08ccd58eb6
definitions are unfolded during elaboration Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
92 lines
2.3 KiB
Text
92 lines
2.3 KiB
Text
import logic
|
||
|
||
inductive nat : Type :=
|
||
zero : nat,
|
||
succ : nat → nat
|
||
|
||
open eq
|
||
|
||
namespace nat
|
||
definition add (x y : nat)
|
||
:= nat.rec x (λ n r, succ r) y
|
||
|
||
infixl `+`:65 := add
|
||
|
||
theorem add_zero_left (x : nat) : x + zero = x
|
||
:= refl _
|
||
|
||
theorem add_succ_left (x y : nat) : x + (succ y) = succ (x + y)
|
||
:= refl _
|
||
|
||
definition is_zero (x : nat)
|
||
:= nat.rec true (λ n r, false) x
|
||
|
||
theorem is_zero_zero : is_zero zero
|
||
:= eq_true_elim (refl _)
|
||
|
||
theorem not_is_zero_succ (x : nat) : ¬ is_zero (succ x)
|
||
:= eq_false_elim (refl _)
|
||
|
||
theorem dichotomy (m : nat) : m = zero ∨ (∃ n, m = succ n)
|
||
:= nat.rec
|
||
(or.intro_left _ (refl zero))
|
||
(λ m H, or.intro_right _ (exists_intro m (refl (succ m))))
|
||
m
|
||
|
||
theorem is_zero_to_eq (x : nat) (H : is_zero x) : x = zero
|
||
:= or.elim (dichotomy x)
|
||
(assume Hz : x = zero, Hz)
|
||
(assume Hs : (∃ n, x = succ n),
|
||
exists_elim Hs (λ (w : nat) (Hw : x = succ w),
|
||
absurd H (eq.subst (symm Hw) (not_is_zero_succ w))))
|
||
|
||
theorem is_not_zero_to_eq {x : nat} (H : ¬ is_zero x) : ∃ n, x = succ n
|
||
:= or.elim (dichotomy x)
|
||
(assume Hz : x = zero,
|
||
absurd (eq.subst (symm Hz) is_zero_zero) H)
|
||
(assume Hs, Hs)
|
||
|
||
theorem not_zero_add (x y : nat) (H : ¬ is_zero y) : ¬ is_zero (x + y)
|
||
:= exists_elim (is_not_zero_to_eq H)
|
||
(λ (w : nat) (Hw : y = succ w),
|
||
have H1 : x + y = succ (x + w), from
|
||
calc x + y = x + succ w : {Hw}
|
||
... = succ (x + w) : refl _,
|
||
have H2 : ¬ is_zero (succ (x + w)), from
|
||
not_is_zero_succ (x+w),
|
||
subst (symm H1) H2)
|
||
|
||
inductive not_zero (x : nat) : Prop :=
|
||
intro : ¬ is_zero x → not_zero x
|
||
|
||
theorem not_zero_not_is_zero {x : nat} (H : not_zero x) : ¬ is_zero x
|
||
:= not_zero.rec (λ H1, H1) H
|
||
|
||
theorem not_zero_add_right [instance] (x y : nat) (H : not_zero y) : not_zero (x + y)
|
||
:= not_zero.intro (not_zero_add x y (not_zero_not_is_zero H))
|
||
|
||
theorem not_zero_succ [instance] (x : nat) : not_zero (succ x)
|
||
:= not_zero.intro (not_is_zero_succ x)
|
||
|
||
variable dvd : Π (x y : nat) {H : not_zero y}, nat
|
||
|
||
variables a b : nat
|
||
|
||
set_option pp.implicit true
|
||
reducible add
|
||
check dvd a (succ b)
|
||
check (λ H : not_zero b, dvd a b)
|
||
check (succ zero)
|
||
check a + (succ zero)
|
||
check dvd a (a + (succ b))
|
||
|
||
reducible [off] add
|
||
check dvd a (a + (succ b))
|
||
|
||
reducible add
|
||
check dvd a (a + (succ b))
|
||
|
||
reducible [off] add
|
||
check dvd a (a + (succ b))
|
||
|
||
end nat
|