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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