additional_operations.v

Require Import HoTT.Classes.interfaces.canonical_names.


Generalizable Variables A R.


Local Set Universe Minimization ToSet.


Global Instance decide_eqb `{DecidablePaths A} : Eqb A
  := fun a b => if decide_rel paths a b then true else false.


Lemma decide_eqb_ok@{i} {A:Type@{i} } `{DecidablePaths A} :
  forall a b, iff@{Set i i} (eqb a b = true) (a = b).

Proof.

unfold eqb,decide_eqb.

intros a b;destruct (decide_rel paths a b) as [E1|E1];split;intros E2;auto.

 destruct (false_ne_true E2).

 destruct (E1 E2).

Qed.


Lemma LT_EQ : LT <> EQ.

Proof.

intros E.

change ((fun r => match r with LT => Unit | _ => Empty end) EQ).

rewrite <-E.

 split.

Qed.


Lemma LT_GT : LT <> GT.

Proof.

intros E.

change ((fun r => match r with LT => Unit | _ => Empty end) GT).

rewrite <-E.

 split.

Qed.


Lemma EQ_LT : EQ <> LT.

Proof.

apply symmetric_neq, LT_EQ.

Qed.


Lemma EQ_GT : EQ <> GT.

Proof.

intros E.

change ((fun r => match r with EQ => Unit | _ => Empty end) GT).

rewrite <-E.

 split.

Qed.


Lemma GT_EQ : GT <> EQ.

Proof.

apply symmetric_neq, EQ_GT.

Qed.


Global Instance compare_eqb `{Compare A} : Eqb A | 2 := fun a b =>
  match a ?= b with
  | EQ => true
  | _ => false
  end.


Lemma compare_eqb_eq `{Compare A} : forall a b : A, a =? b = true -> a ?= b = EQ.

Proof.

unfold eqb,compare_eqb;simpl.

intros a b.

 destruct (a ?= b);trivial;intros E;destruct (false_ne_true E).

Qed.


Global Instance tricho_compare `{Trichotomy A R} : Compare A | 2
  := fun a b =>
  match trichotomy R a b with
  | inl _ => LT
  | inr (inl _) => EQ
  | inr (inr _) => GT
  end.


Lemma tricho_compare_eq `{Trichotomy A R}
  : forall a b : A, compare a b = EQ -> a = b.

Proof.

unfold compare,tricho_compare.

intros a b;destruct (trichotomy R a b) as [E|[E|E]];auto.

 intros E1;destruct (LT_EQ E1).

 intros E1;destruct (GT_EQ E1).

Qed.


Lemma tricho_compare_ok `{Trichotomy A R} `{Irreflexive A R}
  : forall a b : A, compare a b = EQ <-> a = b.

Proof.

unfold compare,tricho_compare.

intros a b;destruct (trichotomy R a b) as [E1|[E1|E1]];split;auto.

 intros E2;destruct (LT_EQ E2).

 intros E2;rewrite E2 in E1.

 destruct (irreflexivity R _ E1).

 intros E2;destruct (GT_EQ E2).

 intros E2;rewrite E2 in E1.

 destruct (irreflexivity R _ E1).

Qed.


Lemma total_abs_either `{Abs A} `{!TotalRelation le}
  : forall x : A, (0 <= x /\ abs x = x) |_| (x <= 0 /\ abs x = - x).

Proof.

intros x.

destruct (total le 0 x) as [E|E].

 left.

 split;trivial.

 apply ((abs_sig x).2);trivial.

 right.

 split;trivial.

 apply ((abs_sig x).2);trivial.

Qed.

Timings for additional_operations.v