Thms myhill nerode Sections AutomataTheory Doc

lang_rel Def L-induced Equiv(x,y) == z:A*. L(z @ x) L(z @ y)

Thm* A:Type, L:LangOver(A). L-induced Equiv A*A*Prop

lquo_rel Def Rg(x,y) == z:A*. (g(z@x)) (g(z@y))

Thm* A:Type, R:(A*A*Prop). (EquivRel x,y:A*. x R y) (x,y,z:A*. (x R y) ((z @ x) R (z @ y))) (g:((x,y:A*//(x R y))). Rg (x,y:A*//(x R y))(x,y:A*//(x R y))Prop)

mn_quo_append Def z@x == z @ x

Thm* A:Type, R:(A*A*Prop). (EquivRel x,y:A*. x R y) (x,y,z:A*. (x R y) ((z @ x) R (z @ y))) (z:A*, y:x,y:A*//(x R y). z@y x,y:A*//(x R y))

append Def as @ bs == Case of as; nil bs ; a.as' a.(as' @ bs) (recursive)

Thm* T:Type, as,bs:T*. (as @ bs) T*

assert Def b == if b True else False fi

Thm* b:. b Prop

decidable Def Dec(P) == P P

Thm* A:Prop. Dec(A) Prop

equiv_rel Def EquivRel x,y:T. E(x;y) == Refl(T;x,y.E(x;y)) & Sym x,y:T. E(x;y) & Trans x,y:T. E(x;y)

Thm* T:Type, E:(TTProp). (EquivRel x,y:T. E(x,y)) Prop

finite Def Fin(s) == n:, f:(ns). Bij(n; s; f)

Thm* T:Type. Fin(T) Prop

iff Def P Q == (P Q) & (P Q)

Thm* A,B:Prop. (A B) Prop

int_seg Def {i..j} == {k:| i k < j }

Thm* m,n:. {m..n} Type

int_upper Def {i...} == {j:| ij }

Thm* n:. {n...} Type

languages Def LangOver(Alph) == Alph*Prop

Thm* Alph:Type{i}. LangOver(Alph) Type{i'}

nat Def == {i:| 0i }

Thm* Type

one_one_corr Def A ~ B == f:(AB), g:(BA). InvFuns(A; B; f; g)

Thm* A,B:Type. (A ~ B) Prop

lelt Def i j < k == ij & j < k

le Def AB == B < A

Thm* i,j:. ij Prop

not Def A == A False

Thm* A:Prop. (A) Prop

trans Def Trans x,y:T. E(x;y) == a,b,c:T. E(a;b) E(b;c) E(a;c)

Thm* T:Type, E:(TTProp). Trans x,y:T. E(x,y) Prop

sym Def Sym x,y:T. E(x;y) == a,b:T. E(a;b) E(b;a)

Thm* T:Type, E:(TTProp). Sym x,y:T. E(x,y) Prop

refl Def Refl(T;x,y.E(x;y)) == a:T. E(a;a)

Thm* T:Type, E:(TTProp). Refl(T;x,y.E(x,y)) Prop

biject Def Bij(A; B; f) == Inj(A; B; f) & Surj(A; B; f)

Thm* A,B:Type, f:(AB). Bij(A; B; f) Prop

rev_implies Def P Q == Q P

Thm* A,B:Prop. (A B) Prop

inv_funs Def InvFuns(A; B; f; g) == g o f = Id & f o g = Id

Thm* A,B:Type, f:(AB), g:(BA). InvFuns(A; B; f; g) Prop

surject Def Surj(A; B; f) == b:B. a:A. f(a) = b

Thm* A,B:Type, f:(AB). Surj(A; B; f) Prop

inject Def Inj(A; B; f) == a1,a2:A. f(a1) = f(a2) B a1 = a2

Thm* A,B:Type, f:(AB). Inj(A; B; f) Prop

tidentity Def Id == Id

Thm* A:Type. Id AA

compose Def (f o g)(x) == f(g(x))

Thm* A,B,C:Type, f:(BC), g:(AB). f o g AC

identity Def Id(x) == x

Thm* A:Type. Id AA

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