*)
(* the same as ⋄ for a basic pair *)
-definition image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^U ⇒_1 Ω^V.
+definition image: ∀U,V:REL. (U ⇒_\r1 V) ⇒_2 (Ω^U ⇒_2 Ω^V).
intros; constructor 1;
- [ apply (λr:U ⇒_\r1 V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S });
- intros; simplify; split; intro; cases e1; exists [1,3: apply w]
+ [ intro r; constructor 1;
+ [ apply (λS: Ω^U. {y | ∃x:U. x ♮r y ∧ x ∈ S });
+ intros; simplify; split; intro; cases e1; exists [1,3: apply w]
[ apply (. (#‡e^-1)‡#); assumption
| apply (. (#‡e)‡#); assumption]
- | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
- [ apply (. #‡(#‡e1^-1)); cases x; split; try assumption;
- apply (if ?? (e ??)); assumption
- | apply (. #‡(#‡e1)); cases x; split; try assumption;
- apply (if ?? (e ^ -1 ??)); assumption]]
+ | intros; split;
+ [ intro y; simplify; intro yA; cases yA; exists; [ apply w ];
+ apply (. #‡(#‡e^-1)); assumption;
+ | intro y; simplify; intro yA; cases yA; exists; [ apply w ];
+ apply (. #‡(#‡e)); assumption;]]
+ | simplify; intros; intro y; simplify; split; simplify; intros (b H); cases H;
+ exists; [1,3: apply w]; cases x; split; try assumption;
+ [ apply (if ?? (e ??)); | apply (fi ?? (e ??)); ] assumption;]
qed.
(* the same as □ for a basic pair *)
-definition minus_star_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^U ⇒_1 Ω^V.
- intros; constructor 1;
- [ apply (λr:U ⇒_\r1 V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
- intros; simplify; split; intros; apply f;
- [ apply (. #‡e); assumption
- | apply (. #‡e ^ -1); assumption]
- | intros; split; simplify; intros; [ apply (. #‡e1^ -1); | apply (. #‡e1 )]
- apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
+definition minus_star_image: ∀U,V:REL. (U ⇒_\r1 V) ⇒_2 (Ω^U ⇒_2 Ω^V).
+ intros; constructor 1; intros;
+ [ constructor 1;
+ [ apply (λS: Ω^U. {y | ∀x:U. x ♮c y → x ∈ S});
+ intros; simplify; split; intros; apply f;
+ [ apply (. #‡e); | apply (. #‡e ^ -1)] assumption;
+ | intros; split; intro; simplify; intros;
+ [ apply (. #‡e^-1);| apply (. #‡e); ] apply f; assumption;]
+ | intros; intro; simplify; split; simplify; intros; apply f;
+ [ apply (. (e x a2)); assumption | apply (. (e^-1 x a2)); assumption]]
qed.
(* the same as Rest for a basic pair *)
-definition star_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^V ⇒_1 Ω^U.
+definition star_image: ∀U,V:REL. (U ⇒_\r1 V) ⇒_2 (Ω^V ⇒_2 Ω^U).
intros; constructor 1;
- [ apply (λr:U ⇒_\r1 V.λS: Ω \sup V. {x | ∀y:V. x ♮r y → y ∈ S});
- intros; simplify; split; intros; apply f;
- [ apply (. e ‡#); assumption
- | apply (. e^ -1‡#); assumption]
- | intros; split; simplify; intros; [ apply (. #‡e1 ^ -1); | apply (. #‡e1)]
- apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
+ [ intro r; constructor 1;
+ [ apply (λS: Ω \sup V. {x | ∀y:V. x ♮r y → y ∈ S});
+ intros; simplify; split; intros; apply f;
+ [ apply (. e ‡#);| apply (. e^ -1‡#);] assumption;
+ | intros; split; simplify; intros;
+ [ apply (. #‡e^-1);| apply (. #‡e); ] apply f; assumption;]
+ | intros; intro; simplify; split; simplify; intros; apply f;
+ [ apply (. e a2 y); | apply (. e^-1 a2 y)] assumption;]
qed.
(* the same as Ext for a basic pair *)
-definition minus_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^V ⇒_1 Ω^U.
+definition minus_image: ∀U,V:REL. (U ⇒_\r1 V) ⇒_2 (Ω^V ⇒_2 Ω^U).
intros; constructor 1;
- [ apply (λr:U ⇒_\r1 V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
- exT ? (λy:V.x ♮r y ∧ y ∈ S) });
- intros; simplify; split; intro; cases e1; exists [1,3: apply w]
- [ apply (. (e ^ -1‡#)‡#); assumption
- | apply (. (e‡#)‡#); assumption]
- | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
- [ apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
- apply (if ?? (e ??)); assumption
- | apply (. #‡(#‡e1)); cases x; split; try assumption;
- apply (if ?? (e ^ -1 ??)); assumption]]
+ [ intro r; constructor 1;
+ [ apply (λS: Ω^V. {x | ∃y:V. x ♮r y ∧ y ∈ S }).
+ intros; simplify; split; intros; cases e1; cases x; exists; [1,3: apply w]
+ split; try assumption; [ apply (. (e^-1‡#)); | apply (. (e‡#));] assumption;
+ | intros; simplify; split; simplify; intros; cases e1; cases x;
+ exists [1,3: apply w] split; try assumption;
+ [ apply (. (#‡e^-1)); | apply (. (#‡e));] assumption]
+ | intros; intro; simplify; split; simplify; intros; cases e1; exists [1,3: apply w]
+ cases x; split; try assumption;
+ [ apply (. e^-1 a2 w); | apply (. e a2 w)] assumption;]
qed.
+interpretation "relation f⎻*" 'OR_f_minus_star r = (fun12 ?? (minus_star_image ? ?) r).
+interpretation "relation f⎻" 'OR_f_minus r = (fun12 ?? (minus_image ? ?) r).
+interpretation "relation f*" 'OR_f_star r = (fun12 ?? (star_image ? ?) r).
+
+definition image_coercion: ∀U,V:REL. (U ⇒_\r1 V) → Ω^U ⇒_2 Ω^V.
+intros (U V r Us); apply (image U V r); qed.
+coercion image_coercion.
+
(* minus_image is the same as ext *)
-theorem image_id: ∀o,U. image o o (id1 REL o) U = U.
- intros; unfold image; simplify; split; simplify; intros;
+theorem image_id: ∀o. (id1 REL o : carr2 (Ω^o ⇒_2 Ω^o)) =_1 (id2 SET1 Ω^o).
+ intros; unfold image_coercion; unfold image; simplify;
+ whd in match (?:carr2 ?);
+ intro U; simplify; split; simplify; intros;
[ change with (a ∈ U);
- cases e; cases x; change in f with (eq1 ? w a); apply (. f^-1‡#); assumption
+ cases e; cases x; change in e1 with (w =_1 a); apply (. e1^-1‡#); assumption
| change in f with (a ∈ U);
exists; [apply a] split; [ change with (a = a); apply refl1 | assumption]]
qed.
-theorem minus_star_image_id: ∀o,U. minus_star_image o o (id1 REL o) U = U.
- intros; unfold minus_star_image; simplify; split; simplify; intros;
+theorem minus_star_image_id: ∀o:REL. (fun12 ?? (minus_star_image o o) (id1 REL o) : carr2 (Ω^o ⇒_2 Ω^o)) =_1 (id2 SET1 Ω^o).
+ intros; unfold minus_star_image; simplify; intro U; simplify;
+ split; simplify; intros;
[ change with (a ∈ U); apply f; change with (a=a); apply refl1
| change in f1 with (eq1 ? x a); apply (. f1‡#); apply f]
qed.
-alias symbol "compose" (instance 2) = "category1 composition".
-theorem image_comp: ∀A,B,C,r,s,X. image A C (r ∘ s) X = image B C r (image A B s X).
- intros; unfold image; simplify; split; simplify; intros; cases e; clear e; cases x;
- clear x; [ cases f; clear f; | cases f1; clear f1 ]
- exists; try assumption; cases x; clear x; split; try assumption;
- exists; try assumption; split; assumption.
+alias symbol "compose" (instance 5) = "category2 composition".
+alias symbol "compose" (instance 4) = "category1 composition".
+theorem image_comp: ∀A,B,C.∀r:B ⇒_\r1 C.∀s:A ⇒_\r1 B.
+ ((r ∘ s) : carr2 (Ω^A ⇒_2 Ω^C)) =_1 r ∘ s.
+ intros; intro U; split; intro x; (unfold image; unfold SET1; simplify);
+ intro H; cases H;
+ cases x1; [cases f|cases f1]; exists; [1,3: apply w1] cases x2; split; try assumption;
+ exists; try assumption; split; assumption;
qed.
theorem minus_star_image_comp:
- ∀A,B,C,r,s,X.
- minus_star_image A C (r ∘ s) X = minus_star_image B C r (minus_star_image A B s X).
- intros; unfold minus_star_image; simplify; split; simplify; intros; whd; intros;
- [ apply f; exists; try assumption; split; assumption
- | change with (x ∈ X); cases f1; cases x1; apply f; assumption]
+ ∀A,B,C.∀r:B ⇒_\r1 C.∀s:A ⇒_\r1 B.
+ minus_star_image A C (r ∘ s) =_1 minus_star_image B C r ∘ (minus_star_image A B s).
+ intros; unfold minus_star_image; intro X; simplify; split; simplify; intros;
+ [ whd; intros; simplify; whd; intros; apply f; exists; try assumption; split; assumption;
+ | cases f1; cases x1; apply f; assumption]
qed.
(*
projects / helm.git / blobdiff