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