Nuprl - Some Definitions

Formula	`Def Formula == rec(formula.Var+formula+(formulaformula)+(formulaformula)+(formulaformula))` `Thm* Formula Type`
sequent_rank	`Def (S) == (S.H)+(S.C)` `Thm* Sequent`
list_rank	`Def (L) == reduce(x,y. (x)+y;0;L)` `Thm* (Formula List)`
formula_rank	`Def == (letrec formula_rank f = case f: x 0; p (formula_rank(p)+1); pq (formula_rank(p)+formula_rank(q)+1); pq (formula_rank(p)+formula_rank(q)+1); pq (formula_rank(p)+formula_rank(q)+1); )` `Thm* Formula`
gt	`Def i > j == j < i` `Thm* i,j:. i > j Prop`
list_exists	`Def xL.P(x) == (letrec list_exists L = (Case of L; nil False ; h.t P(h) list_exists(t)) ) (L)` `Thm* T:Type, P:(TProp), L:T List. xL.P(x) Type`
not	`Def A == A False` `Thm* A:Prop. (A) Prop`
Var	`Def Var == Atom` `Thm* Var Type`
formula_case	`Def case F: x varC(x); p1 notC(p1); p2p3 andC(p2;p3); p4p5 orC(p4;p5); p6p7 impC(p6;p7); == InjCase(F; x. varC(x); F. InjCase(F; p1. notC(p1); F. InjCase(F; x. x/p2,p3.andC(p2;p3); F. InjCase(F; x. x/p4,p5.orC(p4;p5), x/p6,p7.impC(p6;p7)))))`
letrec_body	`Def = b == b`
letrec_arg	`Def x b(x) (x) == b(x)`
letrec	`Def (letrec f b(f)) == b((letrec f b(f)) ) (recursive)`
C	`Def s.C == s.2` `Thm* s:Sequent. s.C Formula List`
H	`Def s.H == s.1` `Thm* s:Sequent. s.H Formula List`
reduce	`Def reduce(f;k;as) == Case of as; nil k ; a.as' f(a,reduce(f;k;as')) (recursive)` `Thm* A,B:Type, f:(ABB), k:B, as:A List. reduce(f;k;as) B`

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