Thm* 'a,'b:S. all( x:hsum('a; 'b). implies(isr(x),equal(inr(outr(x)),x))) | [hinr_houtr] |
Thm* 'a,'b:S. all( x:hsum('a; 'b). implies(isl(x),equal(inl(outl(x)),x))) | [hinl_houtl] |
Thm* 'a,'b:S. all( x:hsum('a; 'b). or(isl(x),isr(x))) | [hisl_or_isr] |
Thm* 'c,'b,'a:S.
Thm* all
Thm* ( f:'a  'c. all
Thm* ( f:'a  'c. ( g:'b  'c. exists_unique
Thm* ( f:'a  'c. ( g:'b  'c. ( h:hsum('a; 'b)  'c. and
Thm* ( f:'a  'c. ( g:'b  'c. ( h:hsum('a; 'b)  'c. (all
Thm* ( f:'a  'c. ( g:'b  'c. ( h:hsum('a; 'b)  'c. (( x:'a. equal
Thm* ( f:'a  'c. ( g:'b  'c. ( h:hsum('a; 'b)  'c. (( x:'a. (h(inl(x))
Thm* ( f:'a  'c. ( g:'b  'c. ( h:hsum('a; 'b)  'c. (( x:'a. ,f(x)))
Thm* ( f:'a  'c. ( g:'b  'c. ( h:hsum('a; 'b)  'c. ,all
Thm* ( f:'a  'c. ( g:'b  'c. ( h:hsum('a; 'b)  'c. ,( x:'b. equal
Thm* ( f:'a  'c. ( g:'b  'c. ( h:hsum('a; 'b)  'c. ,( x:'b. (h(inr(x))
Thm* ( f:'a  'c. ( g:'b  'c. ( h:hsum('a; 'b)  'c. ,( x:'b. ,g(x))))))) | [hsum_axiom_2] |
Thm* 'c,'b,'a:S.
Thm* all
Thm* ( f:'a  'c. all
Thm* ( f:'a  'c. ( g:'b  'c. exists_unique
Thm* ( f:'a  'c. ( g:'b  'c. ( h:hsum('a; 'b)  'c. and
Thm* ( f:'a  'c. ( g:'b  'c. ( h:hsum('a; 'b)  'c. (equal(o(h,inl),f)
Thm* ( f:'a  'c. ( g:'b  'c. ( h:hsum('a; 'b)  'c. ,equal(o(h,inr),g))))) | [hsum_axiom] |
Thm* 'a,'b:S.
Thm* all
Thm* ( e:'b. equal
Thm* ( e:'b. (inr(e)
Thm* ( e:'b. ,abs_sum( b:hbool. x:'a. y:'b. and(equal(y,e),not(b))))) | [hinr_def] |
Thm* 'b,'a:S.
Thm* all
Thm* ( e:'a. equal(inl(e),abs_sum( b:hbool. x:'a. y:'b. and(equal(x,e),b)))) | [hinl_def] |
Thm* 'a,'b:S.
Thm* exists
Thm* ( rep:hsum('a; 'b)  hbool  'a  'b  hbool. type_definition
Thm* ( rep:hsum('a; 'b)  hbool  'a  'b  hbool. (is_sum_rep
Thm* ( rep:hsum('a; 'b)  hbool  'a  'b  hbool. ,rep)) | [hsum_ty_def] |
Thm* 'a,'b:S.
Thm* and
Thm* (all( a:hsum('a; 'b). equal(abs_sum(rep_sum(a)),a))
Thm* ,all
Thm* ,( r:hbool  'a  'b  hbool. equal
Thm* ,( r:hbool  'a  'b  hbool. (is_sum_rep(r)
Thm* ,( r:hbool  'a  'b  hbool. ,equal(rep_sum(abs_sum(r)),r)))) | [hsum_iso_def] |