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authorGravatar Pierre Letouzey <pierre.letouzey@inria.fr>2014-06-26 11:04:34 +0200
committerGravatar Pierre Letouzey <pierre.letouzey@inria.fr>2014-07-09 18:47:26 +0200
commit8836eae5d52fbbadf7722548052da3f7ceb5b260 (patch)
treeff067362a375c7c5e9539bb230378ca8bc0cf1ee /theories/Strings
parent6e9a1c4c71f58aba8bb0bb5942c5063a5984a1bc (diff)
Arith: full integration of the "Numbers" modular framework
- The earlier proof-of-concept file NPeano (which instantiates the "Numbers" framework for nat) becomes now the entry point in the Arith lib, and gets renamed PeanoNat. It still provides an inner module "Nat" which sums up everything about type nat (functions, predicates and properties of them). This inner module Nat is usable as soon as you Require Import Arith, or just Arith_base, or simply PeanoNat. - Definitions of operations over type nat are now grouped in a new file Init/Nat.v. This file is meant to be used without "Import", hence providing for instance Nat.add or Nat.sqrt as soon as coqtop starts (but no proofs about them). - The definitions that used to be in Init/Peano.v (pred, plus, minus, mult) are now compatibility notations (for Nat.pred, Nat.add, Nat.sub, Nat.mul where here Nat is Init/Nat.v). - This Coq.Init.Nat module (with only pure definitions) is Include'd in the aforementioned Coq.Arith.PeanoNat.Nat. You might see Init.Nat sometimes instead of just Nat (for instance when doing "Print plus"). Normally it should be ok to just ignore these "Init" since Init.Nat is included in the full PeanoNat.Nat. I'm investigating if it's possible to get rid of these "Init" prefixes. - Concerning predicates, orders le and lt are still defined in Init/Peano.v, with their notations "<=" and "<". Properties in PeanoNat.Nat directly refer to these predicates in Peano. For instantation reasons, PeanoNat.Nat also contains a Nat.le and Nat.lt (defined via "Definition le := Peano.le", we cannot yet include an Inductive to implement a Parameter), but these aliased predicates won't probably be very convenient to use. - Technical remark: I've split the previous property functor NProp in two parts (NBasicProp and NExtraProp), it helps a lot for building PeanoNat.Nat incrementally. Roughly speaking, we have the following schema: Module Nat. Include Coq.Init.Nat. (* definition of operations : add ... sqrt ... *) ... (** proofs of specifications for basic ops such as + * - *) Include NBasicProp. (** generic properties of these basic ops *) ... (** proofs of specifications for advanced ops (pow sqrt log2...) that may rely on proofs for + * - *) Include NExtraProp. (** all remaining properties *) End Nat. - All other files in directory Arith are now taking advantage of PeanoNat : they are now filled with compatibility notations (when earlier lemmas have exact counterpart in the Nat module) or lemmas with one-line proofs based on the Nat module. All hints for database "arith" remain declared in these old-style file (such as Plus.v, Lt.v, etc). All the old-style files are still Require'd (or not) by Arith.v, just as before. - Compatibility should be almost complete. For instance in the stdlib, the only adaptations were due to .ml code referring to some Coq constant name such as Coq.Init.Peano.pred, which doesn't live well with the new compatibility notations.
Diffstat (limited to 'theories/Strings')
-rw-r--r--theories/Strings/Ascii.v4
1 files changed, 2 insertions, 2 deletions
diff --git a/theories/Strings/Ascii.v b/theories/Strings/Ascii.v
index a89b888ec..7d7dcc6d0 100644
--- a/theories/Strings/Ascii.v
+++ b/theories/Strings/Ascii.v
@@ -10,7 +10,7 @@
(** Contributed by Laurent Théry (INRIA);
Adapted to Coq V8 by the Coq Development Team *)
-Require Import Bool BinPos BinNat Nnat.
+Require Import Bool BinPos BinNat PeanoNat Nnat.
Declare ML Module "ascii_syntax_plugin".
(** * Definition of ascii characters *)
@@ -115,7 +115,7 @@ Proof.
unfold N.lt.
change 256%N with (N.of_nat 256).
rewrite <- Nat2N.inj_compare.
- rewrite <- Compare_dec.nat_compare_lt. auto.
+ now apply Nat.compare_lt_iff.
Qed.