88 lines
3.5 KiB
Text
88 lines
3.5 KiB
Text
import types.trunc types.bool
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open eq bool equiv sigma sigma.ops trunc is_trunc pi
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namespace choice
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universe variable u
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-- 3.8.1. The axiom of choice.
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definition AC [reducible] := Π (X : Type.{u}) (A : X -> Type.{u}) (P : Π x, A x -> Type.{u}),
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is_hset X -> (Π x, is_hset (A x)) -> (Π x a, is_hprop (P x a)) ->
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(Π x, ∥ Σ a, P x a ∥) -> ∥ Σ f, Π x, P x (f x) ∥
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-- 3.8.3. Corresponds to the assertion that
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-- "the cartesian product of a family of nonempty sets is nonempty".
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definition AC_cart [reducible] := Π (X : Type.{u}) (A : X -> Type.{u}),
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is_hset X -> (Π x, is_hset (A x)) -> (Π x, ∥ A x ∥) -> ∥ Π x, A x ∥
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-- A slight variant of AC with a modified (equivalent) codomain.
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definition AC' [reducible] := Π (X : Type.{u}) (A : X -> Type.{u}) (P : Π x, A x -> Type.{u}),
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is_hset X -> (Π x, is_hset (A x)) -> (Π x a, is_hprop (P x a))
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-> (Π x, ∥ Σ a, P x a ∥) -> ∥ Π x, Σ a : A x, P x a ∥
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-- The equivalence of AC and AC' follows from the equivalence of their codomains.
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definition AC_equiv_AC' : AC.{u} ≃ AC'.{u} :=
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equiv_of_is_hprop
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(λ H X A P HX HA HP HI, trunc_functor _ (to_fun !sigma_pi_equiv_pi_sigma) (H X A P HX HA HP HI))
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(λ H X A P HX HA HP HI, trunc_functor _ (to_inv !sigma_pi_equiv_pi_sigma) (H X A P HX HA HP HI))
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-- AC_cart can be derived from AC' by setting P := λ _ _ , unit.
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definition AC_cart_of_AC' : AC'.{u} -> AC_cart.{u} :=
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λ H X A HX HA HI, trunc_functor _ (λ H0 x, (H0 x).1)
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(H X A (λ x a, lift.{0 u} unit) HX HA (λ x a, !is_trunc_lift)
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(λ x, trunc_functor _ (λ a, ⟨a, lift.up.{0 u} unit.star⟩) (HI x)))
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-- And the converse, by setting A := λ x, Σ a, P x a.
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definition AC'_of_AC_cart : AC_cart.{u} -> AC'.{u} :=
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by intro H X A P HX HA HP HI;
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apply H X (λ x, Σ a, P x a) HX (λ x, !is_trunc_sigma) HI
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-- Which is enough to show AC' ≃ AC_cart, since both are hprops.
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definition AC'_equiv_AC_cart : AC'.{u} ≃ AC_cart.{u} :=
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equiv_of_is_hprop AC_cart_of_AC'.{u} AC'_of_AC_cart.{u}
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-- 3.8.2. AC ≃ AC_cart follows by transitivity.
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definition AC_equiv_AC_cart : AC.{u} ≃ AC_cart.{u} :=
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equiv.trans AC_equiv_AC' AC'_equiv_AC_cart
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namespace example385
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definition X : Type.{1} := Σ A : Type.{0}, ∥ A = bool ∥
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definition x0 : X := ⟨bool, merely.intro _ rfl⟩
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definition Y : X -> Type.{1} := λ x, x0 = x
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definition not_is_hset_X : ¬ is_hset X :=
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begin
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intro H, apply not_is_hprop_bool_eq_bool,
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apply @is_trunc_equiv_closed (x0 = x0),
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apply equiv.symm !equiv_subtype
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end
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definition is_hset_x1 (x : X) : is_hset x.1 :=
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by cases x; induction a_1; cases a_1; exact _
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definition is_hset_Yx (x : X) : is_hset (Y x) :=
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begin
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apply @is_trunc_equiv_closed _ _ _ !equiv_subtype,
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apply @is_trunc_equiv_closed _ _ _ (equiv.symm !eq_equiv_equiv),
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apply is_trunc_equiv; repeat (apply is_hset_x1)
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end
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definition trunc_Yx (x : X) : ∥ Y x ∥ :=
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begin
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cases x, induction a_1, apply merely.intro,
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apply to_fun !equiv_subtype, rewrite a_1
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end
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end example385
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open example385
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-- 3.8.5. There exists a type X and a family Y : X → U such that each Y(x) is a set,
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-- but such that (3.8.3) is false.
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definition X_must_be_hset : Σ (X : Type.{1}) (Y : X -> Type.{1})
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(HA : Π x : X, is_hset (Y x)), ¬ ((Π x : X, ∥ Y x ∥) -> ∥ Π x : X, Y x ∥) :=
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⟨X, Y, is_hset_Yx, λ H, trunc.rec_on (H trunc_Yx)
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(λ H0, not_is_hset_X (@is_trunc_of_is_contr _ _ (is_contr.mk x0 H0)))⟩
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end choice
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