include "sets/sets.ma".
nrecord magma_type : Type[1] ≝
- { carr:> Type;
- op: carr → carr → carr
+ { mcarr:> setoid;
+ op: mcarr → mcarr → mcarr
}.
nrecord magma (A: magma_type) : Type[1] ≝
- { mcarr:> Ω \sup A;
+ { mcarr:> powerset_setoid1 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)
+ mmprop: ∀x,y. mmcarr (op … x y) = op … (mmcarr x) (mmcarr y)
}.
nrecord magma_morphism (A) (B) (Ma: magma A) (Mb: magma B) : Type ≝
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: magma A. ∀Mb: magma B. magma_morphism … Ma Mb → magma B.
#A; #B; #Ma; #Mb; #f;
- napply (mk_magma …)
+ napply mk_magma
[ napply (image … f Ma)
| #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
- napply (ex_intro …)
+ napply ex_intro
[ napply (op … x0 y0)
- | napply (conj …)
- [ napply (op_closed …); nassumption
+ | napply conj
+ [ napply op_closed; nassumption
| nrewrite < Hx1;
nrewrite < Hy1;
- napply (mmprop … 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: magma A. ∀Mb: magma B. magma_morphism … Ma Mb → magma A.
#A; #B; #Ma; #Mb; #f;
- napply (mk_magma …)
+ napply mk_magma
[ napply (counter_image … f Mb)
| #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
- napply (ex_intro …)
+ napply ex_intro
[ napply (op … x0 y0)
- | napply (conj …)
- [ napply (op_closed …); nassumption
+ | napply conj
+ [ napply op_closed; nassumption
| nrewrite < Hx1;
nrewrite < Hy1;
- napply (mmprop … f …)]##]
+ napply (mmprop … f)]##]
nqed.
ndefinition m_intersect: ∀A. magma A → magma A → magma A.
napply (mk_magma …)
[ napply (M1 ∩ M2)
| #x; #y; nwhd in ⊢ (% → % → %); *; #Hx1; #Hx2; *; #Hy1; #Hy2;
- napply (conj …); napply (op_closed …); nassumption ]
+ napply conj; napply op_closed; nassumption ]
nqed.
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