55 lines
1.9 KiB
Text
55 lines
1.9 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: Leonardo de Moura, Jeremy Avigad
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import logic.classes.inhabited logic.connectives.eq logic.classes.decidable
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using decidable
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inductive subtype {A : Type} (P : A → Prop) : Type :=
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tag : Πx : A, P x → subtype P
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notation `{` binders `,` r:(scoped P, subtype P) `}` := r
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namespace subtype
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section
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parameter {A : Type}
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parameter {P : A → Prop}
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-- TODO: make this a coercion?
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definition elt_of (a : {x, P x}) : A := subtype_rec (λ x y, x) a
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theorem has_property (a : {x, P x}) : P (elt_of a) := subtype_rec (λ x y, y) a
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theorem elt_of_tag (a : A) (H : P a) : elt_of (tag a H) = a := refl a
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theorem subtype_destruct {Q : {x, P x} → Prop} (a : {x, P x})
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(H : ∀(x : A) (H1 : P x), Q (tag x H1)) : Q a :=
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subtype_rec H a
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theorem tag_irrelevant {a : A} (H1 H2 : P a) : tag a H1 = tag a H2 := refl (tag a H1)
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theorem tag_elt_of (a : subtype P) : Π(H : P (elt_of a)), tag (elt_of a) H = a :=
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subtype_destruct a (take (x : A) (H1 : P x) (H2 : P x), refl _)
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theorem tag_eq {a1 a2 : A} {H1 : P a1} {H2 : P a2} (H3 : a1 = a2) : tag a1 H1 = tag a2 H2 :=
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(show ∀(H2 : P a2), tag a1 H1 = tag a2 H2, from subst H3 (take H2, tag_irrelevant H1 H2)) H2
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theorem subtype_eq {a1 a2 : {x, P x}} : ∀(H : elt_of a1 = elt_of a2), a1 = a2 :=
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subtype_destruct a1 (take x1 H1, subtype_destruct a2 (take x2 H2 H, tag_eq H))
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theorem subtype_inhabited {a : A} (H : P a) : inhabited {x, P x} :=
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inhabited_mk (tag a H)
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theorem subtype_eq_decidable (a1 a2 : {x, P x})
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(H : decidable (elt_of a1 = elt_of a2)) : decidable (a1 = a2) :=
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have H1 : (a1 = a2) ↔ (elt_of a1 = elt_of a2), from
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iff_intro (assume H, subst H rfl) (assume H, subtype_eq H),
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decidable_iff_equiv _ (iff_symm H1)
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end
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instance subtype_inhabited
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instance subtype_eq_decidable
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end subtype
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