X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Fo-algebra.ma;h=ce9583da36098ffd0861c9b517aa3c84561be823;hb=b93b2e4f499c30b01b838f75a1e95df43920ffcc;hp=f04b0a70ca86ed5c33e8638aeeb1011a49b2a7b5;hpb=62e9e8296d172d6497f9ad29bad402fbad2014c3;p=helm.git

diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
index f04b0a70c..ce9583da3 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
@@ -12,160 +12,324 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "datatypes/bool.ma".
-include "datatypes/categories.ma".
+include "categories.ma".
 include "logic/cprop_connectives.ma". 
 
-lemma ums : setoid → setoid → setoid.
-intros (S T);
-constructor 1;
-[ apply (unary_morphism S T);
-| constructor 1;
-  [ intros (f1 f2); apply (∀a,b:S.eq1 ? a b → eq1 ? (f1 a) (f2 b));
-  | whd; simplify; intros; apply (.= (†H)); apply refl1;
-  | whd; simplify; intros; apply (.= (†H1)); apply sym1; apply H; apply refl1;
-  | whd; simplify; intros; apply (.= (†H2)); apply (.= (H ?? #)); apply (.= (H1 ?? #)); apply rule #;]] 
-qed.
+inductive bool : Type0 := true : bool | false : bool.
 
-lemma BOOL : setoid.
+lemma BOOL : objs1 SET.
 constructor 1; [apply bool] constructor 1;
 [ intros (x y); apply (match x with [ true ⇒ match y with [ true ⇒ True | _ ⇒ False] | false ⇒ match y with [ true ⇒ False | false ⇒ True ]]);
 | whd; simplify; intros; cases x; apply I;
 | whd; simplify; intros 2; cases x; cases y; simplify; intros; assumption;
-| whd; simplify; intros 3; cases x; cases y; cases z; simplify; intros; try assumption; apply I]
+| whd; simplify; intros 3; cases x; cases y; cases z; simplify; intros; 
+  try assumption; apply I]
 qed.
 
 lemma IF_THEN_ELSE_p :
-  ∀S:setoid.∀a,b:S.∀x,y:BOOL.x = y → 
-  let f ≝ λm.match m with [ true ⇒ a | false ⇒ b ] in f x = f y.
-intros; cases x in H; cases y; simplify; intros; try apply refl; whd in H; cases H;
-qed. 
-
-lemma if_then_else : ∀T:setoid. ∀a,b:T. ums BOOL T.
-intros; constructor 1; intros; 
-[ apply (match c2 with [ true ⇒ c | false ⇒ c1 ]);
-| apply (IF_THEN_ELSE_p T c c1 a a' H);]
+  ∀S:setoid1.∀a,b:S.∀x,y:BOOL.x = y → 
+    (λm.match m with [ true ⇒ a | false ⇒ b ]) x =
+    (λm.match m with [ true ⇒ a | false ⇒ b ]) y.
+whd in ⊢ (?→?→?→%→?);
+intros; cases x in e; cases y; simplify; intros; try apply refl1; whd in e; cases e;
 qed.
 
-record OAlgebra : Type := {
-  oa_P :> setoid;
-  oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1 *)
+interpretation "unary morphism comprehension with no proof" 'comprehension T P = 
+  (mk_unary_morphism T _ P _).
+interpretation "unary morphism1 comprehension with no proof" 'comprehension T P = 
+  (mk_unary_morphism1 T _ P _).
+
+notation > "hvbox({ ident i ∈ s | term 19 p | by })" with precedence 90
+for @{ 'comprehension_by $s (λ${ident i}. $p) $by}.
+notation < "hvbox({ ident i ∈ s | term 19 p })" with precedence 90
+for @{ 'comprehension_by $s (λ${ident i}:$_. $p) $by}.
+
+interpretation "unary morphism comprehension with proof" 'comprehension_by s \eta.f p = 
+  (mk_unary_morphism s _ f p).
+interpretation "unary morphism1 comprehension with proof" 'comprehension_by s \eta.f p = 
+  (mk_unary_morphism1 s _ f p).
+
+definition hint: Type_OF_category2 SET1 → setoid2.
+ intro; apply (setoid2_of_setoid1 t); qed.
+coercion hint.
+
+definition hint2: Type_OF_category1 SET → objs2 SET1.
+ intro; apply (setoid1_of_setoid t); qed.
+coercion hint2.
+
+(* per il set-indexing vedere capitolo BPTools (foundational tools), Sect. 0.3.4 complete
+   lattices, Definizione 0.9 *)
+(* USARE L'ESISTENZIALE DEBOLE *)
+(*alias symbol "comprehension_by" = "unary morphism comprehension with proof".*)
+record OAlgebra : Type2 := {
+  oa_P :> SET1;
+  oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1, CPROP importante che sia small *)
   oa_overlap: binary_morphism1 oa_P oa_P CPROP;
-  oa_meet: ∀I:setoid.unary_morphism (ums I oa_P) oa_P;
-  oa_join: ∀I:setoid.unary_morphism (ums I oa_P) oa_P;
+  oa_meet: ∀I:SET.unary_morphism2 (arrows2 SET1 I oa_P) oa_P;
+  oa_join: ∀I:SET.unary_morphism2 (arrows2 SET1 I oa_P) oa_P;
   oa_one: oa_P;
   oa_zero: oa_P;
   oa_leq_refl: ∀a:oa_P. oa_leq a a; 
   oa_leq_antisym: ∀a,b:oa_P.oa_leq a b → oa_leq b a → a = b;
   oa_leq_trans: ∀a,b,c:oa_P.oa_leq a b → oa_leq b c → oa_leq a c;
   oa_overlap_sym: ∀a,b:oa_P.oa_overlap a b → oa_overlap b a;
+  (* Errore: = in oa_meet_inf e oa_join_sup *)
   oa_meet_inf: ∀I.∀p_i.∀p:oa_P.oa_leq p (oa_meet I p_i) → ∀i:I.oa_leq p (p_i i);
   oa_join_sup: ∀I.∀p_i.∀p:oa_P.oa_leq (oa_join I p_i) p → ∀i:I.oa_leq (p_i i) p;
   oa_zero_bot: ∀p:oa_P.oa_leq oa_zero p;
   oa_one_top: ∀p:oa_P.oa_leq p oa_one;
-  oa_overlap_preservers_meet: 
-      ∀p,q.oa_overlap p q → oa_overlap p 
-     (oa_meet BOOL (if_then_else oa_P p q));
-  oa_join_split: (* ha I → oa_P da castare a funX (ums A oa_P) *)
-      ∀I:setoid.∀p.∀q:ums I oa_P.oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
+  oa_overlap_preserves_meet_: 
+      ∀p,q:oa_P.oa_overlap p q → oa_overlap p 
+       (oa_meet ? { x ∈ BOOL | match x with [ true ⇒ p | false ⇒ q ] | IF_THEN_ELSE_p oa_P p q });
+  (* ⇔ deve essere =, l'esiste debole *)
+  oa_join_split:
+      ∀I:SET.∀p.∀q:arrows2 SET1 I oa_P.
+       oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
   (*oa_base : setoid;
+  1) enum non e' il nome giusto perche' non e' suriettiva
+  2) manca (vedere altro capitolo) la "suriettivita'" come immagine di insiemi di oa_base
   oa_enum : ums oa_base oa_P;
-  oa_density: ∀p,q.(∀i.oa_overlap p (oa_enum i) → oa_overlap q (oa_enum i)) → oa_leq p q*)
+  oa_density: ∀p,q.(∀i.oa_overlap p (oa_enum i) → oa_overlap q (oa_enum i)) → oa_leq p q
+  *)
   oa_density: 
       ∀p,q.(∀r.oa_overlap p r → oa_overlap q r) → oa_leq p q
 }.
 
-(*
-axiom Al : OAlgebra.
-axiom x : carr (oa_P Al).
-definition wwww := (oa_density Al x x).
-definition X := ((λx:Type.λa:x.True) ? wwww).
-*)
-
-interpretation "o-algebra leq" 'leq a b = (fun1 ___ (oa_leq _) a b).
+interpretation "o-algebra leq" 'leq a b = (fun21 ___ (oa_leq _) a b).
 
 notation "hovbox(a mpadded width -150% (>)< b)" non associative with precedence 45
 for @{ 'overlap $a $b}.
-interpretation "o-algebra overlap" 'overlap a b = (fun1 ___ (oa_overlap _) a b).
+interpretation "o-algebra overlap" 'overlap a b = (fun21 ___ (oa_overlap _) a b).
 
-notation > "hovbox(a ∧ b)" left associative with precedence 50
-for @{ 'oa_meet2 $a $b }.
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (\emsp) \nbsp term 90 p)" 
+non associative with precedence 50 for @{ 'oa_meet $p }.
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (ident i ∈  I) break term 90 p)" 
+non associative with precedence 50 for @{ 'oa_meet_mk (λ${ident i}:$I.$p) }.
+
+(*
+notation < "hovbox(a ∧ b)" left associative with precedence 35
+for @{ 'oa_meet_mk (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ]) }.
+*)
 notation > "hovbox(∧ f)" non associative with precedence 60
 for @{ 'oa_meet $f }.
-notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (ident i ∈  I) break term 90 p)" non associative with precedence 50
-for @{ 'oa_meet (λ${ident i}:$I.$p) }.
-notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (\emsp) \nbsp term 90 p)" non associative with precedence 50
-for @{ 'oa_meet (λ${ident i}.($p $_)) }.
-notation < "hovbox(a ∧ b)" left associative with precedence 50
-for @{ 'oa_meet2 $a $b }.
-
-interpretation "o-algebra meet" 'oa_meet \eta.f = (fun_1 __ (oa_meet __) f).
-interpretation "o-algebra binary meet" 'oa_meet2 x y = (fun_1 __ (oa_meet _ BOOL) (if_then_else _ x y)).
 (*
-notation > "hovbox(a ∨ b)" left associative with precedence 49
-for @{ 'oa_join (λx__:bool.match x__ with [ true ⇒ $a | false ⇒ $b ]) }.
+notation > "hovbox(a ∧ b)" left associative with precedence 50
+for @{ 'oa_meet (mk_unary_morphism BOOL ? (λx__:bool.match x__ with [ true ⇒ $a | false ⇒ $b ]) (IF_THEN_ELSE_p ? $a $b)) }.
+*)
+interpretation "o-algebra meet" 'oa_meet f = 
+  (fun12 __ (oa_meet __) f).
+interpretation "o-algebra meet with explicit function" 'oa_meet_mk f = 
+  (fun12 __ (oa_meet __) (mk_unary_morphism _ _ f _)).
+
+definition hint3: OAlgebra → setoid1.
+ intro; apply (oa_P o);
+qed.
+coercion hint3.
+
+definition hint4: ∀A. setoid2_OF_OAlgebra A → hint3 A.
+ intros; apply t;
+qed.
+coercion hint4.
+
+definition binary_meet : ∀O:OAlgebra. binary_morphism1 O O O.
+intros; split;
+[ intros (p q); 
+  apply (∧ { x ∈ BOOL | match x with [ true ⇒ p | false ⇒ q ] | IF_THEN_ELSE_p ? p q });
+| intros; lapply (prop12 ? O (oa_meet O BOOL));
+   [2: apply ({ x ∈ BOOL | match x with [ true ⇒ a | false ⇒ b ] | IF_THEN_ELSE_p ? a b });
+   |3: apply ({ x ∈ BOOL | match x with [ true ⇒ a' | false ⇒ b' ] | IF_THEN_ELSE_p ? a' b' });
+   | apply Hletin;]
+  intro x; simplify; cases x; simplify; assumption;]
+qed.
+
+notation "hovbox(a ∧ b)" left associative with precedence 35
+for @{ 'oa_meet_bin $a $b }.
+interpretation "o-algebra binary meet" 'oa_meet_bin a b = 
+  (fun21 ___ (binary_meet _) a b).
+
+lemma oa_overlap_preservers_meet: ∀O:OAlgebra.∀p,q:O.p >< q → p >< (p ∧ q).
+intros;  lapply (oa_overlap_preserves_meet_ O p q f);
+lapply (prop21 O O CPROP (oa_overlap O) p p ? (p ∧ q) # ?);
+[3: apply (if ?? (Hletin1)); apply Hletin;|skip] apply refl1;
+qed.
+
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (\emsp) \nbsp term 90 p)" 
+non associative with precedence 49 for @{ 'oa_join $p }.
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (ident i ∈  I) break term 90 p)" 
+non associative with precedence 49 for @{ 'oa_join_mk (λ${ident i}:$I.$p) }.
+notation < "hovbox(a ∨ b)" left associative with precedence 49
+for @{ 'oa_join_mk (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ]) }.
+
 notation > "hovbox(∨ f)" non associative with precedence 59
 for @{ 'oa_join $f }.
-notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (ident i ∈  I) break term 90 p)" non associative with precedence 49
-for @{ 'oa_join (λ${ident i}:$I.$p) }.
-notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (\emsp) \nbsp term 90 p)" non associative with precedence 49
-for @{ 'oa_join (λ${ident i}.($p $_)) }.
-notation < "hovbox(a ∨ b)" left associative with precedence 49
-for @{ 'oa_join (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ]) }.
+notation > "hovbox(a ∨ b)" left associative with precedence 49
+for @{ 'oa_join (mk_unary_morphism BOOL ? (λx__:bool.match x__ with [ true ⇒ $a | false ⇒ $b ]) (IF_THEN_ELSE_p ? $a $b)) }.
 
-interpretation "o-algebra join" 'oa_join \eta.f = (oa_join _ _ f).
-*)
+interpretation "o-algebra join" 'oa_join f = 
+  (fun12 __ (oa_join __) f).
+interpretation "o-algebra join with explicit function" 'oa_join_mk f = 
+  (fun12 __ (oa_join __) (mk_unary_morphism _ _ f _)).
+
+definition hint5: OAlgebra → objs2 SET1.
+ intro; apply (oa_P o);
+qed.
+coercion hint5.
 
 record ORelation (P,Q : OAlgebra) : Type ≝ {
-  or_f : P → Q;
-  or_f_minus_star : P → Q;
-  or_f_star : Q → P;
-  or_f_minus : Q → P;
-  or_prop1 : ∀p,q. or_f p ≤ q ⇔ p ≤ or_f_star q;
-  or_prop2 : ∀p,q. or_f_minus p ≤ q ⇔ p ≤ or_f_minus_star q;
-  or_prop3 : ∀p,q. or_f p >< q ⇔ p >< or_f_minus q
+  or_f_ : P ⇒ Q;
+  or_f_minus_star_ : P ⇒ Q;
+  or_f_star_ : Q ⇒ P;
+  or_f_minus_ : Q ⇒ P;
+  or_prop1_ : ∀p,q. (or_f_ p ≤ q) = (p ≤ or_f_star_ q);
+  or_prop2_ : ∀p,q. (or_f_minus_ p ≤ q) = (p ≤ or_f_minus_star_ q);
+  or_prop3_ : ∀p,q. (or_f_ p >< q) = (p >< or_f_minus_ q)
 }.
 
-notation < "⨍ \sub (term 90 r)" non associative with precedence 90 for @{'OR_f $r}.
-notation < "⨍ \sub (term 90 r) term 90 a" non associative with precedence 70 for @{'OR_f_app1 $r $a}.
-notation > "⨍_(term 90 r)" non associative with precedence 90 for @{'OR_f $r}.
-interpretation "o-relation f" 'OR_f r = (or_f _ _ r).
-interpretation "o-relation f x" 'OR_f_app1 r a = (or_f _ _ r a).
-
-
-definition ORelation_setoid : OAlgebra → OAlgebra → setoid1.
+definition ORelation_setoid : OAlgebra → OAlgebra → setoid2.
 intros (P Q);
 constructor 1;
 [ apply (ORelation P Q);
 | constructor 1;
-   [ apply (λp,q. ∀a.⨍_p a = ⨍_q a (* ∧ f^-1 a = .... *));
-   | whd; simplify; intros; apply refl;
-   | whd; simplify; intros; apply (H ? \sup -1);
-   | whd; simplify; intros; apply trans; [2: apply H;|3: apply H1]]]
-qed.  
+   (* tenere solo una uguaglianza e usare la proposizione 9.9 per
+      le altre (unicita' degli aggiunti e del simmetrico) *)
+   [ apply (λp,q. And4 (eq2 ? (or_f_minus_star_ ?? p) (or_f_minus_star_ ?? q)) 
+             (eq2 ? (or_f_minus_ ?? p) (or_f_minus_ ?? q)) 
+             (eq2 ? (or_f_ ?? p) (or_f_ ?? q)) 
+             (eq2 ? (or_f_star_ ?? p) (or_f_star_ ?? q))); 
+   | whd; simplify; intros; repeat split; intros; apply refl2;
+   | whd; simplify; intros; cases H; clear H; split; 
+     intro a; apply sym1; generalize in match a;assumption;
+   | whd; simplify; intros; cases H; cases H1; clear H H1; split; intro a;
+     [ apply (.= (e a)); apply e4;
+     | apply (.= (e1 a)); apply e5;
+     | apply (.= (e2 a)); apply e6;
+     | apply (.= (e3 a)); apply e7;]]]
+qed.
+
+definition or_f_minus_star:
+ ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (P ⇒ Q).
+ intros; constructor 1;
+  [ apply or_f_minus_star_;
+  | intros; cases e; assumption]
+qed.
+
+definition or_f: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (P ⇒ Q).
+ intros; constructor 1;
+  [ apply or_f_;
+  | intros; cases e; assumption]
+qed.
+
+coercion or_f.
+
+definition or_f_minus: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (Q ⇒ P).
+ intros; constructor 1;
+  [ apply or_f_minus_;
+  | intros; cases e; assumption]
+qed.
+
+definition or_f_star: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (Q ⇒ P).
+ intros; constructor 1;
+  [ apply or_f_star_;
+  | intros; cases e; assumption]
+qed.
+
+lemma arrows1_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → (P ⇒ Q).
+intros; apply (or_f ?? t);
+qed.
+
+coercion arrows1_OF_ORelation_setoid.
+
+lemma umorphism_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → P ⇒ Q.
+intros; apply (or_f ?? t);
+qed.
+
+coercion umorphism_OF_ORelation_setoid.
+
+
+lemma uncurry_arrows : ∀B,C. arrows1 SET B C → B → C. 
+intros; apply ((fun1 ?? t) t1);
+qed.
 
-axiom DAEMON: False.
+coercion uncurry_arrows 1.
 
-definition composition : ∀P,Q,R. 
+lemma hint6 : ∀P,Q. arrows1 SET P Q → P ⇒ Q. intros; apply t;qed.
+coercion hint6.
+
+(*
+lemma hint2: OAlgebra → setoid. intros; apply (oa_P o). qed.
+coercion hint2 nocomposites.
+*)
+
+
+notation "r \sup *" non associative with precedence 90 for @{'OR_f_star $r}.
+notation > "r *" non associative with precedence 90 for @{'OR_f_star $r}.
+
+notation "r \sup (⎻* )" non associative with precedence 90 for @{'OR_f_minus_star $r}.
+notation > "r⎻*" non associative with precedence 90 for @{'OR_f_minus_star $r}.
+
+notation "r \sup ⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
+notation > "r⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
+
+interpretation "o-relation f⎻*" 'OR_f_minus_star r = (fun12 __ (or_f_minus_star _ _) r).
+interpretation "o-relation f⎻" 'OR_f_minus r = (fun12 __ (or_f_minus _ _) r).
+interpretation "o-relation f*" 'OR_f_star r = (fun12 __ (or_f_star _ _) r).
+
+definition or_prop1 : ∀P,Q:OAlgebra.∀F:ORelation_setoid P Q.∀p,q.
+   (F p ≤ q) = (p ≤ F* q).
+intros; apply (or_prop1_ ?? F p q);
+qed.
+
+definition or_prop2 : ∀P,Q:OAlgebra.∀F:ORelation_setoid P Q.∀p,q.
+   (F⎻ p ≤ q) = (p ≤ F⎻* q).
+intros; apply (or_prop2_ ?? F p q);
+qed.
+
+definition or_prop3 : ∀P,Q:OAlgebra.∀F:ORelation_setoid P Q.∀p,q.
+   (F p >< q) = (p >< F⎻ q).
+intros; apply (or_prop3_ ?? F p q);
+qed.
+
+definition ORelation_composition : ∀P,Q,R. 
   binary_morphism1 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
 intros;
 constructor 1;
 [ intros (F G);
   constructor 1;
-  [ apply (λx.⨍_G (⨍_F x));
-  |2,3,4,5,6,7: cases DAEMON;]
-| intros; cases DAEMON;]
+  [ apply (G ∘ F);
+  | apply (G⎻* ∘ F⎻* );
+  | apply (F* ∘ G* );
+  | apply (F⎻ ∘ G⎻);
+  | intros; 
+    change with ((G (F p) ≤ q) = (p ≤ (F* (G* q))));
+    apply (.= (or_prop1 :?));
+    apply (or_prop1 :?);
+  | intros;
+    change with ((F⎻ (G⎻ p) ≤ q) = (p ≤ (G⎻* (F⎻* q))));
+    apply (.= (or_prop2 :?));
+    apply or_prop2 ; 
+  | intros; change with ((G (F (p)) >< q) = (p >< (F⎻ (G⎻ q))));
+    apply (.= (or_prop3 :?));
+    apply or_prop3;
+  ]
+| intros; split; simplify; 
+   [1,3: unfold arrows1_OF_ORelation_setoid; apply ((†H)‡(†H1));
+   |2,4: apply ((†H1)‡(†H));]]
 qed.
 
-definition OA : category1. (* category2 *)
+definition OA : category1.
 split;
 [ apply (OAlgebra);
 | intros; apply (ORelation_setoid o o1);
 | intro O; split;
-  [1,2,3,4: apply (λx.x);
-  |*:intros;split;intros; assumption; ] 
-|*: elim DAEMON;]
-qed. 
-
-
-
+  [1,2,3,4: apply id1;
+  |5,6,7:intros; apply refl1;] 
+| apply ORelation_composition;
+| intros (P Q R S F G H); split;
+   [ change with (H⎻* ∘ G⎻* ∘ F⎻* = H⎻* ∘ (G⎻* ∘ F⎻* ));
+     apply (comp_assoc1 ????? (F⎻* ) (G⎻* ) (H⎻* ));
+   | apply ((comp_assoc1 ????? (H⎻) (G⎻) (F⎻))^-1);
+   | apply ((comp_assoc1 ????? F G H)^-1);
+   | apply ((comp_assoc1 ????? H* G* F* ));]
+| intros; split; unfold ORelation_composition; simplify; apply id_neutral_left1;
+| intros; split; unfold ORelation_composition; simplify; apply id_neutral_right1;]
+qed.
\ No newline at end of file