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Diffstat (limited to 'theories/Numbers/Natural/SpecViaZ/NSig.v')
| -rw-r--r-- | theories/Numbers/Natural/SpecViaZ/NSig.v | 124 |
1 files changed, 0 insertions, 124 deletions
diff --git a/theories/Numbers/Natural/SpecViaZ/NSig.v b/theories/Numbers/Natural/SpecViaZ/NSig.v deleted file mode 100644 index 258e0315..00000000 --- a/theories/Numbers/Natural/SpecViaZ/NSig.v +++ /dev/null @@ -1,124 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Require Import BinInt. - -Open Scope Z_scope. - -(** * NSig *) - -(** Interface of a rich structure about natural numbers. - Specifications are written via translation to Z. -*) - -Module Type NType. - - Parameter t : Type. - - Parameter to_Z : t -> Z. - Local Notation "[ x ]" := (to_Z x). - Parameter spec_pos: forall x, 0 <= [x]. - - Parameter of_N : N -> t. - Parameter spec_of_N: forall x, to_Z (of_N x) = Z.of_N x. - Definition to_N n := Z.to_N (to_Z n). - - Definition eq n m := [n] = [m]. - Definition lt n m := [n] < [m]. - Definition le n m := [n] <= [m]. - - Parameter compare : t -> t -> comparison. - Parameter eqb : t -> t -> bool. - Parameter ltb : t -> t -> bool. - Parameter leb : t -> t -> bool. - Parameter max : t -> t -> t. - Parameter min : t -> t -> t. - Parameter zero : t. - Parameter one : t. - Parameter two : t. - Parameter succ : t -> t. - Parameter pred : t -> t. - Parameter add : t -> t -> t. - Parameter sub : t -> t -> t. - Parameter mul : t -> t -> t. - Parameter square : t -> t. - Parameter pow_pos : t -> positive -> t. - Parameter pow_N : t -> N -> t. - Parameter pow : t -> t -> t. - Parameter sqrt : t -> t. - Parameter log2 : t -> t. - Parameter div_eucl : t -> t -> t * t. - Parameter div : t -> t -> t. - Parameter modulo : t -> t -> t. - Parameter gcd : t -> t -> t. - Parameter even : t -> bool. - Parameter odd : t -> bool. - Parameter testbit : t -> t -> bool. - Parameter shiftr : t -> t -> t. - Parameter shiftl : t -> t -> t. - Parameter land : t -> t -> t. - Parameter lor : t -> t -> t. - Parameter ldiff : t -> t -> t. - Parameter lxor : t -> t -> t. - Parameter div2 : t -> t. - - Parameter spec_compare: forall x y, compare x y = ([x] ?= [y]). - Parameter spec_eqb : forall x y, eqb x y = ([x] =? [y]). - Parameter spec_ltb : forall x y, ltb x y = ([x] <? [y]). - Parameter spec_leb : forall x y, leb x y = ([x] <=? [y]). - Parameter spec_max : forall x y, [max x y] = Z.max [x] [y]. - Parameter spec_min : forall x y, [min x y] = Z.min [x] [y]. - Parameter spec_0: [zero] = 0. - Parameter spec_1: [one] = 1. - Parameter spec_2: [two] = 2. - Parameter spec_succ: forall n, [succ n] = [n] + 1. - Parameter spec_add: forall x y, [add x y] = [x] + [y]. - Parameter spec_pred: forall x, [pred x] = Z.max 0 ([x] - 1). - Parameter spec_sub: forall x y, [sub x y] = Z.max 0 ([x] - [y]). - Parameter spec_mul: forall x y, [mul x y] = [x] * [y]. - Parameter spec_square: forall x, [square x] = [x] * [x]. - Parameter spec_pow_pos: forall x n, [pow_pos x n] = [x] ^ Zpos n. - Parameter spec_pow_N: forall x n, [pow_N x n] = [x] ^ Z.of_N n. - Parameter spec_pow: forall x n, [pow x n] = [x] ^ [n]. - Parameter spec_sqrt: forall x, [sqrt x] = Z.sqrt [x]. - Parameter spec_log2: forall x, [log2 x] = Z.log2 [x]. - Parameter spec_div_eucl: forall x y, - let (q,r) := div_eucl x y in ([q], [r]) = Z.div_eucl [x] [y]. - Parameter spec_div: forall x y, [div x y] = [x] / [y]. - Parameter spec_modulo: forall x y, [modulo x y] = [x] mod [y]. - Parameter spec_gcd: forall a b, [gcd a b] = Z.gcd [a] [b]. - Parameter spec_even: forall x, even x = Z.even [x]. - Parameter spec_odd: forall x, odd x = Z.odd [x]. - Parameter spec_testbit: forall x p, testbit x p = Z.testbit [x] [p]. - Parameter spec_shiftr: forall x p, [shiftr x p] = Z.shiftr [x] [p]. - Parameter spec_shiftl: forall x p, [shiftl x p] = Z.shiftl [x] [p]. - Parameter spec_land: forall x y, [land x y] = Z.land [x] [y]. - Parameter spec_lor: forall x y, [lor x y] = Z.lor [x] [y]. - Parameter spec_ldiff: forall x y, [ldiff x y] = Z.ldiff [x] [y]. - Parameter spec_lxor: forall x y, [lxor x y] = Z.lxor [x] [y]. - Parameter spec_div2: forall x, [div2 x] = Z.div2 [x]. - -End NType. - -Module Type NType_Notation (Import N:NType). - Notation "[ x ]" := (to_Z x). - Infix "==" := eq (at level 70). - Notation "0" := zero. - Notation "1" := one. - Notation "2" := two. - Infix "+" := add. - Infix "-" := sub. - Infix "*" := mul. - Infix "^" := pow. - Infix "<=" := le. - Infix "<" := lt. -End NType_Notation. - -Module Type NType' := NType <+ NType_Notation. |