(* *)
(**************************************************************************)
-include "logic/cprop_connectives.ma".
include "categories.ma".
-record powerset_carrier (A: objs1 SET) : Type1 ≝ { mem_operator: unary_morphism1 A CPROP }.
+record powerset_carrier (A: objs1 SET) : Type1 ≝ { mem_operator: A ⇒_1 CPROP }.
+interpretation "powerset low" 'powerset A = (powerset_carrier A).
+notation "hvbox(a break ∈. b)" non associative with precedence 45 for @{ 'mem_low $a $b }.
+interpretation "memlow" 'mem_low a S = (fun11 ?? (mem_operator ? S) a).
-definition subseteq_operator: ∀A: SET. powerset_carrier A → powerset_carrier A → CProp1 ≝
- λA:objs1 SET.λU,V.∀a:A. mem_operator ? U a → mem_operator ? V a.
+definition subseteq_operator: ∀A: objs1 SET. Ω^A → Ω^A → CProp0 ≝
+ λA:objs1 SET.λU,V.∀a:A. a ∈. U → a ∈. V.
theorem transitive_subseteq_operator: ∀A. transitive2 ? (subseteq_operator A).
intros 6; intros 2;
interpretation "powerset" 'powerset A = (powerset_setoid1 A).
interpretation "subset construction" 'subset \eta.x =
- (mk_powerset_carrier _ (mk_unary_morphism1 _ CPROP x _)).
+ (mk_powerset_carrier ? (mk_unary_morphism1 ? CPROP x ?)).
-definition mem: ∀A. binary_morphism1 A (Ω \sup A) CPROP.
+definition mem: ∀A. A × Ω^A ⇒_1 CPROP.
intros;
constructor 1;
[ apply (λx,S. mem_operator ? S x)
| apply s1; assumption]]
qed.
-interpretation "mem" 'mem a S = (fun21 ___ (mem _) a S).
+interpretation "mem" 'mem a S = (fun21 ??? (mem ?) a S).
-definition subseteq: ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) CPROP.
+definition subseteq: ∀A. Ω^A × Ω^A ⇒_1 CPROP.
intros;
constructor 1;
[ apply (λU,V. subseteq_operator ? U V)
apply (transitive_subseteq_operator ???? s s4) ]]
qed.
-interpretation "subseteq" 'subseteq U V = (fun21 ___ (subseteq _) U V).
+interpretation "subseteq" 'subseteq U V = (fun21 ??? (subseteq ?) U V).
-
-
-theorem subseteq_refl: ∀A.∀S:Ω \sup A.S ⊆ S.
+theorem subseteq_refl: ∀A.∀S:Ω^A.S ⊆ S.
intros 4; assumption.
qed.
-theorem subseteq_trans: ∀A.∀S1,S2,S3: Ω \sup A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
+theorem subseteq_trans: ∀A.∀S1,S2,S3: Ω^A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
intros; apply transitive_subseteq_operator; [apply S2] assumption.
qed.
-definition overlaps: ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) CPROP.
+definition overlaps: ∀A. Ω^A × Ω^A ⇒_1 CPROP.
intros;
constructor 1;
- (* Se metto x al posto di ? ottengo una universe inconsistency *)
- [ apply (λA:objs1 SET.λU,V:Ω \sup A.(exT2 ? (λx:A.?(*x*) ∈ U) (λx:A.?(*x*) ∈ V) : CProp0))
+ [ apply (λA:objs1 SET.λU,V:Ω^A.(exT2 ? (λx:A.x ∈ U) (λx:A.x ∈ V) : CProp0))
| intros;
constructor 1; intro; cases e2; exists; [1,4: apply w]
- [ apply (. #‡e); assumption
- | apply (. #‡e1); assumption
- | apply (. #‡(e \sup -1)); assumption;
- | apply (. #‡(e1 \sup -1)); assumption]]
+ [ apply (. #‡e^-1); assumption
+ | apply (. #‡e1^-1); assumption
+ | apply (. #‡e); assumption;
+ | apply (. #‡e1); assumption]]
qed.
-interpretation "overlaps" 'overlaps U V = (fun21 ___ (overlaps _) U V).
+interpretation "overlaps" 'overlaps U V = (fun21 ??? (overlaps ?) U V).
-definition intersects:
- ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) (Ω \sup A).
+definition intersects: ∀A. Ω^A × Ω^A ⇒_1 Ω^A.
intros;
constructor 1;
[ apply rule (λU,V. {x | x ∈ U ∧ x ∈ V });
intros; simplify; apply (.= (e‡#)‡(e‡#)); apply refl1;
| intros;
split; intros 2; simplify in f ⊢ %;
- [ apply (. (#‡e)‡(#‡e1)); assumption
- | apply (. (#‡(e \sup -1))‡(#‡(e1 \sup -1))); assumption]]
+ [ apply (. (#‡e^-1)‡(#‡e1^-1)); assumption
+ | apply (. (#‡e)‡(#‡e1)); assumption]]
qed.
-interpretation "intersects" 'intersects U V = (fun21 ___ (intersects _) U V).
+interpretation "intersects" 'intersects U V = (fun21 ??? (intersects ?) U V).
-definition union:
- ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) (Ω \sup A).
+definition union: ∀A. Ω^A × Ω^A ⇒_1 Ω^A.
intros;
constructor 1;
[ apply (λU,V. {x | x ∈ U ∨ x ∈ V });
intros; simplify; apply (.= (e‡#)‡(e‡#)); apply refl1
| intros;
split; intros 2; simplify in f ⊢ %;
- [ apply (. (#‡e)‡(#‡e1)); assumption
- | apply (. (#‡(e \sup -1))‡(#‡(e1 \sup -1))); assumption]]
+ [ apply (. (#‡e^-1)‡(#‡e1^-1)); assumption
+ | apply (. (#‡e)‡(#‡e1)); assumption]]
qed.
-interpretation "union" 'union U V = (fun21 ___ (union _) U V).
+interpretation "union" 'union U V = (fun21 ??? (union ?) U V).
(* qua non riesco a mettere set *)
-definition singleton: ∀A:setoid. unary_morphism1 A (Ω \sup A).
+definition singleton: ∀A:setoid. A ⇒_1 Ω^A.
intros; constructor 1;
- [ apply (λa:A.{b | eq ? a b}); unfold setoid1_of_setoid; simplify;
+ [ apply (λa:A.{b | a =_0 b}); unfold setoid1_of_setoid; simplify;
intros; simplify;
split; intro;
apply (.= e1);
[ apply a |4: apply a'] try assumption; apply sym; assumption]
qed.
-interpretation "singleton" 'singl a = (fun11 __ (singleton _) a).
\ No newline at end of file
+interpretation "singleton" 'singl a = (fun11 ?? (singleton ?) a).
+notation > "{ term 19 a : S }" with precedence 90 for @{fun11 ?? (singleton $S) $a}.
+
+definition big_intersects: ∀A:SET.∀I:SET.(I ⇒_2 Ω^A) ⇒_2 Ω^A.
+ intros; constructor 1;
+ [ intro; whd; whd in I;
+ apply ({x | ∀i:I. x ∈ c i});
+ simplify; intros; split; intros; [ apply (. (e^-1‡#)); | apply (. e‡#); ]
+ apply f;
+ | intros; split; intros 2; simplify in f ⊢ %; intro;
+ [ apply (. (#‡(e i)^-1)); apply f;
+ | apply (. (#‡e i)); apply f]]
+qed.
+
+definition big_union: ∀A:SET.∀I:SET.(I ⇒_2 Ω^A) ⇒_2 Ω^A.
+ intros; constructor 1;
+ [ intro; whd; whd in A; whd in I;
+ apply ({x | ∃i:I. x ∈ c i });
+ simplify; intros; split; intros; cases e1; clear e1; exists; [1,3:apply w]
+ [ apply (. (e^-1‡#)); | apply (. (e‡#)); ]
+ apply x;
+ | intros; split; intros 2; simplify in f ⊢ %; cases f; clear f; exists; [1,3:apply w]
+ [ apply (. (#‡(e w)^-1)); apply x;
+ | apply (. (#‡e w)); apply x]]
+qed.
+
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (⋃) \below (\emsp) term 90 p)"
+non associative with precedence 50 for @{ 'bigcup $p }.
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (⋃) \below (ident i ∈ I) break term 90 p)"
+non associative with precedence 50 for @{ 'bigcup_mk (λ${ident i}:$I.$p) }.
+notation > "hovbox(⋃ f)" non associative with precedence 60 for @{ 'bigcup $f }.
+
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (⋂) \below (\emsp) term 90 p)"
+non associative with precedence 50 for @{ 'bigcap $p }.
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (⋂) \below (ident i ∈ I) break term 90 p)"
+non associative with precedence 50 for @{ 'bigcap_mk (λ${ident i}:$I.$p) }.
+notation > "hovbox(⋂ f)" non associative with precedence 60 for @{ 'bigcap $f }.
+
+interpretation "bigcup" 'bigcup f = (fun12 ?? (big_union ??) f).
+interpretation "bigcap" 'bigcap f = (fun12 ?? (big_intersects ??) f).
+interpretation "bigcup mk" 'bigcup_mk f = (fun12 ?? (big_union ??) (mk_unary_morphism2 ?? f ?)).
+interpretation "bigcap mk" 'bigcap_mk f = (fun12 ?? (big_intersects ??) (mk_unary_morphism2 ?? f ?)).
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