+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "subsets.ma".
-
-record binary_relation (A,B: SET) : Type1 ≝
- { satisfy:> binary_morphism1 A B CPROP }.
-
-notation < "hvbox (x \nbsp \natur term 90 r \nbsp y)" with precedence 45 for @{'satisfy $r $x $y}.
-notation > "hvbox (x \natur term 90 r y)" with precedence 45 for @{'satisfy $r $x $y}.
-interpretation "relation applied" 'satisfy r x y = (fun21 ??? (satisfy ?? r) x y).
-
-definition binary_relation_setoid: SET → SET → setoid1.
- intros (A B);
- constructor 1;
- [ apply (binary_relation A B)
- | constructor 1;
- [ 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 ?? (f ??)) | apply (if ?? (f ??))] assumption
- | simplify; intros 7; split; intro;
- [ apply (if ?? (f1 ??)) | apply (fi ?? (f ??)) ]
- [ apply (if ?? (f ??)) | apply (fi ?? (f1 ??)) ]
- assumption]]
-qed.
-
-definition binary_relation_of_binary_relation_setoid :
- ∀A,B.binary_relation_setoid A B → binary_relation A B ≝ λA,B,c.c.
-coercion binary_relation_of_binary_relation_setoid.
-
-definition composition:
- ∀A,B,C.
- (binary_relation_setoid A B) × (binary_relation_setoid B C) ⇒_1 (binary_relation_setoid A C).
- intros;
- constructor 1;
- [ intros (R12 R23);
- constructor 1;
- constructor 1;
- [ apply (λs1:A.λs3:C.∃s2:B. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
- | intros;
- split; intro; cases e2 (w H3); clear e2; exists; [1,3: apply w ]
- [ apply (. (e^-1‡#)‡(#‡e1^-1)); assumption
- | apply (. (e‡#)‡(#‡e1)); assumption]]
- | intros 8; split; intro H2; simplify in H2 ⊢ %;
- cases H2 (w H3); clear H2; exists [1,3: apply w] cases H3 (H2 H4); clear H3;
- [ lapply (if ?? (e x w) H2) | lapply (fi ?? (e x w) H2) ]
- [ lapply (if ?? (e1 w y) H4)| lapply (fi ?? (e1 w y) H4) ]
- exists; try assumption;
- split; assumption]
-qed.
-
-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;
- [ (* 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
- | apply (.= e);
- apply (.= e2);
- apply (e1 ^ -1)]]
- | apply composition
- | intros 9;
- split; intro;
- cases f (w H); clear f; cases H; clear H;
- [cases f (w1 H); clear f | cases f1 (w1 H); clear f1]
- cases H; clear H;
- exists; try assumption;
- split; try assumption;
- exists; try assumption;
- 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 e (w H1); clear e; cases H1; clear H1; unfold;
- [ apply (. (e : eq1 ? x w)‡#); assumption
- | apply (. #‡(e : eq1 ? w y)^-1); 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 setoid_of_REL : objs1 REL → setoid ≝ λx.x.
-coercion setoid_of_REL.
-
-definition binary_relation_setoid_of_arrow1_REL :
- ∀P,Q. arrows1 REL P Q → binary_relation_setoid P Q ≝ λP,Q,x.x.
-coercion binary_relation_setoid_of_arrow1_REL.
-
-
-notation > "B ⇒_\r1 C" right associative with precedence 72 for @{'arrows1_REL $B $C}.
-notation "B ⇒\sub (\r 1) C" right associative with precedence 72 for @{'arrows1_REL $B $C}.
-interpretation "'arrows1_SET" 'arrows1_REL A B = (arrows1 REL A B).
-
-
-definition full_subset: ∀s:REL. Ω^s.
- apply (λs.{x | True});
- intros; simplify; split; intro; assumption.
-qed.
-
-coercion full_subset.
-
-definition comprehension: ∀b:REL. (b ⇒_1. CPROP) → Ω^b.
- apply (λb:REL. λP: b ⇒_1 CPROP. {x | P x});
- intros; simplify;
- apply (.= †e); apply refl1.
-qed.
-
-interpretation "subset comprehension" 'comprehension s p =
- (comprehension s (mk_unary_morphism1 ?? p ?)).
-
-definition ext: ∀X,S:REL. (X ⇒_\r1 S) × S ⇒_1 (Ω^X).
- intros (X S); constructor 1;
- [ apply (λr:X ⇒_\r1 S.λf:S.{x ∈ X | x ♮r f}); intros; simplify; apply (.= (e‡#)); apply refl1
- | intros; simplify; split; intros; simplify;
- [ change with (∀x. x ♮a b → x ♮a' b'); intros;
- apply (. (#‡e1^-1)); whd in e; apply (if ?? (e ??)); assumption
- | change with (∀x. x ♮a' b' → x ♮a b); intros;
- apply (. (#‡e1)); whd in e; apply (fi ?? (e ??));assumption]]
-qed.
-
-(*
-definition extS: ∀X,S:REL. ∀r: arrows1 ? X S. Ω \sup S ⇒ Ω \sup X.
- (* ∃ is not yet a morphism apply (λX,S,r,F.{x ∈ X | ∃a. a ∈ F ∧ x ♮r a});*)
- intros (X S r); constructor 1;
- [ intro F; constructor 1; constructor 1;
- [ apply (λx. x ∈ X ∧ ∃a:S. a ∈ F ∧ x ♮r a);
- | intros; split; intro; cases f (H1 H2); clear f; split;
- [ apply (. (H‡#)); assumption
- |3: apply (. (H\sup -1‡#)); assumption
- |2,4: cases H2 (w H3); exists; [1,3: apply w]
- [ apply (. (#‡(H‡#))); assumption
- | apply (. (#‡(H \sup -1‡#))); assumption]]]
- | intros; split; simplify; intros; cases f; cases H1; split;
- [1,3: assumption
- |2,4: exists; [1,3: apply w]
- [ apply (. (#‡H)‡#); assumption
- | apply (. (#‡H\sup -1)‡#); assumption]]]
-qed.
-
-lemma equalset_extS_id_X_X: ∀o:REL.∀X.extS ?? (id1 ? o) X = X.
- intros;
- unfold extS; simplify;
- split; simplify;
- [ intros 2; change with (a ∈ X);
- cases f; clear f;
- cases H; clear H;
- cases x; clear x;
- change in f2 with (eq1 ? a w);
- apply (. (f2\sup -1‡#));
- assumption
- | intros 2; change in f with (a ∈ X);
- split;
- [ whd; exact I
- | exists; [ apply a ]
- split;
- [ assumption
- | change with (a = a); apply refl]]]
-qed.
-
-lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c2 ∘ c1) S = extS o1 o2 c1 (extS o2 o3 c2 S).
- intros; unfold extS; simplify; split; intros; simplify; intros;
- [ cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
- cases H3 (H4 H5); cases H5 (w1 H6); clear H3 H5; cases H6 (H7 H8); clear H6;
- exists; [apply w1] split [2: assumption] constructor 1; [assumption]
- exists; [apply w] split; assumption
- | cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
- cases H3 (H4 H5); cases H4 (w1 H6); clear H3 H4; cases H6 (w2 H7); clear H6;
- cases H7; clear H7; exists; [apply w2] split; [assumption] exists [apply w] split;
- assumption]
-qed.
-*)
-
-(* the same as ⋄ for a basic pair *)
-definition 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; 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]]
-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]
-qed.
-
-(* the same as Rest for a basic pair *)
-definition star_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^V ⇒_1 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]
-qed.
-
-(* the same as Ext for a basic pair *)
-definition minus_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^V ⇒_1 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]]
-qed.
-
-(* 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;
- [ change with (a ∈ U);
- cases e; cases x; change in f with (eq1 ? w a); apply (. f^-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;
- [ 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.
-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]
-qed.
-
-(*
-(*CSC: unused! *)
-theorem ext_comp:
- ∀o1,o2,o3: REL.
- ∀a: arrows1 ? o1 o2.
- ∀b: arrows1 ? o2 o3.
- ∀x. ext ?? (b∘a) x = extS ?? a (ext ?? b x).
- intros;
- unfold ext; unfold extS; simplify; split; intro; simplify; intros;
- cases f; clear f; split; try assumption;
- [ cases f2; clear f2; cases x1; clear x1; exists; [apply w] split;
- [1: split] assumption;
- | cases H; clear H; cases x1; clear x1; exists [apply w]; split;
- [2: cases f] assumption]
-qed.
-
-theorem extS_singleton:
- ∀o1,o2.∀a:arrows1 ? o1 o2.∀x.extS o1 o2 a (singleton o2 x) = ext o1 o2 a x.
- intros; unfold extS; unfold ext; unfold singleton; simplify;
- split; intros 2; simplify; cases f; split; try assumption;
- [ 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.
-*)
projects / helm.git / blobdiff