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https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_edges_takeUntil_eq_zero | [900, 1] | [919, 19] | simp only [List.count_prod_eq, List.count_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) ((u', v') :: edges (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
⊢ (if (u, x) = (u', v') then Nat.succ (List.count (u, x) (edges (takeUntil p' u (_ : u ∈ support p'))))
else List.count (u, x) (edges (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.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
⊢ (if (u, x) = (u', v') then Nat.succ (List.count (u, x) (edges (takeUntil p' u (_ : u ∈ support p'))))
else List.count (u, x) (edges (takeUntil p' u (_ : u ∈ support p')))) =
0 | case cons.tail.inr.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
h'' : (u, x) = (u', v')
⊢ False
case cons.tail.inr.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
h'' : ¬(u, x) = (u', v')
⊢ List.count (u, x) (edges (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] | simp at h'' | case cons.tail.inr.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
h'' : (u, x) = (u', v')
⊢ False | case cons.tail.inr.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
h'' : u = u' ∧ x = v'
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_edges_takeUntil_eq_zero | [900, 1] | [919, 19] | obtain ⟨rfl, rfl⟩ := h'' | case cons.tail.inr.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
h'' : u = u' ∧ x = v'
⊢ False | case cons.tail.inr.inl.intro
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 w' : V
h' : ¬u = u
p' : Walk G x w'
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
a✝ : List.Mem u (support p')
ha : G.Adj u x
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.count_edges_takeUntil_eq_zero | [900, 1] | [919, 19] | exact (h' rfl).elim | case cons.tail.inr.inl.intro
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 w' : V
h' : ¬u = u
p' : Walk G x w'
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
a✝ : List.Mem u (support p')
ha : G.Adj u x
⊢ False | 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 [List.count_prod_eq] at ih | case cons.tail.inr.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
h'' : ¬(u, x) = (u', v')
⊢ List.count (u, x) (edges (takeUntil p' u (_ : u ∈ support p'))) = 0 | case cons.tail.inr.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'
a✝ : List.Mem u (support p')
h' : ¬u' = u
h'' : ¬(u, x) = (u', v')
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
⊢ List.count (u, x) (edges (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] | apply ih | case cons.tail.inr.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'
a✝ : List.Mem u (support p')
h' : ¬u' = u
h'' : ¬(u, x) = (u', v')
ih : ∀ (h : u ∈ support p'), List.count (u, x) (edges (takeUntil p' u h)) = 0
⊢ List.count (u, x) (edges (takeUntil p' u (_ : u ∈ support p'))) = 0 | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.takeUntil_copy | [922, 1] | [926, 6] | subst_vars | 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
hv : v = v'
hw : w = w'
h : u ∈ support (Walk.copy p hv hw)
⊢ u ∈ support 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 (Walk.copy p (_ : v' = v') (_ : w' = w'))
⊢ u ∈ support p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.takeUntil_copy | [922, 1] | [926, 6] | exact h | 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 (Walk.copy p (_ : v' = v') (_ : w' = w'))
⊢ u ∈ support p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.takeUntil_copy | [922, 1] | [926, 6] | subst_vars | 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
hv : v = v'
hw : w = w'
h : u ∈ support (Walk.copy p hv hw)
⊢ takeUntil (Walk.copy p hv hw) u h = Walk.copy (takeUntil p u (_ : u ∈ support p)) hv (_ : u = u) | 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 (Walk.copy p (_ : v' = v') (_ : w' = w'))
⊢ takeUntil (Walk.copy p (_ : v' = v') (_ : w' = w')) u h =
Walk.copy (takeUntil p u (_ : u ∈ support p)) (_ : v' = v') (_ : u = u) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.takeUntil_copy | [922, 1] | [926, 6] | rfl | 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 (Walk.copy p (_ : v' = v') (_ : w' = w'))
⊢ takeUntil (Walk.copy p (_ : v' = v') (_ : w' = w')) u h =
Walk.copy (takeUntil p u (_ : u ∈ support p)) (_ : v' = v') (_ : u = u) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.dropUntil_copy | [930, 1] | [934, 6] | subst_vars | 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
hv : v = v'
hw : w = w'
h : u ∈ support (Walk.copy p hv hw)
⊢ u ∈ support 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 (Walk.copy p (_ : v' = v') (_ : w' = w'))
⊢ u ∈ support p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.dropUntil_copy | [930, 1] | [934, 6] | exact h | 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 (Walk.copy p (_ : v' = v') (_ : w' = w'))
⊢ u ∈ support p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.dropUntil_copy | [930, 1] | [934, 6] | subst_vars | 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
hv : v = v'
hw : w = w'
h : u ∈ support (Walk.copy p hv hw)
⊢ dropUntil (Walk.copy p hv hw) u h = Walk.copy (dropUntil p u (_ : u ∈ support p)) (_ : u = u) hw | 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 (Walk.copy p (_ : v' = v') (_ : w' = w'))
⊢ dropUntil (Walk.copy p (_ : v' = v') (_ : w' = w')) u h =
Walk.copy (dropUntil p u (_ : u ∈ support p)) (_ : u = u) (_ : w' = w') |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.dropUntil_copy | [930, 1] | [934, 6] | rfl | 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 (Walk.copy p (_ : v' = v') (_ : w' = w'))
⊢ dropUntil (Walk.copy p (_ : v' = v') (_ : w' = w')) u h =
Walk.copy (dropUntil p u (_ : u ∈ support p)) (_ : u = u) (_ : w' = w') | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_takeUntil_subset | [937, 1] | [940, 18] | rw [← take_spec p h, mem_support_append_iff] | 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
hx : x ∈ support (takeUntil p u h)
⊢ x ∈ support 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
x : V
hx : x ∈ support (takeUntil p u h)
⊢ x ∈ support (takeUntil p u h) ∨ x ∈ support (dropUntil p u h) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_takeUntil_subset | [937, 1] | [940, 18] | exact Or.inl hx | 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
hx : x ∈ support (takeUntil p u h)
⊢ x ∈ support (takeUntil p u h) ∨ x ∈ support (dropUntil p u h) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_dropUntil_subset | [943, 1] | [946, 18] | rw [← take_spec p h, mem_support_append_iff] | 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
hx : x ∈ support (dropUntil p u h)
⊢ x ∈ support 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
x : V
hx : x ∈ support (dropUntil p u h)
⊢ x ∈ support (takeUntil p u h) ∨ x ∈ support (dropUntil p u h) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_dropUntil_subset | [943, 1] | [946, 18] | exact Or.inr hx | 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
hx : x ∈ support (dropUntil p u h)
⊢ x ∈ support (takeUntil p u h) ∨ x ∈ support (dropUntil p u h) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_takeUntil_subset | [949, 1] | [952, 18] | rw [← take_spec p h, edges_append, List.mem_append] | 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 × V
hx : x ∈ edges (takeUntil p u h)
⊢ x ∈ edges 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
x : V × V
hx : x ∈ edges (takeUntil p u h)
⊢ x ∈ edges (takeUntil p u h) ∨ x ∈ edges (dropUntil p u h) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_takeUntil_subset | [949, 1] | [952, 18] | exact Or.inl hx | 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 × V
hx : x ∈ edges (takeUntil p u h)
⊢ x ∈ edges (takeUntil p u h) ∨ x ∈ edges (dropUntil p u h) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_dropUntil_subset | [954, 1] | [957, 18] | rw [← take_spec p h, edges_append, List.mem_append] | 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 × V
hx : x ∈ edges (dropUntil p u h)
⊢ x ∈ edges 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
x : V × V
hx : x ∈ edges (dropUntil p u h)
⊢ x ∈ edges (takeUntil p u h) ∨ x ∈ edges (dropUntil p u h) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_dropUntil_subset | [954, 1] | [957, 18] | exact Or.inr hx | 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 × V
hx : x ∈ edges (dropUntil p u h)
⊢ x ∈ edges (takeUntil p u h) ∨ x ∈ edges (dropUntil p u h) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_takeUntil_le | [959, 1] | [963, 26] | have := congr_arg Walk.length (p.take_spec h) | 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
⊢ length (takeUntil p u h) ≤ length 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
this : length (append (takeUntil p u h) (dropUntil p u h)) = length p
⊢ length (takeUntil p u h) ≤ length p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_takeUntil_le | [959, 1] | [963, 26] | rw [length_append] at this | 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
this : length (append (takeUntil p u h) (dropUntil p u h)) = length p
⊢ length (takeUntil p u h) ≤ length 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
this : length (takeUntil p u h) + length (dropUntil p u h) = length p
⊢ length (takeUntil p u h) ≤ length p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_takeUntil_le | [959, 1] | [963, 26] | exact Nat.le.intro this | 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
this : length (takeUntil p u h) + length (dropUntil p u h) = length p
⊢ length (takeUntil p u h) ≤ length p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_dropUntil_le | [966, 1] | [970, 26] | have := congr_arg Walk.length (p.take_spec h) | 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
⊢ length (dropUntil p u h) ≤ length 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
this : length (append (takeUntil p u h) (dropUntil p u h)) = length p
⊢ length (dropUntil p u h) ≤ length p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_dropUntil_le | [966, 1] | [970, 26] | rw [length_append, add_comm] at this | 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
this : length (append (takeUntil p u h) (dropUntil p u h)) = length p
⊢ length (dropUntil p u h) ≤ length 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
this : length (dropUntil p u h) + length (takeUntil p u h) = length p
⊢ length (dropUntil p u h) ≤ length p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_dropUntil_le | [966, 1] | [970, 26] | exact Nat.le.intro this | 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
this : length (dropUntil p u h) + length (takeUntil p u h) = length p
⊢ length (dropUntil p u h) ≤ length p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsTrail.takeUntil | [973, 11] | [975, 58] | rwa [← take_spec _ h] at hc | 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
hc : IsTrail p
h : u ∈ support p
⊢ IsTrail (append (takeUntil p u h) (?m.221792 hc h)) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsTrail.dropUntil | [978, 11] | [980, 59] | rwa [← take_spec _ h] at hc | 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
hc : IsTrail p
h : u ∈ support p
⊢ IsTrail (append (?m.222054 hc h) (dropUntil p u h)) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.takeUntil | [983, 11] | [985, 57] | rwa [← take_spec _ h] at hc | 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
hc : IsPath p
h : u ∈ support p
⊢ IsPath (append (takeUntil p u h) (?m.222316 hc h)) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.dropUntil | [989, 11] | [991, 58] | rwa [← take_spec _ h] at hc | 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
hc : IsPath p
h : u ∈ support p
⊢ IsPath (append (?m.222578 hc h) (dropUntil p u h)) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_rotate | [1000, 1] | [1004, 40] | simp only [rotate, tail_support_append] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
h : u ∈ support c
⊢ List.tail (support (rotate c h)) ~r List.tail (support c) | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
h : u ∈ support c
⊢ (List.tail (support (dropUntil c u h)) ++ List.tail (support (takeUntil c u h))) ~r List.tail (support c) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_rotate | [1000, 1] | [1004, 40] | apply List.IsRotated.trans List.isRotated_append | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
h : u ∈ support c
⊢ (List.tail (support (dropUntil c u h)) ++ List.tail (support (takeUntil c u h))) ~r List.tail (support c) | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
h : u ∈ support c
⊢ (List.tail (support (takeUntil c u h)) ++ List.tail (support (dropUntil c u h))) ~r List.tail (support c) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_rotate | [1000, 1] | [1004, 40] | rw [← tail_support_append, take_spec] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
h : u ∈ support c
⊢ (List.tail (support (takeUntil c u h)) ++ List.tail (support (dropUntil c u h))) ~r List.tail (support c) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.rotate_edges | [1007, 1] | [1011, 33] | simp only [rotate, edges_append] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
h : u ∈ support c
⊢ edges (rotate c h) ~r edges c | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
h : u ∈ support c
⊢ (edges (dropUntil c u h) ++ edges (takeUntil c u h)) ~r edges c |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.rotate_edges | [1007, 1] | [1011, 33] | apply List.IsRotated.trans List.isRotated_append | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
h : u ∈ support c
⊢ (edges (dropUntil c u h) ++ edges (takeUntil c u h)) ~r edges c | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
h : u ∈ support c
⊢ (edges (takeUntil c u h) ++ edges (dropUntil c u h)) ~r edges c |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.rotate_edges | [1007, 1] | [1011, 33] | rw [← edges_append, take_spec] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
h : u ∈ support c
⊢ (edges (takeUntil c u h) ++ edges (dropUntil c u h)) ~r edges c | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsTrail.rotate | [1013, 11] | [1016, 23] | rw [isTrail_def, (c.rotate_edges h).perm.nodup_iff] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
hc : IsTrail c
h : u ∈ support c
⊢ IsTrail (rotate c h) | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
hc : IsTrail c
h : u ∈ support c
⊢ List.Nodup (edges c) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsTrail.rotate | [1013, 11] | [1016, 23] | exact hc.edges_nodup | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
hc : IsTrail c
h : u ∈ support c
⊢ List.Nodup (edges c) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCircuit.rotate | [1019, 11] | [1027, 16] | refine ⟨hc.isTrail.rotate _, ?_⟩ | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
hc : IsCircuit c
h : u ∈ support c
⊢ IsCircuit (rotate c h) | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
hc : IsCircuit c
h : u ∈ support c
⊢ rotate c h ≠ nil |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCircuit.rotate | [1019, 11] | [1027, 16] | cases c | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
hc : IsCircuit c
h : u ∈ support c
⊢ rotate c 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 : V
hc : IsCircuit nil
h : u ∈ support nil
⊢ rotate nil 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 v✝ : V
h✝ : G.Adj v v✝
p✝ : Walk G v✝ v
hc : IsCircuit (cons h✝ p✝)
h : u ∈ support (cons h✝ p✝)
⊢ rotate (cons h✝ p✝) h ≠ nil |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCircuit.rotate | [1019, 11] | [1027, 16] | exact (hc.ne_nil rfl).elim | 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 : V
hc : IsCircuit nil
h : u ∈ support nil
⊢ rotate nil h ≠ nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCircuit.rotate | [1019, 11] | [1027, 16] | intro hn | 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 v✝ : V
h✝ : G.Adj v v✝
p✝ : Walk G v✝ v
hc : IsCircuit (cons h✝ p✝)
h : u ∈ support (cons h✝ p✝)
⊢ rotate (cons h✝ p✝) 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 v✝ : V
h✝ : G.Adj v v✝
p✝ : Walk G v✝ v
hc : IsCircuit (cons h✝ p✝)
h : u ∈ support (cons h✝ p✝)
hn : rotate (cons h✝ p✝) h = nil
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCircuit.rotate | [1019, 11] | [1027, 16] | have hn' := congr_arg length hn | 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 v✝ : V
h✝ : G.Adj v v✝
p✝ : Walk G v✝ v
hc : IsCircuit (cons h✝ p✝)
h : u ∈ support (cons h✝ p✝)
hn : rotate (cons h✝ p✝) h = nil
⊢ False | 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 v✝ : V
h✝ : G.Adj v v✝
p✝ : Walk G v✝ v
hc : IsCircuit (cons h✝ p✝)
h : u ∈ support (cons h✝ p✝)
hn : rotate (cons h✝ p✝) h = nil
hn' : length (rotate (cons h✝ p✝) h) = length nil
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCircuit.rotate | [1019, 11] | [1027, 16] | rw [rotate, length_append, add_comm, ← length_append, take_spec] at hn' | 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 v✝ : V
h✝ : G.Adj v v✝
p✝ : Walk G v✝ v
hc : IsCircuit (cons h✝ p✝)
h : u ∈ support (cons h✝ p✝)
hn : rotate (cons h✝ p✝) h = nil
hn' : length (rotate (cons h✝ p✝) h) = length nil
⊢ False | 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 v✝ : V
h✝ : G.Adj v v✝
p✝ : Walk G v✝ v
hc : IsCircuit (cons h✝ p✝)
h : u ∈ support (cons h✝ p✝)
hn : rotate (cons h✝ p✝) h = nil
hn' : length (cons h✝ p✝) = length nil
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCircuit.rotate | [1019, 11] | [1027, 16] | simp at hn' | 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 v✝ : V
h✝ : G.Adj v v✝
p✝ : Walk G v✝ v
hc : IsCircuit (cons h✝ p✝)
h : u ∈ support (cons h✝ p✝)
hn : rotate (cons h✝ p✝) h = nil
hn' : length (cons h✝ p✝) = length nil
⊢ False | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCycle.rotate | [1030, 11] | [1034, 25] | refine ⟨hc.isCircuit.rotate _, ?_⟩ | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
hc : IsCycle c
h : u ∈ support c
⊢ IsCycle (rotate c h) | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
hc : IsCycle c
h : u ∈ support c
⊢ List.Nodup (List.tail (support (rotate c h))) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCycle.rotate | [1030, 11] | [1034, 25] | rw [List.IsRotated.nodup_iff (support_rotate _ _)] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
hc : IsCycle c
h : u ∈ support c
⊢ List.Nodup (List.tail (support (rotate c h))) | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
hc : IsCycle c
h : u ∈ support c
⊢ List.Nodup (List.tail (support c)) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCycle.rotate | [1030, 11] | [1034, 25] | exact hc.support_nodup | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v : V
c : Walk G v v
hc : IsCycle c
h : u ∈ support c
⊢ List.Nodup (List.tail (support c)) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_boundary_dart | [1041, 1] | [1048, 45] | induction' p with _ x y w a p' 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
S : Set V
uS : u ∈ S
vS : ¬v ∈ S
⊢ ∃ e, e ∈ edges p ∧ e.fst ∈ S ∧ ¬e.snd ∈ S | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
S : Set V
uS✝ : u ∈ S
vS✝ : ¬v ∈ S
u✝ : V
uS : u✝ ∈ S
vS : ¬u✝ ∈ S
⊢ ∃ e, e ∈ edges nil ∧ e.fst ∈ S ∧ ¬e.snd ∈ S
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
S : Set V
uS✝ : u ∈ S
vS✝ : ¬v ∈ S
x y w : V
a : G.Adj x y
p' : Walk G y w
ih : y ∈ S → ¬w ∈ S → ∃ e, e ∈ edges p' ∧ e.fst ∈ S ∧ ¬e.snd ∈ S
uS : x ∈ S
vS : ¬w ∈ S
⊢ ∃ e, e ∈ edges (cons a p') ∧ e.fst ∈ S ∧ ¬e.snd ∈ S |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_boundary_dart | [1041, 1] | [1048, 45] | cases vS uS | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
S : Set V
uS✝ : u ∈ S
vS✝ : ¬v ∈ S
u✝ : V
uS : u✝ ∈ S
vS : ¬u✝ ∈ S
⊢ ∃ e, e ∈ edges nil ∧ e.fst ∈ S ∧ ¬e.snd ∈ S | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_boundary_dart | [1041, 1] | [1048, 45] | by_cases h : y ∈ S | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
S : Set V
uS✝ : u ∈ S
vS✝ : ¬v ∈ S
x y w : V
a : G.Adj x y
p' : Walk G y w
ih : y ∈ S → ¬w ∈ S → ∃ e, e ∈ edges p' ∧ e.fst ∈ S ∧ ¬e.snd ∈ S
uS : x ∈ S
vS : ¬w ∈ S
⊢ ∃ e, e ∈ edges (cons a p') ∧ e.fst ∈ S ∧ ¬e.snd ∈ S | case pos
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
S : Set V
uS✝ : u ∈ S
vS✝ : ¬v ∈ S
x y w : V
a : G.Adj x y
p' : Walk G y w
ih : y ∈ S → ¬w ∈ S → ∃ e, e ∈ edges p' ∧ e.fst ∈ S ∧ ¬e.snd ∈ S
uS : x ∈ S
vS : ¬w ∈ S
h : y ∈ S
⊢ ∃ e, e ∈ edges (cons a p') ∧ e.fst ∈ S ∧ ¬e.snd ∈ S
case neg
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
S : Set V
uS✝ : u ∈ S
vS✝ : ¬v ∈ S
x y w : V
a : G.Adj x y
p' : Walk G y w
ih : y ∈ S → ¬w ∈ S → ∃ e, e ∈ edges p' ∧ e.fst ∈ S ∧ ¬e.snd ∈ S
uS : x ∈ S
vS : ¬w ∈ S
h : ¬y ∈ S
⊢ ∃ e, e ∈ edges (cons a p') ∧ e.fst ∈ S ∧ ¬e.snd ∈ S |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_boundary_dart | [1041, 1] | [1048, 45] | obtain ⟨d, hd, hcd⟩ := ih h vS | case pos
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
S : Set V
uS✝ : u ∈ S
vS✝ : ¬v ∈ S
x y w : V
a : G.Adj x y
p' : Walk G y w
ih : y ∈ S → ¬w ∈ S → ∃ e, e ∈ edges p' ∧ e.fst ∈ S ∧ ¬e.snd ∈ S
uS : x ∈ S
vS : ¬w ∈ S
h : y ∈ S
⊢ ∃ e, e ∈ edges (cons a p') ∧ e.fst ∈ S ∧ ¬e.snd ∈ S | case pos.intro.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
S : Set V
uS✝ : u ∈ S
vS✝ : ¬v ∈ S
x y w : V
a : G.Adj x y
p' : Walk G y w
ih : y ∈ S → ¬w ∈ S → ∃ e, e ∈ edges p' ∧ e.fst ∈ S ∧ ¬e.snd ∈ S
uS : x ∈ S
vS : ¬w ∈ S
h : y ∈ S
d : V × V
hd : d ∈ edges p'
hcd : d.fst ∈ S ∧ ¬d.snd ∈ S
⊢ ∃ e, e ∈ edges (cons a p') ∧ e.fst ∈ S ∧ ¬e.snd ∈ S |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_boundary_dart | [1041, 1] | [1048, 45] | exact ⟨d, List.Mem.tail _ hd, hcd⟩ | case pos.intro.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
S : Set V
uS✝ : u ∈ S
vS✝ : ¬v ∈ S
x y w : V
a : G.Adj x y
p' : Walk G y w
ih : y ∈ S → ¬w ∈ S → ∃ e, e ∈ edges p' ∧ e.fst ∈ S ∧ ¬e.snd ∈ S
uS : x ∈ S
vS : ¬w ∈ S
h : y ∈ S
d : V × V
hd : d ∈ edges p'
hcd : d.fst ∈ S ∧ ¬d.snd ∈ S
⊢ ∃ e, e ∈ edges (cons a p') ∧ e.fst ∈ S ∧ ¬e.snd ∈ S | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_boundary_dart | [1041, 1] | [1048, 45] | exact ⟨(x, y), List.Mem.head _, uS, h⟩ | case neg
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
S : Set V
uS✝ : u ∈ S
vS✝ : ¬v ∈ S
x y w : V
a : G.Adj x y
p' : Walk G y w
ih : y ∈ S → ¬w ∈ S → ∃ e, e ∈ edges p' ∧ e.fst ∈ S ∧ ¬e.snd ∈ S
uS : x ∈ S
vS : ¬w ∈ S
h : ¬y ∈ S
⊢ ∃ e, e ∈ edges (cons a p') ∧ e.fst ∈ S ∧ ¬e.snd ∈ S | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Path.mk'_mem_edges_singleton | [1083, 1] | [1084, 73] | simp [singleton] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : G.Adj u v
hne : u ≠ v
⊢ (u, v) ∈ Walk.edges ↑(singleton h hne) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Path.count_edges_eq_one | [1090, 1] | [1092, 73] | simpa using List.count_eq_one_of_mem p.property.isTrail.edges_nodup hw | 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 : Path G u v
e : V × V
hw : e ∈ Walk.edges ↑p
⊢ List.count e (Walk.edges ↑p) = 1 | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Path.loop_eq | [1098, 1] | [1101, 14] | obtain ⟨_ | _, h⟩ := p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v : V
p : Path G v v
⊢ p = Path.nil | case mk.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v : V
h : Walk.IsPath Walk.nil
⊢ { val := Walk.nil, property := h } = Path.nil
case mk.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v v✝ : V
h✝ : G.Adj v v✝
p✝ : Walk G v✝ v
h : Walk.IsPath (Walk.cons h✝ p✝)
⊢ { val := Walk.cons h✝ p✝, property := h } = Path.nil |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Path.loop_eq | [1098, 1] | [1101, 14] | rfl | case mk.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v : V
h : Walk.IsPath Walk.nil
⊢ { val := Walk.nil, property := h } = Path.nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Path.loop_eq | [1098, 1] | [1101, 14] | simp at h | case mk.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v v✝ : V
h✝ : G.Adj v v✝
p✝ : Walk G v✝ v
h : Walk.IsPath (Walk.cons h✝ p✝)
⊢ { val := Walk.cons h✝ p✝, property := h } = Path.nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Path.not_mem_edges_of_loop | [1103, 1] | [1104, 22] | simp [p.loop_eq] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v : V
e : V × V
p : Path G v v
⊢ ¬e ∈ Walk.edges ↑p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Path.cons_isCycle | [1106, 1] | [1108, 53] | simp [Walk.isCycle_def, Walk.cons_isTrail_iff, he] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Path G v u
h : G.Adj u v
he : ¬(u, v) ∈ Walk.edges ↑p
⊢ Walk.IsCycle (Walk.cons h ↑p) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.bypass_copy | [1133, 1] | [1136, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u' v' : V
p : Walk G u v
hu : u = u'
hv : v = v'
⊢ bypass (Walk.copy p hu hv) = Walk.copy (bypass p) hu hv | 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'
⊢ bypass (Walk.copy p (_ : u' = u') (_ : v' = v')) = Walk.copy (bypass p) (_ : u' = u') (_ : v' = v') |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.bypass_copy | [1133, 1] | [1136, 6] | rfl | 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'
⊢ bypass (Walk.copy p (_ : u' = u') (_ : v' = v')) = Walk.copy (bypass p) (_ : u' = u') (_ : v' = v') | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.bypass_isPath | [1139, 1] | [1146, 32] | simp! | 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 u✝ : V
⊢ IsPath (bypass nil) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.bypass_isPath | [1139, 1] | [1146, 32] | simp only [bypass] | 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 u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p' : Walk G v✝ w✝
ih : IsPath (bypass p')
⊢ IsPath (bypass (cons h✝ p')) | 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 u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p' : Walk G v✝ w✝
ih : IsPath (bypass p')
⊢ IsPath (if hs : u✝ ∈ support (bypass p') then dropUntil (bypass p') u✝ hs else cons h✝ (bypass p')) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.bypass_isPath | [1139, 1] | [1146, 32] | split_ifs with hs | 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 u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p' : Walk G v✝ w✝
ih : IsPath (bypass p')
⊢ IsPath (if hs : u✝ ∈ support (bypass p') then dropUntil (bypass p') u✝ hs else cons h✝ (bypass p')) | case cons.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p' : Walk G v✝ w✝
ih : IsPath (bypass p')
hs : u✝ ∈ support (bypass p')
⊢ IsPath (dropUntil (bypass p') u✝ hs)
case cons.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p' : Walk G v✝ w✝
ih : IsPath (bypass p')
hs : ¬u✝ ∈ support (bypass p')
⊢ IsPath (cons h✝ (bypass p')) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.bypass_isPath | [1139, 1] | [1146, 32] | exact ih.dropUntil hs | case cons.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p' : Walk G v✝ w✝
ih : IsPath (bypass p')
hs : u✝ ∈ support (bypass p')
⊢ IsPath (dropUntil (bypass p') u✝ hs) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.bypass_isPath | [1139, 1] | [1146, 32] | simp [*, cons_isPath_iff] | case cons.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p' : Walk G v✝ w✝
ih : IsPath (bypass p')
hs : ¬u✝ ∈ support (bypass p')
⊢ IsPath (cons h✝ (bypass p')) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_bypass_le | [1149, 1] | [1160, 34] | rfl | 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 u✝ : V
⊢ length (bypass nil) ≤ length nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_bypass_le | [1149, 1] | [1160, 34] | simp only [bypass] | 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 u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
⊢ length (bypass (cons h✝ p✝)) ≤ length (cons h✝ p✝) | 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 u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
⊢ length (if hs : u✝ ∈ support (bypass p✝) then dropUntil (bypass p✝) u✝ hs else cons h✝ (bypass p✝)) ≤
length (cons h✝ p✝) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_bypass_le | [1149, 1] | [1160, 34] | split_ifs | 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 u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
⊢ length (if hs : u✝ ∈ support (bypass p✝) then dropUntil (bypass p✝) u✝ hs else cons h✝ (bypass p✝)) ≤
length (cons h✝ p✝) | case cons.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ length (dropUntil (bypass p✝) u✝ h✝) ≤ length (cons h✝¹ p✝)
case cons.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : ¬u✝ ∈ support (bypass p✝)
⊢ length (cons h✝¹ (bypass p✝)) ≤ length (cons h✝¹ p✝) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_bypass_le | [1149, 1] | [1160, 34] | trans | case cons.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ length (dropUntil (bypass p✝) u✝ h✝) ≤ length (cons h✝¹ p✝) | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ length (dropUntil (bypass p✝) u✝ h✝) ≤ ?m.249227
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ ?m.249227 ≤ length (cons h✝¹ p✝)
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ ℕ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_bypass_le | [1149, 1] | [1160, 34] | apply length_dropUntil_le | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ length (dropUntil (bypass p✝) u✝ h✝) ≤ ?m.249227
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ ?m.249227 ≤ length (cons h✝¹ p✝)
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ ℕ | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ length (bypass p✝) ≤ length (cons h✝¹ p✝) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_bypass_le | [1149, 1] | [1160, 34] | rw [length_cons] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ length (bypass p✝) ≤ length (cons h✝¹ p✝) | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ length (bypass p✝) ≤ length p✝ + 1 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_bypass_le | [1149, 1] | [1160, 34] | exact le_add_right ih | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ length (bypass p✝) ≤ length p✝ + 1 | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_bypass_le | [1149, 1] | [1160, 34] | rw [length_cons, length_cons] | case cons.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : ¬u✝ ∈ support (bypass p✝)
⊢ length (cons h✝¹ (bypass p✝)) ≤ length (cons h✝¹ p✝) | case cons.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : ¬u✝ ∈ support (bypass p✝)
⊢ length (bypass p✝) + 1 ≤ length p✝ + 1 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_bypass_le | [1149, 1] | [1160, 34] | exact add_le_add_right ih 1 | case cons.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : length (bypass p✝) ≤ length p✝
h✝ : ¬u✝ ∈ support (bypass p✝)
⊢ length (bypass p✝) + 1 ≤ length p✝ + 1 | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_bypass_subset | [1168, 1] | [1179, 17] | simp! | 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 u✝ : V
⊢ support (bypass nil) ⊆ support nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_bypass_subset | [1168, 1] | [1179, 17] | simp! only | 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 u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : support (bypass p✝) ⊆ support p✝
⊢ support (bypass (cons h✝ p✝)) ⊆ support (cons h✝ p✝) | 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 u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : support (bypass p✝) ⊆ support p✝
⊢ support (if hs : u✝ ∈ support (bypass p✝) then dropUntil (bypass p✝) u✝ hs else cons h✝ (bypass p✝)) ⊆
u✝ :: support p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_bypass_subset | [1168, 1] | [1179, 17] | split_ifs | 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 u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : support (bypass p✝) ⊆ support p✝
⊢ support (if hs : u✝ ∈ support (bypass p✝) then dropUntil (bypass p✝) u✝ hs else cons h✝ (bypass p✝)) ⊆
u✝ :: support p✝ | case cons.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : support (bypass p✝) ⊆ support p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ support (dropUntil (bypass p✝) u✝ h✝) ⊆ u✝ :: support p✝
case cons.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : support (bypass p✝) ⊆ support p✝
h✝ : ¬u✝ ∈ support (bypass p✝)
⊢ support (cons h✝¹ (bypass p✝)) ⊆ u✝ :: support p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_bypass_subset | [1168, 1] | [1179, 17] | apply List.Subset.trans (support_dropUntil_subset _ _) | case cons.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : support (bypass p✝) ⊆ support p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ support (dropUntil (bypass p✝) u✝ h✝) ⊆ u✝ :: support p✝ | case cons.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : support (bypass p✝) ⊆ support p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ support (bypass p✝) ⊆ u✝ :: support p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_bypass_subset | [1168, 1] | [1179, 17] | apply List.subset_cons_of_subset | case cons.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : support (bypass p✝) ⊆ support p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ support (bypass p✝) ⊆ u✝ :: support p✝ | case cons.inl.a
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : support (bypass p✝) ⊆ support p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ support (bypass p✝) ⊆ support p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_bypass_subset | [1168, 1] | [1179, 17] | assumption | case cons.inl.a
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : support (bypass p✝) ⊆ support p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ support (bypass p✝) ⊆ support p✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_bypass_subset | [1168, 1] | [1179, 17] | rw [support_cons] | case cons.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : support (bypass p✝) ⊆ support p✝
h✝ : ¬u✝ ∈ support (bypass p✝)
⊢ support (cons h✝¹ (bypass p✝)) ⊆ u✝ :: support p✝ | case cons.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : support (bypass p✝) ⊆ support p✝
h✝ : ¬u✝ ∈ support (bypass p✝)
⊢ u✝ :: support (bypass p✝) ⊆ u✝ :: support p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_bypass_subset | [1168, 1] | [1179, 17] | apply List.cons_subset_cons | case cons.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : support (bypass p✝) ⊆ support p✝
h✝ : ¬u✝ ∈ support (bypass p✝)
⊢ u✝ :: support (bypass p✝) ⊆ u✝ :: support p✝ | case cons.inr.s
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : support (bypass p✝) ⊆ support p✝
h✝ : ¬u✝ ∈ support (bypass p✝)
⊢ support (bypass p✝) ⊆ support p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_bypass_subset | [1168, 1] | [1179, 17] | assumption | case cons.inr.s
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : support (bypass p✝) ⊆ support p✝
h✝ : ¬u✝ ∈ support (bypass p✝)
⊢ support (bypass p✝) ⊆ support p✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_bypass_subset | [1187, 1] | [1196, 39] | simp! | 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 u✝ : V
⊢ edges (bypass nil) ⊆ edges nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_bypass_subset | [1187, 1] | [1196, 39] | simp! only | 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 u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : edges (bypass p✝) ⊆ edges p✝
⊢ edges (bypass (cons h✝ p✝)) ⊆ edges (cons h✝ p✝) | 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 u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : edges (bypass p✝) ⊆ edges p✝
⊢ edges (if hs : u✝ ∈ support (bypass p✝) then dropUntil (bypass p✝) u✝ hs else cons h✝ (bypass p✝)) ⊆
(u✝, v✝) :: edges p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_bypass_subset | [1187, 1] | [1196, 39] | split_ifs | 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 u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : edges (bypass p✝) ⊆ edges p✝
⊢ edges (if hs : u✝ ∈ support (bypass p✝) then dropUntil (bypass p✝) u✝ hs else cons h✝ (bypass p✝)) ⊆
(u✝, v✝) :: edges p✝ | case cons.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : edges (bypass p✝) ⊆ edges p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ edges (dropUntil (bypass p✝) u✝ h✝) ⊆ (u✝, v✝) :: edges p✝
case cons.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : edges (bypass p✝) ⊆ edges p✝
h✝ : ¬u✝ ∈ support (bypass p✝)
⊢ edges (cons h✝¹ (bypass p✝)) ⊆ (u✝, v✝) :: edges p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_bypass_subset | [1187, 1] | [1196, 39] | apply List.Subset.trans (edges_dropUntil_subset _ _) | case cons.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : edges (bypass p✝) ⊆ edges p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ edges (dropUntil (bypass p✝) u✝ h✝) ⊆ (u✝, v✝) :: edges p✝ | case cons.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : edges (bypass p✝) ⊆ edges p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ edges (bypass p✝) ⊆ (u✝, v✝) :: edges p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_bypass_subset | [1187, 1] | [1196, 39] | apply List.subset_cons_of_subset _ ih | case cons.inl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : edges (bypass p✝) ⊆ edges p✝
h✝ : u✝ ∈ support (bypass p✝)
⊢ edges (bypass p✝) ⊆ (u✝, v✝) :: edges p✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_bypass_subset | [1187, 1] | [1196, 39] | rw [edges_cons] | case cons.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : edges (bypass p✝) ⊆ edges p✝
h✝ : ¬u✝ ∈ support (bypass p✝)
⊢ edges (cons h✝¹ (bypass p✝)) ⊆ (u✝, v✝) :: edges p✝ | case cons.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : edges (bypass p✝) ⊆ edges p✝
h✝ : ¬u✝ ∈ support (bypass p✝)
⊢ (u✝, v✝) :: edges (bypass p✝) ⊆ (u✝, v✝) :: edges p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_bypass_subset | [1187, 1] | [1196, 39] | exact List.cons_subset_cons _ ih | case cons.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
inst✝ : DecidableEq V
u v u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : edges (bypass p✝) ⊆ edges p✝
h✝ : ¬u✝ ∈ support (bypass p✝)
⊢ (u✝, v✝) :: edges (bypass p✝) ⊆ (u✝, v✝) :: edges p✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_copy | [1227, 1] | [1230, 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
hu : u = u'
hv : v = v'
⊢ Walk.map f (Walk.copy p hu hv) = 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' : V
p : Walk G u' v'
⊢ Walk.map f (Walk.copy p (_ : u' = u') (_ : v' = v')) =
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_copy | [1227, 1] | [1230, 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' : V
p : Walk G u' v'
⊢ Walk.map f (Walk.copy p (_ : u' = u') (_ : v' = v')) =
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_id | [1234, 1] | [1237, 33] | induction p with
| nil => rfl
| cons _ p' ih => simp [ih p'] | 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✝ p : Walk G u v
⊢ Walk.map Hom.id p = p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_id | [1234, 1] | [1237, 33] | 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' u✝ : V
p : Walk G u✝ u✝
⊢ Walk.map Hom.id nil = nil | no goals |
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