16 lines
589 B
Text
16 lines
589 B
Text
open nat
|
|
|
|
set_option pp.implicit true
|
|
set_option pp.notation false
|
|
|
|
definition lt_trans : ∀ {a b c : nat}, a < b → b < c → a < c,
|
|
lt_trans h (lt.base _) := lt.step h,
|
|
lt_trans h₁ (lt.step h₂) := lt.step (lt_trans h₁ h₂)
|
|
|
|
definition lt_succ : ∀ {a b : nat}, a < b → succ a < succ b,
|
|
lt_succ (lt.base a) := lt.base (succ a),
|
|
lt_succ (lt.step h) := lt.step (lt_succ h)
|
|
|
|
definition lt_of_succ : ∀ {a b : nat}, succ a < b → a < b,
|
|
lt_of_succ (lt.base (succ a)) := lt.trans (lt.base a) (lt.base (succ a)),
|
|
lt_of_succ (lt.step h₂) := lt.step (lt_of_succ h₂)
|