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christodoulou.v
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Set Implicit Arguments.
Unset Strict Implicit.
Require Import QArith.
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp Require Import all_ssreflect.
From mathcomp Require Import all_algebra.
Import GRing.Theory Num.Theory Num.Def.
Require Import games.
Local Open Scope nat_scope.
Local Open Scope ring_scope.
Module Christodoulou.
(* ########################################################### *)
(** * Helper lemmas *)
Lemma subrr' (rty : realFieldType) (a : rty) :
- a + a = 0.
Proof.
rewrite addrC. apply: subrr.
Qed.
Lemma sub_add_0 (rty : realFieldType) (a b : rty) :
a - b + b = a.
Proof.
rewrite -addrA subrr' addrC add0r => //.
Qed.
Lemma leq_case (a b : nat) :
is_true (leq a (S b)) ->
a = S b \/ is_true (leq a b).
Proof.
move => H. rewrite leq_eqVlt in H.
move: H. move => /orP H.
destruct H.
left. move: H. move => /eqP H. apply H.
right. apply H.
Qed.
(* ########################################################### *)
(** * Subresult proof.
yz <= y^2 + (1/3)z^2 *)
(* Multiply both sides by 3 *)
Lemma s1 (y z : nat) :
3%:Q * y%:Q * z%:Q <= 3%:Q * y%:Q ^+ 2 + z%:Q ^+ 2 ->
y%:Q * z%:Q <= (3%:Q / 3%:Q) * y%:Q ^+ 2 + (1%:Q / 3%:Q) * z%:Q ^+ 2.
Proof.
move => H0. apply ler_mull2 with (z := 3%:Q); auto.
rewrite mulrA [3%:~R * (_ + _)] mulrDr mulrA.
have H1: ((3%:Q / 3%:Q) = 1%:Q) by apply: divff.
rewrite H1 mulr1 mulrA mulrA mulrA.
have H2: (3%:Q * (1%:Q / 3%:Q) = 1%:Q) by [].
rewrite H2 mul1r.
have H3: (y%:Q ^+ 2 = y%:Q * y%:Q) by [].
have H4: (z%:Q ^+ 2 = z%:Q * z%:Q) by [].
rewrite H3 H4 mulrA in H0. apply H0.
Qed.
(* Subtract 3*y^2 from both sides *)
Lemma s2 (y z : nat) :
(3%:Q * y%:Q * z%:Q - 3%:Q * y%:Q ^+ 2 <= z%:Q ^+ 2) ->
(3%:Q * y%:Q * z%:Q <= 3%:Q * y%:Q ^+ 2 + z%:Q ^+ 2).
Proof.
move => H.
have H0: (3%:Q * y%:Q * z%:Q - 3%:Q * y%:Q ^+ 2 +
3%:Q * y%:Q ^+ 2 <= z%:Q ^+ 2 + 3%:Q * y%:Q ^+ 2).
{ apply ler_add. apply H. auto. }
rewrite sub_add_0 addrC in H0. apply H0.
Qed.
(* Distrbute 3*y *)
Lemma s3 (y z : nat) :
3%:Q * y%:Q * z%:Q - 3%:Q * y%:Q ^+ 2 =
3%:Q * y%:Q * (z%:Q - y%:Q).
Proof. rewrite [_ * (_ - y%:Q)] mulrBr mulrA => //. Qed.
(* Divide both sides by 3 *)
Lemma s4 (y z : nat) :
y%:Q * (z%:Q - y%:Q) <= (z%:Q / 2%:Q) ^+ 2 ->
3%:Q * y%:Q * (z%:Q - y%:Q) <= 3%:Q * (z%:Q / 2%:Q) ^+ 2.
Proof.
move => H.
have H0: (3%:Q * y%:Q * (z%:Q - y%:Q) = 3%:Q * (y%:Q * (z%:Q - y%:Q))).
{ rewrite mulrA => //. }
rewrite H0.
apply ler_mull => //.
Qed.
Lemma s5 (y z : nat) :
z%:Q = y%:Q + (z%:Q - y%:Q).
Proof. rewrite addrC sub_add_0 => //. Qed.
(* Divide both sides by 3 then apply AGM inequality *)
Lemma s6 (y z : nat) :
3%:Q * y%:Q * (z%:Q - y%:Q) <= 3%:Q * (z%:Q / 2%:Q) ^+ 2.
Proof.
apply: s4.
have H: (z%:Q = y%:~R + (z%:~R - y%:~R)) by apply: s5.
rewrite {2} H.
apply: lerif_AGM2.
Qed.
Lemma s7 (z : nat) :
(z%:Q / 2%:Q) * (z%:Q / 2%:Q) = (z%:Q * z%:Q) / 4%:Q.
Proof.
have H0: (4%:Q = 2%:Q * 2%:Q) by [].
rewrite H0 mulrA [z%:~R * z%:~R / _] mulrC.
rewrite -mulrA -mulrA [2%:~R^-1 * _] mulrC mulrC.
rewrite [z%:~R / 2%:~R] mulrC mulrA mulrA [2%:~R^-1 / 2%:~R] mulrC.
rewrite -invrM; auto.
rewrite mulrC [z%:~R / (2%:~R * 2%:~R) * z%:~R] mulrC mulrA => //.
Qed.
Lemma s8 (z : nat) :
3%:Q * (z%:Q / 2%:Q) * (z%:Q / 2%:Q) =
z%:Q * z%:Q * (3%:Q / 4%:Q).
Proof.
rewrite -[3%:Q * _ * _] mulrA s7 mulrA mulrC.
rewrite [4%:~R^-1 * _] mulrA mulrC mulrC => //.
Qed.
(* 1 <= y -> 0 < y *)
Lemma s9 (rty : numDomainType) (z : nat) :
leq (S O) z ->
0 < (z%:R^-1 : rty).
Proof.
rewrite -(ler_nat rty) => H.
rewrite invr_gt0.
have H2: (0 : rty) < 1%:R by [].
by apply: (ltr_le_trans H2 H).
Qed.
(* 1 <= z -> z != 0 *)
Lemma s10 (rty : numDomainType) (z : nat) :
leq (S O) z ->
(z%:R : rty) != (0%:R : rty).
Proof.
move => H. rewrite -(ler_nat rty) in H.
rewrite lt0r_neq0 => //.
have H0: ((0 : rty) < (1%:R : rty)) by [].
by apply: (ltr_le_trans H0 H).
Qed.
Lemma s11 (y z : nat) :
3%:Q * (z%:Q / 2%:Q) ^+ 2 <= z%:Q ^+ 2.
Proof.
case H0: z.
- (* z = 0 *)
rewrite mul0r.
have H1: (0%:Q ^+ 2 = 0%:Q * 0%:Q) by [].
by rewrite !H1 mul0r mulrC mul0r.
- (* z > 0 *)
have H1: ((z%:Q / 2%:Q) ^+ 2 = (z%:Q / 2%:Q) * (z%:Q / 2%:Q)) by [].
rewrite H0 in H1.
rewrite H1 mulrA s8.
have H2: (z%:Q ^+ 2 = z%:Q * z%:Q) by [].
rewrite H0 in H2. rewrite H2.
apply ler_mull2 with (z := z%:Q^-1). apply s9 => //; last by rewrite H0.
rewrite -H0 [z%:Q^-1 * _] mulrA [z%:~R^-1 * _] mulrA [z%:~R^-1 * z%:~R] mulVf.
rewrite mul1r.
apply ler_mull2 with (z := z%:Q^-1). apply s9 => //.
by rewrite H0.
rewrite [z%:~R^-1 * _] mulrA [z%:~R^-1 * z%:~R] mulVf.
rewrite mul1r => //.
apply s10 => //. by rewrite H0.
apply s10 => //. by rewrite H0.
Qed.
(* this implies the main result when y >= 2 *)
Lemma subresult (y z : nat) :
y%:Q * z%:Q <=
(3%:Q / 3%:Q) * y%:Q ^+ 2 + (1%:Q / 3%:Q) * z%:Q ^+ 2.
Proof.
apply: s1. apply: s2. rewrite s3.
have H1: (3%:Q * y%:Q * (z%:Q - y%:Q) <= 3%:Q * (z%:Q / 2%:Q) ^+ 2)
by apply: s6.
have H2: (3%:Q * (z%:Q / 2%:Q) ^+ 2 <= z%:Q ^+ 2) by apply: s11.
apply: ler_trans. apply H1. apply H2.
Qed.
(* ########################################################### *)
(** * Subresult implies main result proof *)
(* 2 <= y -> 0 < y *)
Lemma sr3 (y : nat) :
leq 2 y ->
0 < y%:Q.
Proof.
move => H0.
rewrite -(ltr1n rat_numDomainType) in H0.
have H1: (0%:Q < 1%:Q) by [].
apply: ltr_trans. apply H1. apply H0.
Qed.
(* 2 <= y -> 2/3 <= y *)
Lemma sr4 (y : nat) :
leq 2 y ->
3%:Q / 2%:Q <= y%:Q.
Proof.
move => H0.
rewrite -(ler_nat rat_numDomainType) in H0.
have H1: (3%:Q / 2%:Q <= 2%:Q) by [].
apply: ler_trans. apply: H1. apply: H0.
Qed.
(* 2 <= y -> y <= (2/3)*y^2 *)
Lemma sr5 (y : nat) :
leq 2 y ->
y%:Q <= 2%:Q / 3%:Q * y%:Q ^+ 2.
Proof.
move => H.
rewrite mulrC.
have H0: (y%:Q ^+ 2 = y%:Q * y%:Q) by [].
rewrite H0 mulrA.
have H1: (y%:Q = y%:Q * 1%:Q). rewrite mulrC mul1r => //.
rewrite {1} H1.
have H2: (y%:Q * y%:Q * (2%:Q / 3%:Q) = y%:Q * (y%:Q * (2%:Q / 3%:Q)))
by rewrite -mulrA.
rewrite -mulrA H2.
apply ler_mull with (x := 1%:Q) (y := y%:Q * (2%:Q / 3%:Q)) (z := y%:Q).
rewrite le0r. apply /orP. right. apply sr3 => //.
apply ler_mull2 with (z := 3%:Q / 2%:Q) => //.
rewrite mulrC mul1r.
have H3: (y%:Q * (2%:Q / 3%:Q) = 2%:Q / 3%:Q * y%:Q) by rewrite mulrC.
rewrite H3 mulrA.
have H4: (2%:Q / 3%:Q = (3%:Q / 2%:Q)^-1) by [].
have H5: (3%:Q / 2%:Q * (2%:Q / 3%:Q) = 1) by rewrite mulrC H4 mulVr.
rewrite H5 mul1r.
apply sr4 => //.
Qed.
(* Combine terms *)
Lemma sr6 (y : nat) (a : rat) :
3%:Q / 3%:Q * y%:Q ^+ 2 + a + 2%:Q / 3%:Q * y%:Q ^+ 2 =
5%:Q / 3%:Q * y%:Q ^+ 2 + a.
Proof.
rewrite addrC addrA -mulrDl.
have H: (2%:Q / 3%:Q + 3%:Q / 3%:Q = 5%:Q / 3%:Q) by [].
rewrite H => //.
Qed.
(* When 2 <= y, subresult -> main result *)
Lemma subresult_implies_result (y z: nat) :
leq 2 y ->
y%:Q * z%:Q <= (3%:Q / 3%:Q) * y%:Q ^+ 2 + (1%:Q / 3%:Q) * z%:Q ^+ 2 ->
y%:Q * (z%:Q + 1) <= (5%:Q / 3%:Q) * y%:Q ^+ 2 + (1%:Q / 3%:Q) * z%:Q ^+ 2.
Proof.
move => H0 H1.
have H2: (y%:Q * z%:Q + y%:Q <=
3%:Q / 3%:Q * y%:Q ^+ 2 +
1%:Q / 3%:Q * z%:Q ^+ 2 + (2%:Q / 3%:Q) * y%:Q ^+ 2).
{ apply ler_add. apply H1. apply: sr5 => //. }
rewrite sr6 in H2.
rewrite mulrDr mulr1. apply: H2.
Qed.
(* 3 <= z -> z <= (1/3)*z^2 *)
Lemma r1_1 (z : nat) :
leq 3 z ->
z%:Q <= 1%:Q / 3%:Q * z%:Q ^+ 2.
Proof.
move => H.
rewrite mulrC.
have H0: (z%:Q ^+ 2 = z%:Q * z%:Q) by [].
rewrite H0 mulrA.
have H1: (z%:Q = z%:Q * 1%:Q). rewrite mulrC mul1r => //.
rewrite {1} H1.
have H2: (z%:Q * z%:Q * (1%:Q / 3%:Q) = z%:Q * (z%:Q * (1%:Q / 3%:Q)))
by rewrite -mulrA.
rewrite -mulrA H2.
apply ler_mull with (x := 1%:Q) (y := z%:Q * (1%:Q / 3%:Q)) (z := z%:Q).
rewrite le0r. apply /orP. right. apply sr3 => //.
have H3: (ltn (S O) 2) by [].
apply: ltn_trans. apply H3. apply H.
apply ler_mull2 with (z := 3%:Q / 1%:Q).
rewrite mulrC mul1r => //.
have H3: (z%:Q * (1%:Q / 3%:Q) = 1%:Q / 3%:Q * z%:Q) by rewrite mulrC.
rewrite H3 mulrA.
have H4: (1%:Q / 3%:Q = (3%:Q / 1%:Q)^-1) by [].
have H5: (3%:Q / 1%:Q * (1%:Q / 3%:Q) = 1) by rewrite mulrC H4 mulVr.
rewrite H5 mul1r mulrC mul1r.
have H6: (3%:Q / 1%:Q = 3%:Q) by []. rewrite H6.
rewrite ler_nat. apply H.
Qed.
(* This is the main result when y = 1 *)
Lemma r1 (z : nat) :
z%:Q + 1 <= 5%:Q / 3%:Q + 1%:Q / 3%:Q * z%:Q ^+ 2.
Proof.
have H0: (z%:Q <= 2%:Q / 3%:Q + 1%:Q / 3%:Q * z%:Q ^+ 2).
{ case H: (leq 3 z).
apply elimT with (P := (le 3 z)%N) in H.
rewrite addrC.
have H0: (z%:Q = z%:Q + 0%:Q) by rewrite addrC add0r => //.
rewrite {1} H0. apply ler_add; auto.
apply r1_1. move: H. move => /leP => //.
apply /leP.
apply negbT in H. rewrite -ltnNge in H.
apply leq_case in H. destruct H.
have H1: (z = 2) by case: H. rewrite H1 => //.
apply leq_case in H. destruct H.
have H1: (z = S O) by case: H. rewrite H1 => //.
apply leq_case in H. destruct H.
have H1: (z = O) by case: H. rewrite H1 => //.
rewrite ltn0 in H. congruence. }
have H1: (5%:Q / 3%:Q = 2%:Q / 3%:Q + 1%:Q) by [].
rewrite H1 -addrA [1%:~R + _] addrC addrA.
apply ler_add; by [].
Qed.
(* ########################################################### *)
(** * Main result *)
Lemma result (y z : nat) :
y%:Q * (z%:Q + 1) <=
(5%:Q / 3%:Q) * y%:Q ^+ 2 + (1%:Q / 3%:Q) * z%:Q ^+ 2.
Proof.
case H: (leq 2 y).
- (* y > 1 *)
apply: subresult_implies_result => //.
apply: subresult.
(* other cases *)
apply negbT in H. rewrite -leqNgt in H.
apply leq_case in H. destruct H.
+ (* y = 1 *)
rewrite H. rewrite mul1r. apply: r1.
+ (* y = 0*)
rewrite leqn0 in H. move: H. move => /eqP H.
rewrite H mul0r mulrC mul0r add0r.
apply: mulr_ge0 => //. apply: sqr_ge0.
Qed.
End Christodoulou.