Timings for Core.v
Require Import Spaces.Pos.Core.
Local Set Universe Minimization ToSet.
Local Close Scope trunc_scope.
Local Close Scope nat_scope.
Local Open Scope positive_scope.
(** ** Definition of the Integers *)
(** We define an integer as being a positive number labelled negative, zero or a positive number labelled positive. *)
Inductive Int : Type0 :=
| neg : Pos -> Int
| zero : Int
| pos : Pos -> Int.
Local Open Scope int_scope.
Delimit Scope int_scope with int.
(** The integers are a pointed type *)
Global Instance ispointed_Int : IsPointed Int := zero.
(** Properties of constructors *)
Definition neg_inj {z w : Pos} (p : neg z = neg w) : z = w
:= transport (fun s => z =
(match s with neg a => a | zero => w | pos a => w end)) p (idpath z).
Definition pos_inj {z w : Pos} (p : pos z = pos w) : z = w
:= transport (fun s => z =
(match s with neg a => w | zero => w | pos a => a end)) p (idpath z).
Definition neg_neq_zero {z : Pos} : ~ (neg z = zero)
:= fun p => transport (fun s =>
match s with neg a => z = a| zero => Empty | pos _ => Empty end) p (idpath z).
Definition pos_neq_zero {z : Pos} : ~ (pos z = zero)
:= fun p => transport (fun s =>
match s with pos a => z = a | zero => Empty | neg _ => Empty end) p (idpath z).
Definition neg_neq_pos {z w : Pos} : ~ (neg z = pos w)
:= fun p => transport (fun s =>
match s with neg a => z = a | zero => Empty | pos _ => Empty end) p (idpath z).
Definition zero_neq_neg {z : Pos} := @neg_neq_zero z o symmetry _ _.
Definition zero_neq_pos {z : Pos} := @pos_neq_zero z o symmetry _ _.
Definition pos_neq_neg {z w : Pos} := @neg_neq_pos z w o symmetry _ _.
(** ** Conversion with a decimal representation for printing/parsing *)
Definition int_to_decimal_int (n : Int) : Decimal.int :=
match n with
| neg m => Decimal.Neg (pos_to_uint m)
| zero => Decimal.Pos Decimal.Nil
| pos m => Decimal.Pos (pos_to_uint m)
end.
Definition int_to_number_int (n : Int) : Numeral.int :=
Numeral.IntDec (int_to_decimal_int n).
Fixpoint int_of_decimal_uint (d : Decimal.uint) : Int :=
match d with
| Decimal.Nil => zero
| Decimal.D0 l => int_of_decimal_uint l
| Decimal.D1 l => pos (pos_of_uint_acc l 1)
| Decimal.D2 l => pos (pos_of_uint_acc l 1~0)
| Decimal.D3 l => pos (pos_of_uint_acc l 1~1)
| Decimal.D4 l => pos (pos_of_uint_acc l 1~0~0)
| Decimal.D5 l => pos (pos_of_uint_acc l 1~0~1)
| Decimal.D6 l => pos (pos_of_uint_acc l 1~1~0)
| Decimal.D7 l => pos (pos_of_uint_acc l 1~1~1)
| Decimal.D8 l => pos (pos_of_uint_acc l 1~0~0~0)
| Decimal.D9 l => pos (pos_of_uint_acc l 1~0~0~1)
end.
Definition int_of_decimal_int (d : Decimal.int) : Int :=
match d with
| Decimal.Pos u => int_of_decimal_uint u
| Decimal.Neg u => let t := int_of_decimal_uint u in
match t with
| pos v => neg v
| _ => zero
end
end.
Definition int_of_number_int (d:Numeral.int) :=
match d with
| Numeral.IntDec d => Some (int_of_decimal_int d)
| Numeral.IntHex _ => None
end.
Number Notation Int int_of_number_int int_to_number_int : int_scope.
(* For some reason 0 can be parsed as an integer, but is printed as [zero]. This notation fixes that. *)
Notation "0" := zero : int_scope.
(** ** Doubling and variants *)
Definition int_double x :=
match x with
| 0 => 0
| pos p => pos p~0
| neg p => neg p~0
end.
Definition int_succ_double x :=
match x with
| 0 => 1
| pos p => pos p~1
| neg p => neg (pos_pred_double p)
end.
Definition int_pred_double x :=
match x with
| 0 => neg 1%pos
| neg p => neg p~1
| pos p => pos (pos_pred_double p)
end.
(** ** Subtraction of positive into Int *)
Fixpoint int_pos_sub (x y : Pos) {struct y} : Int :=
match x, y with
| p~1, q~1 => int_double (int_pos_sub p q)
| p~1, q~0 => int_succ_double (int_pos_sub p q)
| p~1, 1 => pos p~0
| p~0, q~1 => int_pred_double (int_pos_sub p q)
| p~0, q~0 => int_double (int_pos_sub p q)
| p~0, 1 => pos (pos_pred_double p)
| 1, q~1 => neg q~0
| 1, q~0 => neg (pos_pred_double q)
| 1, 1 => zero
end%pos.
Definition int_negation x :=
match x with
| zero => zero
| pos x => neg x
| neg x => pos x
end.
Notation "- x" := (int_negation x) : int_scope.
Lemma int_negation_negation n : --n = n.
Definition int_add x y :=
match x, y with
| 0, y => y
| x, 0 => x
| pos x', pos y' => pos (x' + y')
| pos x', neg y' => int_pos_sub x' y'
| neg x', pos y' => int_pos_sub y' x'
| neg x', neg y' => neg (x' + y')
end.
Infix "+" := int_add : int_scope.
Definition int_succ x := x + 1.
Definition int_pred x := x + neg 1%pos.
Definition int_sub m n := m + -n.
Infix "-" := int_sub : int_scope.
Definition int_mul x y :=
match x, y with
| 0, _ => 0
| _, 0 => 0
| pos x', pos y' => pos (x' * y')
| pos x', neg y' => neg (x' * y')
| neg x', pos y' => neg (x' * y')
| neg x', neg y' => pos (x' * y')
end.
Infix "*" := int_mul : int_scope.
Definition int_pow x y :=
match y with
| pos p => pos_iter (int_mul x) p 1
| 0 => 1
| neg _ => 0
end.
Infix "^" := int_pow : int_scope.
Definition int_square x :=
match x with
| 0 => 0
| pos p => pos (pos_square p)
| neg p => pos (pos_square p)
end.
Definition sgn z :=
match z with
| 0 => 0
| pos p => 1
| neg p => neg 1%pos
end.
(* ** Decidable paths and truncation. *)
Global Instance decpaths_int : DecidablePaths Int.
intros [n | | n] [m | | m].
destruct (dec (n = m)) as [p | q].
by apply inr; intro; apply q, neg_inj.
exact (inr neg_neq_zero).
exact (inr zero_neq_neg).
exact (inr zero_neq_pos).
exact (inr pos_neq_zero).
destruct (dec (n = m)) as [p | q].
by apply inr; intro; apply q, pos_inj.
(** Since integers have decidable paths they are a hset *)
Global Instance hset_int : IsHSet Int | 0 := _.