Nuprl - Pf mn_23_lem_1 1 1 1 2 1 1 2 2 2

PrintForm Definitions myhill nerode Sections AutomataTheory Doc

1. Alph: Type
2. R: Alph*Alph*Prop
3. Fin(Alph)
4. EquivRel x,y:Alph*. x R y
5. Fin(x,y:Alph*//(x R y))
6. x,y,z:Alph*. (x R y) ((z @ x) R (z @ y))
7. g: (x,y:Alph*//(x R y))
8. x: x,y:Alph*//(x R y)
9. y: x,y:Alph*//(x R y)
10. < (x,y:Alph*//(x R y))(x,y:Alph*//(x R y)),a,p. p/x,y. < a.x,a.y > > ActionSet(Alph)
11. Fin((x,y:Alph*//(x R y))(x,y:Alph*//(x R y)))
12. x:((x,y:Alph*//(x R y))(x,y:Alph*//(x R y))), y:Alph*. ( < (x,y:Alph*//(x R y))(x,y:Alph*//(x R y)),a,p. p/x,y. < a.x,a.y > > :yx) = (x/x1,x2. < y@x1,y@x2 > )

Dec(x@0:Alph*. (g(x@0@x)) = (g(x@0@y)) = false)

By: Inst Thm* S:ActionSet(Alph), si:S.car. Fin(S.car) Fin(Alph) (RL:S.car*. s:S.car. (w:Alph*. (S:wsi) = s) mem_f(S.car;s;RL)) [Alph; < (x,y:Alph*//(x R y))(x,y:Alph*//(x R y)),a,p. p/x,y. < a.x,a.y > > ; < x,y > ]

Generated subgoals:

1 Fin( < (x,y:Alph*//(x R y))(x,y:Alph*//(x R y)),a,p. p/x,y. < a.x,a.y > > .car)

2 13. RL: < (x,y:Alph*//(x R y))(x,y:Alph*//(x R y)),a,p. p/x,y. < a.x,a.y > > .car*. s: < (x,y:Alph*//(x R y))(x,y:Alph*//(x R y)),a,p. p/x,y. < a.x,a.y > > .car. (w:Alph*. ( < (x,y:Alph*//(x R y))(x,y:Alph*//(x R y)),a,p. p/x,y. < a.x,a.y > > :w < x,y > ) = s) mem_f( < (x,y:Alph*//(x R y))(x,y:Alph*//(x R y)),a,p. p/x,y. < a.x,a.y > > .car;s;RL)
Dec(x@0:Alph*. (g(x@0@x)) = (g(x@0@y)) = false)

About:

1	`Fin( < (x,y:Alph//(x R y))(x,y:Alph//(x R y)),a,p. p/x,y. < a.x,a.y > > .car)`
2	13. `RL: < (x,y:Alph//(x R y))(x,y:Alph//(x R y)),a,p. p/x,y. < a.x,a.y > > .car. s: < (x,y:Alph//(x R y))(x,y:Alph//(x R y)),a,p. p/x,y. < a.x,a.y > > .car. (w:Alph. ( < (x,y:Alph//(x R y))(x,y:Alph//(x R y)),a,p. p/x,y. < a.x,a.y > > :w < x,y > ) = s) mem_f( < (x,y:Alph//(x R y))(x,y:Alph//(x R y)),a,p. p/x,y. < a.x,a.y > > .car;s;RL)` `Dec(x@0:Alph*. (g(x@0@x)) = (g(x@0@y)) = false)`