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https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphInduction.lean | Digraph.deleteEdges_induction' | [39, 1] | [55, 19] | simp | V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hbot : motive ⊥
hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)})
G : Digraph V
v w : V
hvw : G.Adj v w
ih : motive (deleteEdges G {(v, w)})
⊢ ¬(deleteEdges G {(v, w)}).Adj v w | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphInduction.lean | Digraph.deleteEdges_induction' | [39, 1] | [55, 19] | tauto | case h.e'_1.Adj.h.h.a.mp
V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hbot : motive ⊥
hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)})
G : Digraph V
v w : V
hvw : G.Adj v w
ih : motive (deleteEdges G {(v, w)})
a b : V
⊢ G.Adj a b → G.Adj a b ∧ (a = v → ¬b = w) ∨ a = v ∧ b = w | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphInduction.lean | Digraph.deleteEdges_induction' | [39, 1] | [55, 19] | rintro (h | ⟨rfl, rfl⟩) | case h.e'_1.Adj.h.h.a.mpr
V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hbot : motive ⊥
hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)})
G : Digraph V
v w : V
hvw : G.Adj v w
ih : motive (deleteEdges G {(v, w)})
a b : V
⊢ G.Adj a b ∧ (a = v → ¬b = w) ∨ a = v ∧ b = w → G.Adj a b | case h.e'_1.Adj.h.h.a.mpr.inl
V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hbot : motive ⊥
hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)})
G : Digraph V
v w : V
hvw : G.Adj v w
ih : motive (deleteEdges G {(v, w)})
a b : V
h : G.Adj a b ∧ (a = v → ¬b = w)
⊢ G.Adj a b
case h.e'_1.Adj.h.h.a.mpr.inr.intro
V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hbot : motive ⊥
hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)})
G : Digraph V
a b : V
hvw : G.Adj a b
ih : motive (deleteEdges G {(a, b)})
⊢ G.Adj a b |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphInduction.lean | Digraph.deleteEdges_induction' | [39, 1] | [55, 19] | exact h.1 | case h.e'_1.Adj.h.h.a.mpr.inl
V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hbot : motive ⊥
hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)})
G : Digraph V
v w : V
hvw : G.Adj v w
ih : motive (deleteEdges G {(v, w)})
a b : V
h : G.Adj a b ∧ (a = v → ¬b = w)
⊢ G.Adj a b | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphInduction.lean | Digraph.deleteEdges_induction' | [39, 1] | [55, 19] | assumption | case h.e'_1.Adj.h.h.a.mpr.inr.intro
V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hbot : motive ⊥
hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)})
G : Digraph V
a b : V
hvw : G.Adj a b
ih : motive (deleteEdges G {(a, b)})
⊢ G.Adj a b | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphInduction.lean | Digraph.strong_induction | [59, 11] | [70, 59] | classical
have : Fintype V := Fintype.ofFinite V
generalize hs : G.edgeSet.toFinset = s
induction s using Finset.strongInduction generalizing G with
| _ s ih =>
cases hs
apply hind
intros G' hG
exact ih G'.edgeSet.toFinset (by simpa using hG) _ rfl | V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hind : ∀ (G : Digraph V), (∀ (G' : Digraph V), G' < G → motive G') → motive G
G : Digraph V
⊢ motive G | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphInduction.lean | Digraph.strong_induction | [59, 11] | [70, 59] | have : Fintype V := Fintype.ofFinite V | V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hind : ∀ (G : Digraph V), (∀ (G' : Digraph V), G' < G → motive G') → motive G
G : Digraph V
⊢ motive G | V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hind : ∀ (G : Digraph V), (∀ (G' : Digraph V), G' < G → motive G') → motive G
G : Digraph V
this : Fintype V
⊢ motive G |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphInduction.lean | Digraph.strong_induction | [59, 11] | [70, 59] | generalize hs : G.edgeSet.toFinset = s | V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hind : ∀ (G : Digraph V), (∀ (G' : Digraph V), G' < G → motive G') → motive G
G : Digraph V
this : Fintype V
⊢ motive G | V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hind : ∀ (G : Digraph V), (∀ (G' : Digraph V), G' < G → motive G') → motive G
G : Digraph V
this : Fintype V
s : Finset (V × V)
hs : Set.toFinset (edgeSet G) = s
⊢ motive G |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphInduction.lean | Digraph.strong_induction | [59, 11] | [70, 59] | induction s using Finset.strongInduction generalizing G with
| _ s ih =>
cases hs
apply hind
intros G' hG
exact ih G'.edgeSet.toFinset (by simpa using hG) _ rfl | V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hind : ∀ (G : Digraph V), (∀ (G' : Digraph V), G' < G → motive G') → motive G
G : Digraph V
this : Fintype V
s : Finset (V × V)
hs : Set.toFinset (edgeSet G) = s
⊢ motive G | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphInduction.lean | Digraph.strong_induction | [59, 11] | [70, 59] | cases hs | case H
V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hind : ∀ (G : Digraph V), (∀ (G' : Digraph V), G' < G → motive G') → motive G
this : Fintype V
s : Finset (V × V)
ih : ∀ (t : Finset (V × V)), t ⊂ s → ∀ (G : Digraph V), Set.toFinset (edgeSet G) = t → motive G
G : Digraph V
hs : Set.toFinset (edgeSet G) = s
⊢ motive G | case H.refl
V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hind : ∀ (G : Digraph V), (∀ (G' : Digraph V), G' < G → motive G') → motive G
this : Fintype V
G : Digraph V
ih : ∀ (t : Finset (V × V)), t ⊂ Set.toFinset (edgeSet G) → ∀ (G : Digraph V), Set.toFinset (edgeSet G) = t → motive G
⊢ motive G |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphInduction.lean | Digraph.strong_induction | [59, 11] | [70, 59] | apply hind | case H.refl
V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hind : ∀ (G : Digraph V), (∀ (G' : Digraph V), G' < G → motive G') → motive G
this : Fintype V
G : Digraph V
ih : ∀ (t : Finset (V × V)), t ⊂ Set.toFinset (edgeSet G) → ∀ (G : Digraph V), Set.toFinset (edgeSet G) = t → motive G
⊢ motive G | case H.refl.a
V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hind : ∀ (G : Digraph V), (∀ (G' : Digraph V), G' < G → motive G') → motive G
this : Fintype V
G : Digraph V
ih : ∀ (t : Finset (V × V)), t ⊂ Set.toFinset (edgeSet G) → ∀ (G : Digraph V), Set.toFinset (edgeSet G) = t → motive G
⊢ ∀ (G' : Digraph V), G' < G → motive G' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphInduction.lean | Digraph.strong_induction | [59, 11] | [70, 59] | intros G' hG | case H.refl.a
V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hind : ∀ (G : Digraph V), (∀ (G' : Digraph V), G' < G → motive G') → motive G
this : Fintype V
G : Digraph V
ih : ∀ (t : Finset (V × V)), t ⊂ Set.toFinset (edgeSet G) → ∀ (G : Digraph V), Set.toFinset (edgeSet G) = t → motive G
⊢ ∀ (G' : Digraph V), G' < G → motive G' | case H.refl.a
V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hind : ∀ (G : Digraph V), (∀ (G' : Digraph V), G' < G → motive G') → motive G
this : Fintype V
G : Digraph V
ih : ∀ (t : Finset (V × V)), t ⊂ Set.toFinset (edgeSet G) → ∀ (G : Digraph V), Set.toFinset (edgeSet G) = t → motive G
G' : Digraph V
hG : G' < G
⊢ motive G' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphInduction.lean | Digraph.strong_induction | [59, 11] | [70, 59] | exact ih G'.edgeSet.toFinset (by simpa using hG) _ rfl | case H.refl.a
V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hind : ∀ (G : Digraph V), (∀ (G' : Digraph V), G' < G → motive G') → motive G
this : Fintype V
G : Digraph V
ih : ∀ (t : Finset (V × V)), t ⊂ Set.toFinset (edgeSet G) → ∀ (G : Digraph V), Set.toFinset (edgeSet G) = t → motive G
G' : Digraph V
hG : G' < G
⊢ motive G' | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphInduction.lean | Digraph.strong_induction | [59, 11] | [70, 59] | simpa using hG | V : Type u_1
inst✝ : Finite V
motive : Digraph V → Prop
hind : ∀ (G : Digraph V), (∀ (G' : Digraph V), G' < G → motive G') → motive G
this : Fintype V
G : Digraph V
ih : ∀ (t : Finset (V × V)), t ⊂ Set.toFinset (edgeSet G) → ∀ (G : Digraph V), Set.toFinset (edgeSet G) = t → motive G
G' : Digraph V
hG : G' < G
⊢ Set.toFinset (edgeSet G') ⊂ Set.toFinset (edgeSet G) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.copy_copy | [142, 1] | [146, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u' v' u'' v'' : V
p : Walk G u v
hu : u = u'
hv : v = v'
hu' : u' = u''
hv' : v' = v''
⊢ Walk.copy (Walk.copy p hu hv) hu' hv' = Walk.copy p (_ : u = u'') (_ : v = v'') | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u'' v'' : V
p : Walk G u'' v''
⊢ Walk.copy (Walk.copy p (_ : u'' = u'') (_ : v'' = v'')) (_ : u'' = u'') (_ : v'' = v'') =
Walk.copy p (_ : u'' = u'') (_ : v'' = v'') |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.copy_copy | [142, 1] | [146, 6] | rfl | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u'' v'' : V
p : Walk G u'' v''
⊢ Walk.copy (Walk.copy p (_ : u'' = u'') (_ : v'' = v'')) (_ : u'' = u'') (_ : v'' = v'') =
Walk.copy p (_ : u'' = u'') (_ : v'' = v'') | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.copy_nil | [150, 1] | [152, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u u' : V
hu : u = u'
⊢ Walk.copy nil hu hu = nil | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' : V
⊢ Walk.copy nil (_ : u' = u') (_ : u' = u') = nil |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.copy_nil | [150, 1] | [152, 6] | rfl | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' : V
⊢ Walk.copy nil (_ : u' = u') (_ : u' = u') = nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.copy_cons | [155, 1] | [158, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w u' w' : V
h : G.Adj u v
p : Walk G v w
hu : u = u'
hw : w = w'
⊢ Walk.copy (cons h p) hu hw = cons (_ : G.Adj u' v) (Walk.copy p (_ : v = v) hw) | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v u' w' : V
h : G.Adj u' v
p : Walk G v w'
⊢ Walk.copy (cons h p) (_ : u' = u') (_ : w' = w') = cons (_ : G.Adj u' v) (Walk.copy p (_ : v = v) (_ : w' = w')) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.copy_cons | [155, 1] | [158, 6] | rfl | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v u' w' : V
h : G.Adj u' v
p : Walk G v w'
⊢ Walk.copy (cons h p) (_ : u' = u') (_ : w' = w') = cons (_ : G.Adj u' v) (Walk.copy p (_ : v = v) (_ : w' = w')) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.cons_copy | [162, 1] | [165, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w v' w' : V
h : G.Adj u v
p : Walk G v' w'
hv : v' = v
hw : w' = w
⊢ cons h (Walk.copy p hv hw) = Walk.copy (cons (_ : G.Adj u v') p) (_ : u = u) hw | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v' w' : V
p : Walk G v' w'
h : G.Adj u v'
⊢ cons h (Walk.copy p (_ : v' = v') (_ : w' = w')) = Walk.copy (cons (_ : G.Adj u v') p) (_ : u = u) (_ : w' = w') |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.cons_copy | [162, 1] | [165, 6] | rfl | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v' w' : V
p : Walk G v' w'
h : G.Adj u v'
⊢ cons h (Walk.copy p (_ : v' = v') (_ : w' = w')) = Walk.copy (cons (_ : G.Adj u v') p) (_ : u = u) (_ : w' = w') | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.getVert_zero | [207, 1] | [207, 84] | cases w <;> rfl | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
w : Walk G u v
⊢ getVert w 0 = u | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.getVert_of_length_le | [210, 1] | [217, 43] | rfl | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ : V
i : ℕ
hi : length nil ≤ i
⊢ getVert nil i = u✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.getVert_of_length_le | [210, 1] | [217, 43] | cases i | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ {i : ℕ}, length p✝ ≤ i → getVert p✝ i = w✝
i : ℕ
hi : length (cons h✝ p✝) ≤ i
⊢ getVert (cons h✝ p✝) i = w✝ | case cons.zero
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ {i : ℕ}, length p✝ ≤ i → getVert p✝ i = w✝
hi : length (cons h✝ p✝) ≤ Nat.zero
⊢ getVert (cons h✝ p✝) Nat.zero = w✝
case cons.succ
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ {i : ℕ}, length p✝ ≤ i → getVert p✝ i = w✝
n✝ : ℕ
hi : length (cons h✝ p✝) ≤ Nat.succ n✝
⊢ getVert (cons h✝ p✝) (Nat.succ n✝) = w✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.getVert_of_length_le | [210, 1] | [217, 43] | cases hi | case cons.zero
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ {i : ℕ}, length p✝ ≤ i → getVert p✝ i = w✝
hi : length (cons h✝ p✝) ≤ Nat.zero
⊢ getVert (cons h✝ p✝) Nat.zero = w✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.getVert_of_length_le | [210, 1] | [217, 43] | exact ih (Nat.succ_le_succ_iff.1 hi) | case cons.succ
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ {i : ℕ}, length p✝ ≤ i → getVert p✝ i = w✝
n✝ : ℕ
hi : length (cons h✝ p✝) ≤ Nat.succ n✝
⊢ getVert (cons h✝ p✝) (Nat.succ n✝) = w✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.adj_getVert_succ | [225, 1] | [232, 43] | cases hi | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ : V
i : ℕ
hi : i < length nil
⊢ G.Adj (getVert nil i) (getVert nil (i + 1)) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.adj_getVert_succ | [225, 1] | [232, 43] | cases i | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ v✝ w✝ : V
hxy : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ {i : ℕ}, i < length p✝ → G.Adj (getVert p✝ i) (getVert p✝ (i + 1))
i : ℕ
hi : i < length (cons hxy p✝)
⊢ G.Adj (getVert (cons hxy p✝) i) (getVert (cons hxy p✝) (i + 1)) | case cons.zero
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ v✝ w✝ : V
hxy : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ {i : ℕ}, i < length p✝ → G.Adj (getVert p✝ i) (getVert p✝ (i + 1))
hi : Nat.zero < length (cons hxy p✝)
⊢ G.Adj (getVert (cons hxy p✝) Nat.zero) (getVert (cons hxy p✝) (Nat.zero + 1))
case cons.succ
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ v✝ w✝ : V
hxy : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ {i : ℕ}, i < length p✝ → G.Adj (getVert p✝ i) (getVert p✝ (i + 1))
n✝ : ℕ
hi : Nat.succ n✝ < length (cons hxy p✝)
⊢ G.Adj (getVert (cons hxy p✝) (Nat.succ n✝)) (getVert (cons hxy p✝) (Nat.succ n✝ + 1)) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.adj_getVert_succ | [225, 1] | [232, 43] | simp [getVert, hxy] | case cons.zero
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ v✝ w✝ : V
hxy : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ {i : ℕ}, i < length p✝ → G.Adj (getVert p✝ i) (getVert p✝ (i + 1))
hi : Nat.zero < length (cons hxy p✝)
⊢ G.Adj (getVert (cons hxy p✝) Nat.zero) (getVert (cons hxy p✝) (Nat.zero + 1)) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.adj_getVert_succ | [225, 1] | [232, 43] | exact ih (Nat.succ_lt_succ_iff.1 hi) | case cons.succ
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ v✝ w✝ : V
hxy : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ {i : ℕ}, i < length p✝ → G.Adj (getVert p✝ i) (getVert p✝ (i + 1))
n✝ : ℕ
hi : Nat.succ n✝ < length (cons hxy p✝)
⊢ G.Adj (getVert (cons hxy p✝) (Nat.succ n✝)) (getVert (cons hxy p✝) (Nat.succ n✝ + 1)) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.append_nil | [246, 1] | [249, 40] | induction p with
| nil => rfl
| cons _ _ ih => rw [cons_append, 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
⊢ append p nil = p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.append_nil | [246, 1] | [249, 40] | rfl | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ : V
⊢ append nil nil = nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.append_nil | [246, 1] | [249, 40] | rw [cons_append, ih] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : append p✝ nil = p✝
⊢ append (cons h✝ p✝) nil = cons h✝ p✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.append_assoc | [257, 1] | [263, 12] | induction p with
| nil => rfl
| cons h p' ih =>
dsimp only [append]
rw [ih] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w x : V
p : Walk G u v
q : Walk G v w
r : Walk G w x
⊢ append p (append q r) = append (append p q) r | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.append_assoc | [257, 1] | [263, 12] | rfl | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w x : V
r : Walk G w x
u✝ : V
q : Walk G u✝ w
⊢ append nil (append q r) = append (append nil q) r | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.append_assoc | [257, 1] | [263, 12] | dsimp only [append] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w x : V
r : Walk G w x
u✝ v✝ w✝ : V
h : G.Adj u✝ v✝
p' : Walk G v✝ w✝
ih : ∀ (q : Walk G w✝ w), append p' (append q r) = append (append p' q) r
q : Walk G w✝ w
⊢ append (cons h p') (append q r) = append (append (cons h p') q) r | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w x : V
r : Walk G w x
u✝ v✝ w✝ : V
h : G.Adj u✝ v✝
p' : Walk G v✝ w✝
ih : ∀ (q : Walk G w✝ w), append p' (append q r) = append (append p' q) r
q : Walk G w✝ w
⊢ cons h (append p' (append q r)) = cons h (append (append p' q) r) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.append_assoc | [257, 1] | [263, 12] | rw [ih] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w x : V
r : Walk G w x
u✝ v✝ w✝ : V
h : G.Adj u✝ v✝
p' : Walk G v✝ w✝
ih : ∀ (q : Walk G w✝ w), append p' (append q r) = append (append p' q) r
q : Walk G w✝ w
⊢ cons h (append p' (append q r)) = cons h (append (append p' q) r) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.append_copy_copy | [267, 1] | [271, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w u' v' w' : V
p : Walk G u v
q : Walk G v w
hu : u = u'
hv : v = v'
hw : w = w'
⊢ append (Walk.copy p hu hv) (Walk.copy q hv hw) = Walk.copy (append p q) hu hw | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' w' : V
p : Walk G u' v'
q : Walk G v' w'
⊢ append (Walk.copy p (_ : u' = u') (_ : v' = v')) (Walk.copy q (_ : v' = v') (_ : w' = w')) =
Walk.copy (append p q) (_ : u' = u') (_ : w' = w') |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.append_copy_copy | [267, 1] | [271, 6] | rfl | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' w' : V
p : Walk G u' v'
q : Walk G v' w'
⊢ append (Walk.copy p (_ : u' = u') (_ : v' = v')) (Walk.copy q (_ : v' = v') (_ : w' = w')) =
Walk.copy (append p q) (_ : u' = u') (_ : w' = w') | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_append | [286, 1] | [288, 57] | rw [concat_eq_append, ← append_assoc, cons_nil_append] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w x : V
p : Walk G u v
h : G.Adj v w
q : Walk G w x
⊢ append (concat p h) q = append p (cons h q) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_cons_eq_concat | [292, 1] | [299, 25] | induction p generalizing u with
| nil => exact ⟨_, nil, h, rfl⟩
| cons h' p ih =>
obtain ⟨y, q, h'', hc⟩ := ih h'
refine' ⟨y, cons h q, h'', _⟩
rw [concat_cons, hc] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
h : G.Adj u v
p : Walk G v w
⊢ ∃ x q h', cons h p = concat q h' | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_cons_eq_concat | [292, 1] | [299, 25] | exact ⟨_, nil, h, rfl⟩ | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w u✝ u : V
h : G.Adj u u✝
⊢ ∃ x q h', cons h nil = concat q h' | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_cons_eq_concat | [292, 1] | [299, 25] | obtain ⟨y, q, h'', hc⟩ := ih h' | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w u✝ v✝ w✝ : V
h' : G.Adj u✝ v✝
p : Walk G v✝ w✝
ih : ∀ {u : V} (h : G.Adj u v✝), ∃ x q h', cons h p = concat q h'
u : V
h : G.Adj u u✝
⊢ ∃ x q h'_1, cons h (cons h' p) = concat q h'_1 | case cons.intro.intro.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w u✝ v✝ w✝ : V
h' : G.Adj u✝ v✝
p : Walk G v✝ w✝
ih : ∀ {u : V} (h : G.Adj u v✝), ∃ x q h', cons h p = concat q h'
u : V
h : G.Adj u u✝
y : V
q : Walk G u✝ y
h'' : G.Adj y w✝
hc : cons h' p = concat q h''
⊢ ∃ x q h'_1, cons h (cons h' p) = concat q h'_1 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_cons_eq_concat | [292, 1] | [299, 25] | refine' ⟨y, cons h q, h'', _⟩ | case cons.intro.intro.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w u✝ v✝ w✝ : V
h' : G.Adj u✝ v✝
p : Walk G v✝ w✝
ih : ∀ {u : V} (h : G.Adj u v✝), ∃ x q h', cons h p = concat q h'
u : V
h : G.Adj u u✝
y : V
q : Walk G u✝ y
h'' : G.Adj y w✝
hc : cons h' p = concat q h''
⊢ ∃ x q h'_1, cons h (cons h' p) = concat q h'_1 | case cons.intro.intro.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w u✝ v✝ w✝ : V
h' : G.Adj u✝ v✝
p : Walk G v✝ w✝
ih : ∀ {u : V} (h : G.Adj u v✝), ∃ x q h', cons h p = concat q h'
u : V
h : G.Adj u u✝
y : V
q : Walk G u✝ y
h'' : G.Adj y w✝
hc : cons h' p = concat q h''
⊢ cons h (cons h' p) = concat (cons h q) h'' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_cons_eq_concat | [292, 1] | [299, 25] | rw [concat_cons, hc] | case cons.intro.intro.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w u✝ v✝ w✝ : V
h' : G.Adj u✝ v✝
p : Walk G v✝ w✝
ih : ∀ {u : V} (h : G.Adj u v✝), ∃ x q h', cons h p = concat q h'
u : V
h : G.Adj u u✝
y : V
q : Walk G u✝ y
h'' : G.Adj y w✝
hc : cons h' p = concat q h''
⊢ cons h (cons h' p) = concat (cons h q) h'' | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_copy | [320, 1] | [323, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u' v' : V
p : Walk G u v
hu : u = u'
hv : v = v'
⊢ length (Walk.copy p hu hv) = length p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' : V
p : Walk G u' v'
⊢ length (Walk.copy p (_ : u' = u') (_ : v' = v')) = length p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_copy | [320, 1] | [323, 6] | rfl | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' : V
p : Walk G u' v'
⊢ length (Walk.copy p (_ : u' = u') (_ : v' = v')) = length p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_append | [327, 1] | [331, 65] | induction p with
| nil => simp
| cons _ _ ih => simp [ih, add_comm, add_left_comm, add_assoc] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
q : Walk G v w
⊢ length (append p q) = length p + length q | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_append | [327, 1] | [331, 65] | simp | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w u✝ : V
q : Walk G u✝ w
⊢ length (append nil q) = length nil + length q | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_append | [327, 1] | [331, 65] | simp [ih, add_comm, add_left_comm, add_assoc] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ (q : Walk G w✝ w), length (append p✝ q) = length p✝ + length q
q : Walk G w✝ w
⊢ length (append (cons h✝ p✝) q) = length (cons h✝ p✝) + length q | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_length_eq_zero_iff | [344, 1] | [349, 21] | constructor | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
⊢ (∃ p, length p = 0) ↔ u = v | case mp
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
⊢ (∃ p, length p = 0) → u = v
case mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
⊢ u = v → ∃ p, length p = 0 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_length_eq_zero_iff | [344, 1] | [349, 21] | rintro ⟨p, hp⟩ | case mp
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
⊢ (∃ p, length p = 0) → u = v | case mp.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
hp : length p = 0
⊢ u = v |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_length_eq_zero_iff | [344, 1] | [349, 21] | exact eq_of_length_eq_zero hp | case mp.intro
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
hp : length p = 0
⊢ u = v | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_length_eq_zero_iff | [344, 1] | [349, 21] | rintro rfl | case mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
⊢ u = v → ∃ p, length p = 0 | case mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
⊢ ∃ p, length p = 0 |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.exists_length_eq_zero_iff | [344, 1] | [349, 21] | exact ⟨nil, rfl⟩ | case mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
⊢ ∃ p, length p = 0 | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_eq_zero_iff | [353, 1] | [353, 100] | cases p <;> simp | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
p : Walk G u u
⊢ length p = 0 ↔ p = nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_ne_nil | [397, 1] | [398, 28] | cases p <;> simp [concat] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
h : G.Adj v u
⊢ concat p h ≠ nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | cases p' | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ : V
h : G.Adj u✝ w
p' : Walk G u✝ v'
he : concat nil h = concat p' h'
⊢ ∃ hv, Walk.copy nil (_ : u✝ = u✝) hv = p' | case nil.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' h : G.Adj v' w
he : concat nil h = concat nil h'
⊢ ∃ hv, Walk.copy nil (_ : v' = v') hv = nil
case nil.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ : V
h : G.Adj u✝ w
v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ v'
he : concat nil h = concat (cons h✝ p✝) h'
⊢ ∃ hv, Walk.copy nil (_ : u✝ = u✝) hv = cons h✝ p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | exact ⟨rfl, rfl⟩ | case nil.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' h : G.Adj v' w
he : concat nil h = concat nil h'
⊢ ∃ hv, Walk.copy nil (_ : v' = v') hv = nil | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | exfalso | case nil.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ : V
h : G.Adj u✝ w
v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ v'
he : concat nil h = concat (cons h✝ p✝) h'
⊢ ∃ hv, Walk.copy nil (_ : u✝ = u✝) hv = cons h✝ p✝ | case nil.cons.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ : V
h : G.Adj u✝ w
v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ v'
he : concat nil h = concat (cons h✝ p✝) h'
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | simp only [concat_nil, concat_cons, cons.injEq] at he | case nil.cons.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ : V
h : G.Adj u✝ w
v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ v'
he : concat nil h = concat (cons h✝ p✝) h'
⊢ False | case nil.cons.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ : V
h : G.Adj u✝ w
v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ v'
he : w = v✝ ∧ HEq nil (concat p✝ h')
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | obtain ⟨rfl, he⟩ := he | case nil.cons.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ : V
h : G.Adj u✝ w
v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ v'
he : w = v✝ ∧ HEq nil (concat p✝ h')
⊢ False | case nil.cons.h.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ : V
h h✝ : G.Adj u✝ w
p✝ : Walk G w v'
he : HEq nil (concat p✝ h')
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | simp only [heq_iff_eq] at he | case nil.cons.h.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ : V
h h✝ : G.Adj u✝ w
p✝ : Walk G w v'
he : HEq nil (concat p✝ h')
⊢ False | case nil.cons.h.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ : V
h h✝ : G.Adj u✝ w
p✝ : Walk G w v'
he : nil = concat p✝ h'
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | exact concat_ne_nil _ _ he.symm | case nil.cons.h.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ : V
h h✝ : G.Adj u✝ w
p✝ : Walk G w v'
he : nil = concat p✝ h'
⊢ False | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | rw [concat_cons] at he | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝ v'}, concat p✝ h = concat p' h' → ∃ hv, Walk.copy p✝ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ w
p' : Walk G u✝ v'
he : concat (cons h✝ p✝) h = concat p' h'
⊢ ∃ hv, Walk.copy (cons h✝ p✝) (_ : u✝ = u✝) hv = p' | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝ v'}, concat p✝ h = concat p' h' → ∃ hv, Walk.copy p✝ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ w
p' : Walk G u✝ v'
he : cons h✝ (concat p✝ h) = concat p' h'
⊢ ∃ hv, Walk.copy (cons h✝ p✝) (_ : u✝ = u✝) hv = p' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | cases p' | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝ v'}, concat p✝ h = concat p' h' → ∃ hv, Walk.copy p✝ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ w
p' : Walk G u✝ v'
he : cons h✝ (concat p✝ h) = concat p' h'
⊢ ∃ hv, Walk.copy (cons h✝ p✝) (_ : u✝ = u✝) hv = p' | case cons.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
v✝ w✝ : V
p✝ : Walk G v✝ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝ v'}, concat p✝ h = concat p' h' → ∃ hv, Walk.copy p✝ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ w
h✝ : G.Adj v' v✝
he : cons h✝ (concat p✝ h) = concat nil h'
⊢ ∃ hv, Walk.copy (cons h✝ p✝) (_ : v' = v') hv = nil
case cons.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ v✝¹ w✝ : V
h✝¹ : G.Adj u✝ v✝¹
p✝¹ : Walk G v✝¹ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝¹ v'}, concat p✝¹ h = concat p' h' → ∃ hv, Walk.copy p✝¹ (_ : v✝¹ = v✝¹) hv = p'
h : G.Adj w✝ w
v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ v'
he : cons h✝¹ (concat p✝¹ h) = concat (cons h✝ p✝) h'
⊢ ∃ hv, Walk.copy (cons h✝¹ p✝¹) (_ : u✝ = u✝) hv = cons h✝ p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | exfalso | case cons.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
v✝ w✝ : V
p✝ : Walk G v✝ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝ v'}, concat p✝ h = concat p' h' → ∃ hv, Walk.copy p✝ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ w
h✝ : G.Adj v' v✝
he : cons h✝ (concat p✝ h) = concat nil h'
⊢ ∃ hv, Walk.copy (cons h✝ p✝) (_ : v' = v') hv = nil | case cons.nil.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
v✝ w✝ : V
p✝ : Walk G v✝ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝ v'}, concat p✝ h = concat p' h' → ∃ hv, Walk.copy p✝ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ w
h✝ : G.Adj v' v✝
he : cons h✝ (concat p✝ h) = concat nil h'
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | simp only [concat_nil, cons.injEq] at he | case cons.nil.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
v✝ w✝ : V
p✝ : Walk G v✝ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝ v'}, concat p✝ h = concat p' h' → ∃ hv, Walk.copy p✝ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ w
h✝ : G.Adj v' v✝
he : cons h✝ (concat p✝ h) = concat nil h'
⊢ False | case cons.nil.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
v✝ w✝ : V
p✝ : Walk G v✝ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝ v'}, concat p✝ h = concat p' h' → ∃ hv, Walk.copy p✝ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ w
h✝ : G.Adj v' v✝
he : v✝ = w ∧ HEq (concat p✝ h) nil
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | obtain ⟨rfl, he⟩ := he | case cons.nil.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
v✝ w✝ : V
p✝ : Walk G v✝ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝ v'}, concat p✝ h = concat p' h' → ∃ hv, Walk.copy p✝ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ w
h✝ : G.Adj v' v✝
he : v✝ = w ∧ HEq (concat p✝ h) nil
⊢ False | case cons.nil.h.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' v✝ w✝ : V
p✝ : Walk G v✝ w✝
h✝ h' : G.Adj v' v✝
ih : ∀ {h : G.Adj w✝ v✝} {p' : Walk G v✝ v'}, concat p✝ h = concat p' h' → ∃ hv, Walk.copy p✝ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ v✝
he : HEq (concat p✝ h) nil
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | rw [heq_iff_eq] at he | case cons.nil.h.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' v✝ w✝ : V
p✝ : Walk G v✝ w✝
h✝ h' : G.Adj v' v✝
ih : ∀ {h : G.Adj w✝ v✝} {p' : Walk G v✝ v'}, concat p✝ h = concat p' h' → ∃ hv, Walk.copy p✝ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ v✝
he : HEq (concat p✝ h) nil
⊢ False | case cons.nil.h.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' v✝ w✝ : V
p✝ : Walk G v✝ w✝
h✝ h' : G.Adj v' v✝
ih : ∀ {h : G.Adj w✝ v✝} {p' : Walk G v✝ v'}, concat p✝ h = concat p' h' → ∃ hv, Walk.copy p✝ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ v✝
he : concat p✝ h = nil
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | exact concat_ne_nil _ _ he | case cons.nil.h.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' v✝ w✝ : V
p✝ : Walk G v✝ w✝
h✝ h' : G.Adj v' v✝
ih : ∀ {h : G.Adj w✝ v✝} {p' : Walk G v✝ v'}, concat p✝ h = concat p' h' → ∃ hv, Walk.copy p✝ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ v✝
he : concat p✝ h = nil
⊢ False | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | rw [concat_cons, cons.injEq] at he | case cons.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ v✝¹ w✝ : V
h✝¹ : G.Adj u✝ v✝¹
p✝¹ : Walk G v✝¹ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝¹ v'}, concat p✝¹ h = concat p' h' → ∃ hv, Walk.copy p✝¹ (_ : v✝¹ = v✝¹) hv = p'
h : G.Adj w✝ w
v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ v'
he : cons h✝¹ (concat p✝¹ h) = concat (cons h✝ p✝) h'
⊢ ∃ hv, Walk.copy (cons h✝¹ p✝¹) (_ : u✝ = u✝) hv = cons h✝ p✝ | case cons.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ v✝¹ w✝ : V
h✝¹ : G.Adj u✝ v✝¹
p✝¹ : Walk G v✝¹ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝¹ v'}, concat p✝¹ h = concat p' h' → ∃ hv, Walk.copy p✝¹ (_ : v✝¹ = v✝¹) hv = p'
h : G.Adj w✝ w
v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ v'
he : v✝¹ = v✝ ∧ HEq (concat p✝¹ h) (concat p✝ h')
⊢ ∃ hv, Walk.copy (cons h✝¹ p✝¹) (_ : u✝ = u✝) hv = cons h✝ p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | obtain ⟨rfl, he⟩ := he | case cons.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ v✝¹ w✝ : V
h✝¹ : G.Adj u✝ v✝¹
p✝¹ : Walk G v✝¹ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝¹ v'}, concat p✝¹ h = concat p' h' → ∃ hv, Walk.copy p✝¹ (_ : v✝¹ = v✝¹) hv = p'
h : G.Adj w✝ w
v✝ : V
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ v'
he : v✝¹ = v✝ ∧ HEq (concat p✝¹ h) (concat p✝ h')
⊢ ∃ hv, Walk.copy (cons h✝¹ p✝¹) (_ : u✝ = u✝) hv = cons h✝ p✝ | case cons.cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝¹ : Walk G v✝ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝ v'}, concat p✝¹ h = concat p' h' → ∃ hv, Walk.copy p✝¹ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ w
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ v'
he : HEq (concat p✝¹ h) (concat p✝ h')
⊢ ∃ hv, Walk.copy (cons h✝¹ p✝¹) (_ : u✝ = u✝) hv = cons h✝ p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | rw [heq_iff_eq] at he | case cons.cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝¹ : Walk G v✝ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝ v'}, concat p✝¹ h = concat p' h' → ∃ hv, Walk.copy p✝¹ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ w
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ v'
he : HEq (concat p✝¹ h) (concat p✝ h')
⊢ ∃ hv, Walk.copy (cons h✝¹ p✝¹) (_ : u✝ = u✝) hv = cons h✝ p✝ | case cons.cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝¹ : Walk G v✝ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝ v'}, concat p✝¹ h = concat p' h' → ∃ hv, Walk.copy p✝¹ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ w
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ v'
he : concat p✝¹ h = concat p✝ h'
⊢ ∃ hv, Walk.copy (cons h✝¹ p✝¹) (_ : u✝ = u✝) hv = cons h✝ p✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | obtain ⟨rfl, rfl⟩ := ih he | case cons.cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v v' w : V
h' : G.Adj v' w
u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝¹ : Walk G v✝ w✝
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝ v'}, concat p✝¹ h = concat p' h' → ∃ hv, Walk.copy p✝¹ (_ : v✝ = v✝) hv = p'
h : G.Adj w✝ w
h✝ : G.Adj u✝ v✝
p✝ : Walk G v✝ v'
he : concat p✝¹ h = concat p✝ h'
⊢ ∃ hv, Walk.copy (cons h✝¹ p✝¹) (_ : u✝ = u✝) hv = cons h✝ p✝ | case cons.cons.intro.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
h : G.Adj w✝ w
h✝ : G.Adj u✝ v✝
h' : G.Adj w✝ w
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝ w✝}, concat p✝ h = concat p' h' → ∃ hv, Walk.copy p✝ (_ : v✝ = v✝) hv = p'
he : concat p✝ h = concat (Walk.copy p✝ (_ : v✝ = v✝) (_ : w✝ = w✝)) h'
⊢ ∃ hv, Walk.copy (cons h✝¹ p✝) (_ : u✝ = u✝) hv = cons h✝ (Walk.copy p✝ (_ : v✝ = v✝) (_ : w✝ = w✝)) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.concat_inj | [401, 1] | [424, 23] | exact ⟨rfl, rfl⟩ | case cons.cons.intro.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w u✝ v✝ w✝ : V
h✝¹ : G.Adj u✝ v✝
p✝ : Walk G v✝ w✝
h : G.Adj w✝ w
h✝ : G.Adj u✝ v✝
h' : G.Adj w✝ w
ih : ∀ {h : G.Adj w✝ w} {p' : Walk G v✝ w✝}, concat p✝ h = concat p' h' → ∃ hv, Walk.copy p✝ (_ : v✝ = v✝) hv = p'
he : concat p✝ h = concat (Walk.copy p✝ (_ : v✝ = v✝) (_ : w✝ = w✝)) h'
⊢ ∃ hv, Walk.copy (cons h✝¹ p✝) (_ : u✝ = u✝) hv = cons h✝ (Walk.copy p✝ (_ : v✝ = v✝) (_ : w✝ = w✝)) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_concat | [448, 1] | [450, 39] | induction p <;> simp [*, concat_nil] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
h : G.Adj v w
⊢ support (concat p h) = List.concat (support p) w | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_copy | [454, 1] | [457, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u' v' : V
p : Walk G u v
hu : u = u'
hv : v = v'
⊢ support (Walk.copy p hu hv) = support p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' : V
p : Walk G u' v'
⊢ support (Walk.copy p (_ : u' = u') (_ : v' = v')) = support p |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_copy | [454, 1] | [457, 6] | rfl | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' : V
p : Walk G u' v'
⊢ support (Walk.copy p (_ : u' = u') (_ : v' = v')) = support p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_append | [460, 1] | [462, 40] | induction p <;> cases p' <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
p' : Walk G v w
⊢ support (append p p') = support p ++ List.tail (support p') | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_ne_nil | [465, 1] | [465, 90] | cases p <;> simp | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
⊢ support p ≠ [] | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.tail_support_append | [468, 1] | [470, 73] | rw [support_append, List.tail_append_of_ne_nil _ _ (support_ne_nil _)] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
p' : Walk G v w
⊢ List.tail (support (append p p')) = List.tail (support p) ++ List.tail (support p') | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_eq_cons | [473, 1] | [474, 19] | cases p <;> simp | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
⊢ support p = u :: List.tail (support p) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.start_mem_support | [478, 1] | [478, 92] | cases p <;> simp | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
⊢ u ∈ support p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.end_mem_support | [482, 1] | [482, 98] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
⊢ v ∈ support p | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_nonempty | [486, 1] | [487, 15] | simp | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
⊢ u ∈ {w | w ∈ support p} | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.mem_support_iff | [490, 1] | [491, 22] | cases p <;> simp | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
p : Walk G u v
⊢ w ∈ support p ↔ w = u ∨ w ∈ List.tail (support p) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.mem_support_nil_iff | [494, 1] | [494, 90] | simp | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
⊢ u ∈ support nil ↔ u = v | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.mem_tail_support_append_iff | [498, 1] | [500, 44] | rw [tail_support_append, List.mem_append] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
t u v w : V
p : Walk G u v
p' : Walk G v w
⊢ t ∈ List.tail (support (append p p')) ↔ t ∈ List.tail (support p) ∨ t ∈ List.tail (support p') | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.end_mem_tail_support_of_ne | [504, 1] | [506, 7] | obtain ⟨_, _, _, rfl⟩ := exists_eq_cons_of_ne h p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u ≠ v
p : Walk G u v
⊢ v ∈ List.tail (support p) | case intro.intro.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u ≠ v
w✝² : V
w✝¹ : G.Adj u w✝²
w✝ : Walk G w✝² v
⊢ v ∈ List.tail (support (cons w✝¹ w✝)) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.end_mem_tail_support_of_ne | [504, 1] | [506, 7] | simp | case intro.intro.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u ≠ v
w✝² : V
w✝¹ : G.Adj u w✝²
w✝ : Walk G w✝² v
⊢ v ∈ List.tail (support (cons w✝¹ w✝)) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.mem_support_append_iff | [510, 1] | [515, 39] | simp only [mem_support_iff, mem_tail_support_append_iff] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
t u v w : V
p : Walk G u v
p' : Walk G v w
⊢ t ∈ support (append p p') ↔ t ∈ support p ∨ t ∈ support p' | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
t u v w : V
p : Walk G u v
p' : Walk G v w
⊢ t = u ∨ t ∈ List.tail (support p) ∨ t ∈ List.tail (support p') ↔
(t = u ∨ t ∈ List.tail (support p)) ∨ t = v ∨ t ∈ List.tail (support p') |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.mem_support_append_iff | [510, 1] | [515, 39] | obtain rfl | h := eq_or_ne t v <;> obtain rfl | h' := eq_or_ne t u <;>
(try have := h'.symm) <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
t u v w : V
p : Walk G u v
p' : Walk G v w
⊢ t = u ∨ t ∈ List.tail (support p) ∨ t ∈ List.tail (support p') ↔
(t = u ∨ t ∈ List.tail (support p)) ∨ t = v ∨ t ∈ List.tail (support p') | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.mem_support_append_iff | [510, 1] | [515, 39] | try have := h'.symm | case inr.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
t u v w : V
p : Walk G u v
p' : Walk G v w
h : t ≠ v
h' : t ≠ u
⊢ t = u ∨ t ∈ List.tail (support p) ∨ t ∈ List.tail (support p') ↔
(t = u ∨ t ∈ List.tail (support p)) ∨ t = v ∨ t ∈ List.tail (support p') | case inr.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
t u v w : V
p : Walk G u v
p' : Walk G v w
h : t ≠ v
h' : t ≠ u
this : u ≠ t
⊢ t = u ∨ t ∈ List.tail (support p) ∨ t ∈ List.tail (support p') ↔
(t = u ∨ t ∈ List.tail (support p)) ∨ t = v ∨ t ∈ List.tail (support p') |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.mem_support_append_iff | [510, 1] | [515, 39] | have := h'.symm | case inr.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
t u v w : V
p : Walk G u v
p' : Walk G v w
h : t ≠ v
h' : t ≠ u
⊢ t = u ∨ t ∈ List.tail (support p) ∨ t ∈ List.tail (support p') ↔
(t = u ∨ t ∈ List.tail (support p)) ∨ t = v ∨ t ∈ List.tail (support p') | case inr.inr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
t u v w : V
p : Walk G u v
p' : Walk G v w
h : t ≠ v
h' : t ≠ u
this : u ≠ t
⊢ t = u ∨ t ∈ List.tail (support p) ∨ t ∈ List.tail (support p') ↔
(t = u ∨ t ∈ List.tail (support p)) ∨ t = v ∨ t ∈ List.tail (support p') |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.subset_support_append_left | [519, 1] | [521, 59] | simp only [Walk.support_append, List.subset_append_left] | V✝ : Type u
V' : Type v
V'' : Type w
G✝ : Digraph V✝
G' : Digraph V'
G'' : Digraph V''
V : Type u
G : Digraph V
u v w : V
p : Walk G u v
q : Walk G v w
⊢ support p ⊆ support (append p q) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.subset_support_append_right | [525, 1] | [528, 99] | intro h | V✝ : Type u
V' : Type v
V'' : Type w
G✝ : Digraph V✝
G' : Digraph V'
G'' : Digraph V''
V : Type u
G : Digraph V
u v w : V
p : Walk G u v
q : Walk G v w
⊢ support q ⊆ support (append p q) | V✝ : Type u
V' : Type v
V'' : Type w
G✝ : Digraph V✝
G' : Digraph V'
G'' : Digraph V''
V : Type u
G : Digraph V
u v w : V
p : Walk G u v
q : Walk G v w
h : V
⊢ h ∈ support q → h ∈ support (append p q) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.subset_support_append_right | [525, 1] | [528, 99] | simp (config := { contextual := true }) only [mem_support_append_iff, or_true_iff, imp_true_iff] | V✝ : Type u
V' : Type v
V'' : Type w
G✝ : Digraph V✝
G' : Digraph V'
G'' : Digraph V''
V : Type u
G : Digraph V
u v w : V
p : Walk G u v
q : Walk G v w
h : V
⊢ h ∈ support q → h ∈ support (append p q) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.coe_support | [531, 1] | [532, 21] | cases p <;> rfl | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
⊢ ↑(support p) = {u} + ↑(List.tail (support p)) | no goals |
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