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
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
languages	Def LangOver(Alph) == Alph*Prop Thm* Alph:Type{i}. LangOver(Alph) Type{i'}
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
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
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
append	Def as @ bs == Case of as; nil bs ; a.as' a.(as' @ bs) (recursive) Thm* T:Type, as,bs:T*. (as @ bs) T*
iff	Def P Q == (P Q) & (P Q) Thm* A,B:Prop. (A B) Prop
rev_implies	Def P Q == Q P Thm* A,B:Prop. (A B) Prop

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