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|
Require Import Bool.
Require Import ZArith.
Require Import Arith.
Require Export BigN.
Require Export BigZ.
Inductive q_type : Set :=
| Qz : BigZ.t -> q_type
| Qq : BigZ.t -> BigN.t -> q_type.
Definition print_type x :=
match x with
| Qz _ => Z
| _ => (Z*Z)%type
end.
Definition print x :=
match x return print_type x with
| Qz zx => BigZ.to_Z zx
| Qq nx dx => (BigZ.to_Z nx, BigN.to_Z dx)
end.
Module Qp.
Definition t := q_type.
Definition zero := Qz BigZ.zero.
Definition one := Qz BigZ.one.
Definition minus_one := Qz BigZ.minus_one.
Definition of_Z x := Qz (BigZ.of_Z x).
Definition d_to_Z d := BigZ.Pos (BigN.succ d).
Definition compare x y :=
match x, y with
| Qz zx, Qz zy => BigZ.compare zx zy
| Qz zx, Qq ny dy => BigZ.compare (BigZ.mul zx (d_to_Z dy)) ny
| Qq nx dy, Qz zy => BigZ.compare nx (BigZ.mul zy (d_to_Z dy))
| Qq nx dx, Qq ny dy =>
BigZ.compare (BigZ.mul nx (d_to_Z dy)) (BigZ.mul ny (d_to_Z dx))
end.
Definition opp x :=
match x with
| Qz zx => Qz (BigZ.opp zx)
| Qq nx dx => Qq (BigZ.opp nx) dx
end.
(* Inv d > 0, Pour la forme normal unique on veut d > 1 *)
Definition norm n d :=
if BigZ.eq_bool n BigZ.zero then zero
else
let gcd := BigN.gcd (BigZ.to_N n) d in
if BigN.eq_bool gcd BigN.one then Qq n (BigN.pred d)
else
let n := BigZ.div n (BigZ.Pos gcd) in
let d := BigN.div d gcd in
if BigN.eq_bool d BigN.one then Qz n
else Qq n (BigN.pred d).
Definition add x y :=
match x, y with
| Qz zx, Qz zy => Qz (BigZ.add zx zy)
| Qz zx, Qq ny dy => Qq (BigZ.add (BigZ.mul zx (d_to_Z dy)) ny) dy
| Qq nx dx, Qz zy => Qq (BigZ.add nx (BigZ.mul zy (d_to_Z dx))) dx
| Qq nx dx, Qq ny dy =>
let dx' := BigN.succ dx in
let dy' := BigN.succ dy in
let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy')) (BigZ.mul ny (BigZ.Pos dx')) in
let d := BigN.pred (BigN.mul dx' dy') in
Qq n d
end.
Definition add_norm x y :=
match x, y with
| Qz zx, Qz zy => Qz (BigZ.add zx zy)
| Qz zx, Qq ny dy =>
let d := BigN.succ dy in
norm (BigZ.add (BigZ.mul zx (BigZ.Pos d)) ny) d
| Qq nx dx, Qz zy =>
let d := BigN.succ dx in
norm (BigZ.add (BigZ.mul zy (BigZ.Pos d)) nx) d
| Qq nx dx, Qq ny dy =>
let dx' := BigN.succ dx in
let dy' := BigN.succ dy in
let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy')) (BigZ.mul ny (BigZ.Pos dx')) in
let d := BigN.mul dx' dy' in
norm n d
end.
Definition sub x y := add x (opp y).
Definition sub_norm x y := add_norm x (opp y).
Definition mul x y :=
match x, y with
| Qz zx, Qz zy => Qz (BigZ.mul zx zy)
| Qz zx, Qq ny dy => Qq (BigZ.mul zx ny) dy
| Qq nx dx, Qz zy => Qq (BigZ.mul nx zy) dx
| Qq nx dx, Qq ny dy =>
Qq (BigZ.mul nx ny) (BigN.pred (BigN.mul (BigN.succ dx) (BigN.succ dy)))
end.
Definition mul_norm x y :=
match x, y with
| Qz zx, Qz zy => Qz (BigZ.mul zx zy)
| Qz zx, Qq ny dy =>
if BigZ.eq_bool zx BigZ.zero then zero
else
let d := BigN.succ dy in
let gcd := BigN.gcd (BigZ.to_N zx) d in
if BigN.eq_bool gcd BigN.one then Qq (BigZ.mul zx ny) dy
else
let zx := BigZ.div zx (BigZ.Pos gcd) in
let d := BigN.div d gcd in
if BigN.eq_bool d BigN.one then Qz (BigZ.mul zx ny)
else Qq (BigZ.mul zx ny) (BigN.pred d)
| Qq nx dx, Qz zy =>
if BigZ.eq_bool zy BigZ.zero then zero
else
let d := BigN.succ dx in
let gcd := BigN.gcd (BigZ.to_N zy) d in
if BigN.eq_bool gcd BigN.one then Qq (BigZ.mul zy nx) dx
else
let zy := BigZ.div zy (BigZ.Pos gcd) in
let d := BigN.div d gcd in
if BigN.eq_bool d BigN.one then Qz (BigZ.mul zy nx)
else Qq (BigZ.mul zy nx) (BigN.pred d)
| Qq nx dx, Qq ny dy =>
norm (BigZ.mul nx ny) (BigN.mul (BigN.succ dx) (BigN.succ dy))
end.
Definition inv x :=
match x with
| Qz (BigZ.Pos n) =>
if BigN.eq_bool n BigN.zero then zero else Qq BigZ.one (BigN.pred n)
| Qz (BigZ.Neg n) =>
if BigN.eq_bool n BigN.zero then zero else Qq BigZ.minus_one (BigN.pred n)
| Qq (BigZ.Pos n) d =>
if BigN.eq_bool n BigN.zero then zero else Qq (BigZ.Pos (BigN.succ d)) (BigN.pred n)
| Qq (BigZ.Neg n) d =>
if BigN.eq_bool n BigN.zero then zero else Qq (BigZ.Neg (BigN.succ d)) (BigN.pred n)
end.
Definition inv_norm x :=
match x with
| Qz (BigZ.Pos n) =>
if BigN.eq_bool n BigN.zero then zero else
if BigN.eq_bool n BigN.one then x else Qq BigZ.one (BigN.pred n)
| Qz (BigZ.Neg n) =>
if BigN.eq_bool n BigN.zero then zero else
if BigN.eq_bool n BigN.one then x else Qq BigZ.minus_one n
| Qq (BigZ.Pos n) d => let d := BigN.succ d in
if BigN.eq_bool n BigN.one then Qz (BigZ.Pos d)
else Qq (BigZ.Pos d) (BigN.pred n)
| Qq (BigZ.Neg n) d => let d := BigN.succ d in
if BigN.eq_bool n BigN.one then Qz (BigZ.Neg d)
else Qq (BigZ.Pos d) (BigN.pred n)
end.
Definition square x :=
match x with
| Qz zx => Qz (BigZ.square zx)
| Qq nx dx => Qq (BigZ.square nx) (BigN.pred (BigN.square (BigN.succ dx)))
end.
Definition power_pos x p :=
match x with
| Qz zx => Qz (BigZ.power_pos zx p)
| Qq nx dx => Qq (BigZ.power_pos nx p) (BigN.pred (BigN.power_pos (BigN.succ dx) p))
end.
End Qp.
Module Qv.
(* /!\ Invariant maintenu par les fonctions :
- le denominateur n'est jamais nul *)
Definition t := q_type.
Definition zero := Qz BigZ.zero.
Definition one := Qz BigZ.one.
Definition minus_one := Qz BigZ.minus_one.
Definition of_Z x := Qz (BigZ.of_Z x).
Definition is_valid x :=
match x with
| Qz _ => True
| Qq n d => if BigN.eq_bool d BigN.zero then False else True
end.
(* Les fonctions doivent assurer que si leur arguments sont valides alors
le resultat est correct et valide (si c'est un Q)
*)
Definition compare x y :=
match x, y with
| Qz zx, Qz zy => BigZ.compare zx zy
| Qz zx, Qq ny dy => BigZ.compare (BigZ.mul zx (BigZ.Pos dy)) ny
| Qq nx dx, Qz zy => BigZ.compare BigZ.zero zy
| Qq nx dx, Qq ny dy => BigZ.compare (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx))
end.
Definition opp x :=
match x with
| Qz zx => Qz (BigZ.opp zx)
| Qq nx dx => Qq (BigZ.opp nx) dx
end.
Definition norm n d :=
if BigZ.eq_bool n BigZ.zero then zero
else
let gcd := BigN.gcd (BigZ.to_N n) d in
if BigN.eq_bool gcd BigN.one then Qq n d
else
let n := BigZ.div n (BigZ.Pos gcd) in
let d := BigN.div d gcd in
if BigN.eq_bool d BigN.one then Qz n
else Qq n d.
Definition add x y :=
match x, y with
| Qz zx, Qz zy => Qz (BigZ.add zx zy)
| Qz zx, Qq ny dy => Qq (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy
| Qq nx dx, Qz zy => Qq (BigZ.add nx (BigZ.mul zy (BigZ.Pos dx))) dx
| Qq nx dx, Qq ny dy =>
let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in
let d := BigN.mul dx dy in
Qq n d
end.
Definition add_norm x y :=
match x, y with
| Qz zx, Qz zy => Qz (BigZ.add zx zy)
| Qz zx, Qq ny dy =>
norm (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy
| Qq nx dx, Qz zy =>
norm (BigZ.add (BigZ.mul zy (BigZ.Pos dx)) nx) dx
| Qq nx dx, Qq ny dy =>
let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in
let d := BigN.mul dx dy in
norm n d
end.
Definition sub x y := add x (opp y).
Definition sub_norm x y := add_norm x (opp y).
Definition mul x y :=
match x, y with
| Qz zx, Qz zy => Qz (BigZ.mul zx zy)
| Qz zx, Qq ny dy => Qq (BigZ.mul zx ny) dy
| Qq nx dx, Qz zy => Qq (BigZ.mul nx zy) dx
| Qq nx dx, Qq ny dy =>
Qq (BigZ.mul nx ny) (BigN.mul dx dy)
end.
Definition mul_norm x y :=
match x, y with
| Qz zx, Qz zy => Qz (BigZ.mul zx zy)
| Qz zx, Qq ny dy =>
if BigZ.eq_bool zx BigZ.zero then zero
else
let gcd := BigN.gcd (BigZ.to_N zx) dy in
if BigN.eq_bool gcd BigN.one then Qq (BigZ.mul zx ny) dy
else
let zx := BigZ.div zx (BigZ.Pos gcd) in
let d := BigN.div dy gcd in
if BigN.eq_bool d BigN.one then Qz (BigZ.mul zx ny)
else Qq (BigZ.mul zx ny) d
| Qq nx dx, Qz zy =>
if BigZ.eq_bool zy BigZ.zero then zero
else
let gcd := BigN.gcd (BigZ.to_N zy) dx in
if BigN.eq_bool gcd BigN.one then Qq (BigZ.mul zy nx) dx
else
let zy := BigZ.div zy (BigZ.Pos gcd) in
let d := BigN.div dx gcd in
if BigN.eq_bool d BigN.one then Qz (BigZ.mul zy nx)
else Qq (BigZ.mul zy nx) d
| Qq nx dx, Qq ny dy => norm (BigZ.mul nx ny) (BigN.mul dx dy)
end.
Definition inv x :=
match x with
| Qz (BigZ.Pos n) =>
if BigN.eq_bool n BigN.zero then zero else Qq BigZ.one n
| Qz (BigZ.Neg n) =>
if BigN.eq_bool n BigN.zero then zero else Qq BigZ.minus_one n
| Qq (BigZ.Pos n) d =>
if BigN.eq_bool n BigN.zero then zero else Qq (BigZ.Pos d) n
| Qq (BigZ.Neg n) d =>
if BigN.eq_bool n BigN.zero then zero else Qq (BigZ.Neg d) n
end.
Definition square x :=
match x with
| Qz zx => Qz (BigZ.square zx)
| Qq nx dx => Qq (BigZ.square nx) (BigN.square dx)
end.
Definition power_pos x p :=
match x with
| Qz zx => Qz (BigZ.power_pos zx p)
| Qq nx dx => Qq (BigZ.power_pos nx p) (BigN.power_pos dx p)
end.
End Qv.
Module Q.
(* Troisieme solution :
0 a de nombreuse representation :
0, -0, 1/0, ... n/0,
il faut alors faire attention avec la comparaison et l'addition
*)
Definition t := q_type.
Definition zero := Qz BigZ.zero.
Definition one := Qz BigZ.one.
Definition minus_one := Qz BigZ.minus_one.
Definition of_Z x := Qz (BigZ.of_Z x).
Definition compare x y :=
match x, y with
| Qz zx, Qz zy => BigZ.compare zx zy
| Qz zx, Qq ny dy =>
if BigN.eq_bool dy BigN.zero then BigZ.compare zx BigZ.zero
else BigZ.compare (BigZ.mul zx (BigZ.Pos dy)) ny
| Qq nx dx, Qz zy =>
if BigN.eq_bool dx BigN.zero then BigZ.compare BigZ.zero zy
else BigZ.compare nx (BigZ.mul zy (BigZ.Pos dx))
| Qq nx dx, Qq ny dy =>
match BigN.eq_bool dx BigN.zero, BigN.eq_bool dy BigN.zero with
| true, true => Eq
| true, false => BigZ.compare BigZ.zero ny
| false, true => BigZ.compare nx BigZ.zero
| false, false => BigZ.compare (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx))
end
end.
Definition opp x :=
match x with
| Qz zx => Qz (BigZ.opp zx)
| Qq nx dx => Qq (BigZ.opp nx) dx
end.
(* Je pense que cette fonction normalise bien ... *)
Definition norm n d :=
let gcd := BigN.gcd (BigZ.to_N n) d in
match BigN.compare BigN.one gcd with
| Lt =>
let n := BigZ.div n (BigZ.Pos gcd) in
let d := BigN.div d gcd in
match BigN.compare d BigN.one with
| Gt => Qq n d
| Eq => Qz n
| Lt => zero
end
| Eq => Qq n d
| Gt => zero (* gcd = 0 => both numbers are 0 *)
end.
Definition add x y :=
match x with
| Qz zx =>
match y with
| Qz zy => Qz (BigZ.add zx zy)
| Qq ny dy =>
if BigN.eq_bool dy BigN.zero then x
else Qq (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy
end
| Qq nx dx =>
if BigN.eq_bool dx BigN.zero then y
else match y with
| Qz zy => Qq (BigZ.add nx (BigZ.mul zy (BigZ.Pos dx))) dx
| Qq ny dy =>
if BigN.eq_bool dy BigN.zero then x
else
let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in
let d := BigN.mul dx dy in
Qq n d
end
end.
Definition add_norm x y :=
match x with
| Qz zx =>
match y with
| Qz zy => Qz (BigZ.add zx zy)
| Qq ny dy =>
if BigN.eq_bool dy BigN.zero then x
else norm (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy
end
| Qq nx dx =>
if BigN.eq_bool dx BigN.zero then y
else match y with
| Qz zy => norm (BigZ.add nx (BigZ.mul zy (BigZ.Pos dx))) dx
| Qq ny dy =>
if BigN.eq_bool dy BigN.zero then x
else
let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in
let d := BigN.mul dx dy in
norm n d
end
end.
Definition sub x y := add x (opp y).
Definition sub_norm x y := add_norm x (opp y).
Definition mul x y :=
match x, y with
| Qz zx, Qz zy => Qz (BigZ.mul zx zy)
| Qz zx, Qq ny dy => Qq (BigZ.mul zx ny) dy
| Qq nx dx, Qz zy => Qq (BigZ.mul nx zy) dx
| Qq nx dx, Qq ny dy => Qq (BigZ.mul nx ny) (BigN.mul dx dy)
end.
Definition mul_norm x y :=
match x, y with
| Qz zx, Qz zy => Qz (BigZ.mul zx zy)
| Qz zx, Qq ny dy =>
if BigZ.eq_bool zx BigZ.zero then zero
else
let d := BigN.succ dy in
let gcd := BigN.gcd (BigZ.to_N zx) d in
if BigN.eq_bool gcd BigN.one then Qq (BigZ.mul zx ny) dy
else
let zx := BigZ.div zx (BigZ.Pos gcd) in
let d := BigN.div d gcd in
if BigN.eq_bool d BigN.one then Qz (BigZ.mul zx ny)
else Qq (BigZ.mul zx ny) d
| Qq nx dx, Qz zy =>
if BigZ.eq_bool zy BigZ.zero then zero
else
let d := BigN.succ dx in
let gcd := BigN.gcd (BigZ.to_N zy) d in
if BigN.eq_bool gcd BigN.one then Qq (BigZ.mul zy nx) dx
else
let zy := BigZ.div zy (BigZ.Pos gcd) in
let d := BigN.div d gcd in
if BigN.eq_bool d BigN.one then Qz (BigZ.mul zy nx)
else Qq (BigZ.mul zy nx) d
| Qq nx dx, Qq ny dy =>
let (nx, dy) :=
let gcd := BigN.gcd (BigZ.to_N nx) dy in
if BigN.eq_bool gcd BigN.one then (nx, dy)
else (BigZ.div nx (BigZ.Pos gcd), BigN.div dy gcd) in
let (ny, dx) :=
let gcd := BigN.gcd (BigZ.to_N ny) dx in
if BigN.eq_bool gcd BigN.one then (ny, dx)
else (BigZ.div ny (BigZ.Pos gcd), BigN.div dx gcd) in
let d := (BigN.mul dx dy) in
if BigN.eq_bool d BigN.one then Qz (BigZ.mul ny nx)
else Qq (BigZ.mul ny nx) d
end.
Definition inv x :=
match x with
| Qz (BigZ.Pos n) => Qq BigZ.one (BigN.pred n)
| Qz (BigZ.Neg n) => Qq BigZ.minus_one (BigN.pred n)
| Qq (BigZ.Pos n) d => Qq (BigZ.Pos (BigN.succ d)) (BigN.pred n)
| Qq (BigZ.Neg n) d => Qq (BigZ.Neg (BigN.succ d)) (BigN.pred n)
end.
Definition inv_norm x :=
match x with
| Qz (BigZ.Pos n) => if BigN.eq_bool n BigN.one then x else Qq BigZ.one (BigN.pred n)
| Qz (BigZ.Neg n) => if BigN.eq_bool n BigN.one then x else Qq BigZ.minus_one n
| Qq (BigZ.Pos n) d => let d := BigN.succ d in
if BigN.eq_bool n BigN.one then Qz (BigZ.Pos d)
else Qq (BigZ.Pos d) (BigN.pred n)
| Qq (BigZ.Neg n) d => let d := BigN.succ d in
if BigN.eq_bool n BigN.one then Qz (BigZ.Neg d)
else Qq (BigZ.Pos d) (BigN.pred n)
end.
Definition square x :=
match x with
| Qz zx => Qz (BigZ.square zx)
| Qq nx dx => Qq (BigZ.square nx) (BigN.square dx)
end.
Definition power_pos x p :=
match x with
| Qz zx => Qz (BigZ.power_pos zx p)
| Qq nx dx => Qq (BigZ.power_pos nx p) (BigN.power_pos dx p)
end.
End Q.
Module Qif.
(* Troisieme solution :
0 a de nombreuse representation :
0, -0, 1/0, ... n/0,
il faut alors faire attention avec la comparaison et l'addition
Les fonctions de normalization s'effectue seulement si les
nombres sont grands.
*)
Definition t := q_type.
Definition zero := Qz BigZ.zero.
Definition one := Qz BigZ.one.
Definition minus_one := Qz BigZ.minus_one.
Definition of_Z x := Qz (BigZ.of_Z x).
Definition compare x y :=
match x, y with
| Qz zx, Qz zy => BigZ.compare zx zy
| Qz zx, Qq ny dy =>
if BigN.eq_bool dy BigN.zero then BigZ.compare zx BigZ.zero
else BigZ.compare (BigZ.mul zx (BigZ.Pos dy)) ny
| Qq nx dx, Qz zy =>
if BigN.eq_bool dx BigN.zero then BigZ.compare BigZ.zero zy
else BigZ.compare nx (BigZ.mul zy (BigZ.Pos dx))
| Qq nx dx, Qq ny dy =>
match BigN.eq_bool dx BigN.zero, BigN.eq_bool dy BigN.zero with
| true, true => Eq
| true, false => BigZ.compare BigZ.zero ny
| false, true => BigZ.compare nx BigZ.zero
| false, false => BigZ.compare (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx))
end
end.
Definition opp x :=
match x with
| Qz zx => Qz (BigZ.opp zx)
| Qq nx dx => Qq (BigZ.opp nx) dx
end.
Definition do_norm_n n :=
match n with
| BigN.N0 _ => false
| BigN.N1 _ => false
| BigN.N2 _ => false
| BigN.N3 _ => false
| BigN.N4 _ => false
| BigN.N5 _ => false
| BigN.N6 _ => false
| BigN.N7 _ => false
| BigN.N8 _ => false
| BigN.N9 _ => true
| BigN.N10 _ => true
| BigN.N11 _ => true
| BigN.N12 _ => true
| BigN.Nn n _ => true
end.
Definition do_norm_z z :=
match z with
| BigZ.Pos n => do_norm_n n
| BigZ.Neg n => do_norm_n n
end.
Require Import Bool.
(* Je pense que cette fonction normalise bien ... *)
Definition norm n d :=
if andb (do_norm_z n) (do_norm_n d) then
let gcd := BigN.gcd (BigZ.to_N n) d in
match BigN.compare BigN.one gcd with
| Lt =>
let n := BigZ.div n (BigZ.Pos gcd) in
let d := BigN.div d gcd in
match BigN.compare d BigN.one with
| Gt => Qq n d
| Eq => Qz n
| Lt => zero
end
| Eq => Qq n d
| Gt => zero (* gcd = 0 => both numbers are 0 *)
end
else Qq n d.
Definition add x y :=
match x with
| Qz zx =>
match y with
| Qz zy => Qz (BigZ.add zx zy)
| Qq ny dy =>
if BigN.eq_bool dy BigN.zero then x
else Qq (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy
end
| Qq nx dx =>
if BigN.eq_bool dx BigN.zero then y
else match y with
| Qz zy => Qq (BigZ.add nx (BigZ.mul zy (BigZ.Pos dx))) dx
| Qq ny dy =>
if BigN.eq_bool dy BigN.zero then x
else
let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in
let d := BigN.mul dx dy in
Qq n d
end
end.
Definition add_norm x y :=
match x with
| Qz zx =>
match y with
| Qz zy => Qz (BigZ.add zx zy)
| Qq ny dy =>
if BigN.eq_bool dy BigN.zero then x
else norm (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy
end
| Qq nx dx =>
if BigN.eq_bool dx BigN.zero then y
else match y with
| Qz zy => norm (BigZ.add nx (BigZ.mul zy (BigZ.Pos dx))) dx
| Qq ny dy =>
if BigN.eq_bool dy BigN.zero then x
else
let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in
let d := BigN.mul dx dy in
norm n d
end
end.
Definition sub x y := add x (opp y).
Definition sub_norm x y := add_norm x (opp y).
Definition mul x y :=
match x, y with
| Qz zx, Qz zy => Qz (BigZ.mul zx zy)
| Qz zx, Qq ny dy => Qq (BigZ.mul zx ny) dy
| Qq nx dx, Qz zy => Qq (BigZ.mul nx zy) dx
| Qq nx dx, Qq ny dy => Qq (BigZ.mul nx ny) (BigN.mul dx dy)
end.
Definition mul_norm x y :=
match x, y with
| Qz zx, Qz zy => Qz (BigZ.mul zx zy)
| Qz zx, Qq ny dy => norm (BigZ.mul zx ny) dy
| Qq nx dx, Qz zy => norm (BigZ.mul nx zy) dx
| Qq nx dx, Qq ny dy => norm (BigZ.mul nx ny) (BigN.mul dx dy)
end.
Definition inv x :=
match x with
| Qz (BigZ.Pos n) => Qq BigZ.one n
| Qz (BigZ.Neg n) => Qq BigZ.minus_one n
| Qq (BigZ.Pos n) d => Qq (BigZ.Pos d) n
| Qq (BigZ.Neg n) d => Qq (BigZ.Neg d) n
end.
Definition inv_norm x :=
match x with
| Qz (BigZ.Pos n) =>
if BigN.eq_bool n BigN.zero then zero else Qq BigZ.one n
| Qz (BigZ.Neg n) =>
if BigN.eq_bool n BigN.zero then zero else Qq BigZ.minus_one n
| Qq (BigZ.Pos n) d =>
if BigN.eq_bool n BigN.zero then zero else Qq (BigZ.Pos d) n
| Qq (BigZ.Neg n) d =>
if BigN.eq_bool n BigN.zero then zero else Qq (BigZ.Neg d) n
end.
Definition square x :=
match x with
| Qz zx => Qz (BigZ.square zx)
| Qq nx dx => Qq (BigZ.square nx) (BigN.square dx)
end.
Definition power_pos x p :=
match x with
| Qz zx => Qz (BigZ.power_pos zx p)
| Qq nx dx => Qq (BigZ.power_pos nx p) (BigN.power_pos dx p)
end.
End Qif.
Module Qbi.
(* Troisieme solution :
0 a de nombreuse representation :
0, -0, 1/0, ... n/0,
il faut alors faire attention avec la comparaison et l'addition
Les fonctions de normalization s'effectue seulement si les
nombres sont grands.
*)
Definition t := q_type.
Definition zero := Qz BigZ.zero.
Definition one := Qz BigZ.one.
Definition minus_one := Qz BigZ.minus_one.
Definition of_Z x := Qz (BigZ.of_Z x).
Definition compare x y :=
match x, y with
| Qz zx, Qz zy => BigZ.compare zx zy
| Qz zx, Qq ny dy =>
if BigN.eq_bool dy BigN.zero then BigZ.compare zx BigZ.zero
else
match BigZ.cmp_sign zx ny with
| Lt => Lt
| Gt => Gt
| Eq => BigZ.compare (BigZ.mul zx (BigZ.Pos dy)) ny
end
| Qq nx dx, Qz zy =>
if BigN.eq_bool dx BigN.zero then BigZ.compare BigZ.zero zy
else
match BigZ.cmp_sign nx zy with
| Lt => Lt
| Gt => Gt
| Eq => BigZ.compare nx (BigZ.mul zy (BigZ.Pos dx))
end
| Qq nx dx, Qq ny dy =>
match BigN.eq_bool dx BigN.zero, BigN.eq_bool dy BigN.zero with
| true, true => Eq
| true, false => BigZ.compare BigZ.zero ny
| false, true => BigZ.compare nx BigZ.zero
| false, false =>
match BigZ.cmp_sign nx ny with
| Lt => Lt
| Gt => Gt
| Eq => BigZ.compare (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx))
end
end
end.
Definition opp x :=
match x with
| Qz zx => Qz (BigZ.opp zx)
| Qq nx dx => Qq (BigZ.opp nx) dx
end.
Definition do_norm_n n :=
match n with
| BigN.N0 _ => false
| BigN.N1 _ => false
| BigN.N2 _ => false
| BigN.N3 _ => false
| BigN.N4 _ => false
| BigN.N5 _ => false
| BigN.N6 _ => false
| BigN.N7 _ => false
| BigN.N8 _ => false
| BigN.N9 _ => true
| BigN.N10 _ => true
| BigN.N11 _ => true
| BigN.N12 _ => true
| BigN.Nn n _ => true
end.
Definition do_norm_z z :=
match z with
| BigZ.Pos n => do_norm_n n
| BigZ.Neg n => do_norm_n n
end.
(* Je pense que cette fonction normalise bien ... *)
Definition norm n d :=
if andb (do_norm_z n) (do_norm_n d) then
let gcd := BigN.gcd (BigZ.to_N n) d in
match BigN.compare BigN.one gcd with
| Lt =>
let n := BigZ.div n (BigZ.Pos gcd) in
let d := BigN.div d gcd in
match BigN.compare d BigN.one with
| Gt => Qq n d
| Eq => Qz n
| Lt => zero
end
| Eq => Qq n d
| Gt => zero (* gcd = 0 => both numbers are 0 *)
end
else Qq n d.
Definition add x y :=
match x with
| Qz zx =>
match y with
| Qz zy => Qz (BigZ.add zx zy)
| Qq ny dy =>
if BigN.eq_bool dy BigN.zero then x
else Qq (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy
end
| Qq nx dx =>
if BigN.eq_bool dx BigN.zero then y
else match y with
| Qz zy => Qq (BigZ.add nx (BigZ.mul zy (BigZ.Pos dx))) dx
| Qq ny dy =>
if BigN.eq_bool dy BigN.zero then x
else
if BigN.eq_bool dx dy then
let n := BigZ.add nx ny in
Qq n dx
else
let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in
let d := BigN.mul dx dy in
Qq n d
end
end.
Definition add_norm x y :=
match x with
| Qz zx =>
match y with
| Qz zy => Qz (BigZ.add zx zy)
| Qq ny dy =>
if BigN.eq_bool dy BigN.zero then x
else
norm (BigZ.add (BigZ.mul zx (BigZ.Pos dy)) ny) dy
end
| Qq nx dx =>
if BigN.eq_bool dx BigN.zero then y
else match y with
| Qz zy => norm (BigZ.add nx (BigZ.mul zy (BigZ.Pos dx))) dx
| Qq ny dy =>
if BigN.eq_bool dy BigN.zero then x
else
if BigN.eq_bool dx dy then
let n := BigZ.add nx ny in
norm n dx
else
let n := BigZ.add (BigZ.mul nx (BigZ.Pos dy)) (BigZ.mul ny (BigZ.Pos dx)) in
let d := BigN.mul dx dy in
norm n d
end
end.
Definition sub x y := add x (opp y).
Definition sub_norm x y := add_norm x (opp y).
Definition mul x y :=
match x, y with
| Qz zx, Qz zy => Qz (BigZ.mul zx zy)
| Qz zx, Qq ny dy => Qq (BigZ.mul zx ny) dy
| Qq nx dx, Qz zy => Qq (BigZ.mul nx zy) dx
| Qq nx dx, Qq ny dy => Qq (BigZ.mul nx ny) (BigN.mul dx dy)
end.
Definition mul_norm x y :=
match x, y with
| Qz zx, Qz zy => Qz (BigZ.mul zx zy)
| Qz zx, Qq ny dy => mul (Qz ny) (norm zx dy)
| Qq nx dx, Qz zy => mul (Qz nx) (norm zy dx)
| Qq nx dx, Qq ny dy => mul (norm nx dy) (norm ny dx)
end.
Definition inv x :=
match x with
| Qz (BigZ.Pos n) => Qq BigZ.one n
| Qz (BigZ.Neg n) => Qq BigZ.minus_one n
| Qq (BigZ.Pos n) d => Qq (BigZ.Pos d) n
| Qq (BigZ.Neg n) d => Qq (BigZ.Neg d) n
end.
Definition inv_norm x :=
match x with
| Qz (BigZ.Pos n) =>
if BigN.eq_bool n BigN.zero then zero else Qq BigZ.one n
| Qz (BigZ.Neg n) =>
if BigN.eq_bool n BigN.zero then zero else Qq BigZ.minus_one n
| Qq (BigZ.Pos n) d =>
if BigN.eq_bool n BigN.zero then zero else Qq (BigZ.Pos d) n
| Qq (BigZ.Neg n) d =>
if BigN.eq_bool n BigN.zero then zero else Qq (BigZ.Neg d) n
end.
Definition square x :=
match x with
| Qz zx => Qz (BigZ.square zx)
| Qq nx dx => Qq (BigZ.square nx) (BigN.square dx)
end.
Definition power_pos x p :=
match x with
| Qz zx => Qz (BigZ.power_pos zx p)
| Qq nx dx => Qq (BigZ.power_pos nx p) (BigN.power_pos dx p)
end.
End Qbi.
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