include "logic/cprop_connectives.ma".
+definition Type0 := Type.
+definition Type1 := Type.
+definition Type2 := Type.
+definition Type0_lt_Type1 := (Type0 : Type1).
+definition Type1_lt_Type2 := (Type1 : Type2).
+
record equivalence_relation (A:Type) : Type ≝
{ eq_rel:2> A → A → CProp;
refl: reflexive ? eq_rel;
trans: transitive ? eq_rel
}.
-record setoid : Type ≝
+record setoid : Type1 ≝
{ carr:> Type;
eq: equivalence_relation carr
}.
-interpretation "setoid eq" 'eq x y = (eq_rel _ (eq _) x y).
-notation ".= r" with precedence 50 for @{trans ????? $r}.
+definition reflexive1 ≝ λA:Type.λR:A→A→CProp.∀x:A.R x x.
+definition symmetric1 ≝ λC:Type.λlt:C→C→CProp. ∀x,y:C.lt x y → lt y x.
+definition transitive1 ≝ λA:Type.λR:A→A→CProp.∀x,y,z:A.R x y → R y z → R x z.
+
+record equivalence_relation1 (A:Type) : Type2 ≝
+ { eq_rel1:2> A → A → CProp;
+ refl1: reflexive1 ? eq_rel1;
+ sym1: symmetric1 ? eq_rel1;
+ trans1: transitive1 ? eq_rel1
+ }.
+
+record setoid1: Type ≝
+ { carr1:> Type;
+ 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.
+
+(*
+definition Leibniz: Type → setoid.
+ intro;
+ constructor 1;
+ [ apply T
+ | constructor 1;
+ [ apply (λx,y:T.cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? x y)
+ | alias id "refl_eq" = "cic:/matita/logic/equality/eq.ind#xpointer(1/1/1)".
+ apply refl_eq
+ | alias id "sym_eq" = "cic:/matita/logic/equality/sym_eq.con".
+ apply sym_eq
+ | alias id "trans_eq" = "cic:/matita/logic/equality/trans_eq.con".
+ apply trans_eq ]]
+qed.
+
+coercion Leibniz.
+*)
+
+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).
+interpretation "setoid1 symmetry" 'invert r = (sym1 ____ r).
interpretation "setoid symmetry" 'invert r = (sym ____ r).
+notation ".= r" with precedence 50 for @{'trans $r}.
+interpretation "trans1" 'trans r = (trans1 _____ r).
+interpretation "trans" 'trans r = (trans _____ r).
-record binary_morphism (A,B,C: setoid) : Type ≝
+record unary_morphism (A,B: setoid1) : Type0 ≝
+ { fun_1:1> A → B;
+ prop_1: ∀a,a'. eq1 ? a a' → eq1 ? (fun_1 a) (fun_1 a')
+ }.
+
+record binary_morphism (A,B,C:setoid) : Type0 ≝
{ fun:2> A → B → C;
prop: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun a b) (fun a' b')
}.
-notation "# r" with precedence 60 for @{prop ???????? (refl ???) $r}.
-notation "r #" with precedence 60 for @{prop ???????? $r (refl ???)}.
+record binary_morphism1 (A,B,C:setoid1) : Type0 ≝
+ { fun1:2> A → B → C;
+ prop1: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun1 a b) (fun1 a' b')
+ }.
-definition CPROP: setoid.
+notation "† c" with precedence 90 for @{'prop1 $c }.
+notation "l ‡ r" with precedence 90 for @{'prop $l $r }.
+notation "#" with precedence 90 for @{'refl}.
+interpretation "prop_1" 'prop1 c = (prop_1 _____ c).
+interpretation "prop1" 'prop l r = (prop1 ________ l r).
+interpretation "prop" 'prop l r = (prop ________ l r).
+interpretation "refl1" 'refl = (refl1 ___).
+interpretation "refl" 'refl = (refl ___).
+
+definition CPROP: setoid1.
constructor 1;
[ apply CProp
| constructor 1;
[ apply (H4 (H2 H6)) | apply (H3 (H5 H6))]]]
qed.
+definition if': ∀A,B:CPROP. A = B → A → B.
+ intros; apply (if ?? H); assumption.
+qed.
+
+notation ". r" with precedence 50 for @{'if $r}.
+interpretation "if" 'if r = (if' __ r).
+
+definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
+ constructor 1;
+ [ apply And
+ | intros; split; intro; cases H2; split;
+ [ apply (if ?? H a1)
+ | apply (if ?? H1 b1)
+ | apply (fi ?? H a1)
+ | apply (fi ?? H1 b1)]]
+qed.
+
+interpretation "and_morphism" 'and a b = (fun1 ___ and_morphism a b).
+
+definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
+ constructor 1;
+ [ apply Or
+ | intros; split; intro; cases H2; [1,3:left |2,4: right]
+ [ apply (if ?? H a1)
+ | apply (fi ?? H a1)
+ | apply (if ?? H1 b1)
+ | apply (fi ?? H1 b1)]]
+qed.
+
+interpretation "or_morphism" 'or a b = (fun1 ___ or_morphism a b).
+
+definition if_morphism: binary_morphism1 CPROP CPROP CPROP.
+ constructor 1;
+ [ apply (λA,B. A → B)
+ | intros; split; intros;
+ [ apply (if ?? H1); apply H2; apply (fi ?? H); assumption
+ | apply (fi ?? H1); apply H2; apply (if ?? H); assumption]]
+qed.
+
+(*
definition eq_morphism: ∀S:setoid. binary_morphism S S CPROP.
intro;
constructor 1;
apply (.= H2);
apply (H1 \sup -1)]]
qed.
+*)
-record category : Type ≝
+record category : Type1 ≝
{ objs:> Type;
arrows: objs → objs → setoid;
id: ∀o:objs. arrows o o;
id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) = a
}.
-interpretation "category composition" 'compose x y = (fun ___ (comp ____) x y).
-notation "'ASSOC'" with precedence 90 for @{comp_assoc ????????}.
\ No newline at end of file
+record category1 : Type2 ≝
+ { objs1:> Type;
+ 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 = comp1 o1 o2 o4 a12 (comp1 o2 o3 o4 a23 a34);
+ id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a = a;
+ id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) = a
+ }.
+
+notation "'ASSOC'" with precedence 90 for @{'assoc}.
+notation "'ASSOC1'" with precedence 90 for @{'assoc1}.
+
+interpretation "category1 composition" 'compose x y = (fun1 ___ (comp1 ____) y x).
+interpretation "category1 assoc" 'assoc1 = (comp_assoc1 ________).
+interpretation "category composition" 'compose x y = (fun ___ (comp ____) y x).
+interpretation "category assoc" 'assoc = (comp_assoc ________).
+
+definition unary_morphism_setoid: setoid → setoid → setoid.
+ intros;
+ constructor 1;
+ [ apply (unary_morphism s s1);
+ | constructor 1;
+ [ intros (f g); apply (∀a. f a = g a);
+ | intros 1; simplify; intros; apply refl;
+ | simplify; intros; apply sym; apply H;
+ | simplify; intros; apply trans; [2: apply H; | skip | apply H1]]]
+qed.
+
+notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
+interpretation "unary morphism" 'Imply a b = (unary_morphism_setoid a b).
+interpretation "unary morphism" 'Imply a b = (unary_morphism a b).
+
+definition SET: category1.
+ constructor 1;
+ [ apply setoid;
+ | apply rule (λS,T.unary_morphism_setoid S T);
+ | intros; constructor 1; [ apply (λx.x); | intros; assumption ]
+ | intros; constructor 1; [ intros; constructor 1; [ apply (λx. c1 (c x)); | intros;
+ apply († (†H));]
+ | intros; whd; intros; simplify; whd in H1; whd in H;
+ apply trans; [ apply (b (a' a1)); | lapply (prop_1 ?? b (a a1) (a' a1));
+ [ apply Hletin | apply (H a1); ] | apply H1; ]]
+ | 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.
+ intros; apply o; qed.
+
+coercion setoid_OF_SET.
+
+
+definition prop_1_SET :
+ ∀A,B:SET.∀w:arrows1 SET A B.∀a,b:A.eq1 ? a b→eq1 ? (w a) (w b).
+intros; apply (prop_1 A B w a b H);
+qed.
+
+interpretation "SET dagger" 'prop1 h = (prop_1_SET _ _ _ _ _ h).
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