(* *)
(**************************************************************************)
-include "datatypes/subsets.ma".
-include "logic/cprop_connectives.ma".
-include "formal_topology/categories.ma".
+include "formal_topology/relations.ma".
+include "datatypes/categories.ma".
record basic_pair: Type ≝
- { carr1: Type;
- carr2: Type;
- concr: Ω \sup carr1;
- form: Ω \sup carr2;
- rel: binary_relation ?? concr form
+ { concr: REL;
+ form: REL;
+ rel: arrows1 ? concr form
}.
notation "x ⊩ y" with precedence 45 for @{'Vdash2 $x $y}.
interpretation "basic pair relation" 'Vdash2 x y = (rel _ x y).
interpretation "basic pair relation (non applied)" 'Vdash = (rel _).
-alias symbol "eq" = "equal relation".
-alias symbol "compose" = "binary relation composition".
record relation_pair (BP1,BP2: basic_pair): Type ≝
- { concr_rel: binary_relation ?? (concr BP1) (concr BP2);
- form_rel: binary_relation ?? (form BP1) (form BP2);
+ { concr_rel: arrows1 ? (concr BP1) (concr BP2);
+ form_rel: arrows1 ? (form BP1) (form BP2);
commute: concr_rel ∘ ⊩ = ⊩ ∘ 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_relation (relation_pair o1 o2).
+ ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
intros;
constructor 1;
[ apply (λr,r'. r \sub\c ∘ ⊩ = r' \sub\c ∘ ⊩);
| simplify;
intros;
- apply refl_equal_relations;
+ apply refl1;
| simplify;
- intros;
- apply sym_equal_relations;
- assumption
+ intros 2;
+ apply sym1;
| simplify;
- intros;
- apply (trans_equal_relations ??????? H);
- assumption
+ intros 3;
+ apply trans1;
]
qed.
-definition relation_pair_setoid: basic_pair → basic_pair → setoid.
+definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
intros;
constructor 1;
[ apply (relation_pair b b1)
]
qed.
-definition eq' ≝
- λo1,o2.λr,r':relation_pair o1 o2.⊩ ∘ r \sub\f = ⊩ ∘ r' \sub\f.
-
-alias symbol "eq" = "setoid eq".
-lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → eq' ?? r r'.
+lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → ⊩ \circ r \sub\f = ⊩ \circ r'\sub\f.
intros 7 (o1 o2 r r' H c1 f2);
- split; intro;
+ split; intro H1;
[ lapply (fi ?? (commute ?? r c1 f2) H1) as H2;
lapply (if ?? (H c1 f2) H2) as H3;
apply (if ?? (commute ?? r' c1 f2) H3);
apply (if ?? (commute ?? r c1 f2) H3);
]
qed.
-
-definition id: ∀o:basic_pair. relation_pair o o.
+definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
intro;
constructor 1;
- [1,2: constructor 1;
- intros;
- apply (s=s1)
- | simplify; intros;
- split;
- intro;
- cases H;
- cases H1; clear H H1;
- [ exists [ apply y ]
- split
- [ rewrite > H2; assumption
- | reflexivity ]
- | exists [ apply x ]
- split
- [2: rewrite < H3; assumption
- | reflexivity ]]]
+ [1,2: apply id1;
+ | lapply (id_neutral_left1 ? (concr o) ? (⊩)) as H;
+ lapply (id_neutral_right1 ?? (form o) (⊩)) as H1;
+ apply (.= H);
+ apply (H1 \sup -1);]
qed.
definition relation_pair_composition:
- ∀o1,o2,o3. binary_morphism (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
+ ∀o1,o2,o3. binary_morphism1 (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
intros;
constructor 1;
[ intros (r r1);
| apply (r \sub\f ∘ r1 \sub\f)
| lapply (commute ?? r) as H;
lapply (commute ?? r1) as H1;
- apply (equal_morphism ???? (r\sub\c ∘ (r1\sub\c ∘ ⊩)) ? ((⊩ ∘ r\sub\f) ∘ r1\sub\f));
- [1,2: apply associative_composition]
- apply (equal_morphism ???? (r\sub\c ∘ (⊩ ∘ r1\sub\f)) ? ((r\sub\c ∘ ⊩) ∘ r1\sub\f));
- [1,2: apply composition_morphism; first [assumption | apply refl_equal_relations]
- | apply sym_equal_relations;
- apply associative_composition
- ]]
+ apply (.= ASSOC1);
+ apply (.= #‡H1);
+ apply (.= ASSOC1\sup -1);
+ apply (.= H‡#);
+ apply ASSOC1]
| intros;
- alias symbol "eq" = "equal relation".
- change with (a\sub\c ∘ b\sub\c ∘ ⊩ = a'\sub\c ∘ b'\sub\c ∘ ⊩);
- apply (equal_morphism ???? (a\sub\c ∘ (b\sub\c ∘ ⊩)) ? (a'\sub\c ∘ (b'\sub\c ∘ ⊩)));
- [ apply associative_composition
- | apply sym_equal_relations; apply associative_composition]
- apply (equal_morphism ???? (a\sub\c ∘ (b'\sub\c ∘ ⊩)) ? (a' \sub \c∘(b' \sub \c∘⊩)));
- [2: apply refl_equal_relations;
- |1: apply composition_morphism;
- [ apply refl_equal_relations
- | assumption]]
- apply (equal_morphism ???? (a\sub\c ∘ (⊩ ∘ b'\sub\f)) ? (a'\sub\c ∘ (⊩ ∘ b'\sub\f)));
- [1,2: apply composition_morphism;
- [1,3: apply refl_equal_relations
- | apply (commute ?? b');
- | apply sym_equal_relations; apply (commute ?? b');]]
- apply (equal_morphism ???? ((a\sub\c ∘ ⊩) ∘ b'\sub\f) ? ((a'\sub\c ∘ ⊩) ∘ b'\sub\f));
- [2: apply associative_composition
- |1: apply sym_equal_relations; apply associative_composition]
- apply composition_morphism;
- [ assumption
- | apply refl_equal_relations]]
+ change with (a\sub\c ∘ b\sub\c ∘ ⊩ = a'\sub\c ∘ b'\sub\c ∘ ⊩);
+ change in H with (a \sub\c ∘ ⊩ = a' \sub\c ∘ ⊩);
+ change in H1 with (b \sub\c ∘ ⊩ = b' \sub\c ∘ ⊩);
+ apply (.= ASSOC1);
+ apply (.= #‡H1);
+ apply (.= #‡(commute ?? b'));
+ apply (.= ASSOC1 \sup -1);
+ apply (.= H‡#);
+ apply (.= ASSOC1);
+ apply (.= #‡(commute ?? b')\sup -1);
+ apply (ASSOC1 \sup -1)]
qed.
-
-definition BP: category.
+
+definition BP: category1.
constructor 1;
[ apply basic_pair
| apply relation_pair_setoid
- | apply id
+ | apply id_relation_pair
| apply relation_pair_composition
| intros;
change with (a12\sub\c ∘ a23\sub\c ∘ a34\sub\c ∘ ⊩ =
(a12\sub\c ∘ (a23\sub\c ∘ a34\sub\c) ∘ ⊩));
- apply composition_morphism;
- [2: apply refl_equal_relations]
- apply associative_composition
+ apply (ASSOC1‡#);
| intros;
- change with ((id o1)\sub\c ∘ a\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
- apply composition_morphism;
- [2: apply refl_equal_relations]
- intros 2; unfold id; simplify;
- split; intro;
- [ cases H; cases H1; rewrite > H2; assumption
- | exists; [assumption] split; [reflexivity| assumption]]
+ change with ((id_relation_pair o1)\sub\c ∘ a\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
+ apply ((id_neutral_left1 ????)‡#);
| intros;
- change with (a\sub\c ∘ (id o2)\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
- apply composition_morphism;
- [2: apply refl_equal_relations]
- intros 2; unfold id; simplify;
- split; intro;
- [ cases H; cases H1; rewrite < H3; assumption
- | exists; [assumption] split; [assumption|reflexivity]]
+ change with (a\sub\c ∘ (id_relation_pair o2)\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
+ apply ((id_neutral_right1 ????)‡#);
]
-qed.
+qed.
\ No newline at end of file
projects / helm.git / blobdiff