37 lines
1.1 KiB
Text
37 lines
1.1 KiB
Text
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Jeremy Avigad
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-- Ported from Coq HoTT
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import .path
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open path
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-- Truncation levels
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-- -----------------
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inductive Contr (A : Type) : Type :=
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Contr_mk : Π
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(center : A)
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(contr : Πy : A, center ≈ y),
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Contr A
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definition center {A : Type} (C : Contr A) : A := Contr.rec (λcenter contr, center) C
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definition contr {A : Type} (C : Contr A) : Πy : A, center C ≈ y :=
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Contr.rec (λcenter contr, contr) C
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inductive trunc_index : Type :=
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minus_two : trunc_index,
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trunc_S : trunc_index → trunc_index
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-- TODO: add coercions to / from nat
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-- TODO: note in the Coq version, there is an internal version
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definition IsTrunc (n : trunc_index) : Type → Type :=
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trunc_index.rec (λA, Contr A) (λn trunc_n A, (Π(x y : A), trunc_n (x ≈ y))) n
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-- TODO: in the Coq version, this is notation
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definition minus_one := trunc_index.trunc_S trunc_index.minus_two
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definition IsHProp := IsTrunc minus_one
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definition IsHSet := IsTrunc (trunc_index.trunc_S minus_one)
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prefix `!`:75 := center
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