feat(hott) start characterizing truncatedness by sigma and pi types
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1 changed files with 21 additions and 2 deletions
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@ -11,11 +11,30 @@ namespace truncation
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definition is_contr.sigma_char (A : Type) :
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(Σ (center : A), Π (a : A), center = a) ≃ (is_contr A) :=
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sorry
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begin
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fapply equiv.mk,
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intro S, apply is_contr.mk, exact S.2,
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fapply is_equiv.adjointify,
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intro H, apply sigma.mk, exact (@contr A H),
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intro H, apply (is_trunc.rec_on H), intro Hint,
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apply (contr_internal.rec_on Hint), intros (H1, H2),
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apply idp,
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intro S, apply (sigma.rec_on S), intros (H1, H2),
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apply idp,
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end
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set_option pp.implicit true
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definition is_trunc.pi_char (n : trunc_index) (A : Type) :
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(Π (x y : A), is_trunc n (x = y)) ≃ (is_trunc (n .+1) A) :=
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sorry
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begin
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fapply equiv.mk,
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intro H, apply is_trunc_succ, exact H,
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fapply is_equiv.adjointify,
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intros (H, x, y), apply succ_is_trunc, exact H,
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intro H, apply (is_trunc.rec_on H), intro Hint, apply idp,
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intro P,
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exact sorry,
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end
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definition is_trunc_is_hprop {n : trunc_index} :
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Π (A : Type), is_hprop (is_trunc n A) :=
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