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flow.v
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flow.v
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Require Import c_util.
Require Import geometry.
Require Import CRreal.
Require Import CRexp.
Require Import CRln.
Require Import Morphisms.
Set Implicit Arguments.
Open Local Scope CR_scope.
(* We require CR because we use it for Time.
Todo: Take an arbitrary (C)Ring/(C)Field for Time, so that
the classical development does not need a separate concrete module. *)
Let Time := CRasCSetoid.
Let Duration := NonNegCR.
Record Flow (X: CSetoid): Type :=
{ flow_morphism:> morpher (@st_eq X ==> @st_eq Time ==> @st_eq X)%signature
; flow_zero: forall x, flow_morphism x ('0) [=] x
; flow_additive: forall x t t',
flow_morphism x (t + t') [=] flow_morphism (flow_morphism x t) t'
}.
Definition mono (f: Flow CRasCSetoid): Type :=
sum (forall x, strongly_increasing (f x))
(forall x, strongly_decreasing (f x)).
Definition range_flow_inv_spec (f: Flow CRasCSetoid)
(i: OpenRange -> OpenRange -> OpenRange): Prop :=
forall a p, in_orange a p -> forall b t, in_orange b (f p t) -> in_orange (i a b) t.
Obligation Tactic := idtac.
Program Definition product_flow (X Y: CSetoid) (fx: Flow X) (fy: Flow Y):
Flow (ProdCSetoid X Y) :=
Build_Flow _ (fun xy t => (fx (fst xy) t, fy (snd xy) t)) _ _.
Next Obligation.
intros X Y fx fy [s s0] [s1 s2] [A B] x y H0.
split; [rewrite A | rewrite B]; rewrite H0; reflexivity.
Qed.
Next Obligation. destruct x. split; apply flow_zero. Qed.
Next Obligation. destruct x. split; apply flow_additive. Qed.
Module constant.
Section contents.
Program Definition flow: Flow CRasCSetoid := Build_Flow _ (fun x _ => x) _ _.
Next Obligation. exact (fun _ _ H _ _ _ => H). Qed.
Next Obligation. reflexivity. Qed.
Next Obligation. reflexivity. Qed.
Let eps: Qpos := (1#100)%Qpos. (* todo: turn into parameter *)
Definition neg_range: OpenRange.
exists (Some (-'1), Some (-'1)).
unfold uncurry. simpl. auto.
Defined.
Definition inv (a b: OpenRange): OpenRange :=
if overestimate_oranges_overlap eps a b: bool then unbounded_range else neg_range.
Lemma inv_correct: range_flow_inv_spec flow inv.
Proof with auto.
unfold range_flow_inv_spec, inv.
intros.
destruct_call overestimate_oranges_overlap.
destruct x...
elimtype False.
apply n, oranges_share_point with p...
Qed.
End contents.
End constant.
Module scale.
Section contents.
Variables (s: CR) (sp: CRpos s) (f: Flow CRasCSetoid).
Program Definition flow: Flow CRasCSetoid :=
Build_Flow _ (fun x t => f x (s * t)) _ _.
Next Obligation. intros a a' e b b' e'. rewrite e, e'. reflexivity. Qed.
Next Obligation.
intros. unfold morpher_to_func, proj1_sig.
rewrite CRmult_0_r. apply flow_zero.
Qed.
Next Obligation.
intros.
unfold morpher_to_func at 1 3 5, proj1_sig.
rewrite
<- flow_additive,
(Rmul_comm CR_ring_theory),
(Rmul_comm CR_ring_theory s t),
(Rmul_comm CR_ring_theory s t'),
(Rdistr_l CR_ring_theory).
reflexivity.
Qed.
Lemma inc: (forall x, strongly_increasing (f x)) ->
forall x, strongly_increasing (flow x).
Proof.
repeat intro. unfold flow. simpl.
apply X, CRmult_lt_pos_r; assumption.
Qed.
Variable old_inv: OpenRange -> OpenRange -> OpenRange.
Hypothesis old_inv_correct: range_flow_inv_spec f old_inv.
Lemma CRpos_apart_0 x: CRpos x -> x >< '0.
right. apply CRpos_lt_0_rev. assumption.
Defined.
Definition sinv: CR := CRinv s (CRpos_apart_0 sp).
Lemma sinv_nonneg: '0 <= sinv.
Proof.
apply CRlt_le, CRpos_lt_0_rev, CRinv_pos.
assumption.
Qed.
Definition inv (a b: OpenRange): OpenRange := scale_orange sinv_nonneg (old_inv a b).
Lemma inv_correct: range_flow_inv_spec flow inv.
Proof with auto.
unfold range_flow_inv_spec in *.
intros.
unfold inv.
simpl in H0.
set (old_inv_correct H _ H0).
clearbody i.
assert (fst (` (scale_orange sinv_nonneg (old_inv a b))) [=] fst (` (scale_orange sinv_nonneg (old_inv a b)))) by reflexivity.
assert (snd (` (scale_orange sinv_nonneg (old_inv a b))) [=] snd (` (scale_orange sinv_nonneg (old_inv a b)))) by reflexivity.
assert (sinv * (s * t) == t).
rewrite (Rmul_assoc CR_ring_theory).
unfold sinv.
rewrite (Rmul_comm CR_ring_theory (CRinv s (CRpos_apart_0 sp))).
rewrite CRinv_mult.
apply (Rmul_1_l CR_ring_theory).
rewrite <- H3.
apply in_scale_orange...
Qed.
End contents.
End scale.
Hint Resolve scale.inc.
Module positive_linear.
Section contents.
Program Definition f: Flow CRasCSetoid :=
Build_Flow _ (ucFun2 CRplus) CRadd_0_r (Radd_assoc CR_ring_theory).
Next Obligation. do 6 intro. rewrite H, H0. reflexivity. Qed.
Definition inv (x x': CR): Time := x' - x.
Lemma inv_correct x x': f x (inv x x') == x'.
Proof. intros. symmetry. apply t11. Qed.
Lemma increasing: forall x : CRasCSetoid, strongly_increasing (f x).
Proof. repeat intro. apply t1_rev. assumption. Qed.
Definition mono: mono f := inl increasing.
End contents.
End positive_linear.
Hint Immediate positive_linear.increasing.
Module negative_linear.
Section contents.
Let B x t t': x - (t + t') == x - t - t'.
intros.
rewrite (@Ropp_add _ _ _ _ _ _ _ _ t3 CR_ring_eq_ext CR_ring_theory ).
apply (Radd_assoc CR_ring_theory).
Qed.
Program Definition f: Flow CRasCSetoid :=
Build_Flow _ (fun x t => x - t) CRminus_zero B.
Next Obligation. do 6 intro. rewrite H, H0. reflexivity. Qed.
Definition inv (x x': CR): Time := x - x'.
Lemma inv_correct x x': f x (inv x x') == x'.
intros.
unfold f, inv.
simpl morpher_to_func.
rewrite <- diff_opp.
symmetry.
apply t11.
Qed.
Lemma decreasing: forall x : CRasCSetoid, strongly_decreasing (f x).
repeat intro. simpl.
apply t1_rev, CRlt_opp_compat.
assumption.
Qed.
Definition mono: mono f := inr decreasing.
End contents.
End negative_linear.
Require Import vector_setoid.
Require Import VecEq.
Lemma Vforall2n_aux_inv (A B: Type) (R: A -> B -> Prop) n (v: vector A n) m (w: vector B m):
Vforall2n_aux R v w -> forall i (p: lt i n) (q: lt i m), R (Vnth v p) (Vnth w q).
Proof.
induction v; destruct w; simpl; intros; intuition.
elimtype False.
apply (lt_n_O _ p).
destruct i; auto.
Qed.
Definition eq_vec_inv (T: Type) (R: relation T) n (x y: vector T n) (e: eq_vec R x y)
i (p: (i < n)%nat): R (Vnth x p) (Vnth y p)
:= Vforall2n_aux_inv _ _ _ e p p.
(* Todo: Move the above two elsewhere. *)
Section vector_flow.
Variable n : nat.
Variable X : CSetoid.
Variable vec_f : vector (Flow X) n.
Let f (vec_x : vecCSetoid X n) (t : Time) : vecCSetoid X n :=
let flow_dim i ip :=
let f := Vnth vec_f ip in
let x := Vnth vec_x ip in
f x t
in
Vbuild flow_dim.
Program Definition vector_flow : Flow (vecCSetoid X n) :=
Build_Flow _ f _ _.
Next Obligation. (* well-defined-ness *)
do 6 intro.
apply Veq_vec_nth. intros.
unfold f.
repeat rewrite Vbuild_nth.
unfold morpher_to_func.
rewrite (eq_vec_inv (@st_eq _) x y H).
rewrite H0.
reflexivity.
Qed.
Next Obligation. (* flow_zero *)
apply Veq_vec_nth. intros. unfold f.
repeat rewrite Vbuild_nth. apply flow_zero.
Qed.
Next Obligation. (* flow_additive *)
apply Veq_vec_nth. intros. unfold f.
repeat rewrite Vbuild_nth. apply flow_additive.
Qed.
End vector_flow.