(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
Fixpoint domain_equal (A B : Set) (m0 : Map A) (m1 : Map B) {struct m1} :
Prop :=
match m0, m1 with
| M0, M0 => True
| M0, M1 _ _ => False
| M0, M2 _ _ => False
| M1 _ _, M0 => False
| M1 a _, M1 b _ => a = b
| M1 _ _, M2 _ _ => False
| M2 _ _, M0 => False
| M2 _ _, M1 _ _ => False
| M2 a b, M2 c d => domain_equal A B a c /\ domain_equal A B b d
end.
Definition mEnsemble (A : Set) := Map A -> Prop.
Definition mRelation (A : Set) := Map A -> Map A -> Prop.
Definition r_symmetric (A : Set) (r : mRelation A) :=
forall x y : Map A, r x y -> r y x.
Definition r_antisymmetric (A : Set) (r : mRelation A) :=
forall x y : Map A, r x y -> r y x -> x = y.
Definition r_transitive (A : Set) (r : mRelation A) :=
forall x y z : Map A, r x y -> r y z -> r x z.
Definition r_reflexive (A : Set) (r : mRelation A) := forall x : Map A, r x x.
Definition r_order (A : Set) (r : mRelation A) :=
r_reflexive A r /\ r_antisymmetric A r /\ r_transitive A r.
Definition mini (A : Set) (r : mRelation A) (T : mEnsemble A)
(e : Map A) := T e /\ (forall x : Map A, T x -> r e x).
Definition maxi (A : Set) (r : mRelation A) (T : mEnsemble A)
(e : Map A) := T e /\ (forall x : Map A, T x -> r x e).
Definition mLattice (A : Set) (r : mRelation A) (T : mEnsemble A)
(e f : Map A) := r_order A r /\ mini A r T e /\ maxi A r T f.
Inductive prechain (A : Set) : Set :=
| single : Map A -> prechain A
| concat : prechain A -> Map A -> prechain A.
Inductive prechain_dom_ok (A : Set) : mEnsemble A -> prechain A -> Prop :=
| domok_single :
forall (x : Map A) (T : mEnsemble A),
T x -> prechain_dom_ok A T (single A x)
| domok_concat :
forall (x : Map A) (T : mEnsemble A) (p : prechain A),
T x -> prechain_dom_ok A T p -> prechain_dom_ok A T (concat A p x).
Fixpoint chain_length (A : Set) (p : prechain A) {struct p} : nat :=
match p with
| single x => 1
| concat x y => S (chain_length A x)
end.
Definition prechain_last (A : Set) (p : prechain A) :
Map A := match p with
| single x => x
| concat z x => x
end.
Inductive prechain_incr (A : Set) : mRelation A -> prechain A -> Prop :=
| incr_single :
forall (x : Map A) (r : mRelation A), prechain_incr A r (single A x)
| incr_concat :
forall (x : Map A) (r : mRelation A) (p : prechain A),
r (prechain_last A p) x ->
prechain_incr A r p -> prechain_incr A r (concat A p x).
Inductive chain (A : Set) :
mEnsemble A -> mRelation A -> prechain A -> Prop :=
| chain_single :
forall (x : Map A) (T : mEnsemble A) (r : mRelation A),
T x -> chain A T r (single A x)
| chain_concat_s :
forall (x y : Map A) (T : mEnsemble A) (r : mRelation A),
T x -> T y -> r x y -> chain A T r (concat A (single A x) y)
| chain_concat_m :
forall (x y : Map A) (z : prechain A) (T : mEnsemble A)
(r : mRelation A),
T y ->
r x y ->
chain A T r (concat A z x) -> chain A T r (concat A (concat A z x) y).
Definition pre_domok_incr_chain_def (A : Set) (p : prechain A) :=
forall (T : mEnsemble A) (r : mRelation A),
prechain_dom_ok A T p /\ prechain_incr A r p -> chain A T r p.
Inductive dist_chain (A : Set) : prechain A -> Prop :=
| dist_single : forall x : Map A, dist_chain A (single A x)
| dist_concat_s :
forall x y : Map A, x <> y -> dist_chain A (concat A (single A x) y)
| dist_concat_m :
forall (x y : Map A) (z : prechain A),
x <> y ->
dist_chain A (concat A z x) -> dist_chain A (concat A (concat A z x) y).
Inductive non_dist_chain (A : Set) : prechain A -> Prop :=
| non_dist_concat_s :
forall x : Map A, non_dist_chain A (concat A (single A x) x)
| non_dist_concat_m_hd :
forall (x : Map A) (z : prechain A),
non_dist_chain A (concat A (concat A z x) x)
| non_dist_concat_m_tl :
forall (x y : Map A) (z : prechain A),
non_dist_chain A (concat A z x) ->
non_dist_chain A (concat A (concat A z x) y).
Definition sas_chain (A : Set) (T : mEnsemble A) (r : mRelation A)
(p : prechain A) : Prop := chain A T r p /\ dist_chain A p.
Definition dist_compl_def_0 (A : Set) (p : prechain A) : Prop :=
dist_chain A p \/ non_dist_chain A p.
Definition dist_compl_def_1 (A : Set) (p : prechain A) : Prop :=
dist_compl_def_0 A p -> forall m : Map A, dist_compl_def_0 A (concat A p m).
(* propriété de chaine strictement croissante bornée *)
Definition bounded_sas_chain (A : Set) (T : mEnsemble A)
(r : mRelation A) (n : nat) : Prop :=
forall p : prechain A, sas_chain A T r p -> chain_length A p <= n.
Definition def_ok_app (A : Set) (T : mEnsemble A) (f : Map A -> Map A) :
Prop := forall x : Map A, T x -> T (f x).
Definition increasing_app (A : Set) (r : mRelation A)
(f : Map A -> Map A) : Prop := forall x y : Map A, r x y -> r (f x) (f y).
Definition fix_point (A : Set) (T : mEnsemble A) (f : Map A -> Map A)
(x : Map A) : Prop := T x /\ f x = x.
Definition inf_fix_points (A : Set) (T : mEnsemble A)
(r : mRelation A) (f : Map A -> Map A) (x : Map A) : Prop :=
forall y : Map A, fix_point A T f y -> r x y.
Definition lower_fix_point (A : Set) (T : mEnsemble A)
(r : mRelation A) (f : Map A -> Map A) (x : Map A) : Prop :=
fix_point A T f x /\ inf_fix_points A T r f x.
(* itération d'une fonction *)
Fixpoint iteres (A : Set) (f : Map A -> Map A) (x : Map A)
(n : nat) {struct n} : prechain A :=
match n with
| O => single A x
| S p =>
match iteres A f x p with
| single y => concat A (single A y) (f y)
| concat z y => concat A (concat A z y) (f y)
end
end.
Fixpoint power (A : Set) (f : A -> A) (x : A) (n : nat) {struct n} : A :=
match n with
| O => x
| S n => f (power A f x n)
end.
Inductive MapFlag (A : Set) : Set :=
| flag_true : Map A -> MapFlag A
| flag_false : Map A -> MapFlag A.
Fixpoint lazy_power_aux (A : Set) (egalite : Map A -> Map A -> bool)
(f : Map A -> Map A) (x : Map A) (n : nat) {struct n} :
MapFlag A :=
match n with
| O => flag_false A x
| S p =>
match lazy_power_aux A egalite f x p with
| flag_true y => flag_true A y
| flag_false y =>
match f y with
| z => if egalite y z then flag_true A y else flag_false A z
end
end
end.
Definition lazy_power (A : Set) (egalite : Map A -> Map A -> bool)
(f : Map A -> Map A) (x : Map A) (n : nat) : Map A :=
match lazy_power_aux A egalite f x n with
| flag_false z => z
| flag_true z => z
end.
Fixpoint iteres_0 (A : Set) (f : Map A -> Map A) (x : Map A)
(n : nat) {struct n} : prechain A :=
match n with
| O => single A x
| S p =>
match iteres_0 A f x p with
| single y => concat A (single A y) (power (Map A) f x (S p))
| concat z y => concat A (concat A z y) (power (Map A) f x (S p))
end
end.
Definition iteres_ult_const_def_0 (A : Set) (p : prechain A) : Prop :=
forall (f : Map A -> Map A) (x : Map A) (n : nat),
p = iteres A f x n ->
non_dist_chain A p ->
exists q : nat, S q <= n /\ power (Map A) f x q = power (Map A) f x (S q).

Lemma iteres_lower_fix_point :
forall (A : Set) (T : mEnsemble A) (r : mRelation A)
(f : Map A -> Map A) (e : Map A) (n m : nat),
mini A r T e ->
def_ok_app A T f ->
increasing_app A r f ->
bounded_sas_chain A T r m ->
m <= n -> lower_fix_point A T r f (power (Map A) f e n).