96 lines
3 KiB
Text
96 lines
3 KiB
Text
import init.ua
|
|
open nat unit equiv is_trunc
|
|
|
|
inductive vector (A : Type) : nat → Type :=
|
|
| nil {} : vector A zero
|
|
| cons : Π {n}, A → vector A n → vector A (succ n)
|
|
|
|
open vector
|
|
notation a :: b := cons a b
|
|
|
|
definition const {A : Type} : Π (n : nat), A → vector A n
|
|
| zero a := nil
|
|
| (succ n) a := a :: const n a
|
|
|
|
definition head {A : Type} : Π {n : nat}, vector A (succ n) → A
|
|
| n (x :: xs) := x
|
|
|
|
theorem singlenton_vector_unit : ∀ {n : nat} (v w : vector unit n), v = w
|
|
| zero nil nil := rfl
|
|
| (succ n) (star::xs) (star::ys) :=
|
|
begin
|
|
have h₁ : xs = ys, from singlenton_vector_unit xs ys,
|
|
rewrite h₁
|
|
end
|
|
|
|
private definition f (n m : nat) (v : vector unit n) : vector unit m := const m star
|
|
|
|
theorem vn_eqv_vm (n m : nat) : vector unit n ≃ vector unit m :=
|
|
equiv.MK (f n m) (f m n)
|
|
(take v : vector unit m, singlenton_vector_unit (f n m (f m n v)) v)
|
|
(take v : vector unit n, singlenton_vector_unit (f m n (f n m v)) v)
|
|
|
|
theorem vn_eq_vm (n m : nat) : vector unit n = vector unit m :=
|
|
ua (vn_eqv_vm n m)
|
|
|
|
definition vector_inj (A : Type) := ∀ (n m : nat), vector A n = vector A m → n = m
|
|
|
|
theorem not_vector_inj : ¬ vector_inj unit :=
|
|
assume H : vector_inj unit,
|
|
have aux₁ : 0 = 1, from H 0 1 (vn_eq_vm 0 1),
|
|
lift.down (nat.no_confusion aux₁)
|
|
|
|
definition cast {A B : Type} (H : A = B) (a : A) : B :=
|
|
eq.rec_on H a
|
|
|
|
open sigma
|
|
|
|
definition heq {A B : Type} (a : A) (b : B) :=
|
|
Σ (H : A = B), cast H a = b
|
|
|
|
infix `==`:50 := heq
|
|
|
|
definition heq.type_eq {A B : Type} {a : A} {b : B} : a == b → A = B
|
|
| ⟨H, e⟩ := H
|
|
|
|
definition heq.symm : ∀ {A B : Type} {a : A} {b : B}, a == b → b == a
|
|
| A A a a ⟨eq.refl A, eq.refl a⟩ := ⟨eq.refl A, eq.refl a⟩
|
|
|
|
definition heq.trans : ∀ {A B C : Type} {a : A} {b : B} {c : C}, a == b → b == c → a == c
|
|
| A A A a a a ⟨eq.refl A, eq.refl a⟩ ⟨eq.refl A, eq.refl a⟩ := ⟨eq.refl A, eq.refl a⟩
|
|
|
|
theorem cast_heq : ∀ {A B : Type} (H : A = B) (a : A), cast H a == a
|
|
| A A (eq.refl A) a := ⟨eq.refl A, eq.refl a⟩
|
|
|
|
definition default (A : Type) [H : inhabited A] : A :=
|
|
inhabited.rec_on H (λ a, a)
|
|
|
|
definition lem_eq (A : Type) : Type :=
|
|
∀ (n m : nat) (v : vector A n) (w : vector A m), v == w → n = m
|
|
|
|
theorem lem_eq_iff_vector_inj (A : Type) [inh : inhabited A] : lem_eq A ↔ vector_inj A :=
|
|
iff.intro
|
|
(assume Hl : lem_eq A,
|
|
assume n m he,
|
|
assert a : A, from default A,
|
|
assert v : vector A n, from const n a,
|
|
have e₁ : v == cast he v, from heq.symm (cast_heq he v),
|
|
Hl n m v (cast he v) e₁)
|
|
(assume Hr : vector_inj A,
|
|
assume n m v w he,
|
|
Hr n m (heq.type_eq he))
|
|
|
|
theorem lem_eq_of_not_inhabited (A : Type) [ninh : inhabited A → empty] : lem_eq A :=
|
|
take (n m : nat),
|
|
match n with
|
|
| zero :=
|
|
match m with
|
|
| zero := take v w He, rfl
|
|
| (succ m₁) :=
|
|
take (v : vector A zero) (w : vector A (succ m₁)),
|
|
empty.elim _ (ninh (inhabited.mk (head w)))
|
|
end
|
|
| (succ n₁) :=
|
|
take (v : vector A (succ n₁)) (w : vector A m),
|
|
empty.elim _ (ninh (inhabited.mk (head v)))
|
|
end
|