[ apply (λA,B.λr,r': binary_relation A B. ∀x,y. r x y ↔ r' x y)
| simplify; intros 3; split; intro; assumption
| simplify; intros 5; split; intro;
- [ apply (fi ?? (H ??)) | apply (if ?? (H ??))] assumption
+ [ apply (fi ?? (f ??)) | apply (if ?? (f ??))] assumption
| simplify; intros 7; split; intro;
- [ apply (if ?? (H1 ??)) | apply (fi ?? (H ??)) ]
- [ apply (if ?? (H ??)) | apply (fi ?? (H1 ??)) ]
+ [ apply (if ?? (f1 ??)) | apply (fi ?? (f ??)) ]
+ [ apply (if ?? (f ??)) | apply (fi ?? (f1 ??)) ]
assumption]]
qed.
(* carr to avoid universe inconsistency *)
apply (λs1:carr A.λs3:carr C.∃s2:carr B. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
| intros;
- split; intro; cases H (w H3); clear H; exists; [1,3: apply w ]
+ split; intro; cases e2 (w H3); clear e2; exists; [1,3: apply w ]
[ apply (. (e‡#)‡(#‡e1)); assumption
| apply (. ((e \sup -1)‡#)‡(#‡(e1 \sup -1))); assumption]]
| intros 8; split; intro H2; simplify in H2 ⊢ %;
exists; try assumption;
split; assumption]
qed.
-axiom daemon: False.
+
definition REL: category1.
constructor 1;
[ apply setoid
| intros (T T1); apply (binary_relation_setoid T T1)
| intros; constructor 1;
constructor 1; unfold setoid1_of_setoid; simplify;
- [ change with (carr o → carr o → CProp); intros; apply (eq1 ? c c1) ]] cases daemon; qed.
- | intros; split; intro;
+ [ (* changes required to avoid universe inconsistency *)
+ change with (carr o → carr o → CProp); intros; apply (eq ? c c1)
+ | intros; split; intro; change in a a' b b' with (carr o);
+ change in e with (eq ? a a'); change in e1 with (eq ? b b');
[ apply (.= (e ^ -1));
apply (.= e2);
apply e1
split; assumption
|6,7: intros 5; unfold composition; simplify; split; intro;
unfold setoid1_of_setoid in x y; simplify in x y;
- [1,3: cases H (w H1); clear H; cases H1; clear H1; unfold;
+ [1,3: cases e (w H1); clear e; cases H1; clear H1; unfold;
[ apply (. (e ^ -1 : eq1 ? w x)‡#); assumption
| apply (. #‡(e : eq1 ? w y)); assumption]
- |2,4: exists; try assumption; split; first [apply refl1 | assumption]]]
+ |2,4: exists; try assumption; split;
+ (* change required to avoid universe inconsistency *)
+ change in x with (carr o1); change in y with (carr o2);
+ first [apply refl | assumption]]]
qed.
+(*
definition full_subset: ∀s:REL. Ω \sup s.
apply (λs.{x | True});
intros; simplify; split; intro; assumption.
cases H7; clear H7; exists; [apply w2] split; [assumption] exists [apply w] split;
assumption]
qed.
+*)
+(* senza questo exT "fresco", universe inconsistency *)
+inductive exT (A:Type0) (P:A→CProp0) : CProp0 ≝
+ ex_introT: ∀w:A. P w → exT A P.
+
+lemma hint: ∀U. carr U → Type_OF_setoid1 ?(*(setoid1_of_SET U)*).
+ [ apply setoid1_of_SET; apply U
+ | intros; apply c;]
+qed.
+coercion hint.
(* the same as ⋄ for a basic pair *)
definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S});
- intros; simplify; split; intro; cases H1; exists [1,3: apply w]
- [ apply (. (#‡H)‡#); assumption
- | apply (. (#‡H \sup -1)‡#); assumption]
- | intros; split; simplify; intros; cases H2; exists [1,3: apply w]
- [ apply (. #‡(#‡H1)); cases x; split; try assumption;
- apply (if ?? (H ??)); assumption
- | apply (. #‡(#‡H1 \sup -1)); cases x; split; try assumption;
- apply (if ?? (H \sup -1 ??)); assumption]]
+ [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | (*∃x:U. x ♮r y ∧ x ∈ S*)
+ exT ? (λx:carr U.x ♮r y ∧ x ∈ S) });
+ intros; simplify; split; intro; cases e1; exists [1,3: apply w]
+ [ apply (. (#‡e)‡#); assumption
+ | apply (. (#‡e ^ -1)‡#); assumption]
+ | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
+ [ apply (. #‡(#‡e1)); cases x; split; try assumption;
+ apply (if ?? (e ??)); assumption
+ | apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
+ apply (if ?? (e ^ -1 ??)); assumption]]
qed.
(* the same as □ for a basic pair *)
definition minus_star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
- intros; simplify; split; intros; apply H1;
- [ apply (. #‡H \sup -1); assumption
- | apply (. #‡H); assumption]
- | intros; split; simplify; intros; [ apply (. #‡H1); | apply (. #‡H1 \sup -1)]
- apply H2; [ apply (if ?? (H \sup -1 ??)); | apply (if ?? (H ??)) ] assumption]
+ [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:carr U. x ♮r y → x ∈ S});
+ intros; simplify; split; intros; apply f;
+ [ apply (. #‡e ^ -1); assumption
+ | apply (. #‡e); assumption]
+ | intros; split; simplify; intros; [ apply (. #‡e1); | apply (. #‡e1 ^ -1)]
+ apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
+qed.
+
+(* the same as Rest for a basic pair *)
+definition star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
+ intros; constructor 1;
+ [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | ∀y:carr V. x ♮r y → y ∈ S});
+ intros; simplify; split; intros; apply f;
+ [ apply (. e ^ -1‡#); assumption
+ | apply (. e‡#); assumption]
+ | intros; split; simplify; intros; [ apply (. #‡e1); | apply (. #‡e1 ^ -1)]
+ apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
+qed.
+
+(* the same as Ext for a basic pair *)
+definition minus_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
+ intros; constructor 1;
+ [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
+ exT ? (λy:carr V.x ♮r y ∧ y ∈ S) });
+ intros; simplify; split; intro; cases e1; exists [1,3: apply w]
+ [ apply (. (e‡#)‡#); assumption
+ | apply (. (e ^ -1‡#)‡#); assumption]
+ | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
+ [ apply (. #‡(#‡e1)); cases x; split; try assumption;
+ apply (if ?? (e ??)); assumption
+ | apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
+ apply (if ?? (e ^ -1 ??)); assumption]]
qed.
+(*
(* minus_image is the same as ext *)
theorem image_id: ∀o,U. image o o (id1 REL o) U = U.
[ cases H; cases x1; change in f2 with (eq1 ? x w); apply (. #‡f2 \sup -1);
assumption
| exists; try assumption; split; try assumption; change with (x = x); apply refl]
-qed.
\ No newline at end of file
+qed.
+*)
+
+include "o-algebra.ma".
+axiom daemon: False.
+definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → ORelation (SUBSETS o1) (SUBSETS o2).
+ intros;
+ constructor 1;
+ [ constructor 1;
+ [ apply (λU.image ?? t U);
+ | intros; apply (#‡e); ]
+ | constructor 1;
+ [ apply (λU.minus_star_image ?? t U);
+ | intros; apply (#‡e); ]
+ | constructor 1;
+ [ apply (λU.star_image ?? t U);
+ | intros; apply (#‡e); ]
+ | constructor 1;
+ [ apply (λU.minus_image ?? t U);
+ | intros; apply (#‡e); ]
+ | intros; split; intro;
+ [ change in f with (∀a. a ∈ image ?? t p → a ∈ q);
+ change with (∀a:o1. a ∈ p → a ∈ star_image ?? t q);
+ intros 4; apply f; exists; [apply a] split; assumption;
+ | change in f with (∀a:o1. a ∈ p → a ∈ star_image ?? t q);
+ change with (∀a. a ∈ image ?? t p → a ∈ q);
+ intros; cases f1; cases x; clear f1 x; apply (f ? f3); assumption; ]
+ | intros; split; intro;
+ [ change in f with (∀a. a ∈ minus_image ?? t p → a ∈ q);
+ change with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? t q);
+ intros 4; apply f; exists; [apply a] split; assumption;
+ | change in f with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? t q);
+ change with (∀a. a ∈ minus_image ?? t p → a ∈ q);
+ intros; cases f1; cases x; clear f1 x; apply (f ? f3); assumption; ]
+ | intros; split; intro; cases f; clear f;
+ [ cases x; cases x2; clear x x2; exists; [apply w1]
+ [ assumption;
+ | exists; [apply w] split; assumption]
+ | cases x1; cases x2; clear x1 x2; exists; [apply w1]
+ [ exists; [apply w] split; assumption;
+ | assumption; ]]]
+qed. sistemare anche l'hint da un'altra parte e capire l'exT (doppio!)
\ No newline at end of file
projects / helm.git / blobdiff