From 05958b9e55bdbbde3b61211633237ebeaa07bb6d Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Sun, 21 Dec 2008 21:47:57 +0000
Subject: [PATCH] bleah

---
 .../formal_topology/overlap/o-algebra.ma      | 144 ++++++++++++++----
 1 file changed, 115 insertions(+), 29 deletions(-)

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 4dcf99b8b..9e8b473b6 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
@@ -17,7 +17,11 @@ include "logic/cprop_connectives.ma".
 
 inductive bool : Type := true : bool | false : bool.
 
+<<<<<<< .mine
+lemma BOOL : setoid.
+=======
 lemma BOOL : objs1 SET.
+>>>>>>> .r9407
 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;
@@ -33,10 +37,14 @@ coercion hint.
 
 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.
+    (λm.match m with [ true ⇒ a | false ⇒ b ]) x =
+    (λm.match m with [ true ⇒ a | false ⇒ b ]) y.
 intros; cases x in H; cases y; simplify; intros; try apply refl; whd in H; cases H;
 qed. 
 
+<<<<<<< .mine
+interpretation "unary morphism comprehension with no proof" 'comprehension T P = 
+=======
 lemma if_then_else : ∀T:SET. ∀a,b:T. arrows1 SET BOOL T.
 intros; constructor 1; intros; 
 [ apply (match c with [ true ⇒ t | false ⇒ t1 ]);
@@ -44,24 +52,35 @@ intros; constructor 1; intros;
 qed.
 
 interpretation "mk " 'comprehension T P = 
+>>>>>>> .r9407
   (mk_unary_morphism T _ P _).
 
 notation > "hvbox({ ident i ∈ s | term 19 p | by })" with precedence 90
-for @{ 'comprehension_by $s (\lambda ${ident i}. $p) $by}.
+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 f p = 
+interpretation "unary morphism comprehension with proof" 'comprehension_by s \eta.f p = 
   (mk_unary_morphism s _ f p).
 
+<<<<<<< .mine
+=======
 definition A : ∀S:SET.∀a,b:S.arrows1 SET BOOL S.
 apply (λS,a,b.{ x ∈ BOOL | match x with [ true ⇒ a | false ⇒ b] | IF_THEN_ELSE_p S a b}).
 qed.
 
+>>>>>>> .r9407
 record OAlgebra : Type := {
   oa_P :> SET;
   oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1 *)
   oa_overlap: binary_morphism1 oa_P oa_P CPROP;
+<<<<<<< .mine
+  oa_meet: ∀I:setoid.unary_morphism (unary_morphism_setoid I oa_P) oa_P;
+  oa_join: ∀I:setoid.unary_morphism (unary_morphism_setoid I oa_P) oa_P;
+=======
   oa_meet: ∀I:SET.unary_morphism (arrows1 SET I oa_P) oa_P;
   oa_join: ∀I:SET.unary_morphism (arrows1 SET I oa_P) oa_P;
+>>>>>>> .r9407
   oa_one: oa_P;
   oa_zero: oa_P;
   oa_leq_refl: ∀a:oa_P. oa_leq a a; 
@@ -75,12 +94,20 @@ record OAlgebra : Type := {
   oa_overlap_preservers_meet: 
       ∀p,q.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 });
+<<<<<<< .mine
+  oa_join_split:
+      ∀I:setoid.∀p.∀q:I ⇒  oa_P.oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
+  (*
+  oa_base : setoid;
+=======
      (*(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:SET.∀p.∀q:arrows1 SET I oa_P.oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
   (*oa_base : setoid;
+>>>>>>> .r9407
   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
 }.
@@ -91,35 +118,39 @@ notation "hovbox(a mpadded width -150% (>)< b)" non associative with precedence
 for @{ 'overlap $a $b}.
 interpretation "o-algebra overlap" 'overlap a b = (fun1 ___ (oa_overlap _) 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 $_)) }.
-notation < "hovbox(a ∧ b)" left associative with precedence 50
-for @{ 'oa_meet
-  ($foo $bar $baz 
-    (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ]) 
-    $res) }.
-
-interpretation "o-algebra meet" 'oa_meet f = (fun_1 __ (oa_meet __) f).
-(*interpretation "o-algebra binary meet" 'and 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 = 
+  (fun_1 __ (oa_meet __) f).
+interpretation "o-algebra meet with explicit function" 'oa_meet_mk f = 
+  (fun_1 __ (oa_meet __) (mk_unary_morphism _ _ f _)).
+
+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 = 
+  (fun_1 __ (oa_join __) f).
+interpretation "o-algebra join with explicit function" 'oa_join_mk f = 
+  (fun_1 __ (oa_join __) (mk_unary_morphism _ _ f _)).
 
 record ORelation (P,Q : OAlgebra) : Type ≝ {
   or_f :> arrows1 SET P Q;
@@ -150,8 +181,25 @@ intros (P Q);
 constructor 1;
 [ apply (ORelation P Q);
 | constructor 1;
+<<<<<<< .mine
+   [ alias symbol "and" = "constructive and".
+     apply (λp,q. And4 (∀a.p⎻* a = q⎻* a) (∀a.p⎻ a  = q⎻ a)
+                       (∀a.p a   = q a)   (∀a.p* a  = q* a)); 
+   | whd; simplify; intros; repeat split; intros; apply refl;
+=======
    [ apply (λp,q. eq1 ? p⎻* q⎻* ∧ eq1 ? p⎻ q⎻ ∧ eq1 ? p q ∧ eq1 ? p* q* ); 
    | whd; simplify; intros; repeat split; intros; apply refl1;
+>>>>>>> .r9407
+<<<<<<< .mine
+   | whd; simplify; intros; cases H; clear H; split; 
+     intro a; apply sym; generalize in match a;assumption;
+   | whd; simplify; intros; cases H; cases H1; clear H H1; split; intro a;
+     [ apply (.= (H2 a)); apply H6;
+     | apply (.= (H3 a)); apply H7;
+     | apply (.= (H4 a)); apply H8;
+     | apply (.= (H5 a)); apply H9;]]]
+qed.  
+=======
    | whd; simplify; intros; cases H; cases H1; cases H3; clear H H3 H1;
      repeat split; intros; apply sym1; assumption;
    | whd; simplify; intros; cases H; cases H1; cases H2; cases H4; cases H6; cases H8;
@@ -164,7 +212,11 @@ constructor 1;
       |*: assumption
       ]]]
 qed.
+>>>>>>> .r9407
 
+<<<<<<< .mine
+definition ORelation_composition : ∀P,Q,R. 
+=======
 lemma hint1 : ∀P,Q. ORelation_setoid P Q → arrows1 SET P Q. intros; apply (or_f ?? c);qed.
 coercion hint1.
 
@@ -175,11 +227,34 @@ lemma hint2: OAlgebra → setoid. intros; apply (oa_P o). qed.
 coercion hint2.
 
 definition composition : ∀P,Q,R. 
+>>>>>>> .r9407
   binary_morphism1 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
 intros;
 constructor 1;
 [ intros (F G);
   constructor 1;
+<<<<<<< .mine
+    [ apply {x ∈ P | G (F x)}; intros; simplify; apply (†(†H));
+    | apply {x ∈ P | G⎻* (F⎻* x)}; intros; simplify; apply (†(†H));
+    | apply {x ∈ R | F* (G* x)}; intros; simplify; apply (†(†H));
+    | apply {x ∈ R | F⎻ (G⎻ x)}; intros; simplify; apply (†(†H));
+    | intros; simplify; 
+      lapply (or_prop1 ?? G (F p) q) as H1; lapply (or_prop1 ?? F p (G* q)) as H2;
+      split; intro H; 
+        [ apply (if1 ?? H2); apply (if1 ?? H1); apply H;
+        | apply (fi1 ?? H1); apply (fi1 ?? H2); apply H;] 
+    | intros; simplify;
+      lapply (or_prop2 ?? G p (F⎻* q)) as H1; lapply (or_prop2 ?? F (G⎻ p) q) as H2;
+      split; intro H;
+        [ apply (if1 ?? H1); apply (if1 ?? H2); apply H;   
+        | apply (fi1 ?? H2); apply (fi1 ?? H1); apply H;]
+    | intros; simplify;
+      lapply (or_prop3 ?? F p (G⎻ q)) as H1; lapply (or_prop3 ?? G (F p) q) as H2;
+      split; intro H;
+        [ apply (if1 ?? H1); apply (if1 ?? H2); apply H;   
+        | apply (fi1 ?? H2); apply (fi1 ?? H1); apply H;]]
+| intros; simplify; split; simplify; intros; elim DAEMON;]
+=======
   [ apply (G ∘ F);
   | apply (G⎻* ∘ F⎻* );
   | apply (F* ∘ G* );
@@ -204,6 +279,7 @@ constructor 1;
      lapply (.= ((†H1)‡#)); [8: apply Hletin;
    [ apply trans1; [2: lapply (prop1); [apply Hletin;
 *)]
+>>>>>>> .r9407
 qed.
 
 definition OA : category1.
@@ -211,6 +287,16 @@ split;
 [ apply (OAlgebra);
 | intros; apply (ORelation_setoid o o1);
 | intro O; split;
+<<<<<<< .mine
+  [1,2,3,4: constructor 1; [1,3,5,7:apply (λx.x);|*:intros;assumption]
+  |5,6,7:intros;split;intros; assumption;] 
+|4: apply ORelation_composition;
+|*: elim DAEMON;]
+qed. 
+
+
+
+=======
   [1,2,3,4: apply id1;
   |5,6,7:intros; apply refl1;] 
 | apply composition;
@@ -218,4 +304,4 @@ split;
   [1,3: apply (comp_assoc1); | 2,4: apply ((comp_assoc1 ????????) \sup -1);]
 | intros; repeat split; unfold composition; simplify; apply id_neutral_left1;
 | intros; repeat split; unfold composition; simplify; apply id_neutral_right1;]
-qed.
\ No newline at end of file
+qed.>>>>>>> .r9407
-- 
2.39.2