOfAdj G hvw) u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ | /-
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... | split_ifs | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ :=
by | Mathlib.Combinatorics.SimpleGraph.Subgraph.967_0.BlhiAiIDADcXv8t | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case pos
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
inst✝ : DecidableEq V
u v w : V
hvw : Adj G v w
h✝¹ : u = v
h✝ : u = w
⊢ Subgraph.neighborSet (subgraphOfAdj G hvw) u = {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... | subst_vars | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ :=
by split_ifs <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.967_0.BlhiAiIDADcXv8t | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case neg
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
inst✝ : DecidableEq V
u v w : V
hvw : Adj G v w
h✝¹ : u = v
h✝ : ¬u = w
⊢ Subgraph.neighborSet (subgraphOfAdj G hvw) u = {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... | subst_vars | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ :=
by split_ifs <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.967_0.BlhiAiIDADcXv8t | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case pos
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
inst✝ : DecidableEq V
u v w : V
hvw : Adj G v w
h✝¹ : ¬u = v
h✝ : u = w
⊢ Subgraph.neighborSet (subgraphOfAdj G hvw) 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... | subst_vars | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ :=
by split_ifs <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.967_0.BlhiAiIDADcXv8t | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case neg
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
inst✝ : DecidableEq V
u v w : V
hvw : Adj G v w
h✝¹ : ¬u = v
h✝ : ¬u = w
⊢ Subgraph.neighborSet (subgraphOfAdj G hvw) u = ∅ ∪ ∅ | /-
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... | subst_vars | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ :=
by split_ifs <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.967_0.BlhiAiIDADcXv8t | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case pos
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
inst✝ : DecidableEq V
w : V
hvw : Adj G w w
⊢ Subgraph.neighborSet (subgraphOfAdj G hvw) w = {w} ∪ {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 [*] | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ :=
by split_ifs <;> subst_vars <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.967_0.BlhiAiIDADcXv8t | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case neg
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
inst✝ : DecidableEq V
v w : V
hvw : Adj G v w
h✝ : ¬v = w
⊢ Subgraph.neighborSet (subgraphOfAdj G hvw) 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 [*] | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ :=
by split_ifs <;> subst_vars <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.967_0.BlhiAiIDADcXv8t | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case pos
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
inst✝ : DecidableEq V
v w : V
hvw : Adj G v w
h✝ : ¬w = v
⊢ Subgraph.neighborSet (subgraphOfAdj G hvw) 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 [*] | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ :=
by split_ifs <;> subst_vars <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.967_0.BlhiAiIDADcXv8t | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case neg
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
inst✝ : DecidableEq V
u v w : V
hvw : Adj G v w
h✝¹ : ¬u = v
h✝ : ¬u = w
⊢ Subgraph.neighborSet (subgraphOfAdj G hvw) u = ∅ ∪ ∅ | /-
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 neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ :=
by split_ifs <;> subst_vars <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.967_0.BlhiAiIDADcXv8t | theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
(G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
u v : V
h : Adj G u v
⊢ SimpleGraph.singletonSubgraph G u ≤ subgraphOfAdj G h | /-
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 singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph u ≤ G.subgraphOfAdj h := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.972_0.BlhiAiIDADcXv8t | theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph u ≤ G.subgraphOfAdj h | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case left
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
u v : V
h : Adj G u v
⊢ (SimpleGraph.singletonSubgraph G u).verts ⊆ (subgraphOfAdj G h).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.bot_eq_empty] | theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph u ≤ G.subgraphOfAdj h := by
constructor <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.972_0.BlhiAiIDADcXv8t | theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph u ≤ G.subgraphOfAdj h | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
u v : V
h : Adj G u v
⊢ ∀ ⦃v_1 w : V⦄, Subgraph.Adj (SimpleGraph.singletonSubgraph G u) v_1 w → Subgraph.Adj (subgraphOfAdj G h) v_1 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.bot_eq_empty] | theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph u ≤ G.subgraphOfAdj h := by
constructor <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.972_0.BlhiAiIDADcXv8t | theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph u ≤ G.subgraphOfAdj h | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
u v : V
h : Adj G u v
⊢ ∀ ⦃v_1 w : V⦄, ⊥ → u = v_1 ∧ v = w ∨ u = w ∧ v = v_1 | /-
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 _ _ ↦ False.elim | theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph u ≤ G.subgraphOfAdj h := by
constructor <;> simp [-Set.bot_eq_empty]
| Mathlib.Combinatorics.SimpleGraph.Subgraph.972_0.BlhiAiIDADcXv8t | theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph u ≤ G.subgraphOfAdj h | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
u v : V
h : Adj G u v
⊢ SimpleGraph.singletonSubgraph G v ≤ subgraphOfAdj G h | /-
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 singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph v ≤ G.subgraphOfAdj h := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.978_0.BlhiAiIDADcXv8t | theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph v ≤ G.subgraphOfAdj h | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case left
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
u v : V
h : Adj G u v
⊢ (SimpleGraph.singletonSubgraph G v).verts ⊆ (subgraphOfAdj G h).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.bot_eq_empty] | theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph v ≤ G.subgraphOfAdj h := by
constructor <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.978_0.BlhiAiIDADcXv8t | theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph v ≤ G.subgraphOfAdj h | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
u v : V
h : Adj G u v
⊢ ∀ ⦃v_1 w : V⦄, Subgraph.Adj (SimpleGraph.singletonSubgraph G v) v_1 w → Subgraph.Adj (subgraphOfAdj G h) v_1 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.bot_eq_empty] | theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph v ≤ G.subgraphOfAdj h := by
constructor <;> | Mathlib.Combinatorics.SimpleGraph.Subgraph.978_0.BlhiAiIDADcXv8t | theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph v ≤ G.subgraphOfAdj h | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
u v : V
h : Adj G u v
⊢ ∀ ⦃v_1 w : V⦄, ⊥ → u = v_1 ∧ v = w ∨ u = w ∧ v = v_1 | /-
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 _ _ ↦ False.elim | theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph v ≤ G.subgraphOfAdj h := by
constructor <;> simp [-Set.bot_eq_empty]
| Mathlib.Combinatorics.SimpleGraph.Subgraph.978_0.BlhiAiIDADcXv8t | theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
G.singletonSubgraph v ≤ G.subgraphOfAdj h | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
G'' : Subgraph (Subgraph.coe G')
v w : V
⊢ Adj (Subgraph.coeSubgraph G'') v w ↔
∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), Adj G'' { 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... | simp [Relation.Map] | lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) :
(G'.coeSubgraph G'').Adj v w ↔
∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1006_0.BlhiAiIDADcXv8t | lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) :
(G'.coeSubgraph G'').Adj v w ↔
∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
G'' : Subgraph (Subgraph.coe G')
⊢ Subgraph.restrict (Subgraph.coeSubgraph 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 | theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) :
Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1014_0.BlhiAiIDADcXv8t | theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) :
Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : Subgraph G
G'' : Subgraph (Subgraph.coe G')
x✝ : ↑G'.verts
⊢ x✝ ∈ (Subgraph.restrict (Subgraph.coeSubgraph 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 | theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) :
Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by
ext
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.1014_0.BlhiAiIDADcXv8t | theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) :
Subgraph.restrict (Subgraph.coeSubgraph G'') = 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
G'' : Subgraph (Subgraph.coe G')
x✝¹ x✝ : ↑G'.verts
⊢ Adj (Subgraph.restrict (Subgraph.coeSubgraph 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... | rw [restrict_adj, coeSubgraph_adj] | theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) :
Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by
ext
· simp
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.1014_0.BlhiAiIDADcXv8t | theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) :
Subgraph.restrict (Subgraph.coeSubgraph G'') = 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
G'' : Subgraph (Subgraph.coe G')
x✝¹ x✝ : ↑G'.verts
⊢ (Adj G' ↑x✝¹ ↑x✝ ∧
∃ (hv : ↑x✝¹ ∈ G'.verts) (hw : ↑x✝ ∈ G'.verts),
Adj G'' { val := ↑x✝¹, property := hv } { val := ↑x✝, property := hw }) ↔
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... | simpa using G''.adj_sub | theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) :
Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by
ext
· simp
· rw [restrict_adj, coeSubgraph_adj]
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1014_0.BlhiAiIDADcXv8t | theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) :
Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
H : Subgraph G
H' : Subgraph (Subgraph.coe H)
⊢ Subgraph.coeSubgraph H' ≤ H | /-
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 coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) :
Subgraph.coeSubgraph H' ≤ H := by
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1027_0.BlhiAiIDADcXv8t | lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) :
Subgraph.coeSubgraph H' ≤ H | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case left
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
H : Subgraph G
H' : Subgraph (Subgraph.coe H)
⊢ (Subgraph.coeSubgraph H').verts ⊆ H.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 | lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) :
Subgraph.coeSubgraph H' ≤ H := by
constructor
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.1027_0.BlhiAiIDADcXv8t | lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) :
Subgraph.coeSubgraph H' ≤ H | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
H : Subgraph G
H' : Subgraph (Subgraph.coe H)
⊢ ∀ ⦃v w : V⦄, Adj (Subgraph.coeSubgraph H') v w → Adj H 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, rfl, rfl⟩ | lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) :
Subgraph.coeSubgraph H' ≤ H := by
constructor
· simp
· | Mathlib.Combinatorics.SimpleGraph.Subgraph.1027_0.BlhiAiIDADcXv8t | lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) :
Subgraph.coeSubgraph H' ≤ H | Mathlib_Combinatorics_SimpleGraph_Subgraph |
case right.intro.intro.intro.intro
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
H : Subgraph G
H' : Subgraph (Subgraph.coe H)
w✝¹ w✝ : ↑H.verts
h : Adj H' w✝¹ w✝
⊢ Adj H ((Subgraph.hom H) w✝¹) ((Subgraph.hom H) 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 H'.adj_sub h | lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) :
Subgraph.coeSubgraph H' ≤ H := by
constructor
· simp
· rintro v w ⟨_, _, h, rfl, rfl⟩
| Mathlib.Combinatorics.SimpleGraph.Subgraph.1027_0.BlhiAiIDADcXv8t | lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) :
Subgraph.coeSubgraph H' ≤ H | Mathlib_Combinatorics_SimpleGraph_Subgraph |
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
H H' : Subgraph G
⊢ Subgraph.coeSubgraph (Subgraph.restrict H') = H ⊓ H' | /-
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 | lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) :
Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by
| 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 |
case verts.h
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
H H' : Subgraph G
x✝ : V
⊢ x✝ ∈ (Subgraph.coeSubgraph (Subgraph.restrict H')).verts ↔ x✝ ∈ (H ⊓ H').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_comm] | lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) :
Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by
ext
· | 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 |
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
⊢ Adj (Subgraph.coeSubgraph (Subgraph.restrict H')) x✝¹ x✝ ↔ Adj (H ⊓ 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_rw [coeSubgraph_adj, restrict_adj] | lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) :
Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by
ext
· simp [and_comm]
· | 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 |
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
⊢ (∃ (_ : x✝¹ ∈ H.verts) (_ : x✝ ∈ H.verts), Adj H x✝¹ x✝ ∧ Adj H' x✝¹ x✝) ↔ Adj (H ⊓ 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 only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] | 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]
| 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 |
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
⊢ 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... | intro h | 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]
| 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 |
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