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https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_id | [1234, 1] | [1237, 33] | simp [ih p'] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p' : Walk G v✝ w✝
ih : Walk G v✝ w✝ → Walk.map Hom.id p' = p'
p : Walk G u✝ w✝
⊢ Walk.map Hom.id (cons h✝ p') = cons h✝ p' | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_map | [1241, 1] | [1244, 29] | induction p with
| nil => rfl
| cons _ _ ih => simp [ih] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
⊢ Walk.map f' (Walk.map f p) = Walk.map (Hom.comp f' f) p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_map | [1241, 1] | [1244, 29] | rfl | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
u✝ : V
⊢ Walk.map f' (Walk.map f nil) = Walk.map (Hom.comp f' f) nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_map | [1241, 1] | [1244, 29] | simp [ih] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : Walk.map f' (Walk.map f p✝) = Walk.map (Hom.comp f' f) p✝
⊢ Walk.map f' (Walk.map f (cons h✝ p✝)) = Walk.map (Hom.comp f' f) (cons h✝ p✝) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_eq_of_eq | [1249, 1] | [1252, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f'✝ : G' →g G''
u v u' v' : V
p : Walk G u v
f f' : G →g G'
h : f = f'
⊢ Walk.map f p = Walk.copy (Walk.map f' p) (_ : ↑f' u = ↑f u) (_ : ↑f' v = ↑f v) | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f'✝ : G' →g G''
u v u' v' : V
p : Walk G u v
f' : G →g G'
⊢ Walk.map f' p = Walk.copy (Walk.map f' p) (_ : ↑f' u = ↑f' u) (_ : ↑f' v = ↑f' v) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_eq_of_eq | [1249, 1] | [1252, 6] | rfl | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f'✝ : G' →g G''
u v u' v' : V
p : Walk G u v
f' : G →g G'
⊢ Walk.map f' p = Walk.copy (Walk.map f' p) (_ : ↑f' u = ↑f' u) (_ : ↑f' v = ↑f' v) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_eq_nil_iff | [1256, 1] | [1256, 89] | cases p <;> simp | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p✝ : Walk G u v
p : Walk G u u
⊢ Walk.map f p = nil ↔ p = nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_map | [1260, 1] | [1260, 80] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
⊢ length (Walk.map f p) = length p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_append | [1263, 1] | [1264, 83] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u✝ v✝ u' v' : V
p✝ : Walk G u✝ v✝
u v w : V
p : Walk G u v
q : Walk G v w
⊢ Walk.map f (append p q) = append (Walk.map f p) (Walk.map f q) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_map | [1268, 1] | [1268, 89] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
⊢ support (Walk.map f p) = List.map (↑f) (support p) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_map | [1272, 1] | [1272, 96] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
⊢ edges (Walk.map f p) = List.map (Prod.map ↑f ↑f) (edges p) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isPath_of_injective | [1276, 1] | [1285, 18] | induction p with
| nil => simp
| cons _ _ ih =>
rw [Walk.cons_isPath_iff] at hp
simp [ih hp.1]
intro x hx hf
cases hinj hf
exact hp.2 hx | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
hp : IsPath p
⊢ IsPath (Walk.map f p) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isPath_of_injective | [1276, 1] | [1285, 18] | simp | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ : V
hp : IsPath nil
⊢ IsPath (Walk.map f nil) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isPath_of_injective | [1276, 1] | [1285, 18] | rw [Walk.cons_isPath_iff] at hp | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath p✝ → IsPath (Walk.map f p✝)
hp : IsPath (cons h✝ p✝)
⊢ IsPath (Walk.map f (cons h✝ p✝)) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath p✝ → IsPath (Walk.map f p✝)
hp : IsPath p✝ ∧ ¬u✝ ∈ support p✝
⊢ IsPath (Walk.map f (cons h✝ p✝)) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isPath_of_injective | [1276, 1] | [1285, 18] | simp [ih hp.1] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath p✝ → IsPath (Walk.map f p✝)
hp : IsPath p✝ ∧ ¬u✝ ∈ support p✝
⊢ IsPath (Walk.map f (cons h✝ p✝)) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath p✝ → IsPath (Walk.map f p✝)
hp : IsPath p✝ ∧ ¬u✝ ∈ support p✝
⊢ ∀ (x : V), x ∈ support p✝ → ¬↑f x = ↑f u✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isPath_of_injective | [1276, 1] | [1285, 18] | intro x hx hf | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath p✝ → IsPath (Walk.map f p✝)
hp : IsPath p✝ ∧ ¬u✝ ∈ support p✝
⊢ ∀ (x : V), x ∈ support p✝ → ¬↑f x = ↑f u✝ | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath p✝ → IsPath (Walk.map f p✝)
hp : IsPath p✝ ∧ ¬u✝ ∈ support p✝
x : V
hx : x ∈ support p✝
hf : ↑f x = ↑f u✝
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isPath_of_injective | [1276, 1] | [1285, 18] | cases hinj hf | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath p✝ → IsPath (Walk.map f p✝)
hp : IsPath p✝ ∧ ¬u✝ ∈ support p✝
x : V
hx : x ∈ support p✝
hf : ↑f x = ↑f u✝
⊢ False | case cons.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath p✝ → IsPath (Walk.map f p✝)
hp : IsPath p✝ ∧ ¬u✝ ∈ support p✝
hx : u✝ ∈ support p✝
hf : ↑f u✝ = ↑f u✝
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isPath_of_injective | [1276, 1] | [1285, 18] | exact hp.2 hx | case cons.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath p✝ → IsPath (Walk.map f p✝)
hp : IsPath p✝ ∧ ¬u✝ ∈ support p✝
hx : u✝ ∈ support p✝
hf : ↑f u✝ = ↑f u✝
⊢ False | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | induction p with
| nil => simp
| cons _ _ ih =>
rw [map_cons, Walk.cons_isPath_iff, support_map] at hp
rw [Walk.cons_isPath_iff]
cases' hp with hp1 hp2
refine' ⟨ih hp1, _⟩
contrapose! hp2
exact List.mem_map_of_mem f hp2 | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
f : G →g G'
hp : IsPath (Walk.map f p)
⊢ IsPath p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | simp | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
f : G →g G'
u✝ : V
hp : IsPath (Walk.map f Walk.nil)
⊢ IsPath Walk.nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | rw [map_cons, Walk.cons_isPath_iff, support_map] at hp | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
f : G →g G'
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath (Walk.map f p✝) → IsPath p✝
hp : IsPath (Walk.map f (cons h✝ p✝))
⊢ IsPath (cons h✝ p✝) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
f : G →g G'
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath (Walk.map f p✝) → IsPath p✝
hp : IsPath (Walk.map f p✝) ∧ ¬↑f u✝ ∈ List.map (↑f) (support p✝)
⊢ IsPath (cons h✝ p✝) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | rw [Walk.cons_isPath_iff] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
f : G →g G'
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath (Walk.map f p✝) → IsPath p✝
hp : IsPath (Walk.map f p✝) ∧ ¬↑f u✝ ∈ List.map (↑f) (support p✝)
⊢ IsPath (cons h✝ p✝) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
f : G →g G'
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath (Walk.map f p✝) → IsPath p✝
hp : IsPath (Walk.map f p✝) ∧ ¬↑f u✝ ∈ List.map (↑f) (support p✝)
⊢ IsPath p✝ ∧ ¬u✝ ∈ support p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | cases' hp with hp1 hp2 | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
f : G →g G'
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath (Walk.map f p✝) → IsPath p✝
hp : IsPath (Walk.map f p✝) ∧ ¬↑f u✝ ∈ List.map (↑f) (support p✝)
⊢ IsPath p✝ ∧ ¬u✝ ∈ support p✝ | case cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
f : G →g G'
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath (Walk.map f p✝) → IsPath p✝
hp1 : IsPath (Walk.map f p✝)
hp2 : ¬↑f u✝ ∈ List.map (↑f) (support p✝)
⊢ IsPath p✝ ∧ ¬u✝ ∈ support p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | refine' ⟨ih hp1, _⟩ | case cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
f : G →g G'
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath (Walk.map f p✝) → IsPath p✝
hp1 : IsPath (Walk.map f p✝)
hp2 : ¬↑f u✝ ∈ List.map (↑f) (support p✝)
⊢ IsPath p✝ ∧ ¬u✝ ∈ support p✝ | case cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
f : G →g G'
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath (Walk.map f p✝) → IsPath p✝
hp1 : IsPath (Walk.map f p✝)
hp2 : ¬↑f u✝ ∈ List.map (↑f) (support p✝)
⊢ ¬u✝ ∈ support p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | contrapose! hp2 | case cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
f : G →g G'
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath (Walk.map f p✝) → IsPath p✝
hp1 : IsPath (Walk.map f p✝)
hp2 : ¬↑f u✝ ∈ List.map (↑f) (support p✝)
⊢ ¬u✝ ∈ support p✝ | case cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
f : G →g G'
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath (Walk.map f p✝) → IsPath p✝
hp1 : IsPath (Walk.map f p✝)
hp2 : u✝ ∈ support p✝
⊢ ↑f u✝ ∈ List.map (↑f) (support p✝) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | exact List.mem_map_of_mem f hp2 | case cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
f : G →g G'
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsPath (Walk.map f p✝) → IsPath p✝
hp1 : IsPath (Walk.map f p✝)
hp2 : u✝ ∈ support p✝
⊢ ↑f u✝ ∈ List.map (↑f) (support p✝) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isTrail_iff_of_injective | [1304, 1] | [1313, 18] | induction p with
| nil => simp
| cons _ _ ih =>
rw [map_cons, cons_isTrail_iff, ih, cons_isTrail_iff]
apply and_congr_right'
rw [edges_map, ← List.mem_map_of_injective (Injective.Prod_map hinj hinj)]
exact Iff.rfl | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
⊢ IsTrail (Walk.map f p) ↔ IsTrail p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isTrail_iff_of_injective | [1304, 1] | [1313, 18] | simp | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ : V
⊢ IsTrail (Walk.map f nil) ↔ IsTrail nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isTrail_iff_of_injective | [1304, 1] | [1313, 18] | rw [map_cons, cons_isTrail_iff, ih, cons_isTrail_iff] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsTrail (Walk.map f p✝) ↔ IsTrail p✝
⊢ IsTrail (Walk.map f (cons h✝ p✝)) ↔ IsTrail (cons h✝ p✝) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsTrail (Walk.map f p✝) ↔ IsTrail p✝
⊢ IsTrail p✝ ∧ ¬(↑f u✝, ↑f v✝) ∈ edges (Walk.map f p✝) ↔ IsTrail p✝ ∧ ¬(u✝, v✝) ∈ edges p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isTrail_iff_of_injective | [1304, 1] | [1313, 18] | apply and_congr_right' | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsTrail (Walk.map f p✝) ↔ IsTrail p✝
⊢ IsTrail p✝ ∧ ¬(↑f u✝, ↑f v✝) ∈ edges (Walk.map f p✝) ↔ IsTrail p✝ ∧ ¬(u✝, v✝) ∈ edges p✝ | case cons.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsTrail (Walk.map f p✝) ↔ IsTrail p✝
⊢ ¬(↑f u✝, ↑f v✝) ∈ edges (Walk.map f p✝) ↔ ¬(u✝, v✝) ∈ edges p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isTrail_iff_of_injective | [1304, 1] | [1313, 18] | rw [edges_map, ← List.mem_map_of_injective (Injective.Prod_map hinj hinj)] | case cons.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsTrail (Walk.map f p✝) ↔ IsTrail p✝
⊢ ¬(↑f u✝, ↑f v✝) ∈ edges (Walk.map f p✝) ↔ ¬(u✝, v✝) ∈ edges p✝ | case cons.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsTrail (Walk.map f p✝) ↔ IsTrail p✝
⊢ ¬(↑f u✝, ↑f v✝) ∈ List.map (Prod.map ↑f ↑f) (edges p✝) ↔
¬Prod.map ↑f ↑f (u✝, v✝) ∈ List.map (Prod.map ↑f ↑f) (edges p✝) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isTrail_iff_of_injective | [1304, 1] | [1313, 18] | exact Iff.rfl | case cons.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p : Walk G u v
hinj : Injective ↑f
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : IsTrail (Walk.map f p✝) ↔ IsTrail p✝
⊢ ¬(↑f u✝, ↑f v✝) ∈ List.map (Prod.map ↑f ↑f) (edges p✝) ↔
¬Prod.map ↑f ↑f (u✝, v✝) ∈ List.map (Prod.map ↑f ↑f) (edges p✝) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isCycle_iff_of_injective | [1318, 1] | [1321, 59] | rw [isCycle_def, isCycle_def, map_isTrail_iff_of_injective hinj, Ne.def, map_eq_nil_iff,
support_map, ← List.map_tail, List.nodup_map_iff hinj] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
f' : G' →g G''
u v u' v' : V
p✝ : Walk G u v
p : Walk G u u
hinj : Injective ↑f
⊢ IsCycle (Walk.map f p) ↔ IsCycle p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | intro p p' h | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝ v✝ u' v' : V
p : Walk G u✝ v✝
f : G →g G'
hinj : Injective ↑f
u v : V
⊢ Injective (Walk.map f) | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝ v✝ u' v' : V
p✝ : Walk G u✝ v✝
f : G →g G'
hinj : Injective ↑f
u v : V
p p' : Walk G u v
h : Walk.map f p = Walk.map f p'
⊢ p = p' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | cases p' | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝ u' v' : V
p : Walk G u✝¹ v✝
f : G →g G'
hinj : Injective ↑f
u v u✝ : V
p' : Walk G u✝ u✝
h : Walk.map f nil = Walk.map f p'
⊢ nil = p' | case nil.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝ u' v' : V
p : Walk G u✝¹ v✝
f : G →g G'
hinj : Injective ↑f
u v u✝ : V
h : Walk.map f nil = Walk.map f nil
⊢ nil = nil
case nil.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝¹ u' v' : V
p : Walk G u✝¹ v✝¹
f : G →g G'
hinj : Injective ↑f
u v u✝ v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ u✝
h : Walk.map f nil = Walk.map f (cons h✝ p✝)
⊢ nil = cons h✝ p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | rfl | case nil.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝ u' v' : V
p : Walk G u✝¹ v✝
f : G →g G'
hinj : Injective ↑f
u v u✝ : V
h : Walk.map f nil = Walk.map f nil
⊢ nil = nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | simp at h | case nil.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝¹ u' v' : V
p : Walk G u✝¹ v✝¹
f : G →g G'
hinj : Injective ↑f
u v u✝ v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ u✝
h : Walk.map f nil = Walk.map f (cons h✝ p✝)
⊢ nil = cons h✝ p✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | cases p' with
| nil => simp at h
| cons _ _ =>
simp only [map_cons, cons.injEq] at h
cases hinj h.1
simp only [cons.injEq, heq_iff_eq, true_and_iff]
apply ih
simpa using h.2 | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝¹ u' v' : V
p : Walk G u✝¹ v✝¹
f : G →g G'
hinj : Injective ↑f
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ ⦃p' : Walk G v✝ w✝⦄, Walk.map f p✝ = Walk.map f p' → p✝ = p'
p' : Walk G u✝ w✝
h : Walk.map f (cons h✝ p✝) = Walk.map f p'
⊢ cons h✝ p✝ = p' | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | simp at h | case cons.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝¹ u' v' : V
p : Walk G u✝¹ v✝¹
f : G →g G'
hinj : Injective ↑f
u v u✝ v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ u✝
ih : ∀ ⦃p' : Walk G v✝ u✝⦄, Walk.map f p✝ = Walk.map f p' → p✝ = p'
h : Walk.map f (cons h✝ p✝) = Walk.map f nil
⊢ cons h✝ p✝ = nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | simp only [map_cons, cons.injEq] at h | case cons.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝² u' v' : V
p : Walk G u✝¹ v✝²
f : G →g G'
hinj : Injective ↑f
u v u✝ v✝¹ w✝ : V
h✝¹ : G.Adj u✝ v✝¹
p✝¹ : Walk G v✝¹ w✝
ih : ∀ ⦃p' : Walk G v✝¹ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'
v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
h : Walk.map f (cons h✝¹ p✝¹) = Walk.map f (cons h✝ p✝)
⊢ cons h✝¹ p✝¹ = cons h✝ p✝ | case cons.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝² u' v' : V
p : Walk G u✝¹ v✝²
f : G →g G'
hinj : Injective ↑f
u v u✝ v✝¹ w✝ : V
h✝¹ : G.Adj u✝ v✝¹
p✝¹ : Walk G v✝¹ w✝
ih : ∀ ⦃p' : Walk G v✝¹ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'
v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
h : ↑f v✝¹ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)
⊢ cons h✝¹ p✝¹ = cons h✝ p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | cases hinj h.1 | case cons.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝² u' v' : V
p : Walk G u✝¹ v✝²
f : G →g G'
hinj : Injective ↑f
u v u✝ v✝¹ w✝ : V
h✝¹ : G.Adj u✝ v✝¹
p✝¹ : Walk G v✝¹ w✝
ih : ∀ ⦃p' : Walk G v✝¹ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'
v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
h : ↑f v✝¹ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)
⊢ cons h✝¹ p✝¹ = cons h✝ p✝ | case cons.cons.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝¹ u' v' : V
p : Walk G u✝¹ v✝¹
f : G →g G'
hinj : Injective ↑f
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝¹ : Walk G v✝ w✝
ih : ∀ ⦃p' : Walk G v✝ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
h : ↑f v✝ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)
⊢ cons h✝¹ p✝¹ = cons h✝ p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | simp only [cons.injEq, heq_iff_eq, true_and_iff] | case cons.cons.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝¹ u' v' : V
p : Walk G u✝¹ v✝¹
f : G →g G'
hinj : Injective ↑f
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝¹ : Walk G v✝ w✝
ih : ∀ ⦃p' : Walk G v✝ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
h : ↑f v✝ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)
⊢ cons h✝¹ p✝¹ = cons h✝ p✝ | case cons.cons.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝¹ u' v' : V
p : Walk G u✝¹ v✝¹
f : G →g G'
hinj : Injective ↑f
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝¹ : Walk G v✝ w✝
ih : ∀ ⦃p' : Walk G v✝ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
h : ↑f v✝ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)
⊢ p✝¹ = p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | apply ih | case cons.cons.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝¹ u' v' : V
p : Walk G u✝¹ v✝¹
f : G →g G'
hinj : Injective ↑f
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝¹ : Walk G v✝ w✝
ih : ∀ ⦃p' : Walk G v✝ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
h : ↑f v✝ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)
⊢ p✝¹ = p✝ | case cons.cons.refl.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝¹ u' v' : V
p : Walk G u✝¹ v✝¹
f : G →g G'
hinj : Injective ↑f
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝¹ : Walk G v✝ w✝
ih : ∀ ⦃p' : Walk G v✝ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
h : ↑f v✝ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)
⊢ Walk.map f p✝¹ = Walk.map f p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | simpa using h.2 | case cons.cons.refl.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f✝ : G →g G'
f' : G' →g G''
u✝¹ v✝¹ u' v' : V
p : Walk G u✝¹ v✝¹
f : G →g G'
hinj : Injective ↑f
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝¹ : Walk G v✝ w✝
ih : ∀ ⦃p' : Walk G v✝ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
h : ↑f v✝ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)
⊢ Walk.map f p✝¹ = Walk.map f p✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Path.map_injective | [1397, 1] | [1401, 52] | rintro ⟨p, hp⟩ ⟨p', hp'⟩ h | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
hinj : Injective ↑f
u v : V
⊢ Injective (Path.map f hinj) | case mk.mk
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
hinj : Injective ↑f
u v : V
p : Walk G u v
hp : Walk.IsPath p
p' : Walk G u v
hp' : Walk.IsPath p'
h : Path.map f hinj { val := p, property := hp } = Path.map f hinj { val := p', property := hp' }
⊢ { val := p, property := hp } = { val := p', property := hp' } |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Path.map_injective | [1397, 1] | [1401, 52] | simp only [Path.map, Subtype.coe_mk, Subtype.mk.injEq] at h | case mk.mk
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
hinj : Injective ↑f
u v : V
p : Walk G u v
hp : Walk.IsPath p
p' : Walk G u v
hp' : Walk.IsPath p'
h : Path.map f hinj { val := p, property := hp } = Path.map f hinj { val := p', property := hp' }
⊢ { val := p, property := hp } = { val := p', property := hp' } | case mk.mk
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
hinj : Injective ↑f
u v : V
p : Walk G u v
hp : Walk.IsPath p
p' : Walk G u v
hp' : Walk.IsPath p'
h : Walk.map f p = Walk.map f p'
⊢ { val := p, property := hp } = { val := p', property := hp' } |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Path.map_injective | [1397, 1] | [1401, 52] | simp [Walk.map_injective_of_injective hinj u v h] | case mk.mk
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G →g G'
hinj : Injective ↑f
u v : V
p : Walk G u v
hp : Walk.IsPath p
p' : Walk G u v
hp' : Walk.IsPath p'
h : Walk.map f p = Walk.map f p'
⊢ { val := p, property := hp } = { val := p', property := hp' } | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_self | [1435, 1] | [1436, 27] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
⊢ Walk.transfer p G (_ : ∀ ⦃e : V × V⦄, e ∈ edges p → e ∈ edgeSet G) = p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_eq_map_of_le | [1439, 1] | [1441, 27] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : ∀ (e : V × V), e ∈ edges p → e ∈ edgeSet H
GH : G ≤ H
⊢ Walk.transfer p H hp = Walk.map (Hom.mapSpanningSubgraphs GH) p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_transfer | [1445, 1] | [1446, 27] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : ∀ (e : V × V), e ∈ edges p → e ∈ edgeSet H
⊢ edges (Walk.transfer p H hp) = edges p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_transfer | [1450, 1] | [1451, 27] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : ∀ (e : V × V), e ∈ edges p → e ∈ edgeSet H
⊢ support (Walk.transfer p H hp) = support p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_transfer | [1455, 1] | [1456, 27] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : ∀ (e : V × V), e ∈ edges p → e ∈ edgeSet H
⊢ length (Walk.transfer p H hp) = length p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.transfer | [1461, 11] | [1467, 28] | induction p with
| nil => simp
| cons _ _ ih =>
simp only [Walk.transfer, cons_isPath_iff, support_transfer _ ] at pp ⊢
exact ⟨ih _ pp.1, pp.2⟩ | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : ∀ (e : V × V), e ∈ edges p → e ∈ edgeSet H
pp : IsPath p
⊢ IsPath (Walk.transfer p H hp) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.transfer | [1461, 11] | [1467, 28] | simp | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
u✝ : V
hp : ∀ (e : V × V), e ∈ edges Walk.nil → e ∈ edgeSet H
pp : IsPath Walk.nil
⊢ IsPath (Walk.transfer Walk.nil H hp) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.transfer | [1461, 11] | [1467, 28] | simp only [Walk.transfer, cons_isPath_iff, support_transfer _ ] at pp ⊢ | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ (hp : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet H), IsPath p✝ → IsPath (Walk.transfer p✝ H hp)
hp : ∀ (e : V × V), e ∈ edges (cons h✝ p✝) → e ∈ edgeSet H
pp : IsPath (cons h✝ p✝)
⊢ IsPath (Walk.transfer (cons h✝ p✝) H hp) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ (hp : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet H), IsPath p✝ → IsPath (Walk.transfer p✝ H hp)
hp : ∀ (e : V × V), e ∈ edges (cons h✝ p✝) → e ∈ edgeSet H
pp : IsPath p✝ ∧ ¬u✝ ∈ support p✝
⊢ IsPath (Walk.transfer p✝ H (_ : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet H)) ∧ ¬u✝ ∈ support p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.transfer | [1461, 11] | [1467, 28] | exact ⟨ih _ pp.1, pp.2⟩ | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ (hp : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet H), IsPath p✝ → IsPath (Walk.transfer p✝ H hp)
hp : ∀ (e : V × V), e ∈ edges (cons h✝ p✝) → e ∈ edgeSet H
pp : IsPath p✝ ∧ ¬u✝ ∈ support p✝
⊢ IsPath (Walk.transfer p✝ H (_ : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet H)) ∧ ¬u✝ ∈ support p✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCycle.transfer | [1470, 11] | [1477, 37] | cases q with
| nil => simp at qc
| cons _ q =>
simp only [edges_cons, List.find?, List.mem_cons, forall_eq_or_imp, mem_edgeSet] at hq
simp only [Walk.transfer, cons_isCycle_iff, edges_transfer q hq.2] at qc ⊢
exact ⟨qc.1.transfer hq.2, qc.2⟩ | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
q : Walk G u u
qc : IsCycle q
hq : ∀ (e : V × V), e ∈ edges q → e ∈ edgeSet H
⊢ IsCycle (Walk.transfer q H hq) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCycle.transfer | [1470, 11] | [1477, 37] | simp at qc | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
qc : IsCycle nil
hq : ∀ (e : V × V), e ∈ edges nil → e ∈ edgeSet H
⊢ IsCycle (Walk.transfer nil H hq) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCycle.transfer | [1470, 11] | [1477, 37] | simp only [edges_cons, List.find?, List.mem_cons, forall_eq_or_imp, mem_edgeSet] at hq | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
v✝ : V
h✝ : G.Adj u v✝
q : Walk G v✝ u
qc : IsCycle (cons h✝ q)
hq : ∀ (e : V × V), e ∈ edges (cons h✝ q) → e ∈ edgeSet H
⊢ IsCycle (Walk.transfer (cons h✝ q) H hq) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
v✝ : V
h✝ : G.Adj u v✝
q : Walk G v✝ u
qc : IsCycle (cons h✝ q)
hq✝ : ∀ (e : V × V), e ∈ edges (cons h✝ q) → e ∈ edgeSet H
hq : H.Adj u v✝ ∧ ∀ (a : V × V), a ∈ edges q → a ∈ edgeSet H
⊢ IsCycle (Walk.transfer (cons h✝ q) H hq✝) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCycle.transfer | [1470, 11] | [1477, 37] | simp only [Walk.transfer, cons_isCycle_iff, edges_transfer q hq.2] at qc ⊢ | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
v✝ : V
h✝ : G.Adj u v✝
q : Walk G v✝ u
qc : IsCycle (cons h✝ q)
hq✝ : ∀ (e : V × V), e ∈ edges (cons h✝ q) → e ∈ edgeSet H
hq : H.Adj u v✝ ∧ ∀ (a : V × V), a ∈ edges q → a ∈ edgeSet H
⊢ IsCycle (Walk.transfer (cons h✝ q) H hq✝) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
v✝ : V
h✝ : G.Adj u v✝
q : Walk G v✝ u
hq✝ : ∀ (e : V × V), e ∈ edges (cons h✝ q) → e ∈ edgeSet H
hq : H.Adj u v✝ ∧ ∀ (a : V × V), a ∈ edges q → a ∈ edgeSet H
qc : IsPath q ∧ ¬(u, v✝) ∈ edges q
⊢ IsPath (Walk.transfer q H (_ : ∀ (e : V × V), e ∈ edges q → e ∈ edgeSet H)) ∧ ¬(u, v✝) ∈ edges q |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCycle.transfer | [1470, 11] | [1477, 37] | exact ⟨qc.1.transfer hq.2, qc.2⟩ | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
v✝ : V
h✝ : G.Adj u v✝
q : Walk G v✝ u
hq✝ : ∀ (e : V × V), e ∈ edges (cons h✝ q) → e ∈ edgeSet H
hq : H.Adj u v✝ ∧ ∀ (a : V × V), a ∈ edges q → a ∈ edgeSet H
qc : IsPath q ∧ ¬(u, v✝) ∈ edges q
⊢ IsPath (Walk.transfer q H (_ : ∀ (e : V × V), e ∈ edges q → e ∈ edgeSet H)) ∧ ¬(u, v✝) ∈ edges q | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_transfer | [1486, 1] | [1492, 13] | induction p with
| nil => simp
| cons _ _ ih =>
simp only [Walk.transfer, cons.injEq, heq_eq_eq, true_and]
apply ih | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : ∀ (e : V × V), e ∈ edges p → e ∈ edgeSet H
K : Digraph V
hp' : ∀ (e : V × V), e ∈ edges (Walk.transfer p H hp) → e ∈ edgeSet K
⊢ Walk.transfer (Walk.transfer p H hp) K hp' = Walk.transfer p K (_ : ∀ (e : V × V), e ∈ edges p → e ∈ edgeSet K) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_transfer | [1486, 1] | [1492, 13] | simp | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H K : Digraph V
u✝ : V
hp : ∀ (e : V × V), e ∈ edges nil → e ∈ edgeSet H
hp' : ∀ (e : V × V), e ∈ edges (Walk.transfer nil H hp) → e ∈ edgeSet K
⊢ Walk.transfer (Walk.transfer nil H hp) K hp' = Walk.transfer nil K (_ : ∀ (e : V × V), e ∈ edges nil → e ∈ edgeSet K) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_transfer | [1486, 1] | [1492, 13] | simp only [Walk.transfer, cons.injEq, heq_eq_eq, true_and] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H K : Digraph V
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih :
∀ (hp : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet H)
(hp' : ∀ (e : V × V), e ∈ edges (Walk.transfer p✝ H hp) → e ∈ edgeSet K),
Walk.transfer (Walk.transfer p✝ H hp) K hp' = Walk.transfer p✝ K (_ : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet K)
hp : ∀ (e : V × V), e ∈ edges (cons h✝ p✝) → e ∈ edgeSet H
hp' : ∀ (e : V × V), e ∈ edges (Walk.transfer (cons h✝ p✝) H hp) → e ∈ edgeSet K
⊢ Walk.transfer (Walk.transfer (cons h✝ p✝) H hp) K hp' =
Walk.transfer (cons h✝ p✝) K (_ : ∀ (e : V × V), e ∈ edges (cons h✝ p✝) → e ∈ edgeSet K) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H K : Digraph V
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih :
∀ (hp : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet H)
(hp' : ∀ (e : V × V), e ∈ edges (Walk.transfer p✝ H hp) → e ∈ edgeSet K),
Walk.transfer (Walk.transfer p✝ H hp) K hp' = Walk.transfer p✝ K (_ : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet K)
hp : ∀ (e : V × V), e ∈ edges (cons h✝ p✝) → e ∈ edgeSet H
hp' : ∀ (e : V × V), e ∈ edges (Walk.transfer (cons h✝ p✝) H hp) → e ∈ edgeSet K
⊢ Walk.transfer (Walk.transfer p✝ H (_ : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet H)) K
(_ :
∀ (e : V × V),
e ∈ edges (Walk.transfer p✝ H (_ : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet H)) → e ∈ edgeSet K) =
Walk.transfer p✝ K (_ : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet K) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_transfer | [1486, 1] | [1492, 13] | apply ih | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H K : Digraph V
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih :
∀ (hp : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet H)
(hp' : ∀ (e : V × V), e ∈ edges (Walk.transfer p✝ H hp) → e ∈ edgeSet K),
Walk.transfer (Walk.transfer p✝ H hp) K hp' = Walk.transfer p✝ K (_ : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet K)
hp : ∀ (e : V × V), e ∈ edges (cons h✝ p✝) → e ∈ edgeSet H
hp' : ∀ (e : V × V), e ∈ edges (Walk.transfer (cons h✝ p✝) H hp) → e ∈ edgeSet K
⊢ Walk.transfer (Walk.transfer p✝ H (_ : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet H)) K
(_ :
∀ (e : V × V),
e ∈ edges (Walk.transfer p✝ H (_ : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet H)) → e ∈ edgeSet K) =
Walk.transfer p✝ K (_ : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet K) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_append | [1496, 1] | [1502, 95] | simp [he] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
w : V
H : Digraph V
q : Walk G v w
hpq : ∀ (e : V × V), e ∈ edges (append p q) → e ∈ edgeSet H
e : V × V
he : e ∈ edges p
⊢ e ∈ edges (append p q) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_append | [1496, 1] | [1502, 95] | simp [he] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
w : V
H : Digraph V
q : Walk G v w
hpq : ∀ (e : V × V), e ∈ edges (append p q) → e ∈ edgeSet H
e : V × V
he : e ∈ edges q
⊢ e ∈ edges (append p q) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_append | [1496, 1] | [1502, 95] | induction p with
| nil => simp
| cons _ _ ih => simp only [Walk.transfer, cons_append, cons.injEq, heq_eq_eq, true_and, ih] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
w : V
H : Digraph V
q : Walk G v w
hpq : ∀ (e : V × V), e ∈ edges (append p q) → e ∈ edgeSet H
⊢ Walk.transfer (append p q) H hpq =
append (Walk.transfer p H (_ : ∀ (e : V × V), e ∈ edges p → e ∈ edgeSet H))
(Walk.transfer q H (_ : ∀ (e : V × V), e ∈ edges q → e ∈ edgeSet H)) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_append | [1496, 1] | [1502, 95] | simp | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
w : V
H : Digraph V
u✝ : V
q : Walk G u✝ w
hpq : ∀ (e : V × V), e ∈ edges (append nil q) → e ∈ edgeSet H
⊢ Walk.transfer (append nil q) H hpq =
append (Walk.transfer nil H (_ : ∀ (e : V × V), e ∈ edges nil → e ∈ edgeSet H))
(Walk.transfer q H (_ : ∀ (e : V × V), e ∈ edges q → e ∈ edgeSet H)) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_append | [1496, 1] | [1502, 95] | simp only [Walk.transfer, cons_append, cons.injEq, heq_eq_eq, true_and, ih] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
w : V
H : Digraph V
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih :
∀ (q : Walk G w✝ w) (hpq : ∀ (e : V × V), e ∈ edges (append p✝ q) → e ∈ edgeSet H),
Walk.transfer (append p✝ q) H hpq =
append (Walk.transfer p✝ H (_ : ∀ (e : V × V), e ∈ edges p✝ → e ∈ edgeSet H))
(Walk.transfer q H (_ : ∀ (e : V × V), e ∈ edges q → e ∈ edgeSet H))
q : Walk G w✝ w
hpq : ∀ (e : V × V), e ∈ edges (append (cons h✝ p✝) q) → e ∈ edgeSet H
⊢ Walk.transfer (append (cons h✝ p✝) q) H hpq =
append (Walk.transfer (cons h✝ p✝) H (_ : ∀ (e : V × V), e ∈ edges (cons h✝ p✝) → e ∈ edgeSet H))
(Walk.transfer q H (_ : ∀ (e : V × V), e ∈ edges q → e ∈ edgeSet H)) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_toDeleteEdges_eq | [1541, 1] | [1546, 33] | rw [← transfer_eq_map_of_le, transfer_transfer, transfer_self] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V × V)
p : Walk G v w
hp : ∀ (e : V × V), e ∈ edges p → ¬e ∈ s
⊢ Walk.map (Hom.mapSpanningSubgraphs (_ : deleteEdges G s ≤ G)) (toDeleteEdges s p hp) = p | case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V × V)
p : Walk G v w
hp : ∀ (e : V × V), e ∈ edges p → ¬e ∈ s
⊢ ∀ (e : V × V),
e ∈ edges (Walk.transfer p (deleteEdges G s) (_ : ∀ (e : V × V), e ∈ edges p → e ∈ edgeSet (deleteEdges G s))) →
e ∈ edgeSet G
case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V × V)
p : Walk G v w
hp : ∀ (e : V × V), e ∈ edges p → ¬e ∈ s
⊢ ∀ (e : V × V),
e ∈ edges (Walk.transfer p (deleteEdges G s) (_ : ∀ (e : V × V), e ∈ edges p → e ∈ edgeSet (deleteEdges G s))) →
e ∈ edgeSet G |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_toDeleteEdges_eq | [1541, 1] | [1546, 33] | intros e | case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V × V)
p : Walk G v w
hp : ∀ (e : V × V), e ∈ edges p → ¬e ∈ s
⊢ ∀ (e : V × V),
e ∈ edges (Walk.transfer p (deleteEdges G s) (_ : ∀ (e : V × V), e ∈ edges p → e ∈ edgeSet (deleteEdges G s))) →
e ∈ edgeSet G | case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V × V)
p : Walk G v w
hp : ∀ (e : V × V), e ∈ edges p → ¬e ∈ s
e : V × V
⊢ e ∈ edges (Walk.transfer p (deleteEdges G s) (_ : ∀ (e : V × V), e ∈ edges p → e ∈ edgeSet (deleteEdges G s))) →
e ∈ edgeSet G |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_toDeleteEdges_eq | [1541, 1] | [1546, 33] | rw [edges_transfer] | case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V × V)
p : Walk G v w
hp : ∀ (e : V × V), e ∈ edges p → ¬e ∈ s
e : V × V
⊢ e ∈ edges (Walk.transfer p (deleteEdges G s) (_ : ∀ (e : V × V), e ∈ edges p → e ∈ edgeSet (deleteEdges G s))) →
e ∈ edgeSet G | case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V × V)
p : Walk G v w
hp : ∀ (e : V × V), e ∈ edges p → ¬e ∈ s
e : V × V
⊢ e ∈ edges p → e ∈ edgeSet G |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_toDeleteEdges_eq | [1541, 1] | [1546, 33] | apply edges_subset_edgeSet p | case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V × V)
p : Walk G v w
hp : ∀ (e : V × V), e ∈ edges p → ¬e ∈ s
e : V × V
⊢ e ∈ edges p → e ∈ edgeSet G | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.toDeleteEdges_copy | [1557, 1] | [1562, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u' v' : V
s : Set (V × V)
p : Walk G u v
hu : u = u'
hv : v = v'
h : ∀ (e : V × V), e ∈ edges (Walk.copy p hu hv) → ¬e ∈ s
⊢ ∀ (e : V × V), e ∈ edges p → ¬e ∈ s | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' : V
s : Set (V × V)
p : Walk G u' v'
h : ∀ (e : V × V), e ∈ edges (Walk.copy p (_ : u' = u') (_ : v' = v')) → ¬e ∈ s
⊢ ∀ (e : V × V), e ∈ edges p → ¬e ∈ s |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.toDeleteEdges_copy | [1557, 1] | [1562, 6] | exact h | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' : V
s : Set (V × V)
p : Walk G u' v'
h : ∀ (e : V × V), e ∈ edges (Walk.copy p (_ : u' = u') (_ : v' = v')) → ¬e ∈ s
⊢ ∀ (e : V × V), e ∈ edges p → ¬e ∈ s | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.toDeleteEdges_copy | [1557, 1] | [1562, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u' v' : V
s : Set (V × V)
p : Walk G u v
hu : u = u'
hv : v = v'
h : ∀ (e : V × V), e ∈ edges (Walk.copy p hu hv) → ¬e ∈ s
⊢ toDeleteEdges s (Walk.copy p hu hv) h = Walk.copy (toDeleteEdges s p (_ : ∀ (e : V × V), e ∈ edges p → ¬e ∈ s)) hu hv | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' : V
s : Set (V × V)
p : Walk G u' v'
h : ∀ (e : V × V), e ∈ edges (Walk.copy p (_ : u' = u') (_ : v' = v')) → ¬e ∈ s
⊢ toDeleteEdges s (Walk.copy p (_ : u' = u') (_ : v' = v')) h =
Walk.copy (toDeleteEdges s p (_ : ∀ (e : V × V), e ∈ edges p → ¬e ∈ s)) (_ : u' = u') (_ : v' = v') |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.toDeleteEdges_copy | [1557, 1] | [1562, 6] | rfl | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' : V
s : Set (V × V)
p : Walk G u' v'
h : ∀ (e : V × V), e ∈ edges (Walk.copy p (_ : u' = u') (_ : v' = v')) → ¬e ∈ s
⊢ toDeleteEdges s (Walk.copy p (_ : u' = u') (_ : v' = v')) h =
Walk.copy (toDeleteEdges s p (_ : ∀ (e : V × V), e ∈ edges p → ¬e ∈ s)) (_ : u' = u') (_ : v' = v') | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Reachable.elim_path | [1587, 11] | [1588, 80] | classical exact h.elim fun q => hp q.toPath | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
p : Prop
u v : V
h : Reachable G u v
hp : Path G u v → p
⊢ p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Reachable.elim_path | [1587, 11] | [1588, 80] | exact h.elim fun q => hp q.toPath | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
p : Prop
u v : V
h : Reachable G u v
hp : Path G u v → p
⊢ p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | constructor | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
⊢ Reachable G u v ↔ Relation.ReflTransGen G.Adj u v | case mp
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
⊢ Reachable G u v → Relation.ReflTransGen G.Adj u v
case mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
⊢ Relation.ReflTransGen G.Adj u v → Reachable G u v |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | rintro ⟨h⟩ | case mp
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
⊢ Reachable G u v → Relation.ReflTransGen G.Adj u v | case mp.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : Walk G u v
⊢ Relation.ReflTransGen G.Adj u v |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | induction h with
| nil => rfl
| cons h' _ ih => exact (Relation.ReflTransGen.single h').trans ih | case mp.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : Walk G u v
⊢ Relation.ReflTransGen G.Adj u v | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | rfl | case mp.intro.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ : V
⊢ Relation.ReflTransGen G.Adj u✝ u✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | exact (Relation.ReflTransGen.single h').trans ih | case mp.intro.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ v✝ w✝ : V
h' : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : Relation.ReflTransGen G.Adj v✝ w✝
⊢ Relation.ReflTransGen G.Adj u✝ w✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | intro h | case mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
⊢ Relation.ReflTransGen G.Adj u v → Reachable G u v | case mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : Relation.ReflTransGen G.Adj u v
⊢ Reachable G u v |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | induction h with
| refl => rfl
| tail _ ha hr => exact Reachable.trans hr ⟨Walk.cons ha Walk.nil⟩ | case mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : Relation.ReflTransGen G.Adj u v
⊢ Reachable G u v | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | rfl | case mpr.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
⊢ Reachable G u u | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | exact Reachable.trans hr ⟨Walk.cons ha Walk.nil⟩ | case mpr.tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v b✝ c✝ : V
a✝ : Relation.ReflTransGen G.Adj u b✝
ha : G.Adj b✝ c✝
hr : Reachable G u b✝
⊢ Reachable G u c✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Iso.symm_apply_reachable | [1644, 1] | [1646, 52] | rw [← Iso.reachable_iff, RelIso.apply_symm_apply] | V : Type u
V' : Type v
V'' : Type w
G✝ : Digraph V
G'✝ : Digraph V'
G'' : Digraph V''
G : Digraph V
G' : Digraph V'
φ : G ≃g G'
u : V
v : V'
⊢ Reachable G (↑(symm φ) v) u ↔ Reachable G' v (↑φ u) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_self_length_zero_eq | [2030, 1] | [2032, 7] | ext p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
⊢ {p | Walk.length p = 0} = {Walk.nil} | case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
p : Walk G u u
⊢ p ∈ {p | Walk.length p = 0} ↔ p ∈ {Walk.nil} |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_self_length_zero_eq | [2030, 1] | [2032, 7] | simp | case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
p : Walk G u u
⊢ p ∈ {p | Walk.length p = 0} ↔ p ∈ {Walk.nil} | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_zero_eq_of_ne | [2035, 1] | [2039, 58] | ext p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u ≠ v
⊢ {p | Walk.length p = 0} = ∅ | case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u ≠ v
p : Walk G u v
⊢ p ∈ {p | Walk.length p = 0} ↔ p ∈ ∅ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_zero_eq_of_ne | [2035, 1] | [2039, 58] | simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff] | case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u ≠ v
p : Walk G u v
⊢ p ∈ {p | Walk.length p = 0} ↔ p ∈ ∅ | case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u ≠ v
p : Walk G u v
⊢ ¬Walk.length p = 0 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_zero_eq_of_ne | [2035, 1] | [2039, 58] | exact fun h' => absurd (Walk.eq_of_length_eq_zero h') h | case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u ≠ v
p : Walk G u v
⊢ ¬Walk.length p = 0 | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_succ_eq | [2042, 1] | [2054, 10] | ext p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : ℕ
⊢ {p | Walk.length p = Nat.succ n} = ⋃ (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n} | case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : ℕ
p : Walk G u v
⊢ p ∈ {p | Walk.length p = Nat.succ n} ↔ p ∈ ⋃ (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n} |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_succ_eq | [2042, 1] | [2054, 10] | cases' p with _ _ w _ huw pwv | case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : ℕ
p : Walk G u v
⊢ p ∈ {p | Walk.length p = Nat.succ n} ↔ p ∈ ⋃ (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n} | case h.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
n : ℕ
⊢ Walk.nil ∈ {p | Walk.length p = Nat.succ n} ↔
Walk.nil ∈ ⋃ (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n}
case h.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : ℕ
w : V
huw : G.Adj u w
pwv : Walk G w v
⊢ Walk.cons huw pwv ∈ {p | Walk.length p = Nat.succ n} ↔
Walk.cons huw pwv ∈ ⋃ (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n} |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_succ_eq | [2042, 1] | [2054, 10] | simp [eq_comm] | case h.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
n : ℕ
⊢ Walk.nil ∈ {p | Walk.length p = Nat.succ n} ↔
Walk.nil ∈ ⋃ (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n} | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_succ_eq | [2042, 1] | [2054, 10] | simp only [Nat.succ_eq_add_one, Set.mem_setOf_eq, Walk.length_cons, add_left_inj,
Set.mem_iUnion, Set.mem_image, exists_prop] | case h.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : ℕ
w : V
huw : G.Adj u w
pwv : Walk G w v
⊢ Walk.cons huw pwv ∈ {p | Walk.length p = Nat.succ n} ↔
Walk.cons huw pwv ∈ ⋃ (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n} | case h.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : ℕ
w : V
huw : G.Adj u w
pwv : Walk G w v
⊢ Walk.length pwv = n ↔ ∃ i h x, Walk.length x = n ∧ Walk.cons (_ : G.Adj u i) x = Walk.cons huw pwv |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_succ_eq | [2042, 1] | [2054, 10] | constructor | case h.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : ℕ
w : V
huw : G.Adj u w
pwv : Walk G w v
⊢ Walk.length pwv = n ↔ ∃ i h x, Walk.length x = n ∧ Walk.cons (_ : G.Adj u i) x = Walk.cons huw pwv | case h.cons.mp
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : ℕ
w : V
huw : G.Adj u w
pwv : Walk G w v
⊢ Walk.length pwv = n → ∃ i h x, Walk.length x = n ∧ Walk.cons (_ : G.Adj u i) x = Walk.cons huw pwv
case h.cons.mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : ℕ
w : V
huw : G.Adj u w
pwv : Walk G w v
⊢ (∃ i h x, Walk.length x = n ∧ Walk.cons (_ : G.Adj u i) x = Walk.cons huw pwv) → Walk.length pwv = n |
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