+++ /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 "relations.ma".
-include "notation.ma".
-
-record basic_pair: Type1 ≝ {
- concr: REL; form: REL; rel: concr ⇒_\r1 form
-}.
-
-interpretation "basic pair relation" 'Vdash2 x y c = (fun21 ??? (rel c) x y).
-interpretation "basic pair relation (non applied)" 'Vdash c = (rel c).
-
-record relation_pair (BP1,BP2: basic_pair): Type1 ≝ {
- concr_rel: (concr BP1) ⇒_\r1 (concr BP2); form_rel: (form BP1) ⇒_\r1 (form BP2);
- commute: ⊩ ∘ concr_rel =_1 form_rel ∘ ⊩
- }.
-
-interpretation "concrete relation" 'concr_rel r = (concr_rel ?? r).
-interpretation "formal relation" 'form_rel r = (form_rel ?? r).
-
-definition relation_pair_equality: ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
- intros; constructor 1; [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
- | simplify; intros; apply refl1;
- | simplify; intros 2; apply sym1;
- | simplify; intros 3; apply trans1; ]
-qed.
-
-definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
- intros;
- constructor 1;
- [ apply (relation_pair b b1)
- | apply relation_pair_equality
- ]
-qed.
-
-definition relation_pair_of_relation_pair_setoid :
- ∀P,Q. relation_pair_setoid P Q → relation_pair P Q ≝ λP,Q,x.x.
-coercion relation_pair_of_relation_pair_setoid.
-
-lemma eq_to_eq':
- ∀o1,o2.∀r,r':relation_pair_setoid o1 o2. r =_1 r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
- intros 5 (o1 o2 r r' H);
- apply (.= (commute ?? r)^-1);
- change in H with (⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
- apply rule (.= H);
- apply (commute ?? r').
-qed.
-
-definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
- intro;
- constructor 1;
- [1,2: apply id1;
- | lapply (id_neutral_right1 ? (concr o) ? (⊩)) as H;
- lapply (id_neutral_left1 ?? (form o) (⊩)) as H1;
- apply (.= H);
- apply (H1 \sup -1);]
-qed.
-
-lemma relation_pair_composition:
- ∀o1,o2,o3: basic_pair.
- relation_pair_setoid o1 o2 → relation_pair_setoid o2 o3 → relation_pair_setoid o1 o3.
-intros 3 (o1 o2 o3);
- intros (r r1);
- constructor 1;
- [ apply (r1 \sub\c ∘ r \sub\c)
- | apply (r1 \sub\f ∘ r \sub\f)
- | lapply (commute ?? r) as H;
- lapply (commute ?? r1) as H1;
- alias symbol "trans" = "trans1".
- alias symbol "assoc" = "category1 assoc".
- apply (.= ASSOC);
- apply (.= #‡H1);
- alias symbol "invert" = "setoid1 symmetry".
- apply (.= ASSOC ^ -1);
- apply (.= H‡#);
- apply ASSOC]
-qed.
-
-lemma relation_pair_composition_is_morphism:
- ∀o1,o2,o3: basic_pair.
- ∀a,a':relation_pair_setoid o1 o2.
- ∀b,b':relation_pair_setoid o2 o3.
- a=a' → b=b' →
- relation_pair_composition o1 o2 o3 a b
- = relation_pair_composition o1 o2 o3 a' b'.
-intros 3 (o1 o2 o3);
- intros;
- change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));
- change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
- change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
- apply (.= ASSOC);
- apply (.= #‡e1);
- apply (.= #‡(commute ?? b'));
- apply (.= ASSOC ^ -1);
- apply (.= e‡#);
- apply (.= ASSOC);
- apply (.= #‡(commute ?? b')\sup -1);
- apply (ASSOC ^ -1);
-qed.
-
-definition relation_pair_composition_morphism:
- ∀o1,o2,o3. binary_morphism1 (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
- intros;
- constructor 1;
- [ apply relation_pair_composition;
- | apply relation_pair_composition_is_morphism;]
-qed.
-
-lemma relation_pair_composition_morphism_assoc:
-Πo1:basic_pair
-.Πo2:basic_pair
- .Πo3:basic_pair
- .Πo4:basic_pair
- .Πa12:relation_pair_setoid o1 o2
- .Πa23:relation_pair_setoid o2 o3
- .Πa34:relation_pair_setoid o3 o4
- .relation_pair_composition_morphism o1 o3 o4
- (relation_pair_composition_morphism o1 o2 o3 a12 a23) a34
- =relation_pair_composition_morphism o1 o2 o4 a12
- (relation_pair_composition_morphism o2 o3 o4 a23 a34).
- intros;
- change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
- ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
- alias symbol "refl" = "refl1".
- alias symbol "prop2" = "prop21".
- apply (ASSOC‡#);
-qed.
-
-lemma relation_pair_composition_morphism_respects_id:
- ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
- relation_pair_composition_morphism o1 o1 o2 (id_relation_pair o1) a=a.
- intros;
- change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
- apply ((id_neutral_right1 ????)‡#);
-qed.
-
-lemma relation_pair_composition_morphism_respects_id_r:
- ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
- relation_pair_composition_morphism o1 o2 o2 a (id_relation_pair o2)=a.
- intros;
- change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
- apply ((id_neutral_left1 ????)‡#);
-qed.
-
-definition BP: category1.
- constructor 1;
- [ apply basic_pair
- | apply relation_pair_setoid
- | apply id_relation_pair
- | apply relation_pair_composition_morphism
- | apply relation_pair_composition_morphism_assoc;
- | apply relation_pair_composition_morphism_respects_id;
- | apply relation_pair_composition_morphism_respects_id_r;]
-qed.
-
-definition basic_pair_of_BP : objs1 BP → basic_pair ≝ λx.x.
-coercion basic_pair_of_BP.
-
-definition relation_pair_setoid_of_arrows1_BP :
- ∀P,Q. arrows1 BP P Q → relation_pair_setoid P Q ≝ λP,Q,x.x.
-coercion relation_pair_setoid_of_arrows1_BP.
-
-(*
-definition BPext: ∀o: BP. (form o) ⇒_1 Ω^(concr o).
- intros; constructor 1;
- [ apply (ext ? ? (rel o));
- | intros;
- apply (.= #‡e);
- apply refl1]
-qed.
-
-definition BPextS: ∀o: BP. Ω^(form o) ⇒_1 Ω^(concr o).
- intros; constructor 1;
- [ apply (minus_image ?? (rel o));
- | intros; apply (#‡e); ]
-qed.
-
-definition fintersects: ∀o: BP. (form o) × (form o) ⇒_1 Ω^(form o).
- intros (o); constructor 1;
- [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
- intros; simplify; apply (.= (†e)‡#); apply refl1
- | intros; split; simplify; intros;
- [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
- | apply (. #‡((†e)‡(†e1))); assumption]]
-qed.
-
-interpretation "fintersects" 'fintersects U V = (fun21 ??? (fintersects ?) U V).
-
-definition fintersectsS:
- ∀o:BP. Ω^(form o) × Ω^(form o) ⇒_1 Ω^(form o).
- intros (o); constructor 1;
- [ apply (λo: basic_pair.λa,b: Ω^(form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
- intros; simplify; apply (.= (†e)‡#); apply refl1
- | intros; split; simplify; intros;
- [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
- | apply (. #‡((†e)‡(†e1))); assumption]]
-qed.
-
-interpretation "fintersectsS" 'fintersects U V = (fun21 ??? (fintersectsS ?) U V).
-
-definition relS: ∀o: BP. (concr o) × Ω^(form o) ⇒_1 CPROP.
- intros (o); constructor 1;
- [ apply (λx:concr o.λS: Ω^(form o).∃y:form o.y ∈ S ∧ x ⊩⎽o y);
- | intros; split; intros; cases e2; exists [1,3: apply w]
- [ apply (. (#‡e1^-1)‡(e^-1‡#)); assumption
- | apply (. (#‡e1)‡(e‡#)); assumption]]
-qed.
-
-interpretation "basic pair relation for subsets" 'Vdash2 x y c = (fun21 (concr ?) ?? (relS c) x y).
-interpretation "basic pair relation for subsets (non applied)" 'Vdash c = (fun21 ??? (relS c)).
-*)
projects / helm.git / blobdiff