The Coq Proof Assistant

# Library Coq.Reals.Rdefinitions

Require Export ZArith_base.
Require Import QArith_base.
Require Import ConstructiveCauchyReals.
Require Import ConstructiveCauchyRealsMult.
Require Import ConstructiveRcomplete.
Require Import ClassicalDedekindReals.
Declare Scope hex_R_scope.
Declare Scope R_scope.
Delimit Scope hex_R_scope with xR.
Delimit Scope R_scope with R.
Local Open Scope R_scope.
Module Type RbaseSymbolsSig.
Parameter R : Set.
Bind Scope R_scope with R.
Axiom Rabst : CReal -> R.
Axiom Rrepr : R -> CReal.
Axiom Rquot1 : forall x y:R, CRealEq (Rrepr x) (Rrepr y) -> x = y.
Axiom Rquot2 : forall x:CReal, CRealEq (Rrepr (Rabst x)) x.
Parameter R0 : R.
Parameter R1 : R.
Parameter Rplus : R -> R -> R.
Parameter Rmult : R -> R -> R.
Parameter Ropp : R -> R.
Parameter Rlt : R -> R -> Prop.
Parameter R0_def : R0 = Rabst (inject_Q 0).
Parameter R1_def : R1 = Rabst (inject_Q 1).
Parameter Rplus_def : forall x y : R,
Rplus x y = Rabst (CReal_plus (Rrepr x) (Rrepr y)).
Parameter Rmult_def : forall x y : R,
Rmult x y = Rabst (CReal_mult (Rrepr x) (Rrepr y)).
Parameter Ropp_def : forall x : R,
Ropp x = Rabst (CReal_opp (Rrepr x)).
Parameter Rlt_def : forall x y : R,
Rlt x y = CRealLtProp (Rrepr x) (Rrepr y).
End RbaseSymbolsSig.
Module RbaseSymbolsImpl : RbaseSymbolsSig.
Definition R := DReal.
Definition Rabst := DRealAbstr.
Definition Rrepr := DRealRepr.
Definition Rquot1 := DRealQuot1.
Definition Rquot2 := DRealQuot2.
Definition R0 : R := Rabst (inject_Q 0).
Definition R1 : R := Rabst (inject_Q 1).
Definition Rplus : R -> R -> R
:= fun x y : R => Rabst (CReal_plus (Rrepr x) (Rrepr y)).
Definition Rmult : R -> R -> R
:= fun x y : R => Rabst (CReal_mult (Rrepr x) (Rrepr y)).
Definition Ropp : R -> R
:= fun x : R => Rabst (CReal_opp (Rrepr x)).
Definition Rlt : R -> R -> Prop
:= fun x y : R => CRealLtProp (Rrepr x) (Rrepr y).
Definition R0_def := eq_refl R0.
Definition R1_def := eq_refl R1.
Definition Rplus_def := fun x y => eq_refl (Rplus x y).
Definition Rmult_def := fun x y => eq_refl (Rmult x y).
Definition Ropp_def := fun x => eq_refl (Ropp x).
Definition Rlt_def := fun x y => eq_refl (Rlt x y).
End RbaseSymbolsImpl.
Export RbaseSymbolsImpl.
Notation R := RbaseSymbolsImpl.R (only parsing).
Notation R0 := RbaseSymbolsImpl.R0 (only parsing).
Notation R1 := RbaseSymbolsImpl.R1 (only parsing).
Notation Rplus := RbaseSymbolsImpl.Rplus (only parsing).
Notation Rmult := RbaseSymbolsImpl.Rmult (only parsing).
Notation Ropp := RbaseSymbolsImpl.Ropp (only parsing).
Notation Rlt := RbaseSymbolsImpl.Rlt (only parsing).
Bind Scope R_scope with R.
Infix "+" := Rplus : R_scope.
Infix "*" := Rmult : R_scope.
Notation "- x" := (Ropp x) : R_scope.
Infix "<" := Rlt : R_scope.
Definition Rgt (r1 r2:R) : Prop := r2 < r1.
Definition Rle (r1 r2:R) : Prop := r1 < r2 \/ r1 = r2.
Definition Rge (r1 r2:R) : Prop := Rgt r1 r2 \/ r1 = r2.
Definition Rminus (r1 r2:R) : R := r1 + - r2.
Infix "-" := Rminus : R_scope.
Infix "<=" := Rle : R_scope.
Infix ">=" := Rge : R_scope.
Infix ">" := Rgt : R_scope.
Notation "x <= y <= z" := (x <= y /\ y <= z) : R_scope.
Notation "x <= y < z" := (x <= y /\ y < z) : R_scope.
Notation "x < y < z" := (x < y /\ y < z) : R_scope.
Notation "x < y <= z" := (x < y /\ y <= z) : R_scope.

# Injection from Z to R

Fixpoint IPR_2 (p:positive) : R :=
match p with
| xH => R1 + R1
| xO p => (R1 + R1) * IPR_2 p
| xI p => (R1 + R1) * (R1 + IPR_2 p)
end.
Definition IPR (p:positive) : R :=
match p with
| xH => R1
| xO p => IPR_2 p
| xI p => R1 + IPR_2 p
end.
Arguments IPR p%positive : simpl never.
Definition IZR (z:Z) : R :=
match z with
| Z0 => R0
| Zpos n => IPR n
| Zneg n => - IPR n
end.
Arguments IZR z%Z : simpl never.
Lemma total_order_T : forall r1 r2:R, {Rlt r1 r2} + {r1 = r2} + {Rlt r2 r1}.
Lemma Req_appart_dec : forall x y : R,
{ x = y } + { x < y \/ y < x }.
Lemma Rrepr_appart_0 : forall x:R,
(x < R0 \/ R0 < x) -> CReal_appart (Rrepr x) (inject_Q 0).
Module Type RinvSig.
Parameter Rinv : R -> R.
Parameter Rinv_def : forall x : R,
Rinv x = match Req_appart_dec x R0 with
| left _ => R0
| right r => Rabst ((CReal_inv (Rrepr x) (Rrepr_appart_0 x r)))
end.
End RinvSig.
Module RinvImpl : RinvSig.
Definition Rinv : R -> R
:= fun x => match Req_appart_dec x R0 with
| left _ => R0
| right r => Rabst ((CReal_inv (Rrepr x) (Rrepr_appart_0 x r)))
end.
Definition Rinv_def := fun x => eq_refl (Rinv x).
End RinvImpl.
Notation Rinv := RinvImpl.Rinv (only parsing).
Notation "/ x" := (Rinv x) : R_scope.
Definition Rdiv (r1 r2:R) : R := r1 * / r2.
Infix "/" := Rdiv : R_scope.
Definition up (x : R) : Z.
Injection of rational numbers into real numbers.
Definition Q2R (x : Q) : R := (IZR (Qnum x) * / IZR (QDen x))%R.

# Number notation for constants

Inductive IR :=
| IRZ : IZ -> IR
| IRQ : Q -> IR
| IRmult : IR -> IR -> IR
| IRdiv : IR -> IR -> IR.
Definition of_decimal (d : Decimal.decimal) : IR :=
let '(i, f, e) :=
match d with
| Decimal.Decimal i f => (i, f, Decimal.Pos Decimal.Nil)
| Decimal.DecimalExp i f e => (i, f, e)
end in
let zq := match f with
| Decimal.Nil => IRZ (IZ_of_Z (Z.of_int i))
| _ =>
let num := Z.of_int (Decimal.app_int i f) in
let den := Nat.iter (Decimal.nb_digits f) (Pos.mul 10) 1%positive in
IRQ (Qmake num den) end in
let e := Z.of_int e in
match e with
| Z0 => zq
| Zpos e => IRmult zq (IRZ (IZpow_pos 10 e))
| Zneg e => IRdiv zq (IRZ (IZpow_pos 10 e))
end.
Definition of_hexadecimal (d : Hexadecimal.hexadecimal) : IR :=
let '(i, f, e) :=
match d with
| Hexadecimal.Hexadecimal i f => (i, f, Decimal.Pos Decimal.Nil)
| Hexadecimal.HexadecimalExp i f e => (i, f, e)
end in
let zq := match f with
| Hexadecimal.Nil => IRZ (IZ_of_Z (Z.of_hex_int i))
| _ =>
let num := Z.of_hex_int (Hexadecimal.app_int i f) in
let den := Nat.iter (Hexadecimal.nb_digits f) (Pos.mul 16) 1%positive in
IRQ (Qmake num den) end in
let e := Z.of_int e in
match e with
| Z0 => zq
| Zpos e => IRmult zq (IRZ (IZpow_pos 2 e))
| Zneg e => IRdiv zq (IRZ (IZpow_pos 2 e))
end.
Definition of_number (n : Number.number) : IR :=
match n with
| Number.Decimal d => of_decimal d
| Number.Hexadecimal h => of_hexadecimal h
end.
Definition IQmake_to_decimal num den :=
match den with
| 1%positive => None
| _ => IQmake_to_decimal num den
end.
Definition to_decimal (n : IR) : option Decimal.decimal :=
match n with
| IRZ z =>
match IZ_to_Z z with
| Some z => Some (Decimal.Decimal (Z.to_int z) Decimal.Nil)
| None => None
end
| IRQ (Qmake num den) => IQmake_to_decimal num den
| IRmult (IRZ z) (IRZ (IZpow_pos 10 e)) =>
match IZ_to_Z z with
| Some z =>
Some (Decimal.DecimalExp (Z.to_int z) Decimal.Nil (Pos.to_int e))
| None => None
end
| IRmult (IRQ (Qmake num den)) (IRZ (IZpow_pos 10 e)) =>
match IQmake_to_decimal num den with
| Some (Decimal.Decimal i f) =>
Some (Decimal.DecimalExp i f (Pos.to_int e))
| _ => None
end
| IRdiv (IRZ z) (IRZ (IZpow_pos 10 e)) =>
match IZ_to_Z z with
| Some z =>
Some (Decimal.DecimalExp (Z.to_int z) Decimal.Nil (Decimal.Neg (Pos.to_uint e)))
| None => None
end
| IRdiv (IRQ (Qmake num den)) (IRZ (IZpow_pos 10 e)) =>
match IQmake_to_decimal num den with
| Some (Decimal.Decimal i f) =>
Some (Decimal.DecimalExp i f (Decimal.Neg (Pos.to_uint e)))
| _ => None
end
| _ => None
end.
Definition IQmake_to_hexadecimal num den :=
match den with
| 1%positive => None
| _ => IQmake_to_hexadecimal num den
end.
Definition to_hexadecimal (n : IR) : option Hexadecimal.hexadecimal :=
match n with
| IRZ z =>
match IZ_to_Z z with
| Some z => Some (Hexadecimal.Hexadecimal (Z.to_hex_int z) Hexadecimal.Nil)
| None => None
end
| IRQ (Qmake num den) => IQmake_to_hexadecimal num den
| IRmult (IRZ z) (IRZ (IZpow_pos 2 e)) =>
match IZ_to_Z z with
| Some z =>
Some (Hexadecimal.HexadecimalExp (Z.to_hex_int z) Hexadecimal.Nil (Pos.to_int e))
| None => None
end
| IRmult (IRQ (Qmake num den)) (IRZ (IZpow_pos 2 e)) =>
match IQmake_to_hexadecimal num den with
| Some (Hexadecimal.Hexadecimal i f) =>
Some (Hexadecimal.HexadecimalExp i f (Pos.to_int e))
| _ => None
end
| IRdiv (IRZ z) (IRZ (IZpow_pos 2 e)) =>
match IZ_to_Z z with
| Some z =>
Some (Hexadecimal.HexadecimalExp (Z.to_hex_int z) Hexadecimal.Nil (Decimal.Neg (Pos.to_uint e)))
| None => None
end
| IRdiv (IRQ (Qmake num den)) (IRZ (IZpow_pos 2 e)) =>
match IQmake_to_hexadecimal num den with
| Some (Hexadecimal.Hexadecimal i f) =>
Some (Hexadecimal.HexadecimalExp i f (Decimal.Neg (Pos.to_uint e)))
| _ => None
end
| _ => None
end.
Definition to_number q :=
match to_decimal q with
| None => None
| Some q => Some (Number.Decimal q)
end.
Definition to_hex_number q :=
match to_hexadecimal q with
| None => None
| Some q => Some (Number.Hexadecimal q)
end.