include "sets/sets.ma".
-nrecord pre_magma : Type[1] ≝
- { carr: Type;
- op: carr → carr → carr
+nrecord magma_type : Type[1] ≝
+ { mcarr:> setoid;
+ op: mcarr → mcarr → mcarr
}.
-(* this is a projection *)
-ndefinition carr ≝ λM: pre_magma. match M with [ mk_pre_magma carr _ ⇒ carr ].
-ndefinition op ≝
- λM: pre_magma. match M return λM. carr M → carr M → carr M with [ mk_pre_magma _ op ⇒ op ].
-(* ncoercion carr. *)
-nrecord magma (A: pre_magma) : Type[1] ≝
- { mcarr: Ω \sup (carr A);
+nrecord magma (A: magma_type) : Type[1] ≝
+ { mcarr:> powerset_setoid1 A;
op_closed: ∀x,y. x ∈ mcarr → y ∈ mcarr → op A x y ∈ mcarr
}.
-(* this is a projection *)
-ndefinition mcarr ≝ λA.λM: magma A. match M with [ mk_magma mcarr _ ⇒ mcarr ].
-ndefinition op_closed ≝
- λA.λM: magma A.
- match M return λM.∀x,y. x ∈ mcarr ? M → y ∈ mcarr ? M → op A x y ∈ mcarr ? M with
- [ mk_magma _ opc ⇒ opc ].
-nrecord pre_magma_morphism (A,B: pre_magma) : Type ≝
- { mmcarr: carr A → carr B;
- mmprop: ∀x,y. mmcarr (op ? x y) = op ? (mmcarr x) (mmcarr y)
+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)
}.
-(* this is a projection *)
-ndefinition mmcarr ≝
- λA,B.λf: pre_magma_morphism A B. match f with [ mk_pre_magma_morphism f _ ⇒ f ].
nrecord magma_morphism (A) (B) (Ma: magma A) (Mb: magma B) : Type ≝
- { mmmcarr: pre_magma_morphism A B;
- mmclosed: ∀x. x ∈ mcarr ? Ma → mmcarr ?? mmmcarr x ∈ mcarr ? Mb
+ { mmmcarr:> magma_morphism_type A B;
+ mmclosed: ∀x. x ∈ Ma → mmmcarr x ∈ Mb
}.
-(* this is a projection *)
-ndefinition mmmcarr ≝
- λA,B,Ma,Mb.λf: magma_morphism A B Ma Mb. match f with [ mk_magma_morphism f _ ⇒ f ].
-ndefinition mmclosed ≝
- λA,B,Ma,Mb.λf: magma_morphism A B Ma Mb.
- match f return λf.∀x. x ∈ mcarr ? Ma → mmcarr ?? (mmmcarr ???? f) x ∈ mcarr ? Mb with
- [ mk_magma_morphism _ p ⇒ p ].
-
-ndefinition sub_magma ≝
- λA.λM1,M2: magma A. ∀x. x ∈ mcarr ? M1 → x ∈ mcarr ? M2.
ndefinition image: ∀A,B. (A → B) → Ω \sup A → Ω \sup B ≝
λA,B,f,Sa. {y | ∃x. x ∈ Sa ∧ f x = y}.
-naxiom daemon: False.
-
ndefinition mm_image:
- ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism ?? Ma Mb → magma B.
+ ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism … Ma Mb → magma B.
#A; #B; #Ma; #Mb; #f;
- napply (mk_magma ???)
- [ napply (image ?? (mmcarr ?? (mmmcarr ???? f)) (mcarr ? Ma))
- | #x; #y; *; #x0; #Hx0; *; #y0; #Hy0; nwhd;
- napply (ex_intro ????)
- [ napply (op ? x0 y0)
- | nelim daemon ]]
+ 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 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;
+ 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 (M1 ∩ M2)
+ | #x; #y; nwhd in ⊢ (% → % → %); *; #Hx1; #Hx2; *; #Hy1; #Hy2;
+ napply conj; napply op_closed; nassumption ]
nqed.
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