(HCSP-OMEGA VERIFIED DEEP RL CONTROLLER)

(* ================================================================ *)

(* HCSP-OMEGA VERIFIED DEEP RL CONTROLLER                           *)

(* Corrected Coq Version                                            *)

(* ================================================================ *)


Require Import Reals.

Require Import Psatz.

Require Import Lra.


Open Scope R_scope.


(* ================================================================ *)

(* STATE SPACE                                                      *)

(* ================================================================ *)


Record State := mkState {

  psi : R;

  s   : R;

  r   : R;

  n   : R

}.


Definition bounded (x:R) : Prop :=

  0 <= x <= 1.


Definition Valid (st:State) : Prop :=

  bounded (psi st) /\

  bounded (s st) /\

  bounded (r st) /\

  bounded (n st).


(* ================================================================ *)

(* PARAMETERS                                                       *)

(* ================================================================ *)


Parameter alpha gamma eta dt : R.


Hypothesis alpha_range : 0 < alpha <= 1.

Hypothesis gamma_pos   : gamma > 0.

Hypothesis eta_pos     : eta > 0.

Hypothesis dt_small    : 0 < dt <= 0.01.


(* ================================================================ *)

(* ACTIVATION                                                      *)

(* ================================================================ *)


Definition activation (x:R) : R :=

  Rmin 1 (Rmax 0 x).


Lemma activation_bound :

  forall x, 0 <= activation x <= 1.

Proof.

  intro x.

  unfold activation.

  split.

  - apply Rmin_glb.

    + lra.

    + apply Rmax_l.

  - apply Rmin_l.

Qed.


(* ================================================================ *)

(* POLICY                                                          *)

(* ================================================================ *)


Definition hidden (st:State) : R :=

  eta * (s st * r st) - gamma * psi st.


Definition deep_policy (st:State) : R :=

  activation (hidden st - n st).


Lemma policy_bound :

  forall st,

    0 <= deep_policy st <= 1.

Proof.

  intro st.

  apply activation_bound.

Qed.


(* ================================================================ *)

(* LEARNING STEP                                                   *)

(* ================================================================ *)


Definition step (st:State) : R :=

  psi st + alpha * deep_policy st * dt.


Definition Next (st:State) : State :=

  mkState

    (activation (step st))

    (s st)

    (r st)

    (n st).


(* ================================================================ *)

(* SAFETY PROPERTIES                                               *)

(* ================================================================ *)


Theorem Safety :

  forall st,

    Valid st ->

    Valid (Next st).

Proof.

  intros st H.

  destruct H as [Hpsi [Hs [Hr Hn]]].

  unfold Valid, Next.

  simpl.

  repeat split.

  - apply activation_bound.

  - exact Hs.

  - exact Hr.

  - exact Hn.

Qed.


(* ================================================================ *)

(* DETERMINISM                                                     *)

(* ================================================================ *)


Theorem Deterministic :

  forall st,

    exists st', st' = Next st.

Proof.

  intro st.

  exists (Next st).

  reflexivity.

Qed.


(* ================================================================ *)

(* STEP MONOTONICITY                                               *)

(* ================================================================ *)


Theorem step_nonnegative_increment :

  forall st,

    Valid st ->

    psi st <= step st.

Proof.

  intros st H.

  unfold step.


  destruct H as [Hpsi [Hs [Hr Hn]]].

  destruct alpha_range as [Ha _].

  destruct dt_small as [Hd _].

  destruct (policy_bound st) as [Hp _].


  assert (Hprod : 0 <= alpha * deep_policy st * dt).

  {

    apply Rmult_le_pos.

    - apply Rmult_le_pos.

      + lra.

      + exact Hp.

    - lra.

  }


  lra.

Qed.


(* ================================================================ *)

(* LYAPUNOV FUNCTION                                               *)

(* ================================================================ *)


Definition V (st:State) : R :=

  (psi st - 1/2)^2 + n st.


Lemma V_nonnegative :

  forall st,

    Valid st ->

    0 <= V st.

Proof.

  intros st H.

  unfold V.

  destruct H as [_ [_ [_ Hn]]].

  destruct Hn as [Hn0 _].


  assert (0 <= (psi st - 1/2)^2) by apply Rle_0_sqr.

  lra.

Qed.


(* ================================================================ *)

(* ENERGY BOUNDED                                                 *)

(* ================================================================ *)


Theorem V_bounded :

  forall st,

    Valid st ->

    0 <= V st <= 5/4.

Proof.

  intros st H.

  unfold V.


  destruct H as [Hpsi [_ [_ Hn]]].

  destruct Hpsi as [H0 H1].

  destruct Hn as [Hn0 Hn1].


  split.

  - apply Rplus_le_le_0_compat.

    + apply Rle_0_sqr.

    + lra.


  - assert (Habs : Rabs (psi st - 1/2) <= 1/2).

    {

      assert (-1/2 <= psi st - 1/2) by lra.

      assert (psi st - 1/2 <= 1/2) by lra.

      apply Rabs_le; split; assumption.

    }


    assert ((psi st - 1/2)^2 <= (1/2)^2).

    {

      apply Rle_trans with (r2 := (Rabs (psi st - 1/2))^2).

      - apply Rle_0_sqr.

      - rewrite <- Rabs_sqr.

        apply Rmult_le_compat; try lra.

        + apply Rabs_pos.

        + exact Habs.

    }


    replace ((1/2)^2) with (1/4) by field.

    lra.

Qed.


(* ================================================================ *)

(* NEXT STATE PSI BOUND                                            *)

(* ================================================================ *)


Theorem Next_valid_psi :

  forall st,

    0 <= psi (Next st) <= 1.

Proof.

  intro st.

  unfold Next.

  simpl.

  apply activation_bound.

Qed.


(* ================================================================ *)

(* END                                                             *)

(* ================================================================ *)

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