url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_id | [1234, 1] | [1237, 33] | simp [ih p'] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' uβ vβ wβ : V
hβ : G.Adj uβ vβ
p' : Walk G vβ wβ
ih : Walk G vβ wβ β Walk.map Hom.id p' = p'
p : Walk G uβ wβ
β’ Walk.map Hom.id (cons hβ p') = cons hβ p' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' uβ vβ wβ : V
hβ : G.Adj uβ vβ
p' : Walk G vβ wβ
ih : Walk G vβ wβ β Walk.map Hom.id p' = p'
p : Walk G uβ wβ
β’ Walk.map Hom.id (cons hβ p') = cons hβ p'
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_map | [1241, 1] | [1244, 29] | induction p with
| nil => rfl
| cons _ _ ih => simp [ih] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
β’ Walk.map f' (Walk.map f p) = Walk.map (Hom.comp f' f) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
β’ Walk.map f' (Walk.map f p) = Walk.map (Hom.comp f' f) p
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_map | [1241, 1] | [1244, 29] | rfl | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
uβ : V
β’ Walk.map f' (Walk.map f nil) = Walk.map (Hom.comp f' f) nil | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
uβ : V
β’ Walk.map f' (Walk.map f nil) = Walk.map (Hom.comp f' f) nil
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_map | [1241, 1] | [1244, 29] | simp [ih] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : Walk.map f' (Walk.map f pβ) = Walk.map (Hom.comp f' f) pβ
β’ Walk.map f' (Walk.map f (cons hβ pβ)) = Walk.map (Hom.comp f' f) (cons hβ pβ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : Walk.map f' (Walk.map f pβ) = Walk.map (Hom.comp f' f) pβ
β’ Walk.map f' (Walk.map f (cons hβ pβ)) = Walk.map (Hom.comp f' f) (cons hβ pβ)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_eq_of_eq | [1249, 1] | [1252, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f'β : G' βg G''
u v u' v' : V
p : Walk G u v
f f' : G βg G'
h : f = f'
β’ Walk.map f p = Walk.copy (Walk.map f' p) (_ : βf' u = βf u) (_ : βf' v = βf v) | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f'β : G' βg G''
u v u' v' : V
p : Walk G u v
f' : G βg G'
β’ Walk.map f' p = Walk.copy (Walk.map f' p) (_ : βf' u = βf' u) (_ : βf' v = βf' v) | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f'β : G' βg G''
u v u' v' : V
p : Walk G u v
f f' : G βg G'
h : f = f'
β’ Walk.map f p = Walk.copy (Walk.map f' p) (_ : βf' u = βf u) (_ : βf' v = βf v)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_eq_of_eq | [1249, 1] | [1252, 6] | rfl | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f'β : G' βg G''
u v u' v' : V
p : Walk G u v
f' : G βg G'
β’ Walk.map f' p = Walk.copy (Walk.map f' p) (_ : βf' u = βf' u) (_ : βf' v = βf' v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f'β : G' βg G''
u v u' v' : V
p : Walk G u v
f' : G βg G'
β’ Walk.map f' p = Walk.copy (Walk.map f' p) (_ : βf' u = βf' u) (_ : βf' v = βf' v)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_eq_nil_iff | [1256, 1] | [1256, 89] | cases p <;> simp | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
pβ : Walk G u v
p : Walk G u u
β’ Walk.map f p = nil β p = nil | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
pβ : Walk G u v
p : Walk G u u
β’ Walk.map f p = nil β p = nil
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_map | [1260, 1] | [1260, 80] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
β’ length (Walk.map f p) = length p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
β’ length (Walk.map f p) = length p
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_append | [1263, 1] | [1264, 83] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
uβ vβ u' v' : V
pβ : Walk G uβ vβ
u v w : V
p : Walk G u v
q : Walk G v w
β’ Walk.map f (append p q) = append (Walk.map f p) (Walk.map f q) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
uβ vβ u' v' : V
pβ : Walk G uβ vβ
u v w : V
p : Walk G u v
q : Walk G v w
β’ Walk.map f (append p q) = append (Walk.map f p) (Walk.map f q)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_map | [1268, 1] | [1268, 89] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
β’ support (Walk.map f p) = List.map (βf) (support p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
β’ support (Walk.map f p) = List.map (βf) (support p)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_map | [1272, 1] | [1272, 96] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
β’ edges (Walk.map f p) = List.map (Prod.map βf βf) (edges p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
β’ edges (Walk.map f p) = List.map (Prod.map βf βf) (edges p)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isPath_of_injective | [1276, 1] | [1285, 18] | induction p with
| nil => simp
| cons _ _ ih =>
rw [Walk.cons_isPath_iff] at hp
simp [ih hp.1]
intro x hx hf
cases hinj hf
exact hp.2 hx | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
hp : IsPath p
β’ IsPath (Walk.map f p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
hp : IsPath p
β’ IsPath (Walk.map f p)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isPath_of_injective | [1276, 1] | [1285, 18] | simp | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ : V
hp : IsPath nil
β’ IsPath (Walk.map f nil) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ : V
hp : IsPath nil
β’ IsPath (Walk.map f nil)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isPath_of_injective | [1276, 1] | [1285, 18] | rw [Walk.cons_isPath_iff] at hp | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath pβ β IsPath (Walk.map f pβ)
hp : IsPath (cons hβ pβ)
β’ IsPath (Walk.map f (cons hβ pβ)) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath pβ β IsPath (Walk.map f pβ)
hp : IsPath pβ β§ Β¬uβ β support pβ
β’ IsPath (Walk.map f (cons hβ pβ)) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath pβ β IsPath (Walk.map f pβ)
hp : IsPath (cons hβ pβ)
β’ IsPath (Walk.map f (cons hβ pβ))
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isPath_of_injective | [1276, 1] | [1285, 18] | simp [ih hp.1] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath pβ β IsPath (Walk.map f pβ)
hp : IsPath pβ β§ Β¬uβ β support pβ
β’ IsPath (Walk.map f (cons hβ pβ)) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath pβ β IsPath (Walk.map f pβ)
hp : IsPath pβ β§ Β¬uβ β support pβ
β’ β (x : V), x β support pβ β Β¬βf x = βf uβ | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath pβ β IsPath (Walk.map f pβ)
hp : IsPath pβ β§ Β¬uβ β support pβ
β’ IsPath (Walk.map f (cons hβ pβ))
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isPath_of_injective | [1276, 1] | [1285, 18] | intro x hx hf | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath pβ β IsPath (Walk.map f pβ)
hp : IsPath pβ β§ Β¬uβ β support pβ
β’ β (x : V), x β support pβ β Β¬βf x = βf uβ | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath pβ β IsPath (Walk.map f pβ)
hp : IsPath pβ β§ Β¬uβ β support pβ
x : V
hx : x β support pβ
hf : βf x = βf uβ
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath pβ β IsPath (Walk.map f pβ)
hp : IsPath pβ β§ Β¬uβ β support pβ
β’ β (x : V), x β support pβ β Β¬βf x = βf uβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isPath_of_injective | [1276, 1] | [1285, 18] | cases hinj hf | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath pβ β IsPath (Walk.map f pβ)
hp : IsPath pβ β§ Β¬uβ β support pβ
x : V
hx : x β support pβ
hf : βf x = βf uβ
β’ False | case cons.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath pβ β IsPath (Walk.map f pβ)
hp : IsPath pβ β§ Β¬uβ β support pβ
hx : uβ β support pβ
hf : βf uβ = βf uβ
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath pβ β IsPath (Walk.map f pβ)
hp : IsPath pβ β§ Β¬uβ β support pβ
x : V
hx : x β support pβ
hf : βf x = βf uβ
β’ False
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isPath_of_injective | [1276, 1] | [1285, 18] | exact hp.2 hx | case cons.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath pβ β IsPath (Walk.map f pβ)
hp : IsPath pβ β§ Β¬uβ β support pβ
hx : uβ β support pβ
hf : βf uβ = βf uβ
β’ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath pβ β IsPath (Walk.map f pβ)
hp : IsPath pβ β§ Β¬uβ β support pβ
hx : uβ β support pβ
hf : βf uβ = βf uβ
β’ False
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | induction p with
| nil => simp
| cons _ _ ih =>
rw [map_cons, Walk.cons_isPath_iff, support_map] at hp
rw [Walk.cons_isPath_iff]
cases' hp with hp1 hp2
refine' β¨ih hp1, _β©
contrapose! hp2
exact List.mem_map_of_mem f hp2 | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
hp : IsPath (Walk.map f p)
β’ IsPath p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
hp : IsPath (Walk.map f p)
β’ IsPath p
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | simp | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ : V
hp : IsPath (Walk.map f Walk.nil)
β’ IsPath Walk.nil | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ : V
hp : IsPath (Walk.map f Walk.nil)
β’ IsPath Walk.nil
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | rw [map_cons, Walk.cons_isPath_iff, support_map] at hp | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp : IsPath (Walk.map f (cons hβ pβ))
β’ IsPath (cons hβ pβ) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp : IsPath (Walk.map f pβ) β§ Β¬βf uβ β List.map (βf) (support pβ)
β’ IsPath (cons hβ pβ) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp : IsPath (Walk.map f (cons hβ pβ))
β’ IsPath (cons hβ pβ)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | rw [Walk.cons_isPath_iff] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp : IsPath (Walk.map f pβ) β§ Β¬βf uβ β List.map (βf) (support pβ)
β’ IsPath (cons hβ pβ) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp : IsPath (Walk.map f pβ) β§ Β¬βf uβ β List.map (βf) (support pβ)
β’ IsPath pβ β§ Β¬uβ β support pβ | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp : IsPath (Walk.map f pβ) β§ Β¬βf uβ β List.map (βf) (support pβ)
β’ IsPath (cons hβ pβ)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | cases' hp with hp1 hp2 | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp : IsPath (Walk.map f pβ) β§ Β¬βf uβ β List.map (βf) (support pβ)
β’ IsPath pβ β§ Β¬uβ β support pβ | case cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp1 : IsPath (Walk.map f pβ)
hp2 : Β¬βf uβ β List.map (βf) (support pβ)
β’ IsPath pβ β§ Β¬uβ β support pβ | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp : IsPath (Walk.map f pβ) β§ Β¬βf uβ β List.map (βf) (support pβ)
β’ IsPath pβ β§ Β¬uβ β support pβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | refine' β¨ih hp1, _β© | case cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp1 : IsPath (Walk.map f pβ)
hp2 : Β¬βf uβ β List.map (βf) (support pβ)
β’ IsPath pβ β§ Β¬uβ β support pβ | case cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp1 : IsPath (Walk.map f pβ)
hp2 : Β¬βf uβ β List.map (βf) (support pβ)
β’ Β¬uβ β support pβ | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp1 : IsPath (Walk.map f pβ)
hp2 : Β¬βf uβ β List.map (βf) (support pβ)
β’ IsPath pβ β§ Β¬uβ β support pβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | contrapose! hp2 | case cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp1 : IsPath (Walk.map f pβ)
hp2 : Β¬βf uβ β List.map (βf) (support pβ)
β’ Β¬uβ β support pβ | case cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp1 : IsPath (Walk.map f pβ)
hp2 : uβ β support pβ
β’ βf uβ β List.map (βf) (support pβ) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp1 : IsPath (Walk.map f pβ)
hp2 : Β¬βf uβ β List.map (βf) (support pβ)
β’ Β¬uβ β support pβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.of_map | [1288, 11] | [1297, 36] | exact List.mem_map_of_mem f hp2 | case cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp1 : IsPath (Walk.map f pβ)
hp2 : uβ β support pβ
β’ βf uβ β List.map (βf) (support pβ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
f : G βg G'
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsPath (Walk.map f pβ) β IsPath pβ
hp1 : IsPath (Walk.map f pβ)
hp2 : uβ β support pβ
β’ βf uβ β List.map (βf) (support pβ)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isTrail_iff_of_injective | [1304, 1] | [1313, 18] | induction p with
| nil => simp
| cons _ _ ih =>
rw [map_cons, cons_isTrail_iff, ih, cons_isTrail_iff]
apply and_congr_right'
rw [edges_map, β List.mem_map_of_injective (Injective.Prod_map hinj hinj)]
exact Iff.rfl | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
β’ IsTrail (Walk.map f p) β IsTrail p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
β’ IsTrail (Walk.map f p) β IsTrail p
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isTrail_iff_of_injective | [1304, 1] | [1313, 18] | simp | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ : V
β’ IsTrail (Walk.map f nil) β IsTrail nil | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ : V
β’ IsTrail (Walk.map f nil) β IsTrail nil
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isTrail_iff_of_injective | [1304, 1] | [1313, 18] | rw [map_cons, cons_isTrail_iff, ih, cons_isTrail_iff] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsTrail (Walk.map f pβ) β IsTrail pβ
β’ IsTrail (Walk.map f (cons hβ pβ)) β IsTrail (cons hβ pβ) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsTrail (Walk.map f pβ) β IsTrail pβ
β’ IsTrail pβ β§ Β¬(βf uβ, βf vβ) β edges (Walk.map f pβ) β IsTrail pβ β§ Β¬(uβ, vβ) β edges pβ | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsTrail (Walk.map f pβ) β IsTrail pβ
β’ IsTrail (Walk.map f (cons hβ pβ)) β IsTrail (cons hβ pβ)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isTrail_iff_of_injective | [1304, 1] | [1313, 18] | apply and_congr_right' | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsTrail (Walk.map f pβ) β IsTrail pβ
β’ IsTrail pβ β§ Β¬(βf uβ, βf vβ) β edges (Walk.map f pβ) β IsTrail pβ β§ Β¬(uβ, vβ) β edges pβ | case cons.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsTrail (Walk.map f pβ) β IsTrail pβ
β’ Β¬(βf uβ, βf vβ) β edges (Walk.map f pβ) β Β¬(uβ, vβ) β edges pβ | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsTrail (Walk.map f pβ) β IsTrail pβ
β’ IsTrail pβ β§ Β¬(βf uβ, βf vβ) β edges (Walk.map f pβ) β IsTrail pβ β§ Β¬(uβ, vβ) β edges pβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isTrail_iff_of_injective | [1304, 1] | [1313, 18] | rw [edges_map, β List.mem_map_of_injective (Injective.Prod_map hinj hinj)] | case cons.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsTrail (Walk.map f pβ) β IsTrail pβ
β’ Β¬(βf uβ, βf vβ) β edges (Walk.map f pβ) β Β¬(uβ, vβ) β edges pβ | case cons.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsTrail (Walk.map f pβ) β IsTrail pβ
β’ Β¬(βf uβ, βf vβ) β List.map (Prod.map βf βf) (edges pβ) β
Β¬Prod.map βf βf (uβ, vβ) β List.map (Prod.map βf βf) (edges pβ) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsTrail (Walk.map f pβ) β IsTrail pβ
β’ Β¬(βf uβ, βf vβ) β edges (Walk.map f pβ) β Β¬(uβ, vβ) β edges pβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isTrail_iff_of_injective | [1304, 1] | [1313, 18] | exact Iff.rfl | case cons.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsTrail (Walk.map f pβ) β IsTrail pβ
β’ Β¬(βf uβ, βf vβ) β List.map (Prod.map βf βf) (edges pβ) β
Β¬Prod.map βf βf (uβ, vβ) β List.map (Prod.map βf βf) (edges pβ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
p : Walk G u v
hinj : Injective βf
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : IsTrail (Walk.map f pβ) β IsTrail pβ
β’ Β¬(βf uβ, βf vβ) β List.map (Prod.map βf βf) (edges pβ) β
Β¬Prod.map βf βf (uβ, vβ) β List.map (Prod.map βf βf) (edges pβ)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_isCycle_iff_of_injective | [1318, 1] | [1321, 59] | rw [isCycle_def, isCycle_def, map_isTrail_iff_of_injective hinj, Ne.def, map_eq_nil_iff,
support_map, β List.map_tail, List.nodup_map_iff hinj] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
pβ : Walk G u v
p : Walk G u u
hinj : Injective βf
β’ IsCycle (Walk.map f p) β IsCycle p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
f' : G' βg G''
u v u' v' : V
pβ : Walk G u v
p : Walk G u u
hinj : Injective βf
β’ IsCycle (Walk.map f p) β IsCycle p
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | intro p p' h | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβ vβ u' v' : V
p : Walk G uβ vβ
f : G βg G'
hinj : Injective βf
u v : V
β’ Injective (Walk.map f) | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβ vβ u' v' : V
pβ : Walk G uβ vβ
f : G βg G'
hinj : Injective βf
u v : V
p p' : Walk G u v
h : Walk.map f p = Walk.map f p'
β’ p = p' | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβ vβ u' v' : V
p : Walk G uβ vβ
f : G βg G'
hinj : Injective βf
u v : V
β’ Injective (Walk.map f)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | cases p' | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβ u' v' : V
p : Walk G uβΒΉ vβ
f : G βg G'
hinj : Injective βf
u v uβ : V
p' : Walk G uβ uβ
h : Walk.map f nil = Walk.map f p'
β’ nil = p' | case nil.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβ u' v' : V
p : Walk G uβΒΉ vβ
f : G βg G'
hinj : Injective βf
u v uβ : V
h : Walk.map f nil = Walk.map f nil
β’ nil = nil
case nil.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ uβ
h : Walk.map f nil = Walk.map f (cons hβ pβ)
β’ nil = cons hβ pβ | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβ u' v' : V
p : Walk G uβΒΉ vβ
f : G βg G'
hinj : Injective βf
u v uβ : V
p' : Walk G uβ uβ
h : Walk.map f nil = Walk.map f p'
β’ nil = p'
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | rfl | case nil.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβ u' v' : V
p : Walk G uβΒΉ vβ
f : G βg G'
hinj : Injective βf
u v uβ : V
h : Walk.map f nil = Walk.map f nil
β’ nil = nil | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβ u' v' : V
p : Walk G uβΒΉ vβ
f : G βg G'
hinj : Injective βf
u v uβ : V
h : Walk.map f nil = Walk.map f nil
β’ nil = nil
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | simp at h | case nil.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ uβ
h : Walk.map f nil = Walk.map f (cons hβ pβ)
β’ nil = cons hβ pβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ uβ
h : Walk.map f nil = Walk.map f (cons hβ pβ)
β’ nil = cons hβ pβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | cases p' with
| nil => simp at h
| cons _ _ =>
simp only [map_cons, cons.injEq] at h
cases hinj h.1
simp only [cons.injEq, heq_iff_eq, true_and_iff]
apply ih
simpa using h.2 | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : β β¦p' : Walk G vβ wββ¦, Walk.map f pβ = Walk.map f p' β pβ = p'
p' : Walk G uβ wβ
h : Walk.map f (cons hβ pβ) = Walk.map f p'
β’ cons hβ pβ = p' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : β β¦p' : Walk G vβ wββ¦, Walk.map f pβ = Walk.map f p' β pβ = p'
p' : Walk G uβ wβ
h : Walk.map f (cons hβ pβ) = Walk.map f p'
β’ cons hβ pβ = p'
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | simp at h | case cons.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ uβ
ih : β β¦p' : Walk G vβ uββ¦, Walk.map f pβ = Walk.map f p' β pβ = p'
h : Walk.map f (cons hβ pβ) = Walk.map f nil
β’ cons hβ pβ = nil | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ uβ
ih : β β¦p' : Walk G vβ uββ¦, Walk.map f pβ = Walk.map f p' β pβ = p'
h : Walk.map f (cons hβ pβ) = Walk.map f nil
β’ cons hβ pβ = nil
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | simp only [map_cons, cons.injEq] at h | case cons.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒ² u' v' : V
p : Walk G uβΒΉ vβΒ²
f : G βg G'
hinj : Injective βf
u v uβ vβΒΉ wβ : V
hβΒΉ : G.Adj uβ vβΒΉ
pβΒΉ : Walk G vβΒΉ wβ
ih : β β¦p' : Walk G vβΒΉ wββ¦, Walk.map f pβΒΉ = Walk.map f p' β pβΒΉ = p'
vβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
h : Walk.map f (cons hβΒΉ pβΒΉ) = Walk.map f (cons hβ pβ)
β’ cons hβΒΉ pβΒΉ = cons hβ pβ | case cons.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒ² u' v' : V
p : Walk G uβΒΉ vβΒ²
f : G βg G'
hinj : Injective βf
u v uβ vβΒΉ wβ : V
hβΒΉ : G.Adj uβ vβΒΉ
pβΒΉ : Walk G vβΒΉ wβ
ih : β β¦p' : Walk G vβΒΉ wββ¦, Walk.map f pβΒΉ = Walk.map f p' β pβΒΉ = p'
vβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
h : βf vβΒΉ = βf vβ β§ HEq (Walk.map f pβΒΉ) (Walk.map f pβ)
β’ cons hβΒΉ pβΒΉ = cons hβ pβ | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒ² u' v' : V
p : Walk G uβΒΉ vβΒ²
f : G βg G'
hinj : Injective βf
u v uβ vβΒΉ wβ : V
hβΒΉ : G.Adj uβ vβΒΉ
pβΒΉ : Walk G vβΒΉ wβ
ih : β β¦p' : Walk G vβΒΉ wββ¦, Walk.map f pβΒΉ = Walk.map f p' β pβΒΉ = p'
vβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
h : Walk.map f (cons hβΒΉ pβΒΉ) = Walk.map f (cons hβ pβ)
β’ cons hβΒΉ pβΒΉ = cons hβ pβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | cases hinj h.1 | case cons.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒ² u' v' : V
p : Walk G uβΒΉ vβΒ²
f : G βg G'
hinj : Injective βf
u v uβ vβΒΉ wβ : V
hβΒΉ : G.Adj uβ vβΒΉ
pβΒΉ : Walk G vβΒΉ wβ
ih : β β¦p' : Walk G vβΒΉ wββ¦, Walk.map f pβΒΉ = Walk.map f p' β pβΒΉ = p'
vβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
h : βf vβΒΉ = βf vβ β§ HEq (Walk.map f pβΒΉ) (Walk.map f pβ)
β’ cons hβΒΉ pβΒΉ = cons hβ pβ | case cons.cons.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ wβ : V
hβΒΉ : G.Adj uβ vβ
pβΒΉ : Walk G vβ wβ
ih : β β¦p' : Walk G vβ wββ¦, Walk.map f pβΒΉ = Walk.map f p' β pβΒΉ = p'
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
h : βf vβ = βf vβ β§ HEq (Walk.map f pβΒΉ) (Walk.map f pβ)
β’ cons hβΒΉ pβΒΉ = cons hβ pβ | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒ² u' v' : V
p : Walk G uβΒΉ vβΒ²
f : G βg G'
hinj : Injective βf
u v uβ vβΒΉ wβ : V
hβΒΉ : G.Adj uβ vβΒΉ
pβΒΉ : Walk G vβΒΉ wβ
ih : β β¦p' : Walk G vβΒΉ wββ¦, Walk.map f pβΒΉ = Walk.map f p' β pβΒΉ = p'
vβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
h : βf vβΒΉ = βf vβ β§ HEq (Walk.map f pβΒΉ) (Walk.map f pβ)
β’ cons hβΒΉ pβΒΉ = cons hβ pβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | simp only [cons.injEq, heq_iff_eq, true_and_iff] | case cons.cons.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ wβ : V
hβΒΉ : G.Adj uβ vβ
pβΒΉ : Walk G vβ wβ
ih : β β¦p' : Walk G vβ wββ¦, Walk.map f pβΒΉ = Walk.map f p' β pβΒΉ = p'
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
h : βf vβ = βf vβ β§ HEq (Walk.map f pβΒΉ) (Walk.map f pβ)
β’ cons hβΒΉ pβΒΉ = cons hβ pβ | case cons.cons.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ wβ : V
hβΒΉ : G.Adj uβ vβ
pβΒΉ : Walk G vβ wβ
ih : β β¦p' : Walk G vβ wββ¦, Walk.map f pβΒΉ = Walk.map f p' β pβΒΉ = p'
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
h : βf vβ = βf vβ β§ HEq (Walk.map f pβΒΉ) (Walk.map f pβ)
β’ pβΒΉ = pβ | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ wβ : V
hβΒΉ : G.Adj uβ vβ
pβΒΉ : Walk G vβ wβ
ih : β β¦p' : Walk G vβ wββ¦, Walk.map f pβΒΉ = Walk.map f p' β pβΒΉ = p'
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
h : βf vβ = βf vβ β§ HEq (Walk.map f pβΒΉ) (Walk.map f pβ)
β’ cons hβΒΉ pβΒΉ = cons hβ pβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | apply ih | case cons.cons.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ wβ : V
hβΒΉ : G.Adj uβ vβ
pβΒΉ : Walk G vβ wβ
ih : β β¦p' : Walk G vβ wββ¦, Walk.map f pβΒΉ = Walk.map f p' β pβΒΉ = p'
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
h : βf vβ = βf vβ β§ HEq (Walk.map f pβΒΉ) (Walk.map f pβ)
β’ pβΒΉ = pβ | case cons.cons.refl.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ wβ : V
hβΒΉ : G.Adj uβ vβ
pβΒΉ : Walk G vβ wβ
ih : β β¦p' : Walk G vβ wββ¦, Walk.map f pβΒΉ = Walk.map f p' β pβΒΉ = p'
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
h : βf vβ = βf vβ β§ HEq (Walk.map f pβΒΉ) (Walk.map f pβ)
β’ Walk.map f pβΒΉ = Walk.map f pβ | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ wβ : V
hβΒΉ : G.Adj uβ vβ
pβΒΉ : Walk G vβ wβ
ih : β β¦p' : Walk G vβ wββ¦, Walk.map f pβΒΉ = Walk.map f p' β pβΒΉ = p'
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
h : βf vβ = βf vβ β§ HEq (Walk.map f pβΒΉ) (Walk.map f pβ)
β’ pβΒΉ = pβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_injective_of_injective | [1329, 1] | [1345, 22] | simpa using h.2 | case cons.cons.refl.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ wβ : V
hβΒΉ : G.Adj uβ vβ
pβΒΉ : Walk G vβ wβ
ih : β β¦p' : Walk G vβ wββ¦, Walk.map f pβΒΉ = Walk.map f p' β pβΒΉ = p'
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
h : βf vβ = βf vβ β§ HEq (Walk.map f pβΒΉ) (Walk.map f pβ)
β’ Walk.map f pβΒΉ = Walk.map f pβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons.refl.h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
fβ : G βg G'
f' : G' βg G''
uβΒΉ vβΒΉ u' v' : V
p : Walk G uβΒΉ vβΒΉ
f : G βg G'
hinj : Injective βf
u v uβ vβ wβ : V
hβΒΉ : G.Adj uβ vβ
pβΒΉ : Walk G vβ wβ
ih : β β¦p' : Walk G vβ wββ¦, Walk.map f pβΒΉ = Walk.map f p' β pβΒΉ = p'
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
h : βf vβ = βf vβ β§ HEq (Walk.map f pβΒΉ) (Walk.map f pβ)
β’ Walk.map f pβΒΉ = Walk.map f pβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Path.map_injective | [1397, 1] | [1401, 52] | rintro β¨p, hpβ© β¨p', hp'β© h | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
hinj : Injective βf
u v : V
β’ Injective (Path.map f hinj) | case mk.mk
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
hinj : Injective βf
u v : V
p : Walk G u v
hp : Walk.IsPath p
p' : Walk G u v
hp' : Walk.IsPath p'
h : Path.map f hinj { val := p, property := hp } = Path.map f hinj { val := p', property := hp' }
β’ { val := p, property := hp } = { val := p', property := hp' } | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
hinj : Injective βf
u v : V
β’ Injective (Path.map f hinj)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Path.map_injective | [1397, 1] | [1401, 52] | simp only [Path.map, Subtype.coe_mk, Subtype.mk.injEq] at h | case mk.mk
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
hinj : Injective βf
u v : V
p : Walk G u v
hp : Walk.IsPath p
p' : Walk G u v
hp' : Walk.IsPath p'
h : Path.map f hinj { val := p, property := hp } = Path.map f hinj { val := p', property := hp' }
β’ { val := p, property := hp } = { val := p', property := hp' } | case mk.mk
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
hinj : Injective βf
u v : V
p : Walk G u v
hp : Walk.IsPath p
p' : Walk G u v
hp' : Walk.IsPath p'
h : Walk.map f p = Walk.map f p'
β’ { val := p, property := hp } = { val := p', property := hp' } | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
hinj : Injective βf
u v : V
p : Walk G u v
hp : Walk.IsPath p
p' : Walk G u v
hp' : Walk.IsPath p'
h : Path.map f hinj { val := p, property := hp } = Path.map f hinj { val := p', property := hp' }
β’ { val := p, property := hp } = { val := p', property := hp' }
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Path.map_injective | [1397, 1] | [1401, 52] | simp [Walk.map_injective_of_injective hinj u v h] | case mk.mk
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
hinj : Injective βf
u v : V
p : Walk G u v
hp : Walk.IsPath p
p' : Walk G u v
hp' : Walk.IsPath p'
h : Walk.map f p = Walk.map f p'
β’ { val := p, property := hp } = { val := p', property := hp' } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
f : G βg G'
hinj : Injective βf
u v : V
p : Walk G u v
hp : Walk.IsPath p
p' : Walk G u v
hp' : Walk.IsPath p'
h : Walk.map f p = Walk.map f p'
β’ { val := p, property := hp } = { val := p', property := hp' }
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_self | [1435, 1] | [1436, 27] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
β’ Walk.transfer p G (_ : β β¦e : V Γ Vβ¦, e β edges p β e β edgeSet G) = p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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
β’ Walk.transfer p G (_ : β β¦e : V Γ Vβ¦, e β edges p β e β edgeSet G) = p
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_eq_map_of_le | [1439, 1] | [1441, 27] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : β (e : V Γ V), e β edges p β e β edgeSet H
GH : G β€ H
β’ Walk.transfer p H hp = Walk.map (Hom.mapSpanningSubgraphs GH) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : β (e : V Γ V), e β edges p β e β edgeSet H
GH : G β€ H
β’ Walk.transfer p H hp = Walk.map (Hom.mapSpanningSubgraphs GH) p
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.edges_transfer | [1445, 1] | [1446, 27] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : β (e : V Γ V), e β edges p β e β edgeSet H
β’ edges (Walk.transfer p H hp) = edges p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : β (e : V Γ V), e β edges p β e β edgeSet H
β’ edges (Walk.transfer p H hp) = edges p
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.support_transfer | [1450, 1] | [1451, 27] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : β (e : V Γ V), e β edges p β e β edgeSet H
β’ support (Walk.transfer p H hp) = support p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : β (e : V Γ V), e β edges p β e β edgeSet H
β’ support (Walk.transfer p H hp) = support p
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.length_transfer | [1455, 1] | [1456, 27] | induction p <;> simp [*] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : β (e : V Γ V), e β edges p β e β edgeSet H
β’ length (Walk.transfer p H hp) = length p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : β (e : V Γ V), e β edges p β e β edgeSet H
β’ length (Walk.transfer p H hp) = length p
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.transfer | [1461, 11] | [1467, 28] | induction p with
| nil => simp
| cons _ _ ih =>
simp only [Walk.transfer, cons_isPath_iff, support_transfer _ ] at pp β’
exact β¨ih _ pp.1, pp.2β© | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : β (e : V Γ V), e β edges p β e β edgeSet H
pp : IsPath p
β’ IsPath (Walk.transfer p H hp) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : β (e : V Γ V), e β edges p β e β edgeSet H
pp : IsPath p
β’ IsPath (Walk.transfer p H hp)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.transfer | [1461, 11] | [1467, 28] | simp | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
uβ : V
hp : β (e : V Γ V), e β edges Walk.nil β e β edgeSet H
pp : IsPath Walk.nil
β’ IsPath (Walk.transfer Walk.nil H hp) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
uβ : V
hp : β (e : V Γ V), e β edges Walk.nil β e β edgeSet H
pp : IsPath Walk.nil
β’ IsPath (Walk.transfer Walk.nil H hp)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.transfer | [1461, 11] | [1467, 28] | simp only [Walk.transfer, cons_isPath_iff, support_transfer _ ] at pp β’ | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : β (hp : β (e : V Γ V), e β edges pβ β e β edgeSet H), IsPath pβ β IsPath (Walk.transfer pβ H hp)
hp : β (e : V Γ V), e β edges (cons hβ pβ) β e β edgeSet H
pp : IsPath (cons hβ pβ)
β’ IsPath (Walk.transfer (cons hβ pβ) H hp) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : β (hp : β (e : V Γ V), e β edges pβ β e β edgeSet H), IsPath pβ β IsPath (Walk.transfer pβ H hp)
hp : β (e : V Γ V), e β edges (cons hβ pβ) β e β edgeSet H
pp : IsPath pβ β§ Β¬uβ β support pβ
β’ IsPath (Walk.transfer pβ H (_ : β (e : V Γ V), e β edges pβ β e β edgeSet H)) β§ Β¬uβ β support pβ | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : β (hp : β (e : V Γ V), e β edges pβ β e β edgeSet H), IsPath pβ β IsPath (Walk.transfer pβ H hp)
hp : β (e : V Γ V), e β edges (cons hβ pβ) β e β edgeSet H
pp : IsPath (cons hβ pβ)
β’ IsPath (Walk.transfer (cons hβ pβ) H hp)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsPath.transfer | [1461, 11] | [1467, 28] | exact β¨ih _ pp.1, pp.2β© | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : β (hp : β (e : V Γ V), e β edges pβ β e β edgeSet H), IsPath pβ β IsPath (Walk.transfer pβ H hp)
hp : β (e : V Γ V), e β edges (cons hβ pβ) β e β edgeSet H
pp : IsPath pβ β§ Β¬uβ β support pβ
β’ IsPath (Walk.transfer pβ H (_ : β (e : V Γ V), e β edges pβ β e β edgeSet H)) β§ Β¬uβ β support pβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : β (hp : β (e : V Γ V), e β edges pβ β e β edgeSet H), IsPath pβ β IsPath (Walk.transfer pβ H hp)
hp : β (e : V Γ V), e β edges (cons hβ pβ) β e β edgeSet H
pp : IsPath pβ β§ Β¬uβ β support pβ
β’ IsPath (Walk.transfer pβ H (_ : β (e : V Γ V), e β edges pβ β e β edgeSet H)) β§ Β¬uβ β support pβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCycle.transfer | [1470, 11] | [1477, 37] | cases q with
| nil => simp at qc
| cons _ q =>
simp only [edges_cons, List.find?, List.mem_cons, forall_eq_or_imp, mem_edgeSet] at hq
simp only [Walk.transfer, cons_isCycle_iff, edges_transfer q hq.2] at qc β’
exact β¨qc.1.transfer hq.2, qc.2β© | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
q : Walk G u u
qc : IsCycle q
hq : β (e : V Γ V), e β edges q β e β edgeSet H
β’ IsCycle (Walk.transfer q H hq) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
q : Walk G u u
qc : IsCycle q
hq : β (e : V Γ V), e β edges q β e β edgeSet H
β’ IsCycle (Walk.transfer q H hq)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCycle.transfer | [1470, 11] | [1477, 37] | simp at qc | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
qc : IsCycle nil
hq : β (e : V Γ V), e β edges nil β e β edgeSet H
β’ IsCycle (Walk.transfer nil H hq) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
qc : IsCycle nil
hq : β (e : V Γ V), e β edges nil β e β edgeSet H
β’ IsCycle (Walk.transfer nil H hq)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCycle.transfer | [1470, 11] | [1477, 37] | simp only [edges_cons, List.find?, List.mem_cons, forall_eq_or_imp, mem_edgeSet] at hq | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
vβ : V
hβ : G.Adj u vβ
q : Walk G vβ u
qc : IsCycle (cons hβ q)
hq : β (e : V Γ V), e β edges (cons hβ q) β e β edgeSet H
β’ IsCycle (Walk.transfer (cons hβ q) H hq) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
vβ : V
hβ : G.Adj u vβ
q : Walk G vβ u
qc : IsCycle (cons hβ q)
hqβ : β (e : V Γ V), e β edges (cons hβ q) β e β edgeSet H
hq : H.Adj u vβ β§ β (a : V Γ V), a β edges q β a β edgeSet H
β’ IsCycle (Walk.transfer (cons hβ q) H hqβ) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
vβ : V
hβ : G.Adj u vβ
q : Walk G vβ u
qc : IsCycle (cons hβ q)
hq : β (e : V Γ V), e β edges (cons hβ q) β e β edgeSet H
β’ IsCycle (Walk.transfer (cons hβ q) H hq)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCycle.transfer | [1470, 11] | [1477, 37] | simp only [Walk.transfer, cons_isCycle_iff, edges_transfer q hq.2] at qc β’ | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
vβ : V
hβ : G.Adj u vβ
q : Walk G vβ u
qc : IsCycle (cons hβ q)
hqβ : β (e : V Γ V), e β edges (cons hβ q) β e β edgeSet H
hq : H.Adj u vβ β§ β (a : V Γ V), a β edges q β a β edgeSet H
β’ IsCycle (Walk.transfer (cons hβ q) H hqβ) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
vβ : V
hβ : G.Adj u vβ
q : Walk G vβ u
hqβ : β (e : V Γ V), e β edges (cons hβ q) β e β edgeSet H
hq : H.Adj u vβ β§ β (a : V Γ V), a β edges q β a β edgeSet H
qc : IsPath q β§ Β¬(u, vβ) β edges q
β’ IsPath (Walk.transfer q H (_ : β (e : V Γ V), e β edges q β e β edgeSet H)) β§ Β¬(u, vβ) β edges q | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
vβ : V
hβ : G.Adj u vβ
q : Walk G vβ u
qc : IsCycle (cons hβ q)
hqβ : β (e : V Γ V), e β edges (cons hβ q) β e β edgeSet H
hq : H.Adj u vβ β§ β (a : V Γ V), a β edges q β a β edgeSet H
β’ IsCycle (Walk.transfer (cons hβ q) H hqβ)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.IsCycle.transfer | [1470, 11] | [1477, 37] | exact β¨qc.1.transfer hq.2, qc.2β© | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
vβ : V
hβ : G.Adj u vβ
q : Walk G vβ u
hqβ : β (e : V Γ V), e β edges (cons hβ q) β e β edgeSet H
hq : H.Adj u vβ β§ β (a : V Γ V), a β edges q β a β edgeSet H
qc : IsPath q β§ Β¬(u, vβ) β edges q
β’ IsPath (Walk.transfer q H (_ : β (e : V Γ V), e β edges q β e β edgeSet H)) β§ Β¬(u, vβ) β edges q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
vβ : V
hβ : G.Adj u vβ
q : Walk G vβ u
hqβ : β (e : V Γ V), e β edges (cons hβ q) β e β edgeSet H
hq : H.Adj u vβ β§ β (a : V Γ V), a β edges q β a β edgeSet H
qc : IsPath q β§ Β¬(u, vβ) β edges q
β’ IsPath (Walk.transfer q H (_ : β (e : V Γ V), e β edges q β e β edgeSet H)) β§ Β¬(u, vβ) β edges q
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_transfer | [1486, 1] | [1492, 13] | induction p with
| nil => simp
| cons _ _ ih =>
simp only [Walk.transfer, cons.injEq, heq_eq_eq, true_and]
apply ih | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : β (e : V Γ V), e β edges p β e β edgeSet H
K : Digraph V
hp' : β (e : V Γ V), e β edges (Walk.transfer p H hp) β e β edgeSet K
β’ Walk.transfer (Walk.transfer p H hp) K hp' = Walk.transfer p K (_ : β (e : V Γ V), e β edges p β e β edgeSet K) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H : Digraph V
hp : β (e : V Γ V), e β edges p β e β edgeSet H
K : Digraph V
hp' : β (e : V Γ V), e β edges (Walk.transfer p H hp) β e β edgeSet K
β’ Walk.transfer (Walk.transfer p H hp) K hp' = Walk.transfer p K (_ : β (e : V Γ V), e β edges p β e β edgeSet K)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_transfer | [1486, 1] | [1492, 13] | simp | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H K : Digraph V
uβ : V
hp : β (e : V Γ V), e β edges nil β e β edgeSet H
hp' : β (e : V Γ V), e β edges (Walk.transfer nil H hp) β e β edgeSet K
β’ Walk.transfer (Walk.transfer nil H hp) K hp' = Walk.transfer nil K (_ : β (e : V Γ V), e β edges nil β e β edgeSet K) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H K : Digraph V
uβ : V
hp : β (e : V Γ V), e β edges nil β e β edgeSet H
hp' : β (e : V Γ V), e β edges (Walk.transfer nil H hp) β e β edgeSet K
β’ Walk.transfer (Walk.transfer nil H hp) K hp' = Walk.transfer nil K (_ : β (e : V Γ V), e β edges nil β e β edgeSet K)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_transfer | [1486, 1] | [1492, 13] | simp only [Walk.transfer, cons.injEq, heq_eq_eq, true_and] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H K : Digraph V
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih :
β (hp : β (e : V Γ V), e β edges pβ β e β edgeSet H)
(hp' : β (e : V Γ V), e β edges (Walk.transfer pβ H hp) β e β edgeSet K),
Walk.transfer (Walk.transfer pβ H hp) K hp' = Walk.transfer pβ K (_ : β (e : V Γ V), e β edges pβ β e β edgeSet K)
hp : β (e : V Γ V), e β edges (cons hβ pβ) β e β edgeSet H
hp' : β (e : V Γ V), e β edges (Walk.transfer (cons hβ pβ) H hp) β e β edgeSet K
β’ Walk.transfer (Walk.transfer (cons hβ pβ) H hp) K hp' =
Walk.transfer (cons hβ pβ) K (_ : β (e : V Γ V), e β edges (cons hβ pβ) β e β edgeSet K) | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H K : Digraph V
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih :
β (hp : β (e : V Γ V), e β edges pβ β e β edgeSet H)
(hp' : β (e : V Γ V), e β edges (Walk.transfer pβ H hp) β e β edgeSet K),
Walk.transfer (Walk.transfer pβ H hp) K hp' = Walk.transfer pβ K (_ : β (e : V Γ V), e β edges pβ β e β edgeSet K)
hp : β (e : V Γ V), e β edges (cons hβ pβ) β e β edgeSet H
hp' : β (e : V Γ V), e β edges (Walk.transfer (cons hβ pβ) H hp) β e β edgeSet K
β’ Walk.transfer (Walk.transfer pβ H (_ : β (e : V Γ V), e β edges pβ β e β edgeSet H)) K
(_ :
β (e : V Γ V),
e β edges (Walk.transfer pβ H (_ : β (e : V Γ V), e β edges pβ β e β edgeSet H)) β e β edgeSet K) =
Walk.transfer pβ K (_ : β (e : V Γ V), e β edges pβ β e β edgeSet K) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H K : Digraph V
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih :
β (hp : β (e : V Γ V), e β edges pβ β e β edgeSet H)
(hp' : β (e : V Γ V), e β edges (Walk.transfer pβ H hp) β e β edgeSet K),
Walk.transfer (Walk.transfer pβ H hp) K hp' = Walk.transfer pβ K (_ : β (e : V Γ V), e β edges pβ β e β edgeSet K)
hp : β (e : V Γ V), e β edges (cons hβ pβ) β e β edgeSet H
hp' : β (e : V Γ V), e β edges (Walk.transfer (cons hβ pβ) H hp) β e β edgeSet K
β’ Walk.transfer (Walk.transfer (cons hβ pβ) H hp) K hp' =
Walk.transfer (cons hβ pβ) K (_ : β (e : V Γ V), e β edges (cons hβ pβ) β e β edgeSet K)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_transfer | [1486, 1] | [1492, 13] | apply ih | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H K : Digraph V
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih :
β (hp : β (e : V Γ V), e β edges pβ β e β edgeSet H)
(hp' : β (e : V Γ V), e β edges (Walk.transfer pβ H hp) β e β edgeSet K),
Walk.transfer (Walk.transfer pβ H hp) K hp' = Walk.transfer pβ K (_ : β (e : V Γ V), e β edges pβ β e β edgeSet K)
hp : β (e : V Γ V), e β edges (cons hβ pβ) β e β edgeSet H
hp' : β (e : V Γ V), e β edges (Walk.transfer (cons hβ pβ) H hp) β e β edgeSet K
β’ Walk.transfer (Walk.transfer pβ H (_ : β (e : V Γ V), e β edges pβ β e β edgeSet H)) K
(_ :
β (e : V Γ V),
e β edges (Walk.transfer pβ H (_ : β (e : V Γ V), e β edges pβ β e β edgeSet H)) β e β edgeSet K) =
Walk.transfer pβ K (_ : β (e : V Γ V), e β edges pβ β e β edgeSet K) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
H K : Digraph V
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih :
β (hp : β (e : V Γ V), e β edges pβ β e β edgeSet H)
(hp' : β (e : V Γ V), e β edges (Walk.transfer pβ H hp) β e β edgeSet K),
Walk.transfer (Walk.transfer pβ H hp) K hp' = Walk.transfer pβ K (_ : β (e : V Γ V), e β edges pβ β e β edgeSet K)
hp : β (e : V Γ V), e β edges (cons hβ pβ) β e β edgeSet H
hp' : β (e : V Γ V), e β edges (Walk.transfer (cons hβ pβ) H hp) β e β edgeSet K
β’ Walk.transfer (Walk.transfer pβ H (_ : β (e : V Γ V), e β edges pβ β e β edgeSet H)) K
(_ :
β (e : V Γ V),
e β edges (Walk.transfer pβ H (_ : β (e : V Γ V), e β edges pβ β e β edgeSet H)) β e β edgeSet K) =
Walk.transfer pβ K (_ : β (e : V Γ V), e β edges pβ β e β edgeSet K)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_append | [1496, 1] | [1502, 95] | simp [he] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
w : V
H : Digraph V
q : Walk G v w
hpq : β (e : V Γ V), e β edges (append p q) β e β edgeSet H
e : V Γ V
he : e β edges p
β’ e β edges (append p q) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
w : V
H : Digraph V
q : Walk G v w
hpq : β (e : V Γ V), e β edges (append p q) β e β edgeSet H
e : V Γ V
he : e β edges p
β’ e β edges (append p q)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_append | [1496, 1] | [1502, 95] | simp [he] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
w : V
H : Digraph V
q : Walk G v w
hpq : β (e : V Γ V), e β edges (append p q) β e β edgeSet H
e : V Γ V
he : e β edges q
β’ e β edges (append p q) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
w : V
H : Digraph V
q : Walk G v w
hpq : β (e : V Γ V), e β edges (append p q) β e β edgeSet H
e : V Γ V
he : e β edges q
β’ e β edges (append p q)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_append | [1496, 1] | [1502, 95] | induction p with
| nil => simp
| cons _ _ ih => simp only [Walk.transfer, cons_append, cons.injEq, heq_eq_eq, true_and, ih] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
w : V
H : Digraph V
q : Walk G v w
hpq : β (e : V Γ V), e β edges (append p q) β e β edgeSet H
β’ Walk.transfer (append p q) H hpq =
append (Walk.transfer p H (_ : β (e : V Γ V), e β edges p β e β edgeSet H))
(Walk.transfer q H (_ : β (e : V Γ V), e β edges q β e β edgeSet H)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
w : V
H : Digraph V
q : Walk G v w
hpq : β (e : V Γ V), e β edges (append p q) β e β edgeSet H
β’ Walk.transfer (append p q) H hpq =
append (Walk.transfer p H (_ : β (e : V Γ V), e β edges p β e β edgeSet H))
(Walk.transfer q H (_ : β (e : V Γ V), e β edges q β e β edgeSet H))
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_append | [1496, 1] | [1502, 95] | simp | case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
w : V
H : Digraph V
uβ : V
q : Walk G uβ w
hpq : β (e : V Γ V), e β edges (append nil q) β e β edgeSet H
β’ Walk.transfer (append nil q) H hpq =
append (Walk.transfer nil H (_ : β (e : V Γ V), e β edges nil β e β edgeSet H))
(Walk.transfer q H (_ : β (e : V Γ V), e β edges q β e β edgeSet H)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
w : V
H : Digraph V
uβ : V
q : Walk G uβ w
hpq : β (e : V Γ V), e β edges (append nil q) β e β edgeSet H
β’ Walk.transfer (append nil q) H hpq =
append (Walk.transfer nil H (_ : β (e : V Γ V), e β edges nil β e β edgeSet H))
(Walk.transfer q H (_ : β (e : V Γ V), e β edges q β e β edgeSet H))
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.transfer_append | [1496, 1] | [1502, 95] | simp only [Walk.transfer, cons_append, cons.injEq, heq_eq_eq, true_and, ih] | case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
w : V
H : Digraph V
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih :
β (q : Walk G wβ w) (hpq : β (e : V Γ V), e β edges (append pβ q) β e β edgeSet H),
Walk.transfer (append pβ q) H hpq =
append (Walk.transfer pβ H (_ : β (e : V Γ V), e β edges pβ β e β edgeSet H))
(Walk.transfer q H (_ : β (e : V Γ V), e β edges q β e β edgeSet H))
q : Walk G wβ w
hpq : β (e : V Γ V), e β edges (append (cons hβ pβ) q) β e β edgeSet H
β’ Walk.transfer (append (cons hβ pβ) q) H hpq =
append (Walk.transfer (cons hβ pβ) H (_ : β (e : V Γ V), e β edges (cons hβ pβ) β e β edgeSet H))
(Walk.transfer q H (_ : β (e : V Γ V), e β edges q β e β edgeSet H)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
p : Walk G u v
w : V
H : Digraph V
uβ vβ wβ : V
hβ : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih :
β (q : Walk G wβ w) (hpq : β (e : V Γ V), e β edges (append pβ q) β e β edgeSet H),
Walk.transfer (append pβ q) H hpq =
append (Walk.transfer pβ H (_ : β (e : V Γ V), e β edges pβ β e β edgeSet H))
(Walk.transfer q H (_ : β (e : V Γ V), e β edges q β e β edgeSet H))
q : Walk G wβ w
hpq : β (e : V Γ V), e β edges (append (cons hβ pβ) q) β e β edgeSet H
β’ Walk.transfer (append (cons hβ pβ) q) H hpq =
append (Walk.transfer (cons hβ pβ) H (_ : β (e : V Γ V), e β edges (cons hβ pβ) β e β edgeSet H))
(Walk.transfer q H (_ : β (e : V Γ V), e β edges q β e β edgeSet H))
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_toDeleteEdges_eq | [1541, 1] | [1546, 33] | rw [β transfer_eq_map_of_le, transfer_transfer, transfer_self] | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V Γ V)
p : Walk G v w
hp : β (e : V Γ V), e β edges p β Β¬e β s
β’ Walk.map (Hom.mapSpanningSubgraphs (_ : deleteEdges G s β€ G)) (toDeleteEdges s p hp) = p | case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V Γ V)
p : Walk G v w
hp : β (e : V Γ V), e β edges p β Β¬e β s
β’ β (e : V Γ V),
e β edges (Walk.transfer p (deleteEdges G s) (_ : β (e : V Γ V), e β edges p β e β edgeSet (deleteEdges G s))) β
e β edgeSet G
case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V Γ V)
p : Walk G v w
hp : β (e : V Γ V), e β edges p β Β¬e β s
β’ β (e : V Γ V),
e β edges (Walk.transfer p (deleteEdges G s) (_ : β (e : V Γ V), e β edges p β e β edgeSet (deleteEdges G s))) β
e β edgeSet G | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V Γ V)
p : Walk G v w
hp : β (e : V Γ V), e β edges p β Β¬e β s
β’ Walk.map (Hom.mapSpanningSubgraphs (_ : deleteEdges G s β€ G)) (toDeleteEdges s p hp) = p
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_toDeleteEdges_eq | [1541, 1] | [1546, 33] | intros e | case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V Γ V)
p : Walk G v w
hp : β (e : V Γ V), e β edges p β Β¬e β s
β’ β (e : V Γ V),
e β edges (Walk.transfer p (deleteEdges G s) (_ : β (e : V Γ V), e β edges p β e β edgeSet (deleteEdges G s))) β
e β edgeSet G | case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V Γ V)
p : Walk G v w
hp : β (e : V Γ V), e β edges p β Β¬e β s
e : V Γ V
β’ e β edges (Walk.transfer p (deleteEdges G s) (_ : β (e : V Γ V), e β edges p β e β edgeSet (deleteEdges G s))) β
e β edgeSet G | Please generate a tactic in lean4 to solve the state.
STATE:
case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V Γ V)
p : Walk G v w
hp : β (e : V Γ V), e β edges p β Β¬e β s
β’ β (e : V Γ V),
e β edges (Walk.transfer p (deleteEdges G s) (_ : β (e : V Γ V), e β edges p β e β edgeSet (deleteEdges G s))) β
e β edgeSet G
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_toDeleteEdges_eq | [1541, 1] | [1546, 33] | rw [edges_transfer] | case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V Γ V)
p : Walk G v w
hp : β (e : V Γ V), e β edges p β Β¬e β s
e : V Γ V
β’ e β edges (Walk.transfer p (deleteEdges G s) (_ : β (e : V Γ V), e β edges p β e β edgeSet (deleteEdges G s))) β
e β edgeSet G | case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V Γ V)
p : Walk G v w
hp : β (e : V Γ V), e β edges p β Β¬e β s
e : V Γ V
β’ e β edges p β e β edgeSet G | Please generate a tactic in lean4 to solve the state.
STATE:
case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V Γ V)
p : Walk G v w
hp : β (e : V Γ V), e β edges p β Β¬e β s
e : V Γ V
β’ e β edges (Walk.transfer p (deleteEdges G s) (_ : β (e : V Γ V), e β edges p β e β edgeSet (deleteEdges G s))) β
e β edgeSet G
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.map_toDeleteEdges_eq | [1541, 1] | [1546, 33] | apply edges_subset_edgeSet p | case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V Γ V)
p : Walk G v w
hp : β (e : V Γ V), e β edges p β Β¬e β s
e : V Γ V
β’ e β edges p β e β edgeSet G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hp'
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
v w : V
s : Set (V Γ V)
p : Walk G v w
hp : β (e : V Γ V), e β edges p β Β¬e β s
e : V Γ V
β’ e β edges p β e β edgeSet G
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.toDeleteEdges_copy | [1557, 1] | [1562, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u' v' : V
s : Set (V Γ V)
p : Walk G u v
hu : u = u'
hv : v = v'
h : β (e : V Γ V), e β edges (Walk.copy p hu hv) β Β¬e β s
β’ β (e : V Γ V), e β edges p β Β¬e β s | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' : V
s : Set (V Γ V)
p : Walk G u' v'
h : β (e : V Γ V), e β edges (Walk.copy p (_ : u' = u') (_ : v' = v')) β Β¬e β s
β’ β (e : V Γ V), e β edges p β Β¬e β s | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u' v' : V
s : Set (V Γ V)
p : Walk G u v
hu : u = u'
hv : v = v'
h : β (e : V Γ V), e β edges (Walk.copy p hu hv) β Β¬e β s
β’ β (e : V Γ V), e β edges p β Β¬e β s
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.toDeleteEdges_copy | [1557, 1] | [1562, 6] | exact h | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' : V
s : Set (V Γ V)
p : Walk G u' v'
h : β (e : V Γ V), e β edges (Walk.copy p (_ : u' = u') (_ : v' = v')) β Β¬e β s
β’ β (e : V Γ V), e β edges p β Β¬e β s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' : V
s : Set (V Γ V)
p : Walk G u' v'
h : β (e : V Γ V), e β edges (Walk.copy p (_ : u' = u') (_ : v' = v')) β Β¬e β s
β’ β (e : V Γ V), e β edges p β Β¬e β s
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.toDeleteEdges_copy | [1557, 1] | [1562, 6] | subst_vars | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u' v' : V
s : Set (V Γ V)
p : Walk G u v
hu : u = u'
hv : v = v'
h : β (e : V Γ V), e β edges (Walk.copy p hu hv) β Β¬e β s
β’ toDeleteEdges s (Walk.copy p hu hv) h = Walk.copy (toDeleteEdges s p (_ : β (e : V Γ V), e β edges p β Β¬e β s)) hu hv | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' : V
s : Set (V Γ V)
p : Walk G u' v'
h : β (e : V Γ V), e β edges (Walk.copy p (_ : u' = u') (_ : v' = v')) β Β¬e β s
β’ toDeleteEdges s (Walk.copy p (_ : u' = u') (_ : v' = v')) h =
Walk.copy (toDeleteEdges s p (_ : β (e : V Γ V), e β edges p β Β¬e β s)) (_ : u' = u') (_ : v' = v') | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v u' v' : V
s : Set (V Γ V)
p : Walk G u v
hu : u = u'
hv : v = v'
h : β (e : V Γ V), e β edges (Walk.copy p hu hv) β Β¬e β s
β’ toDeleteEdges s (Walk.copy p hu hv) h = Walk.copy (toDeleteEdges s p (_ : β (e : V Γ V), e β edges p β Β¬e β s)) hu hv
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Walk.toDeleteEdges_copy | [1557, 1] | [1562, 6] | rfl | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' : V
s : Set (V Γ V)
p : Walk G u' v'
h : β (e : V Γ V), e β edges (Walk.copy p (_ : u' = u') (_ : v' = v')) β Β¬e β s
β’ toDeleteEdges s (Walk.copy p (_ : u' = u') (_ : v' = v')) h =
Walk.copy (toDeleteEdges s p (_ : β (e : V Γ V), e β edges p β Β¬e β s)) (_ : u' = u') (_ : v' = v') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u' v' : V
s : Set (V Γ V)
p : Walk G u' v'
h : β (e : V Γ V), e β edges (Walk.copy p (_ : u' = u') (_ : v' = v')) β Β¬e β s
β’ toDeleteEdges s (Walk.copy p (_ : u' = u') (_ : v' = v')) h =
Walk.copy (toDeleteEdges s p (_ : β (e : V Γ V), e β edges p β Β¬e β s)) (_ : u' = u') (_ : v' = v')
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Reachable.elim_path | [1587, 11] | [1588, 80] | classical exact h.elim fun q => hp q.toPath | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
p : Prop
u v : V
h : Reachable G u v
hp : Path G u v β p
β’ p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
p : Prop
u v : V
h : Reachable G u v
hp : Path G u v β p
β’ p
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Reachable.elim_path | [1587, 11] | [1588, 80] | exact h.elim fun q => hp q.toPath | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
p : Prop
u v : V
h : Reachable G u v
hp : Path G u v β p
β’ p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
p : Prop
u v : V
h : Reachable G u v
hp : Path G u v β p
β’ p
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | constructor | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
β’ Reachable G u v β Relation.ReflTransGen G.Adj u v | case mp
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
β’ Reachable G u v β Relation.ReflTransGen G.Adj u v
case mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
β’ Relation.ReflTransGen G.Adj u v β Reachable G u v | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
β’ Reachable G u v β Relation.ReflTransGen G.Adj u v
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | rintro β¨hβ© | case mp
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
β’ Reachable G u v β Relation.ReflTransGen G.Adj u v | case mp.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : Walk G u v
β’ Relation.ReflTransGen G.Adj u v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
β’ Reachable G u v β Relation.ReflTransGen G.Adj u v
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | induction h with
| nil => rfl
| cons h' _ ih => exact (Relation.ReflTransGen.single h').trans ih | case mp.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : Walk G u v
β’ Relation.ReflTransGen G.Adj u v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : Walk G u v
β’ Relation.ReflTransGen G.Adj u v
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | rfl | case mp.intro.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v uβ : V
β’ Relation.ReflTransGen G.Adj uβ uβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v uβ : V
β’ Relation.ReflTransGen G.Adj uβ uβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | exact (Relation.ReflTransGen.single h').trans ih | case mp.intro.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v uβ vβ wβ : V
h' : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : Relation.ReflTransGen G.Adj vβ wβ
β’ Relation.ReflTransGen G.Adj uβ wβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v uβ vβ wβ : V
h' : G.Adj uβ vβ
pβ : Walk G vβ wβ
ih : Relation.ReflTransGen G.Adj vβ wβ
β’ Relation.ReflTransGen G.Adj uβ wβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | intro h | case mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
β’ Relation.ReflTransGen G.Adj u v β Reachable G u v | case mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : Relation.ReflTransGen G.Adj u v
β’ Reachable G u v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
β’ Relation.ReflTransGen G.Adj u v β Reachable G u v
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | induction h with
| refl => rfl
| tail _ ha hr => exact Reachable.trans hr β¨Walk.cons ha Walk.nilβ© | case mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : Relation.ReflTransGen G.Adj u v
β’ Reachable G u v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : Relation.ReflTransGen G.Adj u v
β’ Reachable G u v
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | rfl | case mpr.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
β’ Reachable G u u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
β’ Reachable G u u
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.reachable_iff_reflTransGen | [1621, 1] | [1631, 71] | exact Reachable.trans hr β¨Walk.cons ha Walk.nilβ© | case mpr.tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v bβ cβ : V
aβ : Relation.ReflTransGen G.Adj u bβ
ha : G.Adj bβ cβ
hr : Reachable G u bβ
β’ Reachable G u cβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.tail
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v bβ cβ : V
aβ : Relation.ReflTransGen G.Adj u bβ
ha : G.Adj bβ cβ
hr : Reachable G u bβ
β’ Reachable G u cβ
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.Iso.symm_apply_reachable | [1644, 1] | [1646, 52] | rw [β Iso.reachable_iff, RelIso.apply_symm_apply] | V : Type u
V' : Type v
V'' : Type w
Gβ : Digraph V
G'β : Digraph V'
G'' : Digraph V''
G : Digraph V
G' : Digraph V'
Ο : G βg G'
u : V
v : V'
β’ Reachable G (β(symm Ο) v) u β Reachable G' v (βΟ u) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
Gβ : Digraph V
G'β : Digraph V'
G'' : Digraph V''
G : Digraph V
G' : Digraph V'
Ο : G βg G'
u : V
v : V'
β’ Reachable G (β(symm Ο) v) u β Reachable G' v (βΟ u)
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_self_length_zero_eq | [2030, 1] | [2032, 7] | ext p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
β’ {p | Walk.length p = 0} = {Walk.nil} | case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
p : Walk G u u
β’ p β {p | Walk.length p = 0} β p β {Walk.nil} | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
β’ {p | Walk.length p = 0} = {Walk.nil}
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_self_length_zero_eq | [2030, 1] | [2032, 7] | simp | case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
p : Walk G u u
β’ p β {p | Walk.length p = 0} β p β {Walk.nil} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
p : Walk G u u
β’ p β {p | Walk.length p = 0} β p β {Walk.nil}
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_zero_eq_of_ne | [2035, 1] | [2039, 58] | ext p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u β v
β’ {p | Walk.length p = 0} = β
| case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u β v
p : Walk G u v
β’ p β {p | Walk.length p = 0} β p β β
| Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u β v
β’ {p | Walk.length p = 0} = β
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_zero_eq_of_ne | [2035, 1] | [2039, 58] | simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff] | case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u β v
p : Walk G u v
β’ p β {p | Walk.length p = 0} β p β β
| case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u β v
p : Walk G u v
β’ Β¬Walk.length p = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u β v
p : Walk G u v
β’ p β {p | Walk.length p = 0} β p β β
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_zero_eq_of_ne | [2035, 1] | [2039, 58] | exact fun h' => absurd (Walk.eq_of_length_eq_zero h') h | case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u β v
p : Walk G u v
β’ Β¬Walk.length p = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
h : u β v
p : Walk G u v
β’ Β¬Walk.length p = 0
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_succ_eq | [2042, 1] | [2054, 10] | ext p | V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : β
β’ {p | Walk.length p = Nat.succ n} = β (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n} | case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : β
p : Walk G u v
β’ p β {p | Walk.length p = Nat.succ n} β p β β (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n} | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : β
β’ {p | Walk.length p = Nat.succ n} = β (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n}
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_succ_eq | [2042, 1] | [2054, 10] | cases' p with _ _ w _ huw pwv | case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : β
p : Walk G u v
β’ p β {p | Walk.length p = Nat.succ n} β p β β (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n} | case h.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
n : β
β’ Walk.nil β {p | Walk.length p = Nat.succ n} β
Walk.nil β β (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n}
case h.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : β
w : V
huw : G.Adj u w
pwv : Walk G w v
β’ Walk.cons huw pwv β {p | Walk.length p = Nat.succ n} β
Walk.cons huw pwv β β (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : β
p : Walk G u v
β’ p β {p | Walk.length p = Nat.succ n} β p β β (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n}
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_succ_eq | [2042, 1] | [2054, 10] | simp [eq_comm] | case h.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
n : β
β’ Walk.nil β {p | Walk.length p = Nat.succ n} β
Walk.nil β β (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.nil
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u : V
n : β
β’ Walk.nil β {p | Walk.length p = Nat.succ n} β
Walk.nil β β (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n}
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_succ_eq | [2042, 1] | [2054, 10] | simp only [Nat.succ_eq_add_one, Set.mem_setOf_eq, Walk.length_cons, add_left_inj,
Set.mem_iUnion, Set.mem_image, exists_prop] | case h.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : β
w : V
huw : G.Adj u w
pwv : Walk G w v
β’ Walk.cons huw pwv β {p | Walk.length p = Nat.succ n} β
Walk.cons huw pwv β β (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n} | case h.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : β
w : V
huw : G.Adj u w
pwv : Walk G w v
β’ Walk.length pwv = n β β i h x, Walk.length x = n β§ Walk.cons (_ : G.Adj u i) x = Walk.cons huw pwv | Please generate a tactic in lean4 to solve the state.
STATE:
case h.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : β
w : V
huw : G.Adj u w
pwv : Walk G w v
β’ Walk.cons huw pwv β {p | Walk.length p = Nat.succ n} β
Walk.cons huw pwv β β (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n}
TACTIC:
|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_succ_eq | [2042, 1] | [2054, 10] | constructor | case h.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : β
w : V
huw : G.Adj u w
pwv : Walk G w v
β’ Walk.length pwv = n β β i h x, Walk.length x = n β§ Walk.cons (_ : G.Adj u i) x = Walk.cons huw pwv | case h.cons.mp
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : β
w : V
huw : G.Adj u w
pwv : Walk G w v
β’ Walk.length pwv = n β β i h x, Walk.length x = n β§ Walk.cons (_ : G.Adj u i) x = Walk.cons huw pwv
case h.cons.mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : β
w : V
huw : G.Adj u w
pwv : Walk G w v
β’ (β i h x, Walk.length x = n β§ Walk.cons (_ : G.Adj u i) x = Walk.cons huw pwv) β Walk.length pwv = n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : β
w : V
huw : G.Adj u w
pwv : Walk G w v
β’ Walk.length pwv = n β β i h x, Walk.length x = n β§ Walk.cons (_ : G.Adj u i) x = Walk.cons huw pwv
TACTIC:
|
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