Equivalence between a type and the coproduct of a decidable subtype and its complement #1376
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Co-authored-by: Fredrik Bakke <[email protected]>
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| disjoint-decidable-subtype-Prop : Prop (l1 ⊔ l2 ⊔ l3) | ||
| disjoint-decidable-subtype-Prop = |
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This should be called is-disjoint-prop-decidable-subtype, correspondinly, disjoint-subtype-Prop should be called is-disjoint-prop-subtype, sorry that I didn't catch that last time
| ( subtype-decidable-subtype B) | ||
| ( subtype-decidable-subtype C) | ||
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| disjoint-decidable-subtype : UU (l1 ⊔ l2 ⊔ l3) |
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Suggested change
| disjoint-decidable-subtype : UU (l1 ⊔ l2 ⊔ l3) | |
| is-disjoint-decidable-subtype : UU (l1 ⊔ l2 ⊔ l3) |
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| all-elements-equal-coproduct-disjoint-prop : | ||
| (x : X) → all-elements-equal (type-Prop (A x) + type-Prop (B x)) | ||
| all-elements-equal-coproduct-disjoint-prop x (inl _) (inl _) = | ||
| ap inl (eq-type-Prop (A x)) | ||
| all-elements-equal-coproduct-disjoint-prop x (inr _) (inr _) = | ||
| ap inr (eq-type-Prop (B x)) | ||
| all-elements-equal-coproduct-disjoint-prop x (inl ax) (inr bx) = | ||
| ex-falso (H x (ax , bx)) | ||
| all-elements-equal-coproduct-disjoint-prop x (inr bx) (inl ax) = | ||
| ex-falso (H x (ax , bx)) |
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This result should be derived from the corresponding result for disjunctions of propositions: if ¬ (A ∧ B) then A + B is a proposition and equivalent to A ∨ B
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