Thms relation autom Sections AutomataTheory Doc

NOTE: EquivRel(A)(R(_1;_2)) is alpha-equivalent to EquivRel x,y:A. R(x;y).

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

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

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

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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