From 59726c5343613379d38a9409af044d85cca130ed Mon Sep 17 00:00:00 2001 From: letouzey Date: Thu, 18 Nov 2010 18:02:20 +0000 Subject: Some more revision of {P,N,Z}Arith + bitwise ops in Ndigits Initial plan was only to add shiftl/shiftr/land/... to N and other number type, this is only partly done, but this work has diverged into a big reorganisation and improvement session of PArith,NArith,ZArith. Bool/Bool: add lemmas orb_diag (a||a = a) and andb_diag (a&&a = a) PArith/BinPos: - added a power function Ppow - iterator iter_pos moved from Zmisc to here + some lemmas - added Psize_pos, which is 1+log2, used to define Nlog2/Zlog2 - more lemmas on Pcompare and succ/+/* and order, allow to simplify a lot some old proofs elsewhere. - new/revised results on Pminus (including some direct proof of stuff from Pnat) PArith/Pnat: - more direct proofs (limit the need of stuff about Pmult_nat). - provide nicer names for some lemmas (eg. Pplus_plus instead of nat_of_P_plus_morphism), compatibility notations provided. - kill some too-specific lemmas unused in stdlib + contribs NArith/BinNat: - N_of_nat, nat_of_N moved from Nnat to here. - a lemma relating Npred and Nminus - revised definitions and specification proofs of Npow and Nlog2 NArith/Nnat: - shorter proofs. - stuff about Z_of_N is moved to Znat. This way, NArith is entirely independent from ZArith. NArith/Ndigits: - added bitwise operations Nand Nor Ndiff Nshiftl Nshiftr - revised proofs about Nxor, still using functional bit stream - use the same approach to prove properties of Nand Nor Ndiff ZArith/BinInt: huge simplification of Zplus_assoc + cosmetic stuff ZArith/Zcompare: nicer proofs of ugly things like Zcompare_Zplus_compat ZArith/Znat: some nicer proofs and names, received stuff about Z_of_N ZArith/Zmisc: almost empty new, only contain stuff about badly-named iter. Should be reformed more someday. ZArith/Zlog_def: Zlog2 is now based on Psize_pos, this factorizes proofs and avoid slowdown due to adding 1 in Z instead of in positive Zarith/Zpow_def: Zpower_opt is renamed more modestly Zpower_alt as long as I dont't know why it's slower on powers of two. Elsewhere: propagate new names + some nicer proofs NB: Impact on compatibility is probably non-zero, but should be really moderate. We'll see on contribs, but a few Require here and there might be necessary. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13651 85f007b7-540e-0410-9357-904b9bb8a0f7 --- plugins/omega/OmegaLemmas.v | 7 +++++++ 1 file changed, 7 insertions(+) (limited to 'plugins/omega/OmegaLemmas.v') diff --git a/plugins/omega/OmegaLemmas.v b/plugins/omega/OmegaLemmas.v index ff433bbd8..5b6f4670f 100644 --- a/plugins/omega/OmegaLemmas.v +++ b/plugins/omega/OmegaLemmas.v @@ -298,3 +298,10 @@ Definition fast_Zred_factor5 (x y : Z) (P : Z -> Prop) Definition fast_Zred_factor6 (x : Z) (P : Z -> Prop) (H : P (x + 0)) := eq_ind_r P H (Zred_factor6 x). + +Theorem intro_Z : + forall n:nat, exists y : Z, Z_of_nat n = y /\ 0 <= y * 1 + 0. +Proof. + intros n; exists (Z_of_nat n); split; trivial. + rewrite Zmult_1_r, Zplus_0_r. apply Zle_0_nat. +Qed. -- cgit v1.2.3