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https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.coe_support_append | [535, 1] | [537, 55] | rw [support_append, ← Multiset.coe_add, coe_support] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
p' : Walk G v w
⊢ ↑(support (append p p')) = {u} + ↑(List.tail (support p)) + ↑(List.tail (support p')) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.coe_support_append' | [540, 1] | [545, 49] | rw [support_append, ← Multiset.coe_add] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G u v
p' : Walk G v w
⊢ ↑(support (append p p')) = ↑(support p) + ↑(support p') - {v} | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G u v
p' : Walk G v w
⊢ ↑(support p) + ↑(List.tail (support p')) = ↑(support p) + ↑(support p') - {v} |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.coe_support_append' | [540, 1] | [545, 49] | simp only [coe_support] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G u v
p' : Walk G v w
⊢ ↑(support p) + ↑(List.tail (support p')) = ↑(support p) + ↑(support p') - {v} | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G u v
p' : Walk G v w
⊢ {u} + ↑(List.tail (support p)) + ↑(List.tail (support p')) =
{u} + ↑(List.tail (support p)) + ({v} + ↑(List.tail (support p'))) - {v} |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.coe_support_append' | [540, 1] | [545, 49] | rw [add_comm ({v} : Multiset V)] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G u v
p' : Walk G v w
⊢ {u} + ↑(List.tail (support p)) + ↑(List.tail (support p')) =
{u} + ↑(List.tail (support p)) + ({v} + ↑(List.tail (support p'))) - {v} | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G u v
p' : Walk G v w
⊢ {u} + ↑(List.tail (support p)) + ↑(List.tail (support p')) =
{u} + ↑(List.tail (support p)) + (↑(List.tail (support p')) + {v}) - {v} |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.coe_support_append' | [540, 1] | [545, 49] | simp only [← add_assoc, add_tsub_cancel_right] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G u v
p' : Walk G v w
⊢ {u} + ↑(List.tail (support p)) + ↑(List.tail (support p')) =
{u} + ↑(List.tail (support p)) + (↑(List.tail (support p')) + {v}) - {v} | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_subset_edgeSet | [561, 1] | [566, 48] | cases h | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h' : G.Adj u v✝
p' : Walk G v✝ v
e : V × V
h : e ∈ edges (cons h' p')
⊢ e ∈ edgeSet G | case head
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h' : G.Adj u v✝
p' : Walk G v✝ v
⊢ (u, v✝) ∈ edgeSet G
case tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h' : G.Adj u v✝
p' : Walk G v✝ v
e : V × V
a✝ : List.Mem e (edges p')
⊢ e ∈ edgeSet G |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_subset_edgeSet | [561, 1] | [566, 48] | next h' => exact edges_subset_edgeSet p' h' | case tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h' : G.Adj u v✝
p' : Walk G v✝ v
e : V × V
a✝ : List.Mem e (edges p')
⊢ e ∈ edgeSet G | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_subset_edgeSet | [561, 1] | [566, 48] | exact h' | case head
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h' : G.Adj u v✝
p' : Walk G v✝ v
⊢ (u, v✝) ∈ edgeSet G | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_subset_edgeSet | [561, 1] | [566, 48] | exact edges_subset_edgeSet p' h' | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h'✝ : G.Adj u v✝
p' : Walk G v✝ v
e : V × V
h' : List.Mem e (edges p')
⊢ e ∈ edgeSet G | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_concat | [579, 1] | [581, 39] | induction p <;> simp [*, concat_nil] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
h : G.Adj v w
⊢ edges (concat p h) = List.concat (edges p) (v, w) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_copy | [584, 1] | [587, 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
p : Walk G u v
hu : u = u'
hv : v = v'
⊢ edges (Walk.copy p hu hv) = edges p | 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'
⊢ edges (Walk.copy p (_ : u' = u') (_ : v' = v')) = edges p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_copy | [584, 1] | [587, 6] | rfl | 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'
⊢ edges (Walk.copy p (_ : u' = u') (_ : v' = v')) = edges p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_append | [590, 1] | [592, 27] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
p' : Walk G v w
⊢ edges (append p p') = edges p ++ edges p' | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.cons_map_snd_edges | [594, 1] | [595, 28] | 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
⊢ u :: List.map (fun x => x.snd) (edges p) = support p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_snd_edges | [597, 1] | [598, 57] | simpa using congr_arg List.tail (cons_map_snd_edges p) | 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
⊢ List.map (fun x => x.snd) (edges p) = List.tail (support p) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_fst_edges_append | [600, 1] | [602, 28] | 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
⊢ List.map (fun x => x.fst) (edges p) ++ [v] = support p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_fst_darts | [604, 1] | [605, 64] | simpa! using congr_arg List.dropLast (map_fst_edges_append p) | 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
⊢ List.map (fun x => x.fst) (edges p) = List.dropLast (support p) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_support | [609, 1] | [610, 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
⊢ List.length (support p) = length p + 1 | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_edges | [614, 1] | [615, 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
⊢ List.length (edges p) = length p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_fst_mem_support_of_mem_darts | [617, 1] | [623, 61] | simp only [support_cons, edges_cons, List.mem_cons] at hd ⊢ | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h : G.Adj u v✝
p' : Walk G v✝ v
d : V × V
hd : d ∈ edges (cons h p')
⊢ d.fst ∈ support (cons h p') | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h : G.Adj u v✝
p' : Walk G v✝ v
d : V × V
hd : d = (u, v✝) ∨ d ∈ edges p'
⊢ d.fst = u ∨ d.fst ∈ support p' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_fst_mem_support_of_mem_darts | [617, 1] | [623, 61] | rcases hd with (rfl | hd) | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h : G.Adj u v✝
p' : Walk G v✝ v
d : V × V
hd : d = (u, v✝) ∨ d ∈ edges p'
⊢ d.fst = u ∨ d.fst ∈ support p' | case inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h : G.Adj u v✝
p' : Walk G v✝ v
⊢ (u, v✝).fst = u ∨ (u, v✝).fst ∈ support p'
case inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h : G.Adj u v✝
p' : Walk G v✝ v
d : V × V
hd : d ∈ edges p'
⊢ d.fst = u ∨ d.fst ∈ support p' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_fst_mem_support_of_mem_darts | [617, 1] | [623, 61] | exact Or.inl rfl | case inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h : G.Adj u v✝
p' : Walk G v✝ v
⊢ (u, v✝).fst = u ∨ (u, v✝).fst ∈ support p' | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_fst_mem_support_of_mem_darts | [617, 1] | [623, 61] | exact Or.inr (edges_fst_mem_support_of_mem_darts _ hd) | case inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h : G.Adj u v✝
p' : Walk G v✝ v
d : V × V
hd : d ∈ edges p'
⊢ d.fst = u ∨ d.fst ∈ support p' | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_snd_mem_support_of_mem_darts | [625, 1] | [631, 61] | simp only [support_cons, edges_cons, List.mem_cons] at hd ⊢ | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h : G.Adj u v✝
p' : Walk G v✝ v
d : V × V
hd : d ∈ edges (cons h p')
⊢ d.snd ∈ support (cons h p') | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h : G.Adj u v✝
p' : Walk G v✝ v
d : V × V
hd : d = (u, v✝) ∨ d ∈ edges p'
⊢ d.snd = u ∨ d.snd ∈ support p' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_snd_mem_support_of_mem_darts | [625, 1] | [631, 61] | rcases hd with (rfl | hd) | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h : G.Adj u v✝
p' : Walk G v✝ v
d : V × V
hd : d = (u, v✝) ∨ d ∈ edges p'
⊢ d.snd = u ∨ d.snd ∈ support p' | case inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h : G.Adj u v✝
p' : Walk G v✝ v
⊢ (u, v✝).snd = u ∨ (u, v✝).snd ∈ support p'
case inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h : G.Adj u v✝
p' : Walk G v✝ v
d : V × V
hd : d ∈ edges p'
⊢ d.snd = u ∨ d.snd ∈ support p' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_snd_mem_support_of_mem_darts | [625, 1] | [631, 61] | simp | case inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h : G.Adj u v✝
p' : Walk G v✝ v
⊢ (u, v✝).snd = u ∨ (u, v✝).snd ∈ support p' | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_snd_mem_support_of_mem_darts | [625, 1] | [631, 61] | exact Or.inr (edges_snd_mem_support_of_mem_darts _ hd) | case inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v✝ : V
h : G.Adj u v✝
p' : Walk G v✝ v
d : V × V
hd : d ∈ edges p'
⊢ d.snd = u ∨ d.snd ∈ support p' | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.fst_mem_support_of_mem_edges | [633, 1] | [635, 48] | exact edges_fst_mem_support_of_mem_darts _ he | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
t u v w : V
p : Walk G v w
he : (t, u) ∈ edges p
⊢ t ∈ support p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.snd_mem_support_of_mem_edges | [637, 1] | [639, 48] | exact edges_snd_mem_support_of_mem_darts _ he | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
t u v w : V
p : Walk G v w
he : (t, u) ∈ edges p
⊢ u ∈ support p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_nodup_of_support_nodup | [641, 1] | [647, 77] | induction p with
| nil => simp
| cons _ p' ih =>
simp only [edges_cons, support_cons, List.nodup_cons] at h ⊢
exact ⟨fun h' => h.1 (edges_fst_mem_support_of_mem_darts p' h'), ih h.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 : List.Nodup (support p)
⊢ List.Nodup (edges p) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_nodup_of_support_nodup | [641, 1] | [647, 77] | simp | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ : V
h : List.Nodup (support nil)
⊢ List.Nodup (edges nil) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_nodup_of_support_nodup | [641, 1] | [647, 77] | simp only [edges_cons, support_cons, List.nodup_cons] at h ⊢ | case 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 : List.Nodup (support p') → List.Nodup (edges p')
h : List.Nodup (support (cons h✝ p'))
⊢ List.Nodup (edges (cons h✝ p')) | case 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 : List.Nodup (support p') → List.Nodup (edges p')
h : ¬u✝ ∈ support p' ∧ List.Nodup (support p')
⊢ ¬(u✝, v✝) ∈ edges p' ∧ List.Nodup (edges p') |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_nodup_of_support_nodup | [641, 1] | [647, 77] | exact ⟨fun h' => h.1 (edges_fst_mem_support_of_mem_darts p' h'), ih h.2⟩ | case 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 : List.Nodup (support p') → List.Nodup (edges p')
h : ¬u✝ ∈ support p' ∧ List.Nodup (support p')
⊢ ¬(u✝, v✝) ∈ edges p' ∧ List.Nodup (edges p') | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.isTrail_copy | [691, 1] | [694, 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
p : Walk G u v
hu : u = u'
hv : v = v'
⊢ IsTrail (Walk.copy p hu hv) ↔ IsTrail p | 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'
⊢ IsTrail (Walk.copy p (_ : u' = u') (_ : v' = v')) ↔ IsTrail p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.isTrail_copy | [691, 1] | [694, 6] | rfl | 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'
⊢ IsTrail (Walk.copy p (_ : u' = u') (_ : v' = v')) ↔ IsTrail p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.isPath_copy | [706, 1] | [709, 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
p : Walk G u v
hu : u = u'
hv : v = v'
⊢ IsPath (Walk.copy p hu hv) ↔ IsPath p | 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'
⊢ IsPath (Walk.copy p (_ : u' = u') (_ : v' = v')) ↔ IsPath p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.isPath_copy | [706, 1] | [709, 6] | rfl | 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'
⊢ IsPath (Walk.copy p (_ : u' = u') (_ : v' = v')) ↔ IsPath p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.isCircuit_copy | [717, 1] | [720, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u u' : V
p : Walk G u u
hu : u = u'
⊢ IsCircuit (Walk.copy p hu hu) ↔ IsCircuit p | 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'
⊢ IsCircuit (Walk.copy p (_ : u' = u') (_ : u' = u')) ↔ IsCircuit p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.isCircuit_copy | [717, 1] | [720, 6] | rfl | 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'
⊢ IsCircuit (Walk.copy p (_ : u' = u') (_ : u' = u')) ↔ IsCircuit p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.isCycle_copy | [729, 1] | [732, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u u' : V
p : Walk G u u
hu : u = u'
⊢ IsCycle (Walk.copy p hu hu) ↔ IsCycle p | 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'
⊢ IsCycle (Walk.copy p (_ : u' = u') (_ : u' = u')) ↔ IsCycle p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.isCycle_copy | [729, 1] | [732, 6] | rfl | 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'
⊢ IsCycle (Walk.copy p (_ : u' = u') (_ : u' = u')) ↔ IsCycle p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsTrail.nil | [736, 1] | [737, 20] | simp [edges] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
⊢ List.Nodup (edges Walk.nil) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsTrail.of_cons | [740, 1] | [741, 60] | simp [isTrail_def] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
h : G.Adj u v
p : Walk G v w
⊢ IsTrail (cons h p) → IsTrail p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.cons_isTrail_iff | [745, 1] | [746, 89] | simp [isTrail_def, and_comm] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
h : G.Adj u v
p : Walk G v w
⊢ IsTrail (cons h p) ↔ IsTrail p ∧ ¬(u, v) ∈ edges p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsTrail.of_append_left | [749, 1] | [752, 14] | rw [isTrail_def, edges_append, List.nodup_append] at h | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
q : Walk G v w
h : IsTrail (append p q)
⊢ IsTrail p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
q : Walk G v w
h : List.Nodup (edges p) ∧ List.Nodup (edges q) ∧ List.Disjoint (edges p) (edges q)
⊢ IsTrail p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsTrail.of_append_left | [749, 1] | [752, 14] | exact ⟨h.1⟩ | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
q : Walk G v w
h : List.Nodup (edges p) ∧ List.Nodup (edges q) ∧ List.Disjoint (edges p) (edges q)
⊢ IsTrail p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsTrail.of_append_right | [755, 1] | [758, 16] | rw [isTrail_def, edges_append, List.nodup_append] at h | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
q : Walk G v w
h : IsTrail (append p q)
⊢ IsTrail q | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
q : Walk G v w
h : List.Nodup (edges p) ∧ List.Nodup (edges q) ∧ List.Disjoint (edges p) (edges q)
⊢ IsTrail q |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsTrail.of_append_right | [755, 1] | [758, 16] | exact ⟨h.2.1⟩ | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
q : Walk G v w
h : List.Nodup (edges p) ∧ List.Nodup (edges q) ∧ List.Disjoint (edges p) (edges q)
⊢ IsTrail q | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.List.count_prod_eq | [764, 1] | [770, 17] | congr! | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
xs : List (V × V)
e : V × V
⊢ List.count e xs = List.count e xs | case h.e'_2
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
xs : List (V × V)
e : V × V
⊢ instBEqProd = instBEq |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.List.count_prod_eq | [764, 1] | [770, 17] | ext p q | case h.e'_2
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
xs : List (V × V)
e : V × V
⊢ instBEqProd = instBEq | case h.e'_2.beq.h.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
xs : List (V × V)
e p q : V × V
⊢ (p == q) = (p == q) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.List.count_prod_eq | [764, 1] | [770, 17] | cases p | case h.e'_2.beq.h.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
xs : List (V × V)
e p q : V × V
⊢ (p == q) = (p == q) | case h.e'_2.beq.h.h.mk
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
xs : List (V × V)
e q : V × V
fst✝ snd✝ : V
⊢ ((fst✝, snd✝) == q) = ((fst✝, snd✝) == q) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.List.count_prod_eq | [764, 1] | [770, 17] | cases q | case h.e'_2.beq.h.h.mk
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
xs : List (V × V)
e q : V × V
fst✝ snd✝ : V
⊢ ((fst✝, snd✝) == q) = ((fst✝, snd✝) == q) | case h.e'_2.beq.h.h.mk.mk
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
xs : List (V × V)
e : V × V
fst✝¹ snd✝¹ fst✝ snd✝ : V
⊢ ((fst✝¹, snd✝¹) == (fst✝, snd✝)) = ((fst✝¹, snd✝¹) == (fst✝, snd✝)) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.List.count_prod_eq | [764, 1] | [770, 17] | simp [BEq.beq] | case h.e'_2.beq.h.h.mk.mk
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
xs : List (V × V)
e : V × V
fst✝¹ snd✝¹ fst✝ snd✝ : V
⊢ ((fst✝¹, snd✝¹) == (fst✝, snd✝)) = ((fst✝¹, snd✝¹) == (fst✝, snd✝)) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsTrail.count_edges_le_one | [772, 1] | [774, 61] | simpa using List.nodup_iff_count_le_one.mp h.edges_nodup e | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
p : Walk G u v
h : IsTrail p
e : V × V
⊢ List.count e (edges p) ≤ 1 | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsTrail.count_edges_eq_one | [776, 1] | [778, 56] | simpa using List.count_eq_one_of_mem h.edges_nodup he | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
p : Walk G u v
h : IsTrail p
e : V × V
he : e ∈ edges p
⊢ List.count e (edges p) = 1 | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.nil | [780, 1] | [780, 82] | constructor <;> simp | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
⊢ IsPath Walk.nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_cons | [783, 1] | [784, 57] | simp [isPath_def] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
h : G.Adj u v
p : Walk G v w
⊢ IsPath (cons h p) → IsPath p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.cons_isPath_iff | [788, 1] | [790, 71] | constructor <;> simp (config := { contextual := true }) [isPath_def] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
h : G.Adj u v
p : Walk G v w
⊢ IsPath (cons h p) ↔ IsPath p ∧ ¬u ∈ support p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.isPath_iff_eq_nil | [794, 1] | [795, 32] | cases p <;> simp [IsPath.nil] | 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
⊢ IsPath p ↔ p = nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_append_left | [798, 1] | [801, 34] | simp only [isPath_def, support_append] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
q : Walk G v w
⊢ IsPath (append p q) → IsPath p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
q : Walk G v w
⊢ List.Nodup (support p ++ List.tail (support q)) → List.Nodup (support p) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_append_left | [798, 1] | [801, 34] | exact List.Nodup.of_append_left | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
q : Walk G v w
⊢ List.Nodup (support p ++ List.tail (support q)) → List.Nodup (support p) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_append_right | [804, 1] | [808, 46] | induction p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
q : Walk G v w
⊢ IsPath (append p q) → IsPath q | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w u✝ : V
q : Walk G u✝ w
⊢ IsPath (append Walk.nil q) → IsPath q
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ {q : Walk G w✝ w}, IsPath (append p✝ q) → IsPath q
q : Walk G w✝ w
⊢ IsPath (append (cons h✝ p✝) q) → IsPath q |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_append_right | [804, 1] | [808, 46] | exact id | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w u✝ : V
q : Walk G u✝ w
⊢ IsPath (append Walk.nil q) → IsPath q | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_append_right | [804, 1] | [808, 46] | simp (config := {contextual := true}) [*] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ {q : Walk G w✝ w}, IsPath (append p✝ q) → IsPath q
q : Walk G w✝ w
⊢ IsPath (append (cons h✝ p✝) q) → IsPath q | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.cons_isCycle_iff | [814, 1] | [819, 8] | simp only [Walk.isCycle_def, Walk.isPath_def, Walk.isTrail_def, edges_cons, List.nodup_cons,
support_cons, List.tail_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 v u
h : G.Adj u v
⊢ IsCycle (cons h p) ↔ IsPath p ∧ ¬(u, v) ∈ edges p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G v u
h : G.Adj u v
⊢ (¬(u, v) ∈ edges p ∧ List.Nodup (edges p)) ∧ cons h p ≠ nil ∧ List.Nodup (support p) ↔
List.Nodup (support p) ∧ ¬(u, v) ∈ edges p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.cons_isCycle_iff | [814, 1] | [819, 8] | have : p.support.Nodup → p.edges.Nodup := edges_nodup_of_support_nodup | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G v u
h : G.Adj u v
⊢ (¬(u, v) ∈ edges p ∧ List.Nodup (edges p)) ∧ cons h p ≠ nil ∧ List.Nodup (support p) ↔
List.Nodup (support p) ∧ ¬(u, v) ∈ edges p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G v u
h : G.Adj u v
this : List.Nodup (support p) → List.Nodup (edges p)
⊢ (¬(u, v) ∈ edges p ∧ List.Nodup (edges p)) ∧ cons h p ≠ nil ∧ List.Nodup (support p) ↔
List.Nodup (support p) ∧ ¬(u, v) ∈ edges p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.cons_isCycle_iff | [814, 1] | [819, 8] | tauto | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G v u
h : G.Adj u v
this : List.Nodup (support p) → List.Nodup (edges p)
⊢ (¬(u, v) ∈ edges p ∧ List.Nodup (edges p)) ∧ cons h p ≠ nil ∧ List.Nodup (support p) ↔
List.Nodup (support p) ∧ ¬(u, v) ∈ edges p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.length_lt | [828, 1] | [831, 40] | rw [Nat.lt_iff_add_one_le, ← length_support] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : Fintype V
u v : V
p : Walk G u v
hp : IsPath p
⊢ length p < Fintype.card V | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : Fintype V
u v : V
p : Walk G u v
hp : IsPath p
⊢ List.length (support p) ≤ Fintype.card V |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.length_lt | [828, 1] | [831, 40] | exact hp.support_nodup.length_le_card | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : Fintype V
u v : V
p : Walk G u v
hp : IsPath p
⊢ List.length (support p) ≤ Fintype.card V | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.take_spec | [866, 1] | [875, 52] | induction p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h : u ∈ support p
⊢ append (takeUntil p u h) (dropUntil p u h) = p | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ : V
h : u ∈ support nil
⊢ append (takeUntil nil u h) (dropUntil nil u h) = nil
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), append (takeUntil p✝ u h) (dropUntil p✝ u h) = p✝
h : u ∈ support (cons h✝ p✝)
⊢ append (takeUntil (cons h✝ p✝) u h) (dropUntil (cons h✝ p✝) u h) = cons h✝ p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.take_spec | [866, 1] | [875, 52] | rw [mem_support_nil_iff] at h | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ : V
h : u ∈ support nil
⊢ append (takeUntil nil u h) (dropUntil nil u h) = nil | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ : V
h✝ : u ∈ support nil
h : u = u✝
⊢ append (takeUntil nil u h✝) (dropUntil nil u h✝) = nil |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.take_spec | [866, 1] | [875, 52] | subst u | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ : V
h✝ : u ∈ support nil
h : u = u✝
⊢ append (takeUntil nil u h✝) (dropUntil nil u h✝) = nil | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
v w u✝ : V
h : u✝ ∈ support nil
⊢ append (takeUntil nil u✝ h) (dropUntil nil u✝ h) = nil |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.take_spec | [866, 1] | [875, 52] | rfl | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
v w u✝ : V
h : u✝ ∈ support nil
⊢ append (takeUntil nil u✝ h) (dropUntil nil u✝ h) = nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.take_spec | [866, 1] | [875, 52] | cases h | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), append (takeUntil p✝ u h) (dropUntil p✝ u h) = p✝
h : u ∈ support (cons h✝ p✝)
⊢ append (takeUntil (cons h✝ p✝) u h) (dropUntil (cons h✝ p✝) u h) = cons h✝ p✝ | case cons.head
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w v✝ w✝ : V
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), append (takeUntil p✝ u h) (dropUntil p✝ u h) = p✝
h✝ : G.Adj u v✝
⊢ append (takeUntil (cons h✝ p✝) u (_ : List.Mem u (u :: support p✝)))
(dropUntil (cons h✝ p✝) u (_ : List.Mem u (u :: support p✝))) =
cons h✝ p✝
case cons.tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), append (takeUntil p✝ u h) (dropUntil p✝ u h) = p✝
a✝ : List.Mem u (support p✝)
⊢ append (takeUntil (cons h✝ p✝) u (_ : List.Mem u (u✝ :: support p✝)))
(dropUntil (cons h✝ p✝) u (_ : List.Mem u (u✝ :: support p✝))) =
cons h✝ p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.take_spec | [866, 1] | [875, 52] | simp! | case cons.head
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w v✝ w✝ : V
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), append (takeUntil p✝ u h) (dropUntil p✝ u h) = p✝
h✝ : G.Adj u v✝
⊢ append (takeUntil (cons h✝ p✝) u (_ : List.Mem u (u :: support p✝)))
(dropUntil (cons h✝ p✝) u (_ : List.Mem u (u :: support p✝))) =
cons h✝ p✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.take_spec | [866, 1] | [875, 52] | simp! only | case cons.tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), append (takeUntil p✝ u h) (dropUntil p✝ u h) = p✝
a✝ : List.Mem u (support p✝)
⊢ append (takeUntil (cons h✝ p✝) u (_ : List.Mem u (u✝ :: support p✝)))
(dropUntil (cons h✝ p✝) u (_ : List.Mem u (u✝ :: support p✝))) =
cons h✝ p✝ | case cons.tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), append (takeUntil p✝ u h) (dropUntil p✝ u h) = p✝
a✝ : List.Mem u (support p✝)
⊢ append
(if h : u✝ = u then
Eq.rec (motive := fun x x_1 => x ∈ u✝ :: support p✝ → Walk G u✝ x) (fun h => nil) (_ : u✝ = u)
(_ : List.Mem u (u✝ :: support p✝))
else cons h✝ (takeUntil p✝ u (_ : u ∈ support p✝)))
(if h : u✝ = u then
Eq.rec (motive := fun x x_1 => x ∈ u✝ :: support p✝ → Walk G x w✝) (fun h => cons h✝ p✝) (_ : u✝ = u)
(_ : List.Mem u (u✝ :: support p✝))
else dropUntil p✝ u (_ : u ∈ support p✝)) =
cons h✝ p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.take_spec | [866, 1] | [875, 52] | split_ifs with h' <;> subst_vars <;> simp [*] | case cons.tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), append (takeUntil p✝ u h) (dropUntil p✝ u h) = p✝
a✝ : List.Mem u (support p✝)
⊢ append
(if h : u✝ = u then
Eq.rec (motive := fun x x_1 => x ∈ u✝ :: support p✝ → Walk G u✝ x) (fun h => nil) (_ : u✝ = u)
(_ : List.Mem u (u✝ :: support p✝))
else cons h✝ (takeUntil p✝ u (_ : u ∈ support p✝)))
(if h : u✝ = u then
Eq.rec (motive := fun x x_1 => x ∈ u✝ :: support p✝ → Walk G x w✝) (fun h => cons h✝ p✝) (_ : u✝ = u)
(_ : List.Mem u (u✝ :: support p✝))
else dropUntil p✝ u (_ : u ∈ support p✝)) =
cons h✝ p✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.mem_support_iff_exists_append | [878, 1] | [884, 88] | constructor | V✝ : Type u
V' : Type v
V'' : Type w
G✝ : Digraph V✝
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V✝
V : Type u
G : Digraph V
u v w : V
p : Walk G u v
⊢ w ∈ support p ↔ ∃ q r, p = append q r | case mp
V✝ : Type u
V' : Type v
V'' : Type w
G✝ : Digraph V✝
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V✝
V : Type u
G : Digraph V
u v w : V
p : Walk G u v
⊢ w ∈ support p → ∃ q r, p = append q r
case mpr
V✝ : Type u
V' : Type v
V'' : Type w
G✝ : Digraph V✝
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V✝
V : Type u
G : Digraph V
u v w : V
p : Walk G u v
⊢ (∃ q r, p = append q r) → w ∈ support p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.mem_support_iff_exists_append | [878, 1] | [884, 88] | exact fun h => ⟨_, _, (p.take_spec h).symm⟩ | case mp
V✝ : Type u
V' : Type v
V'' : Type w
G✝ : Digraph V✝
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V✝
V : Type u
G : Digraph V
u v w : V
p : Walk G u v
⊢ w ∈ support p → ∃ q r, p = append q r | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.mem_support_iff_exists_append | [878, 1] | [884, 88] | rintro ⟨q, r, rfl⟩ | case mpr
V✝ : Type u
V' : Type v
V'' : Type w
G✝ : Digraph V✝
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V✝
V : Type u
G : Digraph V
u v w : V
p : Walk G u v
⊢ (∃ q r, p = append q r) → w ∈ support p | case mpr.intro.intro
V✝ : Type u
V' : Type v
V'' : Type w
G✝ : Digraph V✝
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V✝
V : Type u
G : Digraph V
u v w : V
q : Walk G u w
r : Walk G w v
⊢ w ∈ support (append q r) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.mem_support_iff_exists_append | [878, 1] | [884, 88] | simp only [mem_support_append_iff, end_mem_support, start_mem_support, or_self_iff] | case mpr.intro.intro
V✝ : Type u
V' : Type v
V'' : Type w
G✝ : Digraph V✝
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V✝
V : Type u
G : Digraph V
u v w : V
q : Walk G u w
r : Walk G w v
⊢ w ∈ support (append q r) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_support_takeUntil_eq_one | [888, 1] | [897, 93] | induction p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h : u ∈ support p
⊢ List.count u (support (takeUntil p u h)) = 1 | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ : V
h : u ∈ support nil
⊢ List.count u (support (takeUntil nil u h)) = 1
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), List.count u (support (takeUntil p✝ u h)) = 1
h : u ∈ support (cons h✝ p✝)
⊢ List.count u (support (takeUntil (cons h✝ p✝) u h)) = 1 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_support_takeUntil_eq_one | [888, 1] | [897, 93] | rw [mem_support_nil_iff] at h | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ : V
h : u ∈ support nil
⊢ List.count u (support (takeUntil nil u h)) = 1 | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ : V
h✝ : u ∈ support nil
h : u = u✝
⊢ List.count u (support (takeUntil nil u h✝)) = 1 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_support_takeUntil_eq_one | [888, 1] | [897, 93] | subst u | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ : V
h✝ : u ∈ support nil
h : u = u✝
⊢ List.count u (support (takeUntil nil u h✝)) = 1 | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
v w u✝ : V
h : u✝ ∈ support nil
⊢ List.count u✝ (support (takeUntil nil u✝ h)) = 1 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_support_takeUntil_eq_one | [888, 1] | [897, 93] | simp! | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
v w u✝ : V
h : u✝ ∈ support nil
⊢ List.count u✝ (support (takeUntil nil u✝ h)) = 1 | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_support_takeUntil_eq_one | [888, 1] | [897, 93] | cases h | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), List.count u (support (takeUntil p✝ u h)) = 1
h : u ∈ support (cons h✝ p✝)
⊢ List.count u (support (takeUntil (cons h✝ p✝) u h)) = 1 | case cons.head
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w v✝ w✝ : V
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), List.count u (support (takeUntil p✝ u h)) = 1
h✝ : G.Adj u v✝
⊢ List.count u (support (takeUntil (cons h✝ p✝) u (_ : List.Mem u (u :: support p✝)))) = 1
case cons.tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), List.count u (support (takeUntil p✝ u h)) = 1
a✝ : List.Mem u (support p✝)
⊢ List.count u (support (takeUntil (cons h✝ p✝) u (_ : List.Mem u (u✝ :: support p✝)))) = 1 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_support_takeUntil_eq_one | [888, 1] | [897, 93] | simp! | case cons.head
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w v✝ w✝ : V
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), List.count u (support (takeUntil p✝ u h)) = 1
h✝ : G.Adj u v✝
⊢ List.count u (support (takeUntil (cons h✝ p✝) u (_ : List.Mem u (u :: support p✝)))) = 1 | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_support_takeUntil_eq_one | [888, 1] | [897, 93] | simp! only | case cons.tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), List.count u (support (takeUntil p✝ u h)) = 1
a✝ : List.Mem u (support p✝)
⊢ List.count u (support (takeUntil (cons h✝ p✝) u (_ : List.Mem u (u✝ :: support p✝)))) = 1 | case cons.tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), List.count u (support (takeUntil p✝ u h)) = 1
a✝ : List.Mem u (support p✝)
⊢ List.count u
(support
(if h : u✝ = u then
Eq.rec (motive := fun x x_1 => x ∈ u✝ :: support p✝ → Walk G u✝ x) (fun h => nil) (_ : u✝ = u)
(_ : List.Mem u (u✝ :: support p✝))
else cons h✝ (takeUntil p✝ u (_ : u ∈ support p✝)))) =
1 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_support_takeUntil_eq_one | [888, 1] | [897, 93] | split_ifs with h' <;> rw [eq_comm] at h' <;> subst_vars <;> simp! [*, List.count_cons] | case cons.tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
p_ih✝ : ∀ (h : u ∈ support p✝), List.count u (support (takeUntil p✝ u h)) = 1
a✝ : List.Mem u (support p✝)
⊢ List.count u
(support
(if h : u✝ = u then
Eq.rec (motive := fun x x_1 => x ∈ u✝ :: support p✝ → Walk G u✝ x) (fun h => nil) (_ : u✝ = u)
(_ : List.Mem u (u✝ :: support p✝))
else cons h✝ (takeUntil p✝ u (_ : u ∈ support p✝)))) =
1 | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_edges_takeUntil_eq_zero | [900, 1] | [919, 19] | induction' p with u' u' v' w' ha p' ih | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h : u ∈ support p
x : V
⊢ List.count (u, x) (edges (takeUntil p u h)) = 0 | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h✝ : u ∈ support p
x u' : V
h : u ∈ support nil
⊢ List.count (u, x) (edges (takeUntil nil u h)) = 0
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h✝ : u ∈ support p
x u' v' w' : V
ha : G.Adj u' v'
p' : Walk G v' w'
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
h : u ∈ support (cons ha p')
⊢ List.count (u, x) (edges (takeUntil (cons ha p') u h)) = 0 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_edges_takeUntil_eq_zero | [900, 1] | [919, 19] | rw [mem_support_nil_iff] at h | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h✝ : u ∈ support p
x u' : V
h : u ∈ support nil
⊢ List.count (u, x) (edges (takeUntil nil u h)) = 0 | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h✝¹ : u ∈ support p
x u' : V
h✝ : u ∈ support nil
h : u = u'
⊢ List.count (u, x) (edges (takeUntil nil u h✝)) = 0 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_edges_takeUntil_eq_zero | [900, 1] | [919, 19] | subst u | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h✝¹ : u ∈ support p
x u' : V
h✝ : u ∈ support nil
h : u = u'
⊢ List.count (u, x) (edges (takeUntil nil u h✝)) = 0 | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
v w : V
p : Walk G v w
x u' : V
h✝ : u' ∈ support p
h : u' ∈ support nil
⊢ List.count (u', x) (edges (takeUntil nil u' h)) = 0 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_edges_takeUntil_eq_zero | [900, 1] | [919, 19] | simp! | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
v w : V
p : Walk G v w
x u' : V
h✝ : u' ∈ support p
h : u' ∈ support nil
⊢ List.count (u', x) (edges (takeUntil nil u' h)) = 0 | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_edges_takeUntil_eq_zero | [900, 1] | [919, 19] | cases h | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h✝ : u ∈ support p
x u' v' w' : V
ha : G.Adj u' v'
p' : Walk G v' w'
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
h : u ∈ support (cons ha p')
⊢ List.count (u, x) (edges (takeUntil (cons ha p') u h)) = 0 | case cons.head
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h : u ∈ support p
x v' w' : V
p' : Walk G v' w'
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
ha : G.Adj u v'
⊢ List.count (u, x) (edges (takeUntil (cons ha p') u (_ : List.Mem u (u :: support p')))) = 0
case cons.tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h : u ∈ support p
x u' v' w' : V
ha : G.Adj u' v'
p' : Walk G v' w'
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
a✝ : List.Mem u (support p')
⊢ List.count (u, x) (edges (takeUntil (cons ha p') u (_ : List.Mem u (u' :: support p')))) = 0 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_edges_takeUntil_eq_zero | [900, 1] | [919, 19] | simp! | case cons.head
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h : u ∈ support p
x v' w' : V
p' : Walk G v' w'
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
ha : G.Adj u v'
⊢ List.count (u, x) (edges (takeUntil (cons ha p') u (_ : List.Mem u (u :: support p')))) = 0 | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_edges_takeUntil_eq_zero | [900, 1] | [919, 19] | simp! only | case cons.tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h : u ∈ support p
x u' v' w' : V
ha : G.Adj u' v'
p' : Walk G v' w'
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
a✝ : List.Mem u (support p')
⊢ List.count (u, x) (edges (takeUntil (cons ha p') u (_ : List.Mem u (u' :: support p')))) = 0 | case cons.tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h : u ∈ support p
x u' v' w' : V
ha : G.Adj u' v'
p' : Walk G v' w'
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
a✝ : List.Mem u (support p')
⊢ List.count (u, x)
(edges
(if h : u' = u then
Eq.rec (motive := fun x x_1 => x ∈ u' :: support p' → Walk G u' x) (fun h => nil) (_ : u' = u)
(_ : List.Mem u (u' :: support p'))
else cons ha (takeUntil p' u (_ : u ∈ support p')))) =
0 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_edges_takeUntil_eq_zero | [900, 1] | [919, 19] | split_ifs with h' | case cons.tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h : u ∈ support p
x u' v' w' : V
ha : G.Adj u' v'
p' : Walk G v' w'
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
a✝ : List.Mem u (support p')
⊢ List.count (u, x)
(edges
(if h : u' = u then
Eq.rec (motive := fun x x_1 => x ∈ u' :: support p' → Walk G u' x) (fun h => nil) (_ : u' = u)
(_ : List.Mem u (u' :: support p'))
else cons ha (takeUntil p' u (_ : u ∈ support p')))) =
0 | case cons.tail.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h : u ∈ support p
x u' v' w' : V
ha : G.Adj u' v'
p' : Walk G v' w'
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
a✝ : List.Mem u (support p')
h' : u' = u
⊢ List.count (u, x)
(edges
(Eq.rec (motive := fun x x_1 => x ∈ u' :: support p' → Walk G u' x) (fun h => nil) (_ : u' = u)
(_ : List.Mem u (u' :: support p')))) =
0
case cons.tail.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h : u ∈ support p
x u' v' w' : V
ha : G.Adj u' v'
p' : Walk G v' w'
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
a✝ : List.Mem u (support p')
h' : ¬u' = u
⊢ List.count (u, x) (edges (cons ha (takeUntil p' u (_ : u ∈ support p')))) = 0 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_edges_takeUntil_eq_zero | [900, 1] | [919, 19] | subst h' | case cons.tail.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h : u ∈ support p
x u' v' w' : V
ha : G.Adj u' v'
p' : Walk G v' w'
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
a✝ : List.Mem u (support p')
h' : u' = u
⊢ List.count (u, x)
(edges
(Eq.rec (motive := fun x x_1 => x ∈ u' :: support p' → Walk G u' x) (fun h => nil) (_ : u' = u)
(_ : List.Mem u (u' :: support p')))) =
0 | case cons.tail.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
v w : V
p : Walk G v w
x u' v' w' : V
ha : G.Adj u' v'
p' : Walk G v' w'
h : u' ∈ support p
ih : ∀ (h : u' ∈ support p'), List.count (u', x) (edges (takeUntil p' u' h)) = 0
a✝ : List.Mem u' (support p')
⊢ List.count (u', x)
(edges
(Eq.rec (motive := fun x x_1 => x ∈ u' :: support p' → Walk G u' x) (fun h => nil) (_ : u' = u')
(_ : List.Mem u' (u' :: support p')))) =
0 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_edges_takeUntil_eq_zero | [900, 1] | [919, 19] | simp | case cons.tail.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
v w : V
p : Walk G v w
x u' v' w' : V
ha : G.Adj u' v'
p' : Walk G v' w'
h : u' ∈ support p
ih : ∀ (h : u' ∈ support p'), List.count (u', x) (edges (takeUntil p' u' h)) = 0
a✝ : List.Mem u' (support p')
⊢ List.count (u', x)
(edges
(Eq.rec (motive := fun x x_1 => x ∈ u' :: support p' → Walk G u' x) (fun h => nil) (_ : u' = u')
(_ : List.Mem u' (u' :: support p')))) =
0 | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_edges_takeUntil_eq_zero | [900, 1] | [919, 19] | rw [edges_cons] | case cons.tail.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h : u ∈ support p
x u' v' w' : V
ha : G.Adj u' v'
p' : Walk G v' w'
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
a✝ : List.Mem u (support p')
h' : ¬u' = u
⊢ List.count (u, x) (edges (cons ha (takeUntil p' u (_ : u ∈ support p')))) = 0 | case cons.tail.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v w : V
p : Walk G v w
h : u ∈ support p
x u' v' w' : V
ha : G.Adj u' v'
p' : Walk G v' w'
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
a✝ : List.Mem u (support p')
h' : ¬u' = u
⊢ List.count (u, x) ((u', v') :: edges (takeUntil p' u (_ : u ∈ support p'))) = 0 |
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