include "sets/sets.ma".
nrecord magma_type : Type[1] ≝
- { carr:> Type;
- op: carr → carr → carr
+ { mtcarr:> setoid;
+ op: unary_morphism mtcarr (unary_morph_setoid mtcarr mtcarr)
}.
nrecord magma (A: magma_type) : Type[1] ≝
- { mcarr:> Ω \sup A;
+ { mcarr:> ext_powerclass A;
op_closed: ∀x,y. x ∈ mcarr → y ∈ mcarr → op A x y ∈ mcarr
}.
-nrecord magma_morphism_type (A,B: magma_type) : Type ≝
- { mmcarr:1> A → B;
- mmprop: ∀x,y. mmcarr (op … x y) = op … (mmcarr x) (mmcarr y)
+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 ≝
+nrecord magma_morphism (A) (B) (Ma: magma A) (Mb: magma B) : Type[0] ≝
{ mmmcarr:> magma_morphism_type A B;
- mmclosed: ∀x. x ∈ Ma → mmmcarr x ∈ Mb
+ mmclosed: ∀x:A. x ∈ mcarr ? Ma → mmmcarr x ∈ mcarr ? Mb
}.
-
-ndefinition image: ∀A,B. (A → B) → Ω \sup A → Ω \sup B ≝
- λA,B,f,Sa. {y | ∃x. x ∈ Sa ∧ f x = y}.
+(*
ndefinition mm_image:
∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism … Ma Mb → magma B.
#A; #B; #Ma; #Mb; #f;
napply (mmprop … f)]##]
nqed.
-ndefinition counter_image: ∀A,B. (A → B) → Ω \sup B → Ω \sup A ≝
- λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
-
ndefinition mm_counter_image:
∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism … Ma Mb → magma A.
#A; #B; #Ma; #Mb; #f;
nrewrite < Hy1;
napply (mmprop … f)]##]
nqed.
+*)
ndefinition m_intersect: ∀A. magma A → magma A → magma A.
#A; #M1; #M2;
napply (mk_magma …)
- [ napply (M1 ∩ M2)
+ [ 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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