+++ /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 "cprop_connectives.ma".
-
-notation "hvbox(a break = \sub \ID b)" non associative with precedence 45
-for @{ 'eqID $a $b }.
-
-notation > "hvbox(a break =_\ID b)" non associative with precedence 45
-for @{ cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? $a $b }.
-
-interpretation "ID eq" 'eqID x y = (cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? x y).
-
-record equivalence_relation (A:Type0) : Type1 ≝
- { eq_rel:2> A → A → CProp0;
- refl: reflexive ? eq_rel;
- sym: symmetric ? eq_rel;
- trans: transitive ? eq_rel
- }.
-
-record setoid : Type1 ≝
- { carr:> Type0;
- eq: equivalence_relation carr
- }.
-
-record equivalence_relation1 (A:Type1) : Type2 ≝
- { eq_rel1:2> A → A → CProp1;
- refl1: reflexive1 ? eq_rel1;
- sym1: symmetric1 ? eq_rel1;
- trans1: transitive1 ? eq_rel1
- }.
-
-record setoid1: Type2 ≝
- { carr1:> Type1;
- eq1: equivalence_relation1 carr1
- }.
-
-definition setoid1_of_setoid: setoid → setoid1.
- intro;
- constructor 1;
- [ apply (carr s)
- | constructor 1;
- [ apply (eq_rel s);
- apply (eq s)
- | apply (refl s)
- | apply (sym s)
- | apply (trans s)]]
-qed.
-
-coercion setoid1_of_setoid.
-prefer coercion Type_OF_setoid.
-
-record equivalence_relation2 (A:Type2) : Type3 ≝
- { eq_rel2:2> A → A → CProp2;
- refl2: reflexive2 ? eq_rel2;
- sym2: symmetric2 ? eq_rel2;
- trans2: transitive2 ? eq_rel2
- }.
-
-record setoid2: Type3 ≝
- { carr2:> Type2;
- eq2: equivalence_relation2 carr2
- }.
-
-definition setoid2_of_setoid1: setoid1 → setoid2.
- intro;
- constructor 1;
- [ apply (carr1 s)
- | constructor 1;
- [ apply (eq_rel1 s);
- apply (eq1 s)
- | apply (refl1 s)
- | apply (sym1 s)
- | apply (trans1 s)]]
-qed.
-
-coercion setoid2_of_setoid1.
-prefer coercion Type_OF_setoid2.
-prefer coercion Type_OF_setoid.
-prefer coercion Type_OF_setoid1.
-(* we prefer 0 < 1 < 2 *)
-
-record equivalence_relation3 (A:Type3) : Type4 ≝
- { eq_rel3:2> A → A → CProp3;
- refl3: reflexive3 ? eq_rel3;
- sym3: symmetric3 ? eq_rel3;
- trans3: transitive3 ? eq_rel3
- }.
-
-record setoid3: Type4 ≝
- { carr3:> Type3;
- eq3: equivalence_relation3 carr3
- }.
-
-interpretation "setoid3 eq" 'eq t x y = (eq_rel3 ? (eq3 t) x y).
-interpretation "setoid2 eq" 'eq t x y = (eq_rel2 ? (eq2 t) x y).
-interpretation "setoid1 eq" 'eq t x y = (eq_rel1 ? (eq1 t) x y).
-interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq t) x y).
-
-notation > "hvbox(a break =_12 b)" non associative with precedence 45
-for @{ eq_rel2 (carr2 (setoid2_of_setoid1 ?)) (eq2 (setoid2_of_setoid1 ?)) $a $b }.
-notation > "hvbox(a break =_0 b)" non associative with precedence 45
-for @{ eq_rel ? (eq ?) $a $b }.
-notation > "hvbox(a break =_1 b)" non associative with precedence 45
-for @{ eq_rel1 ? (eq1 ?) $a $b }.
-notation > "hvbox(a break =_2 b)" non associative with precedence 45
-for @{ eq_rel2 ? (eq2 ?) $a $b }.
-notation > "hvbox(a break =_3 b)" non associative with precedence 45
-for @{ eq_rel3 ? (eq3 ?) $a $b }.
-
-interpretation "setoid3 symmetry" 'invert r = (sym3 ???? r).
-interpretation "setoid2 symmetry" 'invert r = (sym2 ???? r).
-interpretation "setoid1 symmetry" 'invert r = (sym1 ???? r).
-interpretation "setoid symmetry" 'invert r = (sym ???? r).
-notation ".= r" with precedence 50 for @{'trans $r}.
-interpretation "trans3" 'trans r = (trans3 ????? r).
-interpretation "trans2" 'trans r = (trans2 ????? r).
-interpretation "trans1" 'trans r = (trans1 ????? r).
-interpretation "trans" 'trans r = (trans ????? r).
-
-record unary_morphism (A,B: setoid) : Type0 ≝
- { fun1:1> A → B;
- prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 a')
- }.
-
-record unary_morphism1 (A,B: setoid1) : Type1 ≝
- { fun11:1> A → B;
- prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
- }.
-
-record unary_morphism2 (A,B: setoid2) : Type2 ≝
- { fun12:1> A → B;
- prop12: ∀a,a'. eq2 ? a a' → eq2 ? (fun12 a) (fun12 a')
- }.
-
-record unary_morphism3 (A,B: setoid3) : Type3 ≝
- { fun13:1> A → B;
- prop13: ∀a,a'. eq3 ? a a' → eq3 ? (fun13 a) (fun13 a')
- }.
-
-record binary_morphism (A,B,C:setoid) : Type0 ≝
- { fun2:2> A → B → C;
- prop2: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun2 a b) (fun2 a' b')
- }.
-
-record binary_morphism1 (A,B,C:setoid1) : Type1 ≝
- { fun21:2> A → B → C;
- prop21: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun21 a b) (fun21 a' b')
- }.
-
-record binary_morphism2 (A,B,C:setoid2) : Type2 ≝
- { fun22:2> A → B → C;
- prop22: ∀a,a',b,b'. eq2 ? a a' → eq2 ? b b' → eq2 ? (fun22 a b) (fun22 a' b')
- }.
-
-record binary_morphism3 (A,B,C:setoid3) : Type3 ≝
- { fun23:2> A → B → C;
- prop23: ∀a,a',b,b'. eq3 ? a a' → eq3 ? b b' → eq3 ? (fun23 a b) (fun23 a' b')
- }.
-
-notation "† c" with precedence 90 for @{'prop1 $c }.
-notation "l ‡ r" with precedence 90 for @{'prop2 $l $r }.
-notation "#" with precedence 90 for @{'refl}.
-interpretation "prop1" 'prop1 c = (prop1 ????? c).
-interpretation "prop11" 'prop1 c = (prop11 ????? c).
-interpretation "prop12" 'prop1 c = (prop12 ????? c).
-interpretation "prop13" 'prop1 c = (prop13 ????? c).
-interpretation "prop2" 'prop2 l r = (prop2 ???????? l r).
-interpretation "prop21" 'prop2 l r = (prop21 ???????? l r).
-interpretation "prop22" 'prop2 l r = (prop22 ???????? l r).
-interpretation "prop23" 'prop2 l r = (prop23 ???????? l r).
-interpretation "refl" 'refl = (refl ???).
-interpretation "refl1" 'refl = (refl1 ???).
-interpretation "refl2" 'refl = (refl2 ???).
-interpretation "refl3" 'refl = (refl3 ???).
-
-notation > "A × term 74 B ⇒ term 19 C" non associative with precedence 72 for @{'binary_morphism0 $A $B $C}.
-notation > "A × term 74 B ⇒_1 term 19 C" non associative with precedence 72 for @{'binary_morphism1 $A $B $C}.
-notation > "A × term 74 B ⇒_2 term 19 C" non associative with precedence 72 for @{'binary_morphism2 $A $B $C}.
-notation > "A × term 74 B ⇒_3 term 19 C" non associative with precedence 72 for @{'binary_morphism3 $A $B $C}.
-notation > "B ⇒_1 C" right associative with precedence 72 for @{'arrows1_SET $B $C}.
-notation > "B ⇒_1. C" right associative with precedence 72 for @{'arrows1_SETlow $B $C}.
-notation > "B ⇒_2 C" right associative with precedence 72 for @{'arrows2_SET1 $B $C}.
-notation > "B ⇒_2. C" right associative with precedence 72 for @{'arrows2_SET1low $B $C}.
-
-notation "A × term 74 B ⇒ term 19 C" non associative with precedence 72 for @{'binary_morphism0 $A $B $C}.
-notation "A × term 74 B ⇒\sub 1 term 19 C" non associative with precedence 72 for @{'binary_morphism1 $A $B $C}.
-notation "A × term 74 B ⇒\sub 2 term 19 C" non associative with precedence 72 for @{'binary_morphism2 $A $B $C}.
-notation "A × term 74 B ⇒\sub 3 term 19 C" non associative with precedence 72 for @{'binary_morphism3 $A $B $C}.
-notation "B ⇒\sub 1 C" right associative with precedence 72 for @{'arrows1_SET $B $C}.
-notation "B ⇒\sub 2 C" right associative with precedence 72 for @{'arrows2_SET1 $B $C}.
-notation "B ⇒\sub 1. C" right associative with precedence 72 for @{'arrows1_SETlow $B $C}.
-notation "B ⇒\sub 2. C" right associative with precedence 72 for @{'arrows2_SET1low $B $C}.
-
-interpretation "'binary_morphism0" 'binary_morphism0 A B C = (binary_morphism A B C).
-interpretation "'arrows2_SET1 low" 'arrows2_SET1 A B = (unary_morphism2 A B).
-interpretation "'arrows2_SET1low" 'arrows2_SET1low A B = (unary_morphism2 A B).
-interpretation "'binary_morphism1" 'binary_morphism1 A B C = (binary_morphism1 A B C).
-interpretation "'binary_morphism2" 'binary_morphism2 A B C = (binary_morphism2 A B C).
-interpretation "'binary_morphism3" 'binary_morphism3 A B C = (binary_morphism3 A B C).
-interpretation "'arrows1_SET low" 'arrows1_SET A B = (unary_morphism1 A B).
-interpretation "'arrows1_SETlow" 'arrows1_SETlow A B = (unary_morphism1 A B).
-
-
-definition unary_morphism2_of_unary_morphism1:
- ∀S,T:setoid1.unary_morphism1 S T → unary_morphism2 (setoid2_of_setoid1 S) T.
- intros;
- constructor 1;
- [ apply (fun11 ?? u);
- | apply (prop11 ?? u); ]
-qed.
-
-definition CPROP: setoid1.
- constructor 1;
- [ apply CProp0
- | constructor 1;
- [ apply Iff
- | intros 1; split; intro; assumption
- | intros 3; cases i; split; assumption
- | intros 5; cases i; cases i1; split; intro;
- [ apply (f2 (f x1)) | apply (f1 (f3 z1))]]]
-qed.
-
-definition CProp0_of_CPROP: carr1 CPROP → CProp0 ≝ λx.x.
-coercion CProp0_of_CPROP.
-
-alias symbol "eq" = "setoid1 eq".
-definition fi': ∀A,B:CPROP. A = B → B → A.
- intros; apply (fi ?? e); assumption.
-qed.
-
-notation ". r" with precedence 50 for @{'fi $r}.
-interpretation "fi" 'fi r = (fi' ?? r).
-
-definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
- constructor 1;
- [ apply And
- | intros; split; intro; cases a1; split;
- [ apply (if ?? e a2)
- | apply (if ?? e1 b1)
- | apply (fi ?? e a2)
- | apply (fi ?? e1 b1)]]
-qed.
-
-interpretation "and_morphism" 'and a b = (fun21 ??? and_morphism a b).
-
-definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
- constructor 1;
- [ apply Or
- | intros; split; intro; cases o; [1,3:left |2,4: right]
- [ apply (if ?? e a1)
- | apply (fi ?? e a1)
- | apply (if ?? e1 b1)
- | apply (fi ?? e1 b1)]]
-qed.
-
-interpretation "or_morphism" 'or a b = (fun21 ??? or_morphism a b).
-
-definition if_morphism: binary_morphism1 CPROP CPROP CPROP.
- constructor 1;
- [ apply (λA,B. A → B)
- | intros; split; intros;
- [ apply (if ?? e1); apply f; apply (fi ?? e); assumption
- | apply (fi ?? e1); apply f; apply (if ?? e); assumption]]
-qed.
-
-notation > "hvbox(a break ∘ b)" left associative with precedence 55 for @{ comp ??? $a $b }.
-record category : Type1 ≝ {
- objs:> Type0;
- arrows: objs → objs → setoid;
- id: ∀o:objs. arrows o o;
- comp: ∀o1,o2,o3. (arrows o1 o2) × (arrows o2 o3) ⇒ (arrows o1 o3);
- comp_assoc: ∀o1,o2,o3,o4.∀a12:arrows o1 ?.∀a23:arrows o2 ?.∀a34:arrows o3 o4.
- (a12 ∘ a23) ∘ a34 =_0 a12 ∘ (a23 ∘ a34);
- id_neutral_left : ∀o1,o2. ∀a: arrows o1 o2. (id o1) ∘ a =_0 a;
- id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. a ∘ (id o2) =_0 a
-}.
-notation "hvbox(a break ∘ b)" left associative with precedence 55 for @{ 'compose $a $b }.
-
-record category1 : Type2 ≝
- { objs1:> Type1;
- arrows1: objs1 → objs1 → setoid1;
- id1: ∀o:objs1. arrows1 o o;
- comp1: ∀o1,o2,o3. binary_morphism1 (arrows1 o1 o2) (arrows1 o2 o3) (arrows1 o1 o3);
- comp_assoc1: ∀o1,o2,o3,o4. ∀a12,a23,a34.
- comp1 o1 o3 o4 (comp1 o1 o2 o3 a12 a23) a34 =_1 comp1 o1 o2 o4 a12 (comp1 o2 o3 o4 a23 a34);
- id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a =_1 a;
- id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) =_1 a
- }.
-
-record category2 : Type3 ≝
- { objs2:> Type2;
- arrows2: objs2 → objs2 → setoid2;
- id2: ∀o:objs2. arrows2 o o;
- comp2: ∀o1,o2,o3. binary_morphism2 (arrows2 o1 o2) (arrows2 o2 o3) (arrows2 o1 o3);
- comp_assoc2: ∀o1,o2,o3,o4. ∀a12,a23,a34.
- comp2 o1 o3 o4 (comp2 o1 o2 o3 a12 a23) a34 =_2 comp2 o1 o2 o4 a12 (comp2 o2 o3 o4 a23 a34);
- id_neutral_right2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? (id2 o1) a =_2 a;
- id_neutral_left2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? a (id2 o2) =_2 a
- }.
-
-record category3 : Type4 ≝
- { objs3:> Type3;
- arrows3: objs3 → objs3 → setoid3;
- id3: ∀o:objs3. arrows3 o o;
- comp3: ∀o1,o2,o3. binary_morphism3 (arrows3 o1 o2) (arrows3 o2 o3) (arrows3 o1 o3);
- comp_assoc3: ∀o1,o2,o3,o4. ∀a12,a23,a34.
- comp3 o1 o3 o4 (comp3 o1 o2 o3 a12 a23) a34 =_3 comp3 o1 o2 o4 a12 (comp3 o2 o3 o4 a23 a34);
- id_neutral_right3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? (id3 o1) a =_3 a;
- id_neutral_left3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? a (id3 o2) =_3 a
- }.
-
-notation "'ASSOC'" with precedence 90 for @{'assoc}.
-
-interpretation "category2 composition" 'compose x y = (fun22 ??? (comp2 ????) y x).
-interpretation "category2 assoc" 'assoc = (comp_assoc2 ????????).
-interpretation "category1 composition" 'compose x y = (fun21 ??? (comp1 ????) y x).
-interpretation "category1 assoc" 'assoc = (comp_assoc1 ????????).
-interpretation "category composition" 'compose x y = (fun2 ??? (comp ????) y x).
-interpretation "category assoc" 'assoc = (comp_assoc ????????).
-
-definition category2_of_category1: category1 → category2.
- intro;
- constructor 1;
- [ apply (objs1 c);
- | intros; apply (setoid2_of_setoid1 (arrows1 c o o1));
- | apply (id1 c);
- | intros;
- constructor 1;
- [ intros; apply (comp1 c o1 o2 o3 c1 c2);
- | intros; unfold setoid2_of_setoid1 in e e1 a a' b b'; simplify in e e1 a a' b b';
- change with ((b∘a) =_1 (b'∘a')); apply (e‡e1); ]
- | intros; simplify; whd in a12 a23 a34; whd; apply rule (ASSOC);
- | intros; simplify; whd in a; whd; apply id_neutral_right1;
- | intros; simplify; whd in a; whd; apply id_neutral_left1; ]
-qed.
-(*coercion category2_of_category1.*)
-
-record functor2 (C1: category2) (C2: category2) : Type3 ≝
- { map_objs2:1> C1 → C2;
- map_arrows2: ∀S,T. unary_morphism2 (arrows2 ? S T) (arrows2 ? (map_objs2 S) (map_objs2 T));
- respects_id2: ∀o:C1. map_arrows2 ?? (id2 ? o) = id2 ? (map_objs2 o);
- respects_comp2:
- ∀o1,o2,o3.∀f1:arrows2 ? o1 o2.∀f2:arrows2 ? o2 o3.
- map_arrows2 ?? (f2 ∘ f1) = map_arrows2 ?? f2 ∘ map_arrows2 ?? f1}.
-
-notation > "F⎽⇒ x" left associative with precedence 60 for @{'map_arrows2 $F $x}.
-notation "F\sub⇒ x" left associative with precedence 60 for @{'map_arrows2 $F $x}.
-interpretation "map_arrows2" 'map_arrows2 F x = (fun12 ?? (map_arrows2 ?? F ??) x).
-
-definition functor2_setoid: category2 → category2 → setoid3.
- intros (C1 C2);
- constructor 1;
- [ apply (functor2 C1 C2);
- | constructor 1;
- [ intros (f g);
- apply (∀c:C1. cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? (f c) (g c));
- | simplify; intros; apply cic:/matita/logic/equality/eq.ind#xpointer(1/1/1);
- | simplify; intros; apply cic:/matita/logic/equality/sym_eq.con; apply H;
- | simplify; intros; apply cic:/matita/logic/equality/trans_eq.con;
- [2: apply H; | skip | apply H1;]]]
-qed.
-
-definition functor2_of_functor2_setoid: ∀S,T. functor2_setoid S T → functor2 S T ≝ λS,T,x.x.
-coercion functor2_of_functor2_setoid.
-
-definition CAT2: category3.
- constructor 1;
- [ apply category2;
- | apply functor2_setoid;
- | intros; constructor 1;
- [ apply (λx.x);
- | intros; constructor 1;
- [ apply (λx.x);
- | intros; assumption;]
- | intros; apply rule #;
- | intros; apply rule #; ]
- | intros; constructor 1;
- [ intros; constructor 1;
- [ intros; apply (c1 (c o));
- | intros; constructor 1;
- [ intro; apply (map_arrows2 ?? c1 ?? (map_arrows2 ?? c ?? c2));
- | intros; apply (††e); ]
- | intros; simplify;
- apply (.= †(respects_id2 : ?));
- apply (respects_id2 : ?);
- | intros; simplify;
- apply (.= †(respects_comp2 : ?));
- apply (respects_comp2 : ?); ]
- | intros; intro; simplify;
- apply (cic:/matita/logic/equality/eq_ind.con ????? (e ?));
- apply (cic:/matita/logic/equality/eq_ind.con ????? (e1 ?));
- constructor 1; ]
- | intros; intro; simplify; constructor 1;
- | intros; intro; simplify; constructor 1;
- | intros; intro; simplify; constructor 1; ]
-qed.
-
-definition category2_of_objs3_CAT2: objs3 CAT2 → category2 ≝ λx.x.
-coercion category2_of_objs3_CAT2.
-
-definition functor2_setoid_of_arrows3_CAT2: ∀S,T. arrows3 CAT2 S T → functor2_setoid S T ≝ λS,T,x.x.
-coercion functor2_setoid_of_arrows3_CAT2.
-
-definition unary_morphism_setoid: setoid → setoid → setoid.
- intros;
- constructor 1;
- [ apply (unary_morphism s s1);
- | constructor 1;
- [ intros (f g); apply (∀a:s. eq ? (f a) (g a));
- | intros 1; simplify; intros; apply refl;
- | simplify; intros; apply sym; apply f;
- | simplify; intros; apply trans; [2: apply f; | skip | apply f1]]]
-qed.
-
-definition SET: category1.
- constructor 1;
- [ apply setoid;
- | apply rule (λS,T:setoid.setoid1_of_setoid (unary_morphism_setoid S T));
- | intros; constructor 1; [ apply (λx:carr o.x); | intros; assumption ]
- | intros; constructor 1; [ intros; constructor 1; [ apply (λx. c1 (c x)); | intros;
- apply († (†e));]
- | intros; whd; intros; simplify; whd in H1; whd in H;
- apply trans; [ apply (b (a' a1)); | lapply (prop1 ?? b (a a1) (a' a1));
- [ apply Hletin | apply (e a1); ] | apply e1; ]]
- | intros; whd; intros; simplify; apply refl;
- | intros; simplify; whd; intros; simplify; apply refl;
- | intros; simplify; whd; intros; simplify; apply refl;
- ]
-qed.
-
-definition setoid_of_SET: objs1 SET → setoid ≝ λx.x.
-coercion setoid_of_SET.
-
-definition unary_morphism_setoid_of_arrows1_SET:
- ∀P,Q.arrows1 SET P Q → unary_morphism_setoid P Q ≝ λP,Q,x.x.
-coercion unary_morphism_setoid_of_arrows1_SET.
-
-interpretation "'arrows1_SET" 'arrows1_SET A B = (arrows1 SET A B).
-
-definition unary_morphism1_setoid1: setoid1 → setoid1 → setoid1.
- intros;
- constructor 1;
- [ apply (unary_morphism1 s s1);
- | constructor 1;
- [ intros (f g);
- alias symbol "eq" = "setoid1 eq".
- apply (∀a: carr1 s. f a = g a);
- | intros 1; simplify; intros; apply refl1;
- | simplify; intros; apply sym1; apply f;
- | simplify; intros; apply trans1; [2: apply f; | skip | apply f1]]]
-qed.
-
-definition unary_morphism1_of_unary_morphism1_setoid1 :
- ∀S,T. unary_morphism1_setoid1 S T → unary_morphism1 S T ≝ λP,Q,x.x.
-coercion unary_morphism1_of_unary_morphism1_setoid1.
-
-definition SET1: objs3 CAT2.
- constructor 1;
- [ apply setoid1;
- | apply rule (λS,T.setoid2_of_setoid1 (unary_morphism1_setoid1 S T));
- | intros; constructor 1; [ apply (λx.x); | intros; assumption ]
- | intros; constructor 1; [ intros; constructor 1; [ apply (λx. c1 (c x)); | intros;
- apply († (†e));]
- | intros; whd; intros; simplify; whd in H1; whd in H;
- apply trans1; [ apply (b (a' a1)); | lapply (prop11 ?? b (a a1) (a' a1));
- [ apply Hletin | apply (e a1); ] | apply e1; ]]
- | intros; whd; intros; simplify; apply refl1;
- | intros; simplify; whd; intros; simplify; apply refl1;
- | intros; simplify; whd; intros; simplify; apply refl1;
- ]
-qed.
-
-interpretation "'arrows2_SET1" 'arrows2_SET1 A B = (arrows2 SET1 A B).
-
-definition setoid1_of_SET1: objs2 SET1 → setoid1 ≝ λx.x.
-coercion setoid1_of_SET1.
-
-definition unary_morphism1_setoid1_of_arrows2_SET1:
- ∀P,Q.arrows2 SET1 P Q → unary_morphism1_setoid1 P Q ≝ λP,Q,x.x.
-coercion unary_morphism1_setoid1_of_arrows2_SET1.
-
-variant objs2_of_category1: objs1 SET → objs2 SET1 ≝ setoid1_of_setoid.
-coercion objs2_of_category1.
-
-prefer coercion Type_OF_setoid. (* we prefer the lower carrier projection *)
-prefer coercion Type_OF_objs1.
projects / helm.git / blobdiff