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.
intros;
| 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.
intros;
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).
[ apply (λA:objs1 SET.λU,V:Ω \sup 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).
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).
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).
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).
+interpretation "singleton" 'singl a = (fun11 ?? (singleton ?) a).
definition big_intersects:
∀A:SET.∀I:SET.unary_morphism2 (setoid1_of_setoid I ⇒ Ω \sup A) (setoid2_of_setoid1 (Ω \sup A)).
intros; constructor 1;
[ intro; whd; whd in I;
- apply ({x | ∀i:I. x ∈ t i});
- simplify; intros; split; intros; [ apply (. (e‡#)); | apply (. (e \sup -1‡#)); ]
+ 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))); apply f;
- | apply (. (#‡(e i)\sup -1)); apply f]]
+ [ apply (. (#‡(e i)^-1)); apply f;
+ | apply (. (#‡e i)); apply f]]
qed.
definition big_union:
∀A:SET.∀I:SET.unary_morphism2 (setoid1_of_setoid I ⇒ Ω \sup A) (setoid2_of_setoid1 (Ω \sup A)).
intros; constructor 1;
[ intro; whd; whd in A; whd in I;
- apply ({x | ∃i:carr I. x ∈ t i });
+ apply ({x | ∃i:I. x ∈ c i });
simplify; intros; split; intros; cases e1; clear e1; exists; [1,3:apply w]
- [ apply (. (e‡#)); | apply (. (e \sup -1‡#)); ]
+ [ 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))); apply x;
- | apply (. (#‡(e w)\sup -1)); apply x]]
-qed.
-
-(* incluso prima non funziona piu' nulla *)
-include "o-algebra.ma".
-
-definition SUBSETS: objs1 SET → OAlgebra.
- intro A; constructor 1;
- [ apply (Ω \sup A);
- | apply subseteq;
- | apply overlaps;
- | apply big_intersects;
- | apply big_union;
- | apply ({x | True});
- simplify; intros; apply (refl1 ? (eq1 CPROP));
- | apply ({x | False});
- simplify; intros; apply (refl1 ? (eq1 CPROP));
- | intros; whd; intros; assumption
- | intros; whd; split; assumption
- | intros; apply transitive_subseteq_operator; [2: apply f; | skip | assumption]
- | intros; cases f; exists [apply w] assumption
- | intros; intros 2; apply (f ? f1 i);
- | intros; intros 2; apply f;
- (* senza questa change, universe inconsistency *)
- whd; change in ⊢ (? ? (λ_:%.?)) with (carr I);
- exists; [apply i] assumption;
- | intros 3; cases f;
- | intros 3; constructor 1;
- | intros; cases f; exists; [apply w]
- [ assumption
- | whd; intros; cases i; simplify; assumption]
- | intros; split; intro;
- [ cases f; cases x1;
- (* senza questa change, universe inconsistency *)
- change in ⊢ (? ? (λ_:%.?)) with (carr I);
- exists [apply w1] exists [apply w] assumption;
- | cases e; cases x; exists; [apply w1]
- [ assumption
- | (* senza questa change, universe inconsistency *)
- whd; change in ⊢ (? ? (λ_:%.?)) with (carr I);
- exists; [apply w] assumption]]
- | intros; intros 2; cases (f (singleton ? a) ?);
- [ exists; [apply a] [assumption | change with (a = a); apply refl1;]
- | change in x1 with (a = w); change with (mem A a q); apply (. (x1 \sup -1‡#));
- assumption]]
+ [ apply (. (#‡(e w)^-1)); apply x;
+ | apply (. (#‡e w)); apply x]]
qed.
projects / helm.git / blobdiff