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Diffstat (limited to 'plugins/setoid_ring/ArithRing.v')
-rw-r--r-- | plugins/setoid_ring/ArithRing.v | 60 |
1 files changed, 60 insertions, 0 deletions
diff --git a/plugins/setoid_ring/ArithRing.v b/plugins/setoid_ring/ArithRing.v new file mode 100644 index 00000000..e5a4c8d1 --- /dev/null +++ b/plugins/setoid_ring/ArithRing.v @@ -0,0 +1,60 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +Require Import Mult. +Require Import BinNat. +Require Import Nnat. +Require Export Ring. +Set Implicit Arguments. + +Lemma natSRth : semi_ring_theory O (S O) plus mult (@eq nat). + Proof. + constructor. exact plus_0_l. exact plus_comm. exact plus_assoc. + exact mult_1_l. exact mult_0_l. exact mult_comm. exact mult_assoc. + exact mult_plus_distr_r. + Qed. + +Lemma nat_morph_N : + semi_morph 0 1 plus mult (eq (A:=nat)) + 0%N 1%N Nplus Nmult Neq_bool nat_of_N. +Proof. + constructor;trivial. + exact nat_of_Nplus. + exact nat_of_Nmult. + intros x y H;rewrite (Neq_bool_ok _ _ H);trivial. +Qed. + +Ltac natcst t := + match isnatcst t with + true => constr:(N_of_nat t) + | _ => constr:InitialRing.NotConstant + end. + +Ltac Ss_to_add f acc := + match f with + | S ?f1 => Ss_to_add f1 (S acc) + | _ => constr:(acc + f)%nat + end. + +Ltac natprering := + match goal with + |- context C [S ?p] => + match p with + O => fail 1 (* avoid replacing 1 with 1+0 ! *) + | p => match isnatcst p with + | true => fail 1 + | false => let v := Ss_to_add p (S 0) in + fold v; natprering + end + end + | _ => idtac + end. + +Add Ring natr : natSRth + (morphism nat_morph_N, constants [natcst], preprocess [natprering]). + |