include "datatypes/subsets.ma".
-record ssigma (A:Type) (S: powerset A) : Type ≝
- { witness:> A;
- proof:> witness ∈ S
- }.
-
-coercion ssigma.
-
-record binary_relation (A,B: Type) (U: Ω \sup A) (V: Ω \sup B) : Type ≝
- { satisfy:2> U → V → CProp }.
+record binary_relation (A,B: setoid) : Type ≝
+ { 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 = (satisfy ____ r x y).
+interpretation "relation applied" 'satisfy r x y = (fun1 ___ (satisfy __ r) x y).
+
+definition binary_relation_setoid: setoid → setoid → 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 ?? (H ??)) | apply (if ?? (H ??))] assumption
+ | simplify; intros 7; split; intro;
+ [ apply (if ?? (H1 ??)) | apply (fi ?? (H ??)) ]
+ [ apply (if ?? (H ??)) | apply (fi ?? (H1 ??)) ]
+ assumption]]
+qed.
definition composition:
- ∀A,B,C.∀U1: Ω \sup A.∀U2: Ω \sup B.∀U3: Ω \sup C.
- binary_relation ?? U1 U2 → binary_relation ?? U2 U3 →
- binary_relation ?? U1 U3.
- intros (A B C U1 U2 U3 R12 R23);
+ ∀A,B,C.
+ binary_morphism1 (binary_relation_setoid A B) (binary_relation_setoid B C) (binary_relation_setoid A C).
+ intros;
constructor 1;
- intros (s1 s3);
- apply (∃s2. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
+ [ intros (R12 R23);
+ constructor 1;
+ constructor 1;
+ [ apply (λs1:A.λs3:C.∃s2:B. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
+ | intros;
+ split; intro; cases H2 (w H3); clear H2; exists; [1,3: apply w ]
+ [ apply (. (H‡#)‡(#‡H1)); assumption
+ | apply (. ((H \sup -1)‡#)‡(#‡(H1 \sup -1))); 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 ?? (H x w) H2) | lapply (fi ?? (H x w) H2) ]
+ [ lapply (if ?? (H1 w y) H4)| lapply (fi ?? (H1 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;
+ [ intros; apply (c = c1)
+ | intros; split; intro;
+ [ apply (trans ????? (H \sup -1));
+ apply (trans ????? H2);
+ apply H1
+ | apply (trans ????? H);
+ apply (trans ????? H2);
+ apply (H1 \sup -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 H (w H1); clear H; cases H1; clear H1; unfold;
+ [ apply (. (H \sup -1 : eq1 ? w x)‡#); assumption
+ | apply (. #‡(H : eq1 ? w y)); assumption]
+ |2,4: exists; try assumption; split; first [apply refl | assumption]]]
+qed.
+
+definition full_subset: ∀s:REL. Ω \sup s.
+ apply (λs.{x | True});
+ intros; simplify; split; intro; assumption.
qed.
-interpretation "binary relation composition" 'compose x y = (composition ______ x y).
+coercion full_subset.
-definition equal_relations ≝
- λA,B,U,V.λr,r': binary_relation A B U V.
- ∀x,y. r x y ↔ r' x y.
+definition setoid1_of_REL: REL → setoid ≝ λS. S.
-interpretation "equal relation" 'eq x y = (equal_relations ____ x y).
+coercion setoid1_of_REL.
-lemma refl_equal_relations: ∀A,B,U,V. reflexive ? (equal_relations A B U V).
- intros 5; intros 2; split; intro; assumption.
+definition comprehension: ∀b:REL. (b ⇒ CPROP) → Ω \sup b.
+ apply (λb:REL. λP: b ⇒ CPROP. {x | x ∈ b ∧ P x});
+ intros; simplify; apply (.= (H‡#)‡(†H)); apply refl1.
qed.
-lemma sym_equal_relations: ∀A,B,U,V. symmetric ? (equal_relations A B U V).
- intros 7; intros 2; split; intro;
- [ apply (fi ?? (H ??)) | apply (if ?? (H ??))] assumption.
+interpretation "subset comprehension" 'comprehension s p =
+ (comprehension s (mk_unary_morphism __ p _)).
+
+definition ext: ∀X,S:REL. binary_morphism1 (arrows1 ? X S) S (Ω \sup X).
+ apply (λX,S.mk_binary_morphism1 ??? (λr:arrows1 ? X S.λf:S.{x ∈ X | x ♮r f}) ?);
+ [ intros; simplify; apply (.= (H‡#)); apply refl1
+ | intros; simplify; split; intros; simplify; intros; cases f; split; try assumption;
+ [ apply (. (#‡H1)); whd in H; apply (if ?? (H ??)); assumption
+ | apply (. (#‡H1\sup -1)); whd in H; apply (fi ?? (H ??));assumption]]
qed.
-lemma trans_equal_relations: ∀A,B,U,V. transitive ? (equal_relations A B U V).
- intros 9; intros 2; split; intro;
- [ apply (if ?? (H1 ??)) | apply (fi ?? (H ??)) ]
- [ apply (if ?? (H ??)) | apply (fi ?? (H1 ??)) ]
- assumption.
+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 associative_composition:
- ∀A,B,C,D.∀U1,U2,U3,U4.
- ∀r1:binary_relation A B U1 U2.
- ∀r2:binary_relation B C U2 U3.
- ∀r3:binary_relation C D U3 U4.
- (r1 ∘ r2) ∘ r3 = r1 ∘ (r2 ∘ r3).
- intros 13;
- split; intro;
- cases H; clear H; cases H1; clear H1;
- [cases H; clear H | cases H2; clear H2]
- cases H1; clear H1;
- exists; try assumption;
- split; try assumption;
- exists; try assumption;
- split; assumption.
+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 composition_morphism:
- ∀A,B,C.∀U1,U2,U3.
- ∀r1,r1':binary_relation A B U1 U2.
- ∀r2,r2':binary_relation B C U2 U3.
- r1 = r1' → r2 = r2' → r1 ∘ r2 = r1' ∘ r2'.
- intros 14; split; intro;
- cases H2; clear H2; cases H3; clear H3;
- [ lapply (if ?? (H x w) H2) | lapply (fi ?? (H x w) H2) ]
- [ lapply (if ?? (H1 w y) H4)| lapply (fi ?? (H1 w y) H4) ]
- exists; try assumption;
- split; assumption.
+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.
-definition binary_relation_setoid: ∀A,B. Ω \sup A → Ω \sup B → setoid.
- intros (A B U V);
- constructor 1;
- [ apply (binary_relation ?? U V)
- | constructor 1;
- [ apply equal_relations
- | apply refl_equal_relations
- | apply sym_equal_relations
- | apply trans_equal_relations
- ]]
+(* 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]]
qed.
-record sigma (A:Type) (P: A → Type) : Type ≝
- { s_witness:> A;
- s_proof:> P s_witness
- }.
+(* 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]
+qed.
-interpretation "sigma" 'sigma \eta.x = (sigma _ x).
+(* minus_image is the same as ext *)
-definition REL: category.
- constructor 1;
- [ apply (ΣA:Type.Ω \sup A)
- | intros; apply (binary_relation_setoid ?? (s_proof ?? s) (s_proof ?? s1))
- | intros; constructor 1; intros; apply (s=s1)
- | intros; constructor 1;
- [ apply composition
- | apply composition_morphism
- ]
- | intros; unfold mk_binary_morphism; simplify;
- apply associative_composition
- |6,7: intros 5; simplify; split; intro;
- [1,3: cases H; clear H; cases H1; clear H1;
- [ rewrite > H | rewrite < H2 ]
- assumption
- |*: exists; try assumption; split; first [ reflexivity | assumption ]]]
+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 H; cases x; change in f with (eq1 ? w a); apply (. f‡#); assumption
+ | change in f with (a ∈ U);
+ exists; [apply a] split; [ change with (a = a); apply refl | assumption]]
qed.
-definition elements: objs REL → Type ≝
- λb:ΣA.Ω\sup A.ssigma (s_witness ?? b) (s_proof ?? b).
-
-coercion elements.
+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 H; change with (a=a); apply refl
+ | change in f1 with (eq1 ? x a); apply (. f1 \sup -1‡#); apply f]
+qed.
-definition carrier: objs REL → Type ≝
- λb:ΣA.Ω\sup A.s_witness ?? b.
+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 H; clear H; 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.
-interpretation "REL carrier" 'card c = (carrier c).
+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 H; exists; try assumption; split; assumption
+ | change with (x ∈ X); cases f; cases x1; apply H; assumption]
+qed.
-definition subset: ∀b:objs REL. Ω \sup (carrier b) ≝
- λb:ΣA.Ω\sup A.s_proof ?? b.
+(*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.
-coercion subset.
+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.
\ No newline at end of file
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