+(**************************************************************************)
+(* ___ *)
+(* ||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 "logic/connectives.ma".
include "properties/relations.ma".
+include "hints_declaration.ma".
+
+nrecord setoid : Type[1] ≝ {
+ carr:> Type[0];
+ eq0: equivalence_relation carr
+}.
+
+(* activate non uniform coercions on: Type → setoid *)
+unification hint 0 ≔ R : setoid;
+ MR ≟ carr R,
+ lock ≟ mk_lock1 Type[0] MR setoid R
+(* ---------------------------------------- *) ⊢
+ setoid ≡ force1 ? MR lock.
+
+notation < "[\setoid\ensp\of term 19 x]" non associative with precedence 90 for @{'mk_setoid $x}.
+interpretation "mk_setoid" 'mk_setoid x = (mk_setoid x ?).
+
+interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq0 t) x y).
+(* single = is for the abstract equality of setoids, == is for concrete
+ equalities (that may be lifted to the setoid level when needed *)
+notation < "hvbox(a break mpadded width -50% (=)= b)" non associative with precedence 45 for @{ 'eq_low $a $b }.
+notation > "a == b" non associative with precedence 45 for @{ 'eq_low $a $b }.
+
+notation > "hvbox(a break =_0 b)" non associative with precedence 45
+for @{ eq_rel ? (eq0 ?) $a $b }.
+
+interpretation "setoid symmetry" 'invert r = (sym ???? r).
+notation ".= r" with precedence 50 for @{'trans $r}.
+interpretation "trans" 'trans r = (trans ????? r).
+notation > ".=_0 r" with precedence 50 for @{'trans_x0 $r}.
+interpretation "trans_x0" 'trans_x0 r = (trans ????? r).
+
+nrecord unary_morphism (A,B: setoid) : Type[0] ≝ {
+ fun1:1> A → B;
+ prop1: ∀a,a'. a = a' → fun1 a = fun1 a'
+}.
+
+notation > "B ⇒_0 C" right associative with precedence 72 for @{'umorph0 $B $C}.
+notation "hvbox(B break ⇒\sub 0 C)" right associative with precedence 72 for @{'umorph0 $B $C}.
+interpretation "unary morphism 0" 'umorph0 A B = (unary_morphism 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 "refl" 'refl = (refl ???).
+notation "┼_0 c" with precedence 89 for @{'prop1_x0 $c }.
+notation "l ╪_0 r" with precedence 89 for @{'prop2_x0 $l $r }.
+interpretation "prop1_x0" 'prop1_x0 c = (prop1 ????? c).
+
+ndefinition unary_morph_setoid : setoid → setoid → setoid.
+#S1; #S2; @ (S1 ⇒_0 S2); @;
+##[ #f; #g; napply (∀x,x'. x=x' → f x = g x');
+##| #f; #x; #x'; #Hx; napply (.= †Hx); napply #;
+##| #f; #g; #H; #x; #x'; #Hx; napply ((H … Hx^-1)^-1);
+##| #f; #g; #h; #H1; #H2; #x; #x'; #Hx; napply (trans … (H1 …) (H2 …)); //; ##]
+nqed.
+
+alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
+unification hint 0 ≔ o1,o2 ;
+ X ≟ unary_morph_setoid o1 o2
+ (* ----------------------------- *) ⊢
+ carr X ≡ o1 ⇒_0 o2.
+
+interpretation "prop2" 'prop2 l r = (prop1 ? (unary_morph_setoid ??) ? ?? l ?? r).
+interpretation "prop2_x0" 'prop2_x0 l r = (prop1 ? (unary_morph_setoid ??) ? ?? l ?? r).
-nrecord iff (A,B: CProp) : CProp ≝
- { if: A → B;
- fi: B → A
- }.
-
-notation > "hvbox(a break \liff b)"
- left associative with precedence 25
-for @{ 'iff $a $b }.
-
-notation "hvbox(a break \leftrightarrow b)"
- left associative with precedence 25
-for @{ 'iff $a $b }.
-
-interpretation "logical iff" 'iff x y = (iff x y).
-
-nrecord setoid : Type[1] ≝
- { carr:> Type;
- eq: carr → carr → CProp;
- refl: reflexive ? eq;
- sym: symmetric ? eq;
- trans: transitive ? eq
- }.
-
-ndefinition proofs: CProp → setoid.
-#P; napply (mk_setoid ?????);
-##[ napply P;
-##| #x; #y; napply True;
-##|##*: nwhd; nrepeat (#_); napply I;
-##]
+nlemma unary_morph_eq: ∀A,B.∀f,g:A ⇒_0 B. (∀x. f x = g x) → f = g.
+#A B f g H x1 x2 E; napply (.= †E); napply H; nqed.
+
+nlemma mk_binary_morphism:
+ ∀A,B,C: setoid. ∀f: A → B → C. (∀a,a',b,b'. a=a' → b=b' → f a b = f a' b') →
+ A ⇒_0 (unary_morph_setoid B C).
+ #A; #B; #C; #f; #H; @; ##[ #x; @ (f x) ] #a; #a'; #Ha [##2: napply unary_morph_eq; #y]
+ /2/.
nqed.
-(*
-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 setoid1 : Type ≝
- { carr1:> Type;
- eq1: carr1 → carr1 → CProp;
- refl1: reflexive1 ? eq1;
- sym1: symmetric1 ? eq1;
- trans1: transitive1 ? eq1
- }.
-
-definition proofs1: CProp → setoid1.
- intro;
- constructor 1;
- [ apply A
- | intros;
- apply True
- | intro;
- constructor 1
- | intros 3;
- constructor 1
- | intros 5;
- constructor 1]
-qed.
-*)
-
-(*
-ndefinition CCProp: setoid1.
- constructor 1;
- [ apply CProp
- | apply iff
- | intro;
- split;
- intro;
- assumption
- | intros 3;
- cases H; clear H;
- split;
- assumption
- | intros 5;
- cases H; cases H1; clear H H1;
- split;
- intros;
- [ apply (H4 (H2 H))
- | apply (H3 (H5 H))]]
-qed.
-
-*)
-
-(************************CSC
-nrecord function_space (A,B: setoid): Type[1] ≝
- { f:1> carr A → carr B}.;
- f_ok: ∀a,a':A. proofs (eq ? a a') → proofs (eq ? (f a) (f a'))
- }.
+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.o2 ⇒_0 o3 → o1 ⇒_0 o2 → o1 ⇒_0 o3.
+#o1; #o2; #o3; #f; #g; @ (f ∘ g);
+ #a; #a'; #e; nnormalize; napply (.= †(†e)); napply #.
+nqed.
-notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
-
-(*
-record function_space1 (A: setoid1) (B: setoid1): Type ≝
- { f1:1> A → B;
- f1_ok: ∀a,a':A. proofs1 (eq1 ? a a') → proofs1 (eq1 ? (f1 a) (f1 a'))
- }.
-*)
-
-definition function_space_setoid: setoid → setoid → setoid.
- intros (A B);
- constructor 1;
- [ apply (function_space A B);
- | intros;
- apply (∀a:A. proofs (eq ? (f a) (f1 a)));
- | simplify;
- intros;
- apply (f_ok ? ? x);
- unfold carr; unfold proofs; simplify;
- apply (refl A)
- | simplify;
- intros;
- unfold carr; unfold proofs; simplify;
- apply (sym B);
- apply (f a)
- | simplify;
- intros;
- unfold carr; unfold proofs; simplify;
- apply (trans B ? (y a));
- [ apply (f a)
- | apply (f1 a)]]
-qed.
-
-definition function_space_setoid1: setoid1 → setoid1 → setoid1.
- intros (A B);
- constructor 1;
- [ apply (function_space1 A B);
- | intros;
- apply (∀a:A. proofs1 (eq1 ? (f a) (f1 a)));
- |*: cases daemon] (* simplify;
- intros;
- apply (f1_ok ? ? x);
- unfold proofs; simplify;
- apply (refl1 A)
- | simplify;
- intros;
- unfold proofs; simplify;
- apply (sym1 B);
- apply (f a)
- | simplify;
- intros;
- unfold carr; unfold proofs; simplify;
- apply (trans1 B ? (y a));
- [ apply (f a)
- | apply (f1 a)]] *)
-qed.
-
-interpretation "function_space_setoid1" 'Imply a b = (function_space_setoid1 a b).
-
-record isomorphism (A,B: setoid): Type ≝
- { map1:> function_space_setoid A B;
- map2:> function_space_setoid B A;
- inv1: ∀a:A. proofs (eq ? (map2 (map1 a)) a);
- inv2: ∀b:B. proofs (eq ? (map1 (map2 b)) b)
- }.
-
-interpretation "isomorphism" 'iff x y = (isomorphism x y).
-
-definition setoids: setoid1.
- constructor 1;
- [ apply setoid;
- | apply isomorphism;
- | intro;
- split;
- [1,2: constructor 1;
- [1,3: intro; assumption;
- |*: intros; assumption]
- |3,4:
- intros;
- simplify;
- unfold proofs; simplify;
- apply refl;]
- |*: cases daemon]
-qed.
-
-definition setoid1_of_setoid: setoid → setoid1.
- intro;
- constructor 1;
- [ apply (carr s)
- | apply (eq s)
- | apply (refl s)
- | apply (sym s)
- | apply (trans s)]
-qed.
-
-coercion setoid1_of_setoid.
-
-(*
-record dependent_product (A:setoid) (B: A ⇒ setoids): Type ≝
- { dp:> ∀a:A.carr (B a);
- dp_ok: ∀a,a':A. ∀p:proofs1 (eq1 ? a a'). proofs1 (eq1 ? (dp a) (map2 ?? (f1_ok ?? B ?? p) (dp a')))
- }.*)
-
-record forall (A:setoid) (B: A ⇒ CCProp): CProp ≝
- { fo:> ∀a:A.proofs (B a) }.
-
-record subset (A: setoid) : CProp ≝
- { mem: A ⇒ CCProp }.
-
-definition ssubset: setoid → setoid1.
- intro;
- constructor 1;
- [ apply (subset s);
- | apply (λU,V:subset s. ∀a. mem ? U a \liff mem ? V a)
- | simplify;
- intros;
- split;
- intro;
- assumption
- | simplify;
- cases daemon
- | cases daemon]
-qed.
-
-definition mmem: ∀A:setoid. (ssubset A) ⇒ A ⇒ CCProp.
- intros;
- constructor 1;
- [ apply mem;
- | unfold function_space_setoid1; simplify;
- intros (b b');
- change in ⊢ (? (? (?→? (? %)))) with (mem ? b a \liff mem ? b' a);
- unfold proofs1; simplify; intros;
- unfold proofs1 in c; simplify in c;
- unfold ssubset in c; simplify in c;
- cases (c a); clear c;
- split;
- assumption]
-qed.
-
-(*
-definition sand: CCProp ⇒ CCProp.
-
-definition intersection: ∀A. ssubset A ⇒ ssubset A ⇒ ssubset A.
- intro;
- constructor 1;
- [ intro;
- constructor 1;
- [ intro;
- constructor 1;
- constructor 1;
- intro;
- apply (mem ? c c2 ∧ mem ? c1 c2);
- |
- |
- |
-*)
-*******************)
+unification hint 0 ≔ o1,o2,o3:setoid,f:o2 ⇒_0 o3,g:o1 ⇒_0 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.(o2 ⇒_0 o3) ⇒_0 ((o1 ⇒_0 o2) ⇒_0 (o1 ⇒_0 o3)).
+#o1; #o2; #o3; napply mk_binary_morphism
+ [ #f; #g; napply (comp_unary_morphisms ??? f g)
+ (* CSC: why not ∘?
+ GARES: because the coercion to FunClass is not triggered if there
+ are no "extra" arguments. We could fix that in the refiner
+ *)
+ | #a; #a'; #b; #b'; #ea; #eb; #x; #x'; #Hx; nnormalize; /3/ ]
+nqed.
projects / helm.git / blobdiff