(* ------------------------ *) ⊢
fun11 … R r ≡ or_f_minus_star P Q r.
-(*CSC:
ndefinition ORelation_composition : ∀P,Q,R.
binary_morphism1 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
#P; #Q; #R; @
[ #F; #G; @
- [ napply (G ∘ F);
- | apply rule (G⎻* ∘ F⎻* );
- | apply (F* ∘ G* );
- | apply (F⎻ ∘ G⎻);
- | intros;
- change with ((G (F p) ≤ q) = (p ≤ (F* (G* q))));
- apply (.= (or_prop1 :?));
- apply (or_prop1 :?);
- | intros;
- change with ((F⎻ (G⎻ p) ≤ q) = (p ≤ (G⎻* (F⎻* q))));
- apply (.= (or_prop2 :?));
- apply or_prop2 ;
- | intros; change with ((G (F (p)) >< q) = (p >< (F⎻ (G⎻ q))));
- apply (.= (or_prop3 :?));
- apply or_prop3;
+ [ napply (comp1_unary_morphisms … G F) (*CSC: was (G ∘ F);*)
+ | napply (comp1_unary_morphisms … G⎻* F⎻* ) (*CSC: was (G⎻* ∘ F⎻* );*)
+ | napply (comp1_unary_morphisms … F* G* ) (*CSC: was (F* ∘ G* );*)
+ | napply (comp1_unary_morphisms … F⎻ G⎻) (*CSC: was (F⎻ ∘ G⎻);*)
+ | #p; #q; nnormalize;
+ napply (.= (or_prop1 … G …)); (*CSC: it used to understand without G *)
+ napply (or_prop1 …)
+ | #p; #q; nnormalize;
+ napply (.= (or_prop2 … F …));
+ napply or_prop2
+ | #p; #q; nnormalize;
+ napply (.= (or_prop3 … G …));
+ napply or_prop3
]
-| intros; split; simplify;
- [3: unfold arrows1_of_ORelation_setoid; apply ((†e)‡(†e1));
- |1: apply ((†e)‡(†e1));
- |2,4: apply ((†e1)‡(†e));]]
-qed.
+##| #a;#a';#b;#b';#e;#e1;#x;nnormalize;napply (.= †(e x));napply e1]
+nqed.
-definition OA : category2.
+(*
+ndefinition OA : category2.
split;
[ apply (OAlgebra);
| intros; apply (ORelation_setoid o o1);
include "sets/sets.ma".
-ndefinition binary_morph_setoid : setoid → setoid → setoid → setoid.
-#S1; #S2; #T; @ (binary_morphism S1 S2 T); @;
-##[ #f; #g; napply (∀x,y. f x y = g x y);
-##| #f; #x; #y; napply #;
-##| #f; #g; #H; #x; #y; napply ((H x y)^-1);
-##| #f; #g; #h; #H1; #H2; #x; #y; napply (trans … (H1 …) (H2 …)); ##]
-nqed.
-
-ndefinition unary_morph_setoid : setoid → setoid → setoid.
-#S1; #S2; @ (unary_morphism S1 S2); @;
-##[ #f; #g; napply (∀x. f x = g x);
-##| #f; #x; napply #;
-##| #f; #g; #H; #x; napply ((H x)^-1);
-##| #f; #g; #h; #H1; #H2; #x; napply (trans … (H1 …) (H2 …)); ##]
-nqed.
-
nrecord category : Type[2] ≝
{ objs:> Type[1];
arrows: objs → objs → setoid;
id: ∀o:objs. arrows o o;
- comp: ∀o1,o2,o3. binary_morphism (arrows o1 o2) (arrows o2 o3) (arrows o1 o3);
- comp_assoc: ∀o1,o2,o3,o4. ∀a12,a23,a34.
- comp o1 o3 o4 (comp o1 o2 o3 a12 a23) a34 = comp o1 o2 o4 a12 (comp o2 o3 o4 a23 a34);
- id_neutral_left: ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o1) a = a;
- id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) = a
+ comp: ∀o1,o2,o3. binary_morphism (arrows o2 o3) (arrows o1 o2) (arrows o1 o3);
+ comp_assoc: ∀o1,o2,o3,o4. ∀a34,a23,a12.
+ comp o1 o3 o4 a34 (comp o1 o2 o3 a23 a12) = comp o1 o2 o4 (comp o2 o3 o4 a34 a23) a12;
+ id_neutral_left: ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o2) a = a;
+ id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o1) = a
}.
notation "hvbox(A break ⇒ B)" right associative with precedence 50 for @{ 'arrows $A $B }.
##[ napply setoid;
##| napply unary_morph_setoid;
##| #o; @ (λx.x); #a; #b; #H; napply H;
-##| #o1; #o2; #o3; @;
- ##[ #f; #g; @(λx.g (f x)); #a; #b; #H; napply (.= (††H)); napply #;
- ##| #f; #g; #f'; #g'; #H1; #H2; nwhd; #x; napply (.= (H2 (f x)));
- napply (.= (†(H1 x))); napply #; ##]
+##| napply comp_binary_morphisms; (*CSC: why not ∘?*)
##| #o1; #o2; #o3; #o4; #f; #g; #h; nwhd; #x; napply #;
##|##6,7: #o1; #o2; #f; nwhd; #x; napply #; ##]
nqed.
include "logic/connectives.ma".
include "properties/relations.ma".
+include "hints_declaration.ma".
(*
notation "hvbox(a break = \sub \ID b)" non associative with precedence 45
interpretation "prop1" 'prop1 c = (prop1 ????? c).
interpretation "prop2" 'prop2 l r = (prop2 ???????? l r).
interpretation "refl" 'refl = (refl ???).
+
+ndefinition binary_morph_setoid : setoid → setoid → setoid → setoid.
+#S1; #S2; #T; @ (binary_morphism S1 S2 T); @;
+##[ #f; #g; napply (∀x,y. f x y = g x y);
+##| #f; #x; #y; napply #;
+##| #f; #g; #H; #x; #y; napply ((H x y)^-1);
+##| #f; #g; #h; #H1; #H2; #x; #y; napply (trans … (H1 …) (H2 …)); ##]
+nqed.
+
+ndefinition unary_morph_setoid : setoid → setoid → setoid.
+#S1; #S2; @ (unary_morphism S1 S2); @;
+##[ #f; #g; napply (∀x. f x = g x);
+##| #f; #x; napply #;
+##| #f; #g; #H; #x; napply ((H x)^-1);
+##| #f; #g; #h; #H1; #H2; #x; napply (trans … (H1 …) (H2 …)); ##]
+nqed.
+
+(*
+unification hint 0
+ (∀o1,o2. (λx,y:Type[0].True) (carr (unary_morph_setoid o1 o2)) (unary_morphism o1 o2)).
+*)
+
+ndefinition composition ≝
+ λo1,o2,o3:Type[0].λf:o2 → o3.λg: o1 → o2.λx.f (g x).
+
+interpretation "function composition" 'compose f g = (composition ??? f g).
+
+ndefinition comp_unary_morphisms:
+ ∀o1,o2,o3:setoid.
+ unary_morphism o2 o3 → unary_morphism o1 o2 →
+ unary_morphism o1 o3.
+#o1; #o2; #o3; #f; #g; @ (f ∘ g);
+ #a; #a'; #e; nnormalize; napply (.= †(†e)); napply #.
+nqed.
+
+unification hint 0 ≔ o1,o2,o3:setoid,f:unary_morphism o2 o3,g:unary_morphism o1 o2;
+ R ≟ (mk_unary_morphism ?? (composition … f g)
+ (prop1 ?? (comp_unary_morphisms o1 o2 o3 f g)))
+ (* -------------------------------------------------------------------- *) ⊢
+ fun1 ?? R ≡ (composition … f g).
+
+ndefinition comp_binary_morphisms:
+ ∀o1,o2,o3.
+ binary_morphism (unary_morph_setoid o2 o3) (unary_morph_setoid o1 o2)
+ (unary_morph_setoid o1 o3).
+#o1; #o2; #o3; @
+ [ #f; #g; napply (comp_unary_morphisms … f g) (*CSC: why not ∘?*)
+ | #a; #a'; #b; #b'; #ea; #eb; #x; nnormalize;
+ napply (.= †(eb x)); napply ea.
+nqed.
unification hint 0 ≔ S, T ;
R ≟ (unary_morphism1_setoid1 S T)
(* --------------------------------- *) ⊢
- carr1 R ≡ unary_morphism1 S T.
\ No newline at end of file
+ carr1 R ≡ unary_morphism1 S T.
+
+ndefinition composition1 ≝
+ λo1,o2,o3:Type[1].λf:o2 → o3.λg: o1 → o2.λx.f (g x).
+
+interpretation "function composition" 'compose f g = (composition ??? f g).
+interpretation "function composition1" 'compose f g = (composition1 ??? f g).
+
+ndefinition comp1_unary_morphisms:
+ ∀o1,o2,o3:setoid1.
+ unary_morphism1 o2 o3 → unary_morphism1 o1 o2 →
+ unary_morphism1 o1 o3.
+#o1; #o2; #o3; #f; #g; @ (f ∘ g);
+ #a; #a'; #e; nnormalize; napply (.= †(†e)); napply #.
+nqed.
+
+unification hint 0 ≔ o1,o2,o3:setoid1,f:unary_morphism1 o2 o3,g:unary_morphism1 o1 o2;
+ R ≟ (mk_unary_morphism1 ?? (composition1 … f g)
+ (prop11 ?? (comp1_unary_morphisms o1 o2 o3 f g)))
+ (* -------------------------------------------------------------------- *) ⊢
+ fun11 ?? R ≡ (composition1 … f g).
+
+ndefinition comp_binary_morphisms:
+ ∀o1,o2,o3.
+ binary_morphism1 (unary_morphism1_setoid1 o2 o3) (unary_morphism1_setoid1 o1 o2)
+ (unary_morphism1_setoid1 o1 o3).
+#o1; #o2; #o3; @
+ [ #f; #g; napply (comp1_unary_morphisms … f g) (*CSC: why not ∘?*)
+ | #a; #a'; #b; #b'; #ea; #eb; #x; nnormalize;
+ napply (.= †(eb x)); napply ea.
+nqed.
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