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Proofreading SubsetTypes
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@ -123,7 +123,7 @@ Extraction pred_strong1.
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* systematically.
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*
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* We can reimplement our dependently typed [pred] based on _subset types_,
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* defined in the standard library with the type family %[sig]. *)
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* defined in the standard library with the type family [sig]. *)
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Print sig.
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@ -146,7 +146,7 @@ Definition pred_strong2 (s : {n : nat | n > 0} ) : nat :=
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* [P] for [sig]) are not mentioned in pattern matching, but _are_ mentioned in
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* construction of terms (if they are not marked as implicit arguments).
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* (Actually, this behavior changed between Coq versions 8.4 and 8.5, hence the
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* near at the top of the file to revert to the old behavior.) *)
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* command at the top of the file to revert to the old behavior.) *)
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Compute pred_strong2 (exist _ 2 two_gt0).
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@ -172,7 +172,7 @@ Definition pred_strong3 (s : {n : nat | n > 0}) : {m : nat | proj1_sig s = S m}
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Compute pred_strong3 (exist _ 2 two_gt0).
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(* A value in a subset type can be thought of as a _dependent pair_ (or
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* _sigma type_ of a base value and a proof about it. The function [proj1_sig]
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* _sigma type_) of a base value and a proof about it. The function [proj1_sig]
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* extracts the first component of the pair. It turns out that we need to
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* include an explicit [return] clause here, since Coq's heuristics are not
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* smart enough to propagate the result type that we wrote earlier.
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@ -243,7 +243,7 @@ Print pred_strong4.
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Compute pred_strong4 two_gt0.
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(* We are almost done with the ideal implementation of dependent predecessor.
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* We can use Coq's syntax extension facility to arrive at code with almost no
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* We can use Coq's syntax-extension facility to arrive at code with almost no
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* complexity beyond a Haskell or ML program with a complete specification in a
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* comment. In this book, we will not dwell on the details of syntax
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* extensions; the Coq manual gives a straightforward introduction to them. *)
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@ -268,7 +268,7 @@ Compute pred_strong5 two_gt0.
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(** * Decidable Proposition Types *)
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(* There is another type in the standard library that captures the idea of
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* program values that indicate which of two propositions is true. *)
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* a program value indicating which of two propositions is true. *)
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Print sumbool.
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@ -353,8 +353,8 @@ Section In_dec.
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Defined.
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End In_dec.
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Compute In_dec eq_nat_dec 2 (1 :: 2 :: nil).
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Compute In_dec eq_nat_dec 3 (1 :: 2 :: nil).
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Compute In_dec eq_nat_dec 2 [1; 2].
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Compute In_dec eq_nat_dec 3 [1; 2].
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(* The [In_dec] function has a reasonable extraction to OCaml. *)
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