105 lines
4.1 KiB
Text
105 lines
4.1 KiB
Text
/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn, Jakob von Raumer
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The category of sets is complete and cocomplete
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-/
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import .colimits ..constructions.set hit.set_quotient
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open eq functor is_trunc sigma pi sigma.ops trunc set_quotient
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namespace category
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local attribute category.to_precategory [unfold 2]
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definition is_complete_set_cone.{u v w} [constructor]
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(I : Precategory.{v w}) (F : I ⇒ set.{max u v w}) : cone_obj F :=
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begin
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fapply cone_obj.mk,
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{ fapply trunctype.mk,
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{ exact Σ(s : Π(i : I), trunctype.carrier (F i)),
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Π{i j : I} (f : i ⟶ j), F f (s i) = (s j)},
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{ with_options [elaborator.ignore_instances true] -- TODO: fix
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( refine is_trunc_sigma _ _;
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( apply is_trunc_pi);
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( intro s;
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refine is_trunc_pi _ _; intro i;
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refine is_trunc_pi _ _; intro j;
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refine is_trunc_pi _ _; intro f;
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apply is_trunc_eq))}},
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{ fapply nat_trans.mk,
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{ intro i x, esimp at x, exact x.1 i},
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{ intro i j f, esimp, apply eq_of_homotopy, intro x, esimp at x, induction x with s p,
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esimp, apply p}}
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end
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definition is_complete_set.{u v w} [instance] : is_complete.{(max u v w)+1 (max u v w) v w} set :=
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begin
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intro I F, fapply has_terminal_object.mk,
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{ exact is_complete_set_cone.{u v w} I F},
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{ intro c, esimp at *, induction c with X η, induction η with η p, esimp at *,
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fapply is_contr.mk,
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{ fapply cone_hom.mk,
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{ intro x, esimp at *, fapply sigma.mk,
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{ intro i, exact η i x},
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{ intro i j f, exact ap10 (p f) x}},
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{ intro i, reflexivity}},
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{ esimp, intro h, induction h with f q, apply cone_hom_eq, esimp at *,
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apply eq_of_homotopy, intro x, fapply sigma_eq: esimp,
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{ apply eq_of_homotopy, intro i, exact (ap10 (q i) x)⁻¹},
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{ with_options [elaborator.ignore_instances true] -- TODO: fix
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( refine is_prop.elimo _ _ _;
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refine is_trunc_pi _ _; intro i;
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refine is_trunc_pi _ _; intro j;
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refine is_trunc_pi _ _; intro f;
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apply is_trunc_eq)}}}
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end
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definition is_cocomplete_set_cone_rel.{u v w} [unfold 3 4]
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(I : Precategory.{v w}) (F : I ⇒ set.{max u v w}ᵒᵖ) : (Σ(i : I), trunctype.carrier (F i)) →
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(Σ(i : I), trunctype.carrier (F i)) → hprop.{max u v w} :=
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begin
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intro v w, induction v with i x, induction w with j y,
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fapply trunctype.mk,
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{ exact ∃(f : i ⟶ j), to_fun_hom F f y = x},
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{ exact _}
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end
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definition is_cocomplete_set_cone.{u v w} [constructor]
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(I : Precategory.{v w}) (F : I ⇒ set.{max u v w}ᵒᵖ) : cone_obj F :=
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begin
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fapply cone_obj.mk,
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{ fapply trunctype.mk,
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{ apply set_quotient (is_cocomplete_set_cone_rel.{u v w} I F)},
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{ apply is_set_set_quotient}},
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{ fapply nat_trans.mk,
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{ intro i x, esimp, apply class_of, exact ⟨i, x⟩},
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{ intro i j f, esimp, apply eq_of_homotopy, intro y, apply eq_of_rel, esimp,
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exact exists.intro f idp}}
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end
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-- TODO: change this after induction tactic for trunc/set_quotient is implemented
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definition is_cocomplete_set.{u v w} [instance]
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: is_cocomplete.{(max u v w)+1 (max u v w) v w} set :=
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begin
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intro I F, fapply has_terminal_object.mk,
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{ exact is_cocomplete_set_cone.{u v w} I F},
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{ intro c, esimp at *, induction c with X η, induction η with η p, esimp at *,
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fapply is_contr.mk,
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{ fapply cone_hom.mk,
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{ refine set_quotient.elim _ _,
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{ intro v, induction v with i x, exact η i x},
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{ intro v w r, induction v with i x, induction w with j y, esimp at *,
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refine trunc.elim_on r _, clear r,
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intro u, induction u with f q,
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exact ap (η i) q⁻¹ ⬝ ap10 (p f) y}},
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{ intro i, reflexivity}},
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{ esimp, intro h, induction h with f q, apply cone_hom_eq, esimp at *,
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apply eq_of_homotopy, refine set_quotient.rec _ _,
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{ intro v, induction v with i x, esimp, exact (ap10 (q i) x)⁻¹},
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{ intro v w r, apply is_prop.elimo}}},
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end
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end category
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