+nrecord magma_type : Type[1] ≝
+ { mtcarr:> setoid;
+ op: unary_morphism mtcarr (unary_morph_setoid mtcarr mtcarr)
+ }.
+
+nrecord magma (A: magma_type) : Type[1] ≝
+ { mcarr:> ext_powerclass A;
+ op_closed: ∀x,y. x ∈ mcarr → y ∈ mcarr → op A x y ∈ mcarr
+ }.
+
+alias symbol "hint_decl" = "hint_decl_Type2".
+unification hint 0 ≔
+ A : ? ⊢ carr1 (ext_powerclass_setoid A) ≡ ext_powerclass A.
+
+(*
+ncoercion mcarr' : ∀A. ∀M: magma A. carr1 (qpowerclass_setoid (mtcarr A))
+ ≝ λA.λM: magma A.mcarr ? M
+ on _M: magma ? to carr1 (qpowerclass_setoid (mtcarr ?)).
+*)
+
+nrecord magma_morphism_type (A,B: magma_type) : Type[0] ≝
+ { mmcarr:> unary_morphism A B;
+ mmprop: ∀x,y:A. mmcarr (op ? x y) = op … (mmcarr x) (mmcarr y)
+ }.
+
+nrecord magma_morphism (A) (B) (Ma: magma A) (Mb: magma B) : Type[0] ≝
+ { mmmcarr:> magma_morphism_type A B;
+ mmclosed: ∀x:A. x ∈ mcarr ? Ma → (fun1 ?? mmmcarr x) ∈ mcarr ? Mb
+ }. (* XXX bug nelle coercions, fun1 non inserita *)
+
+(*
+ndefinition mm_image:
+ ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism … Ma Mb → magma B.
+ #A; #B; #Ma; #Mb; #f;
+ napply mk_magma
+ [ napply (image … f Ma)
+ | #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
+ napply ex_intro
+ [ napply (op … x0 y0)
+ | napply conj
+ [ napply op_closed; nassumption
+ | nrewrite < Hx1;
+ nrewrite < Hy1;
+ napply (mmprop … f)]##]
+nqed.
+
+ndefinition mm_counter_image:
+ ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism … Ma Mb → magma A.
+ #A; #B; #Ma; #Mb; #f;
+ napply mk_magma
+ [ napply (counter_image … f Mb)
+ | #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
+ napply ex_intro
+ [ napply (op … x0 y0)
+ | napply conj
+ [ napply op_closed; nassumption
+ | nrewrite < Hx1;
+ nrewrite < Hy1;
+ napply (mmprop … f)]##]
+nqed.
+*)
+
+ndefinition m_intersect: ∀A. magma A → magma A → magma A.
+ #A; #M1; #M2;
+ napply (mk_magma …)
+ [ napply (intersect_is_ext_morph ? M1 M2)
+ | #x; #y; nwhd in ⊢ (% → % → %); *; #Hx1; #Hx2; *; #Hy1; #Hy2;
+ napply conj; napply op_closed; nassumption ]
+nqed.
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