Commit 0ed31f27 authored by Felipe Cerqueira's avatar Felipe Cerqueira

Add more lemmas about min/max

parent 76263149
Require Import rt.util.tactics rt.util.notation rt.util.sorting rt.util.nat.
Require Import rt.util.tactics rt.util.notation rt.util.sorting rt.util.nat rt.util.list.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop.
Section MinMaxSeq.
Section Arg.
Context {T: eqType}.
Section ArgGeneric.
Variable F: T -> nat.
Context {T1 T2: eqType}.
Variable rel: T2 -> T2 -> bool.
Variable F: T1 -> T2.
Fixpoint seq_argmax (F: T -> nat) (l: seq T) :=
Fixpoint seq_argmin (l: seq T1) :=
if l is x :: l' then
if seq_argmax F l' is Some y then
if F x >= F y then Some x else Some y
if seq_argmin l' is Some y then
if rel (F x) (F y) then Some x else Some y
else Some x
else None.
Fixpoint seq_argmin (F: T -> nat) (l: seq T) :=
Fixpoint seq_argmax (l: seq T1) :=
if l is x :: l' then
if seq_argmin F l' is Some y then
if F x <= F y then Some x else Some y
if seq_argmax l' is Some y then
if rel (F y) (F x) then Some x else Some y
else Some x
else None.
Section Lemmas.
Lemma seq_max_exists:
Lemma seq_argmin_exists:
forall l x,
x \in l ->
seq_argmax F l != None.
seq_argmin l != None.
Proof.
induction l; first by done.
intros x; rewrite in_cons.
move => /orP [/eqP EQ | IN] /=;
first by subst; destruct (seq_argmax F l); first by case: ifP.
by destruct (seq_argmax F l); first by case: ifP.
first by subst; destruct (seq_argmin l); first by case: ifP.
by destruct (seq_argmin l); first by case: ifP.
Qed.
Lemma mem_seq_max:
Lemma seq_argmin_in_seq:
forall l x,
seq_argmax F l = Some x ->
seq_argmin l = Some x ->
x \in l.
Proof.
induction l; simpl; first by done.
intros x ARG.
destruct (seq_argmax F l);
destruct (seq_argmin l);
last by case: ARG => EQ; subst; rewrite in_cons eq_refl.
destruct (F s <= F a);
destruct (rel (F a) (F s));
first by case: ARG => EQ; subst; rewrite in_cons eq_refl.
case: ARG => EQ; subst.
by rewrite in_cons; apply/orP; right; apply IHl.
Qed.
Lemma seq_max_computes_max:
forall l x y,
seq_argmax F l = Some x ->
y \in l ->
F x >= F y.
Proof.
induction l; first by done.
intros x y EQmax IN; simpl in EQmax.
rewrite in_cons in IN.
move: IN => /orP [/eqP EQ | IN].
{
subst.
destruct (seq_argmax F l) eqn:ARG;
last by case: EQmax => EQ; subst.
destruct (leqP (F s) (F a)) as [LE | GT];
first by case: EQmax => EQ; subst.
apply leq_trans with (n := F s); first by apply ltnW.
apply IHl; first by done.
by apply mem_seq_max.
}
{
destruct (seq_argmax F l) eqn:ARG.
{
destruct (leqP (F s) (F a)) as [LE | GT];
last by case: EQmax => EQ; subst; apply IHl.
case: EQmax => EQ; subst.
by apply: (leq_trans _ LE); apply IHl.
}
{
case: EQmax => EQ; subst.
by apply seq_max_exists in IN; rewrite ARG in IN.
}
}
Qed.
Lemma seq_min_exists:
Lemma seq_argmax_exists:
forall l x,
x \in l ->
seq_argmin F l != None.
seq_argmax l != None.
Proof.
induction l; first by done.
intros x; rewrite in_cons.
move => /orP [/eqP EQ | IN] /=;
first by subst; destruct (seq_argmin F l); first by case: ifP.
by destruct (seq_argmin F l); first by case: ifP.
first by subst; destruct (seq_argmax l); first by case: ifP.
by destruct (seq_argmax l); first by case: ifP.
Qed.
Lemma mem_seq_min:
Lemma seq_argmax_in_seq:
forall l x,
seq_argmin F l = Some x ->
seq_argmax l = Some x ->
x \in l.
Proof.
induction l; simpl; first by done.
intros x ARG.
destruct (seq_argmin F l);
destruct (seq_argmax l);
last by case: ARG => EQ; subst; rewrite in_cons eq_refl.
destruct (F s >= F a);
destruct (rel (F s) (F a));
first by case: ARG => EQ; subst; rewrite in_cons eq_refl.
case: ARG => EQ; subst.
by rewrite in_cons; apply/orP; right; apply IHl.
Qed.
Section TotalOrder.
Lemma seq_min_computes_min:
forall l x y,
seq_argmin F l = Some x ->
y \in l ->
F x <= F y.
Proof.
induction l; first by done.
intros x y EQmin IN; simpl in EQmin.
rewrite in_cons in IN.
move: IN => /orP [/eqP EQ | IN].
{
subst; destruct (seq_argmin F l) eqn:ARG;
last by case: EQmin => EQ; subst.
destruct (ltnP (F s) (F a)) as [LT | GE];
last by case: EQmin => EQ; subst.
apply leq_trans with (n := F s); last by apply ltnW.
apply IHl; first by done.
by apply mem_seq_min.
}
{
destruct (seq_argmin F l) eqn:ARG.
Hypothesis H_transitive: transitive rel.
Variable l: seq T1.
Hypothesis H_total_over_list:
forall x y,
x \in l ->
y \in l ->
rel (F x) (F y) || rel (F y) (F x).
Lemma seq_argmin_computes_min:
forall x y,
seq_argmin l = Some x ->
y \in l ->
rel (F x) (F y).
Proof.
rename H_transitive into TRANS, H_total_over_list into TOT, l into l'.
induction l'; first by done.
intros x y EQmin IN; simpl in EQmin.
rewrite in_cons in IN.
move: IN => /orP [/eqP EQ | IN].
{
destruct (ltnP (F s) (F a)) as [LT | GE];
first by case: EQmin => EQ; subst; apply IHl.
case: EQmin => EQ; subst.
by apply: (leq_trans GE); apply IHl.
subst; destruct (seq_argmin l') eqn:ARG; last first.
{
case: EQmin => EQ; subst.
by exploit (TOT x x); try (by rewrite in_cons eq_refl); rewrite orbb.
}
{
destruct (rel (F a) (F s)) eqn:REL; case: EQmin => EQ; subst;
first by exploit (TOT x x); try (by rewrite in_cons eq_refl); rewrite orbb.
exploit (TOT a x).
- by rewrite in_cons eq_refl.
- by rewrite in_cons; apply/orP; right; apply seq_argmin_in_seq.
- by rewrite REL /=.
}
}
{
destruct (seq_argmin l') eqn:ARG.
{
destruct (rel (F a) (F s)) eqn:REL; case: EQmin => EQ; subst; last first.
{
apply IHl'; [| by done | by done].
by intros x0 y0 INx INy; apply TOT; rewrite in_cons; apply/orP; right.
}
{
apply TRANS with (y := F s); first by done.
apply IHl'; [| by done | by done].
by intros x0 y0 INx INy; apply TOT; rewrite in_cons; apply/orP; right.
}
}
{
case: EQmin => EQ; subst.
by apply seq_argmin_exists in IN; rewrite ARG in IN.
}
}
Qed.
Lemma seq_argmax_computes_max:
forall x y,
seq_argmax l = Some x ->
y \in l ->
rel (F y) (F x).
Proof.
rename H_transitive into TRANS, H_total_over_list into TOT, l into l'.
induction l'; first by done.
intros x y EQmin IN; simpl in EQmin.
rewrite in_cons in IN.
move: IN => /orP [/eqP EQ | IN].
{
case: EQmin => EQ; subst.
by apply seq_min_exists in IN; rewrite ARG in IN.
subst; destruct (seq_argmax l') eqn:ARG; last first.
{
case: EQmin => EQ; subst.
by exploit (TOT x x); try (by rewrite in_cons eq_refl); rewrite orbb.
}
{
destruct (rel (F s) (F a)) eqn:REL; case: EQmin => EQ; subst;
first by exploit (TOT x x); try (by rewrite in_cons eq_refl); rewrite orbb.
exploit (TOT a x).
- by rewrite in_cons eq_refl.
- by rewrite in_cons; apply/orP; right; apply seq_argmax_in_seq.
- by rewrite REL orbF.
}
}
}
{
destruct (seq_argmax l') eqn:ARG.
{
destruct (rel (F s) (F a)) eqn:REL; case: EQmin => EQ; subst; last first.
{
apply IHl'; [| by done | by done].
by intros x0 y0 INx INy; apply TOT; rewrite in_cons; apply/orP; right.
}
{
apply TRANS with (y := F s); last by done.
apply IHl'; [| by done | by done].
by intros x0 y0 INx INy; apply TOT; rewrite in_cons; apply/orP; right.
}
}
{
case: EQmin => EQ; subst.
by apply seq_argmax_exists in IN; rewrite ARG in IN.
}
}
Qed.
End TotalOrder.
End Lemmas.
End ArgGeneric.
Section MinGeneric.
Context {T: eqType}.
Variable rel: rel T.
Definition seq_min := seq_argmin rel id.
Definition seq_max := seq_argmax rel id.
Section Lemmas.
Lemma seq_min_exists:
forall l x,
x \in l ->
seq_min l != None.
Proof.
by apply seq_argmin_exists.
Qed.
Lemma seq_min_in_seq:
forall l x,
seq_min l = Some x ->
x \in l.
Proof.
by apply seq_argmin_in_seq.
Qed.
Lemma seq_max_exists:
forall l x,
x \in l ->
seq_max l != None.
Proof.
by apply seq_argmax_exists.
Qed.
Lemma seq_max_in_seq:
forall l x,
seq_max l = Some x ->
x \in l.
Proof.
by apply seq_argmax_in_seq.
Qed.
Section TotalOrder.
Hypothesis H_transitive: transitive rel.
Variable l: seq T.
Hypothesis H_total_over_list:
forall x y,
x \in l ->
y \in l ->
rel x y || rel y x.
Lemma seq_min_computes_min:
forall x y,
seq_min l = Some x ->
y \in l ->
rel x y.
Proof.
by apply seq_argmin_computes_min.
Qed.
Lemma seq_max_computes_max:
forall x y,
seq_max l = Some x ->
y \in l ->
rel y x.
Proof.
by apply seq_argmax_computes_max.
Qed.
End TotalOrder.
End Lemmas.
End Arg.
Definition seq_max := seq_argmax id.
Definition seq_min := seq_argmin id.
End MinGeneric.
Section ArgNat.
Context {T: eqType}.
Variable F: T -> nat.
Definition seq_argmin_nat := seq_argmin leq F.
Definition seq_argmax_nat := seq_argmax leq F.
Section Lemmas.
Lemma seq_argmin_nat_exists:
forall l x,
x \in l ->
seq_argmin_nat l != None.
Proof.
by apply seq_argmin_exists.
Qed.
Lemma seq_argmin_nat_in_seq:
forall l x,
seq_argmin_nat l = Some x ->
x \in l.
Proof.
by apply seq_argmin_in_seq.
Qed.
Lemma seq_argmax_nat_exists:
forall l x,
x \in l ->
seq_argmax_nat l != None.
Proof.
by apply seq_argmax_exists.
Qed.
Lemma seq_argmax_nat_in_seq:
forall l x,
seq_argmax_nat l = Some x ->
x \in l.
Proof.
by apply seq_argmax_in_seq.
Qed.
Section TotalOrder.
Lemma seq_argmin_nat_computes_min:
forall l x y,
seq_argmin_nat l = Some x ->
y \in l ->
F x <= F y.
Proof.
intros l x y SOME IN.
apply seq_argmin_computes_min with (l0 := l); try (by done).
- by intros x1 x2 x3; apply leq_trans.
- by intros x1 x2 IN1 IN2; apply leq_total.
Qed.
Lemma seq_argmax_nat_computes_max:
forall l x y,
seq_argmax_nat l = Some x ->
y \in l ->
F x >= F y.
Proof.
intros l x y SOME IN.
apply seq_argmax_computes_max with (l0 := l); try (by done).
- by intros x1 x2 x3; apply leq_trans.
- by intros x1 x2 IN1 IN2; apply leq_total.
Qed.
End TotalOrder.
End Lemmas.
End ArgNat.
Section MinNat.
Definition seq_min_nat := seq_argmin leq id.
Definition seq_max_nat := seq_argmax leq id.
Section Lemmas.
Lemma seq_min_nat_exists:
forall l x,
x \in l ->
seq_min_nat l != None.
Proof.
by apply seq_argmin_exists.
Qed.
Lemma seq_min_nat_in_seq:
forall l x,
seq_min_nat l = Some x ->
x \in l.
Proof.
by apply seq_argmin_in_seq.
Qed.
Lemma seq_max_nat_exists:
forall l x,
x \in l ->
seq_max_nat l != None.
Proof.
by apply seq_argmax_exists.
Qed.
Lemma seq_max_nat_in_seq:
forall l x,
seq_max_nat l = Some x ->
x \in l.
Proof.
by apply seq_argmax_in_seq.
Qed.
Section TotalOrder.
Lemma seq_min_nat_computes_min:
forall l x y,
seq_min_nat l = Some x ->
y \in l ->
x <= y.
Proof.
intros l x y SOME IN.
apply seq_min_computes_min with (l0 := l); try (by done).
- by intros x1 x2 x3; apply leq_trans.
- by intros x1 x2 IN1 IN2; apply leq_total.
Qed.
Lemma seq_max_nat_computes_max:
forall l x y,
seq_max_nat l = Some x ->
y \in l ->
x >= y.
Proof.
intros l x y SOME IN.
apply seq_max_computes_max with (l0 := l); try (by done).
- by intros x1 x2 x3; apply leq_trans.
- by intros x1 x2 IN1 IN2; apply leq_total.
Qed.
End TotalOrder.
End Lemmas.
End MinNat.
Section NatRange.
Definition values_between (a b: nat) :=
filter (fun x => x >= a) (map (@nat_of_ord _) (enum 'I_b)).
Lemma mem_values_between a b:
forall x, x \in values_between a b = (a <= x < b).
Proof.
intros x; rewrite mem_filter.
apply/idP/idP.
{
move => /andP [GE IN].
move: IN => /mapP [x' IN] EQ; subst.
rewrite mem_enum in IN.
by apply/andP; split.
}
{
move => /andP [GE LT].
rewrite GE andTb.
apply/mapP; exists (Ordinal LT); last by done.
by rewrite mem_enum.
}
Qed.
Definition min_nat_cond P (a b: nat) :=
seq_min_nat (filter P (values_between a b)).
Definition max_nat_cond P (a b: nat) :=
seq_max_nat (filter P (values_between a b)).
Lemma min_nat_cond_in_seq:
forall P a b x,
min_nat_cond P a b = Some x ->
a <= x < b /\ P x.
Proof.
intros P a b x SOME.
apply seq_min_nat_in_seq in SOME.
rewrite mem_filter in SOME; move: SOME => /andP [Px LE].
by split; first by rewrite mem_values_between in LE.
Qed.
Lemma min_nat_cond_computes_min:
forall P a b x,
min_nat_cond P a b = Some x ->
(forall y, a <= y < b -> P y -> x <= y).
Proof.
intros P a b x SOME y LE Py.
apply seq_min_nat_computes_min with (y := y) in SOME; first by done.
by rewrite mem_filter Py andTb mem_values_between.
Qed.
Lemma max_nat_cond_in_seq:
forall P a b x,
max_nat_cond P a b = Some x ->
a <= x < b /\ P x.
Proof.
intros P a b x SOME.
apply seq_max_nat_in_seq in SOME.
rewrite mem_filter in SOME; move: SOME => /andP [Px LE].
by split; first by rewrite mem_values_between in LE.
Qed.
Lemma max_nat_cond_computes_max:
forall P a b x,
max_nat_cond P a b = Some x ->
(forall y, a <= y < b -> P y -> y <= x).
Proof.
intros P a b x SOME y LE Py.
apply seq_max_nat_computes_max with (y := y) in SOME; first by done.
by rewrite mem_filter Py andTb mem_values_between.
Qed.
End NatRange.
End MinMaxSeq.
(* Additional lemmas about max. *)
......
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