Clean PF version

thesis_lina_coq/eval_event_model.v

-Require Import Arith NPeano List Omega.
-
-Require Import event_model types. 
-
+Require Import Arith NPeano List Omega.
+
+Require Import event_model types. 
+
 (* ########################################################### *)
 (** *          Evaluations                                     *)
-(* ########################################################### *)
-
-Definition sigma1 (t : instant) : nb_occurrences :=
-match t with
-  | 0 => 0
-  | 1 => 0
-  | 2 => 1
-  | 3 => 2
-  | 4 => 1
-  | 5 => 1
-  | 9 => 3
-  | n => 4
- end.
-
-Check delta.
-
-(*
-Example test_delta : @delta sigma1 AE 1 3 = 1.
-unfold delta.
-simpl.
-unfold instant_of.
-assert (H : instant_after sigma1 AE 3 0 = 3).
- - 
-   unfold instant_after.
-   elim sigma1.
-   induction sigma1.
-   unfold instant_after_func.
-   simpl.
-   induction.
-Eval compute in (@next sigma1 AE 1).
-
+(* ########################################################### *)
+
+Definition sigma (t : instant) : nb_occurrences :=
+match t with
+  | 0 => 0
+  | 1 => 0
+  | 2 => 1
+  | 3 => 2
+  | 4 => 0
+  | 5 => 0
+  | 6 => 3
+  | n => 1
+ end.
+
+
+(*
+Example test_delta : @delta sigma1 AE 1 3 = 1.
+unfold delta.
+simpl.
+unfold instant_of.
+assert (H : instant_after sigma1 AE 3 0 = 3).
+ - 
+   unfold instant_after.
+   elim sigma1.
+   induction sigma1.
+   unfold instant_after_func.
+   simpl.
+   induction.
+Eval compute in (@next sigma1 AE 1).
+
 *)
\ No newline at end of file

thesis_lina_coq/event_models.v

-Require Import NPeano Arith Omega.
-
-Require Import event_functions trace types util.
-
+Require Import NPeano Arith Omega.
+
+Require Import event_functions trace types util tactics.
+
 (* #################################################### *)
 (* ** Definitions related to the event load function ** *)
-(* #################################################### *)
-
-Definition max_eta_trace (f : duration -> nb_occurrences) : Prop :=
-  forall t dt, eta t dt <= f dt.
-
-(**   The event load maximum function *)
-Definition eta_max (f : duration -> nb_occurrences) : Prop :=
-  max_eta_trace f /\ subadditive f.
-
-
+(* #################################################### *)
+
+Definition max_eta_trace (f : duration -> nb_occurrences) : Prop :=
+  forall t dt, eta t dt <= f dt.
+
+(**   The event load maximum function *)
+Definition eta_max (f : duration -> nb_occurrences) : Prop :=
+  max_eta_trace f /\ subadditive f.
+
+
 (* ######################################################## *)
 (* ** Definitions related to the event distance function ** *)                                
-(* ######################################################## *)
-
-Definition pseudo_superadditive (f : nat -> nat) : Prop :=
-  forall (x y : nat), f (x + 1) + f (y + 1) <= f (x + (y + 1)).
-Definition pseudo_superadditive (f : nat -> nat) : Prop :=
-  forall (x y : nat), f (x + 1) + f (y + 1) <= f (x + (y + 1)).
-
-Definition min_delta_trace (f : nb_occurrences -> duration) : Prop :=
-  forall n k, f k <= delta n k.
-
-(**   The event distance minimum function *)
-Definition delta_min (f : nb_occurrences -> duration) : Prop :=
-  min_delta_trace f /\ pseudo_superadditive f.
-
-(* Relation between delta and eta_max *)
-Property eta_max_by_delta :
-forall n k (eta_plus : duration -> nb_occurrences), 
-       max_eta_trace eta_plus
-       -> k <= eta_plus ((delta n k) + 1).
-Proof.
-intros.
-unfold max_eta_trace in H.
-transitivity (eta (instant_of n) (delta n k +1)).
-apply delta_to_eta.
-apply H.
-Qed.
+(* ######################################################## *)
+
+Definition pseudo_superadditive (f : nat -> nat) : Prop :=
+  forall (x y : nat), f (x + 1) + f (y + 1) <= f (x + (y + 1)).
+
+
+Definition min_delta_trace (f : nb_occurrences -> duration) : Prop :=
+  forall n k, 1 <= n -> f k <= delta n k.
+
+(**   The event distance minimum function *)
+Definition delta_min (f : nb_occurrences -> duration) : Prop :=
+  min_delta_trace f /\ pseudo_superadditive f.
+
+(* Relation between delta and eta_max *)
+Property eta_max_by_delta :
+forall n k (eta_plus : duration -> nb_occurrences), 
+        1 <= n -> 1 <= k -> max_eta_trace eta_plus
+       -> k <= eta_plus ((delta n k) + 1).
+Proof.
+introv I1 I2 H.
+transitivity (eta (instant_of n) (delta n k +1)).
+eapply delta_to_eta; try easy.
+apply H.
+Qed.
\ No newline at end of file

thesis_lina_coq/tactics.v

 (* ------------------------------------------------------- *)
 (** #<hr> <center> <h1>#
-        Basic and simple tactics and notations
-       (mostly taken form Software Foundations' LibTactics.v 
-        with a few complements) 
-#</h1>#    
+        Basic tactics and notations
+        (mostly taken from Software Foundations' LibTactics.v 
+         with a few complements) 
+#</h1>#
 #</center> <hr>#                                           *)
 (*          Pascal Fradet - 2014_2015                      *)
 (* ------------------------------------------------------- *)
@@ -614,7 +614,7 @@ Tactic Notation "inverts" hyp(H) "as" simple_intropattern(I1)
 (*   and the following small shortcuts                     *)
 (* ####################################################### *)
 
-Ltac easy := solve [simpl; intros; auto].
+Ltac easy := solve [cbn; intros; auto].
 (* Ltac easy := solve [simpl; intros; info_auto]. *)
 
 Ltac dupl_base h1 h2 := generalize h1; intro h2.

thesis_lina_coq/util.v

-Require Import List Coq.Logic.ConstructiveEpsilon Coq.Logic.Decidable Omega.
-
-Require Import tactics.
-
-
+Require Import List Coq.Logic.ConstructiveEpsilon Coq.Logic.Decidable Omega Psatz.
+Require Import tactics.
+
 (* ########################### *)
 (* ** Arithmetic properties ** *)
-(* ########################### *)
-
-Property plus_le_reg_r :
-forall n m p, 
-       n + p <= m + p -> n <= m.
-Proof.
-intros.
-apply plus_le_reg_l with (p := p). omega.
-Qed.
-
-
-Property plus_le_min:
-forall a b c,
- a <= b -> b - c = b - a + a - c .
-Proof.
-intros a b c H; omega.
-Qed.
-
-
+(* ########################### *)
+
+Lemma neq_le_imp_lt : forall a b, a <> b -> a <= b -> a < b.
+Proof. introv N I. omega. Qed.
+
+Lemma le_sub_eq : forall a b, b <= a -> a <= b <-> a=b.
+Proof. intros. omega. Qed.
+
+Lemma le_sub_plus : forall a b c, b <= a -> a - b + c = a + c - b.
+Proof. intros. omega. Qed.
+
+Lemma le_sub_add : forall a b, a <= b -> a + (b - a) = b.
+Proof. intros. omega. Qed.
+
+Lemma  sub_S_1: forall a : nat, S a - 1 = a.
+Proof. intros. omega. Qed.
+
+Lemma plus_sub_S_1 : forall a b, a + S b - 1 = a + b.
+Proof. intros. omega. Qed.
+
+Theorem le_minus_0: forall a b, a <= b <-> a - b = 0.
+Proof. intros. omega. Qed.
+
+Property plus_le_reg_r : forall a b c,
+                         a + c <= b + c -> a <= b.
+Proof. intros. apply plus_le_reg_l with (p := c). omega. Qed.
+
+Property lt_imp_eq : forall a b, a <= b -> b <= a -> a=b.
+Proof. intros. omega. Qed.
+
+Property plus_le_min: forall a b c,
+                      a <= b -> b - c = b - a + a - c .
+Proof. intros a b c H; omega. Qed.
+
+Lemma let_sub_lt : forall a b c, c <= a < b -> a - c < b - c.
+Proof. induction c; introv [A B]; omega. Qed.
+
+Lemma exists_int_if_lt : forall a b, a < b -> exists c, c > 0 /\ b = a + c.
+Proof.
+introv G. induction b.
+- omega.
+- apply lt_n_Sm_le in G.
+  apply le_lt_eq_dec in G.
+  destruct G as [G | G].
+  + apply IHb in G. destruct G as [c [G1 G2]].
+    exists (S c). omega.
+  + exists (S 0). omega.
+Qed.
+
+
+(* ############################################ *)
+(* ** Non-decreasing functions               ** *)
 (* ############################################ *)
-(* ** Non-decreasing functions ** *)
-(* ############################################ *)
-
-Definition non_decreasing (f: nat -> nat) : Prop := 
-forall a b, a < b -> f a <= f b.
-
-Property non_decreasing_lemma :
-forall (f: nat -> nat) a b,
-       non_decreasing f 
-       -> f a + 1 <= f b -> a + 1 <= b.
-Proof.
-intros.
-Admitted.
-
-
+
+Definition non_decreasing (f: nat -> nat) : Prop := forall a b, a < b -> f a <= f b.
+
+Property non_decreasing_lemma : forall (f: nat -> nat) a b,
+                                non_decreasing f -> f a + 1 <= f b
+                             -> a + 1 <= b.
+Proof.
+introv ND G.
+destruct (gt_eq_gt_dec a b) as [[I | I] | I]; try (subst; omega).
+apply ND in I. omega.
+Qed.
+
+Definition strictly_increasing(f: nat -> nat) : Prop := forall a b, a < b -> f a < f b.
+
+Lemma strict_incr_imp_non_decr : forall f, strictly_increasing f -> non_decreasing f.
+Proof. introv SI I. apply SI in I. omega. Qed.
+
+Definition non_increasing (f: nat -> nat) : Prop := forall a b, b < a -> f a <= f b.
+
 (* ###################################### *)
 (* ** Sub- and superadditive functions ** *)
-(* ###################################### *)
-
-Definition superadditive (f : nat -> nat) : Prop :=
-  forall (x y : nat), f x + f y <= f (x + y).
-
-Definition subadditive (f : nat -> nat) : Prop :=
-  forall (x y : nat), f (x + y) <= f x + f y.
-
-Property subadditive_implies_non_decreasing:
-forall (f: nat -> nat), 
-       subadditive f
-       -> non_decreasing f.
-Proof.
-Admitted.
-
-
-(**  General property on P:nat->Prop predicates: 
+(* ###################################### *)
+
+Definition superadditive (f : nat -> nat) : Prop :=
+  forall (x y : nat), f x + f y <= f (x + y).
+
+Definition subadditive (f : nat -> nat) : Prop :=
+  forall (x y : nat), f (x + y) <= f x + f y.
+
+(* Property subadditive_imp_non_decreasing:
+forall (f: nat -> nat), subadditive f -> non_decreasing f.
+Proof.
+False. Counterexample:
+f 0 = 1, f x = 0 if x > 0
+or
+f 0 = 0, f 1 = 2, f x = 1 if x>1
+*)
+
+
+Property superadditive_imp_non_decreasing:
+forall (f: nat -> nat), superadditive f -> non_decreasing  f.
+Proof.
+introv SA I.
+destruct (exists_int_if_lt a b I) as [c [I1 I2]].
+assert (X: (f a) + (f c) <=  f (a+c)) by apply SA.
+rewrite <- I2 in X. omega.
+Qed.
+
+Property superadditive_imp_f0_eq_0 : forall (f: nat -> nat), superadditive f -> f 0 = 0.
+Proof.
+introv G.
+assert (X:f 1 + f 0 <= f (1+0)) by apply G.
+cbn in X. omega.
+Qed.
+
+
+(**  General property on P:nat->Prop predicates:
      if P is decidable and P holds for at least one nat
      then there is a minimal integer for which P holds                  *)
 
-Lemma P_upto_n : forall P n, (forall x, P x \/ ~ P x) 
+Lemma P_upto_n : forall P n, (forall x, P x \/ ~ P x)
                              -> (exists m, m < n /\ P m) \/ (forall m, m < n -> ~ P m).
 Proof.
 introv Pdec. induction n.
@@ -76,9 +130,9 @@ introv Pdec. induction n.
       destruct G; try easy.
       apply IHn. omega.
 Qed.
-
 
-Lemma exists_min : forall n (P:nat->Prop), (forall x, P x \/ ~ P x) -> P n 
+
+Lemma exists_min : forall n (P:nat->Prop), (forall x, P x \/ ~ P x) -> P n
                                         -> exists m, P m /\ forall p, p < m -> ~ P p.
 Proof.
 induction n using lt_wf_ind; intros.