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;
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 ]);
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;
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
}.
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;
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;
|*: 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.
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* );
lapply (.= ((†H1)‡#)); [8: apply Hletin;
[ apply trans1; [2: lapply (prop1); [apply Hletin;
*)]
+>>>>>>> .r9407
qed.
definition OA : category1.
[ 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;
[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
projects / helm.git / commitdiff