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refactor(library/init/nat): cleanup

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Leonardo de Moura 2015-04-30 07:46:52 -07:00
parent 018cbe9447
commit f60dc8ae8f
1 changed files with 10 additions and 12 deletions
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22
library/init/nat.lean
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@ -78,11 +78,11 @@ namespace nat
(λ a (hlt : zero < succ a), lt.step hlt) (λ a (hlt : zero < succ a), lt.step hlt)
definition lt.trans {a b c : nat} (H₁ : a < b) (H₂ : b < c) : a < c := definition lt.trans {a b c : nat} (H₁ : a < b) (H₂ : b < c) : a < c :=
have aux : ∀ {d}, d < c → b = d → a < b → a < c, from have aux : a < b → a < c, from
(λ d H, lt.rec_on H lt.rec_on H₂
(λ h₁ h₂, lt.step (eq.rec_on h₁ h₂)) (λ h₁, lt.step h₁)
(λ b hl ih h₁ h₂, lt.step (ih h₁ h₂))), (λ b₁ bb₁ ih h₁, lt.step (ih h₁)),
aux H₂ rfl H₁ aux H₁
definition succ_lt_succ {a b : nat} (H : a < b) : succ a < succ b := definition succ_lt_succ {a b : nat} (H : a < b) : succ a < succ b :=
lt.rec_on H lt.rec_on H
@ -90,11 +90,9 @@ namespace nat
(λ b hlt ih, lt.trans ih (lt.base (succ b))) (λ b hlt ih, lt.trans ih (lt.base (succ b)))
definition lt_of_succ_lt {a b : nat} (H : succ a < b) : a < b := definition lt_of_succ_lt {a b : nat} (H : succ a < b) : a < b :=
have aux : ∀ {a₁}, a₁ < b → succ a = a₁ → a < b, from lt.rec_on H
λ a₁ H, lt.rec_on H (lt.step (lt.base a))
(λ e₁, eq.rec_on e₁ (lt.step (lt.base a))) (λ b h ih, lt.step ih)
(λ d hlt ih e₁, lt.step (ih e₁)),
aux H rfl
definition lt_of_succ_lt_succ {a b : nat} (H : succ a < succ b) : a < b := definition lt_of_succ_lt_succ {a b : nat} (H : succ a < succ b) : a < b :=
have aux : pred (succ a) < pred (succ b), from have aux : pred (succ a) < pred (succ b), from
@ -276,7 +274,7 @@ namespace nat
section section
local attribute sub [reducible] local attribute sub [reducible]
definition succ_sub_succ_eq_sub (a b : nat) : succ a - succ b = a - b := definition succ_sub_succ_eq_sub (a b : nat) : succ a - succ b = a - b :=
nat.induction_on b nat.rec_on b
rfl rfl
(λ b₁ (ih : succ a - succ b₁ = a - b₁), (λ b₁ (ih : succ a - succ b₁ = a - b₁),
eq.rec_on ih (eq.refl (pred (succ a - succ b₁)))) eq.rec_on ih (eq.refl (pred (succ a - succ b₁))))
@ -286,7 +284,7 @@ namespace nat
eq.rec_on (succ_sub_succ_eq_sub a b) rfl eq.rec_on (succ_sub_succ_eq_sub a b) rfl
definition zero_sub_eq_zero (a : nat) : zero - a = zero := definition zero_sub_eq_zero (a : nat) : zero - a = zero :=
nat.induction_on a nat.rec_on a
rfl rfl
(λ a₁ (ih : zero - a₁ = zero), (λ a₁ (ih : zero - a₁ = zero),
eq.rec_on ih (eq.refl (pred (zero - a₁)))) eq.rec_on ih (eq.refl (pred (zero - a₁))))