60 lines
1.7 KiB
Text
60 lines
1.7 KiB
Text
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import types.eq types.pi hit.colimit
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open eq is_trunc unit quotient seq_colim equiv
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namespace one_step_tr
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section
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parameters {A : Type}
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variables (a a' : A)
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protected definition R (a a' : A) : Type₀ := unit
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parameter (A)
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definition one_step_tr : Type := quotient R
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parameter {A}
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definition tr : one_step_tr :=
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class_of R a
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definition tr_eq : tr a = tr a' :=
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eq_of_rel _ star
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protected definition rec {P : one_step_tr → Type} (Pt : Π(a : A), P (tr a))
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(Pe : Π(a a' : A), Pt a =[tr_eq a a'] Pt a') (x : one_step_tr) : P x :=
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begin
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fapply (quotient.rec_on x),
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{ intro a, apply Pt},
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{ intro a a' H, cases H, apply Pe}
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end
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protected definition elim {P : Type} (Pt : A → P)
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(Pe : Π(a a' : A), Pt a = Pt a') (x : one_step_tr) : P :=
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rec Pt (λa a', pathover_of_eq (Pe a a')) x
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theorem rec_tr_eq {P : one_step_tr → Type} (Pt : Π(a : A), P (tr a))
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(Pe : Π(a a' : A), Pt a =[tr_eq a a'] Pt a') (a a' : A)
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: apdo (rec Pt Pe) (tr_eq a a') = Pe a a' :=
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!rec_eq_of_rel
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theorem elim_tr_eq {P : Type} (Pt : A → P)
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(Pe : Π(a a' : A), Pt a = Pt a') (a a' : A)
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: ap (elim Pt Pe) (tr_eq a a') = Pe a a' :=
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begin
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apply eq_of_fn_eq_fn_inv !(pathover_constant (tr_eq a a')),
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rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑elim,rec_tr_eq],
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end
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end
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end one_step_tr
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attribute one_step_tr.rec one_step_tr.elim [recursor 5]
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open one_step_tr
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definition one_step_tr_up (A B : Type)
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: (one_step_tr A → B) ≃ Σ(f : A → B), Π(x y : A), f x = f y :=
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begin
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fapply equiv.MK,
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{ intro f, fconstructor, intro a, exact f (tr a), intros, exact ap f !tr_eq},
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{ exact sorry},
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{ exact sorry},
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{ exact sorry},
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end
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