Theorem: Graph.reduction_reduced (GroupComp/Graph/ReducedPaths.lean)

State before:
V : Type u
E : Type v
inst✝¹ : DecidableEq V
inst✝ : DecidableEq E
G : Graph V E
v w : V
p : EdgePath G v w
⊢ reduced (reduction p)
Tactic:
induction p with
| nil _ =>
  simp [reduction_nil, reduced_nil]
| cons e p ih =>
  apply reduced_redCons
  assumption
State after:
no goals

State before:
case nil
V : Type u
E : Type v
inst✝¹ : DecidableEq V
inst✝ : DecidableEq E
G : Graph V E
v w v✝ : V
⊢ reduced (reduction (nil v✝))
Tactic:
simp [reduction_nil, reduced_nil]
State after:
no goals

State before:
case cons
V : Type u
E : Type v
inst✝¹ : DecidableEq V
inst✝ : DecidableEq E
G : Graph V E
v w v✝ w✝ u✝ : V
e : EdgeBetween G v✝ w✝
p : EdgePath G w✝ u✝
ih : reduced (reduction p)
⊢ reduced (reduction (cons e p))
Tactic:
apply reduced_redCons
State after:
case cons.hyp
V : Type u
E : Type v
inst✝¹ : DecidableEq V
inst✝ : DecidableEq E
G : Graph V E
v w v✝ w✝ u✝ : V
e : EdgeBetween G v✝ w✝
p : EdgePath G w✝ u✝
ih : reduced (reduction p)
⊢ reduced (reduction p)

State before:
case cons.hyp
V : Type u
E : Type v
inst✝¹ : DecidableEq V
inst✝ : DecidableEq E
G : Graph V E
v w v✝ w✝ u✝ : V
e : EdgeBetween G v✝ w✝
p : EdgePath G w✝ u✝
ih : reduced (reduction p)
⊢ reduced (reduction p)
Tactic:
assumption
State after:
no goals