(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
Require BinPos Ndec.
Require Export Ring.
(******************************************************************************)
(* A version of arithmetic on nat (natural numbers) that is better suited to  *)
(* small scale reflection than the Coq Arith library. It contains an          *)
(* extensive equational theory (including, e.g., the AGM inequality), as well *)
(* as support for the ring tactic, and congruence tactics.                    *)
(*   The following operations and notations are provided:                     *)
(*                                                                            *)
(*   successor and predecessor                                                *)
(*     n.+1, n.+2, n.+3, n.+4 and n.-1, n.-2                                  *)
(*     this frees the names "S" and "pred"                                    *)
(*                                                                            *)
(*   basic arithmetic                                                         *)
(*     m + n, m - n, m * n                                                    *)
(*   Important: m - n denotes TRUNCATED subtraction: m - n = 0 if m <= n.     *)
(*   The definitions use the nosimpl tag to prevent undesirable computation   *)
(*   computation during simplification, but remain compatible with the ones   *)
(*   provided in the Coq.Init.Peano prelude.                                  *)
(*     For computation, a module NatTrec rebinds all arithmetic notations     *)
(*   to less convenient but also less inefficient tail-recursive functions;   *)
(*   the auxiliary functions used by these versions are flagged with %Nrec.   *)
(*     Also, there is support for input and output of large nat values.       *)
(*       Num 3 082 241 inputs the number 3082241                              *)
(*         [Num of n]  outputs the value n                                    *)
(*   There are coercions num >-> BinNat.N >-> nat; ssrnat rebinds the scope   *)
(*   delimiter for BinNat.N to %num, as it uses the shorter %N for its own    *)
(*   notations (Peano notations are flagged with %coq_nat).                   *)
(*                                                                            *)
(*   doubling, halving, and parity                                            *)
(*      n.*2, n./2, odd n, uphalf n,  with uphalf n = n.+1./2                 *)
(*   bool coerces to nat so we can write, e.g., n = odd n + n./2.*2.          *)
(*                                                                            *)
(*   iteration                                                                *)
(*             iter n f x0  == f ( .. (f x0))                                 *)
(*             iteri n g x0 == g n.-1 (g ... (g 0 x0))                        *)
(*         iterop n op x x0 == op x (... op x x) (n x's) or x0 if n = 0       *)
(*                                                                            *)
(*   exponentiation, factorial                                                *)
(*        m ^ n, n`!                                                          *)
(*        m ^ 1 is convertible to m, and m ^ 2 to m * m                       *)
(*                                                                            *)
(*   comparison                                                               *)
(*      m <= n, m < n, m >= n, m > n, m == n, m <= n <= p, etc.,              *)
(*   comparisons are BOOLEAN operators, and m == n is the generic eqType      *)
(*   operation.                                                               *)
(*     Most compatibility lemmas are stated as boolean equalities; this keeps *)
(*   the size of the library down. All the inequalities refer to the same     *)
(*   constant "leq"; in particular m < n is identical to m.+1 <= n.           *)
(*                                                                            *)
(*   conditionally strict inequality `leqif'                                  *)
(*      m <= n ?= iff condition   ==   (m <= n) and ((m == n) = condition)    *)
(*   This is actually a pair of boolean equalities, so rewriting with an      *)
(*   `leqif' lemma can affect several kinds of comparison. The transitivity   *)
(*   lemma for leqif aggregates the conditions, allowing for arguments of     *)
(*   the form ``m <= n <= p <= m, so equality holds throughout''.             *)
(*                                                                            *)
(*   maximum and minimum                                                      *)
(*      maxn m n, minn m n                                                    *)
(*   Note that maxn m n = m + (n - m), due to the truncating subtraction.     *)
(*   Absolute difference (linear distance) between nats is defined in the int *)
(*   library (in the int.IntDist sublibrary), with the syntax `|m - n|. The   *)
(*   '-' in this notation is the signed integer difference.                   *)
(*                                                                            *)
(*   countable choice                                                         *)
(*     ex_minn : forall P : pred nat, (exists n, P n) -> nat                  *)
(*   This returns the smallest n such that P n holds.                         *)
(*     ex_maxn : forall (P : pred nat) m,                                     *)
(*        (exists n, P n) -> (forall n, P n -> n <= m) -> nat                 *)
(*   This returns the largest n such that P n holds (given an explicit upper  *)
(*   bound).                                                                  *)
(*                                                                            *)
(*  This file adds the following suffix conventions to those documented in    *)
(* ssrbool.v and eqtype.v:                                                    *)
(*   A (infix) -- conjunction, as in                                          *)
(*      ltn_neqAle : (m < n) = (m != n) && (m <= n).                          *)
(*   B -- subtraction, as in subBn : (m - n) - p = m - (n + p).               *)
(*   D -- addition, as in mulnDl : (m + n) * p = m * p + n * p.               *)
(*   M -- multiplication, as in expnMn : (m * n) ^ p = m ^ p * n ^ p.         *)
(*   p (prefix) -- positive, as in                                            *)
(*      eqn_pmul2l : m > 0 -> (m * n1 == m * n2) = (n1 == n2).                *)
(*   P  -- greater than 1, as in                                              *)
(*      ltn_Pmull : 1 < n -> 0 < m -> m < n * m.                              *)
(*   S -- successor, as in addSn : n.+1 + m = (n + m).+1.                     *)
(*   V (infix) -- disjunction, as in                                          *)
(*      leq_eqVlt : (m <= n) = (m == n) || (m < n).                           *)
(*   X - exponentiation, as in lognX : logn p (m ^ n) = logn p m * n in       *)
(*         file prime.v (the suffix is not used in this file).                 *)
(* Suffixes that abbreviate operations (D, B, M and X) are used to abbreviate *)
(* second-rank operations in equational lemma names that describe left-hand   *)
(* sides (e.g., mulnDl); they are not used to abbreviate the main operation   *)
(* of relational lemmas (e.g., leq_add2l).                                    *)
(*   For the asymmetrical exponentiation operator expn (m ^ n) a right suffix *)
(* indicates an operation on the exponent, e.g., expnM : m ^ (n1 * n2) = ...; *)
(* a trailing "n" is used to indicate the left operand, e.g.,                 *)
(* expnMn : (m1 * m2) ^ n = ... The operands of other operators are selected  *)
(* using the l/r suffixes.                                                    *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Declare Scope coq_nat_scope.
Declare Scope nat_rec_scope.
(* Disable Coq prelude hints to improve proof script robustness. *)
Remove Hints plus_n_O plus_n_Sm mult_n_O mult_n_Sm : core.
(* Declare legacy Arith operators in new scope. *)
Delimit Scope coq_nat_scope with coq_nat.
Notation "m + n" := (plus m n) : coq_nat_scope.
Notation "m - n" := (minus m n) : coq_nat_scope.
Notation "m * n" := (mult m n) : coq_nat_scope.
Notation "m <= n" := (le m n) : coq_nat_scope.
Notation "m < n" := (lt m n) : coq_nat_scope.
Notation "m >= n" := (ge m n) : coq_nat_scope.
Notation "m > n" := (gt m n) : coq_nat_scope.
(* Rebind scope delimiters, reserving a scope for the "recursive",     *)
(* i.e., unprotected version of operators.                             *)
Delimit Scope N_scope with num.
Delimit Scope nat_scope with N.
Delimit Scope nat_rec_scope with Nrec.
(* Postfix notation for the successor and predecessor functions.  *)
(* SSreflect uses "pred" for the generic predicate type, and S as *)
(* a local bound variable.                                        *)
Notation succn := Datatypes.S.
Notation predn := Peano.pred.
Notation "n .+1" := (succn n) (at level 2, left associativity,
format "n .+1") : nat_scope.
Notation "n .+2" := n.+1.+1 (at level 2, left associativity,
format "n .+2") : nat_scope.
Notation "n .+3" := n.+2.+1 (at level 2, left associativity,
format "n .+3") : nat_scope.
Notation "n .+4" := n.+2.+2 (at level 2, left associativity,
format "n .+4") : nat_scope.
Notation "n .-1" := (predn n) (at level 2, left associativity,
format "n .-1") : nat_scope.
Notation "n .-2" := n.-1.-1 (at level 2, left associativity,
format "n .-2") : nat_scope.
(* Predeclare postfix doubling/halving operators. *)
Reserved Notation "n .*2" (at level 2, format "n .*2").
Reserved Notation "n ./2" (at level 2, format "n ./2").
(* Canonical comparison and eqType for nat.                                *)
Fixpoint eqn m n {struct m} :=
match m, n with
| 0, 0 => true
| m'.+1, n'.+1 => eqn m' n'
| _, _ => false
end.
Canonical nat_eqMixin := EqMixin eqnP.
Canonical nat_eqType := Eval hnf in EqType nat nat_eqMixin.
Arguments eqn !m !n.
Arguments eqnP {x y}.
(* Link to the legacy comparison predicates. *)
Arguments leP {m n}.

Lemma leq_addl m n : n <= m + n. Proof. exact: (leq_add2r n 0). Qed.