\require L

\* Feferman's system T0 *\

\open elements \* [1] 2.1. 2.2. 2.4. *\

   \decl "rule application" App: *Obj => *Obj => *Obj -> *Prop

   \decl "classification predicate" Cl: *Obj -> *Prop

   \decl "classification membership" Eta: *Obj => *Obj -> *Prop

\* we must make an explicit coercion from *Obj to *Term *\
   \decl "object-to-term-coercion" T: *Obj -> *Term

   \decl "term application" At: *Term => *Term -> *Term

   \decl "term-object equivalence" E: *Term => *Obj -> *Prop

\close

\open logical_abbreviations \* [1] 2.3. 2.5. *\

   \def "logical comprehension restricted to classifications"
      CAll = [q:*Obj->*Prop] [x:*Obj] Cl(x) -> q(x)      
           : (*Obj -> *Prop) -> *Prop

   \def "logical existence restricted to classifications"
      CEx = [q:*Obj->*Prop] Ex([x:*Obj] And(Cl(x), q(x)))      
          : (*Obj -> *Prop) -> *Prop

   \def "logical comprehension restricted to a classification"
      EAll = [a:*Obj, q:*Obj->*Prop] [x:*Obj] Eta(x, a) -> q(x)      
           : *Obj => (*Obj -> *Prop) -> *Prop

   \def "logical existence restricted to a classification"
      EEx = [a:*Obj, q:*Obj->*Prop] Ex([x:*Obj] And(Eta(x, a), q(x)))      
          : *Obj => (*Obj -> *Prop) -> *Prop

\close

\open non_logical_abbreviations \* [1] 2.4. 2.7 *\

   \def "object application"
      OAt = [f:*Obj, x:*Obj] At(T(f), T(x)) : *Obj => *Obj -> *Term

   \def "convergence of a term to an object"
      Conv = [t:*Term] EX([y:*Obj] E(t, y)) : *Term -> *Prop

   \def "term-term equivalence"
      Eq = [t1:*Term, t2:*Term] [y:*Obj] Iff(E(t1, y), E(t2, y))
         : *Term => *Term -> *Prop

   \def "classification membership of a term"
      TEta = [t:*Term, a:*Obj] EEx(a, [y:*Obj] E(t, y))
           : *Term => *Obj -> *Prop

   \def "operation (rule with inhabited domain)"
      Op = [f:*Obj] Ex([x:*Obj] Conv(OAt(f, x))) : *Obj -> *Prop

   \def "classification inclusion"
      ESub = [a1:*Obj, a2:*Obj] EAll(a1, [x:*Obj] Eta(x, a2))
           : *Obj => *Obj -> *Prop

    \def "classification morphism"
      ETo = [f:*Obj, a:*Obj, b:*Obj] EAll(a, [x:*Obj] TEta(OAt(f, x), b))
          : *Obj => *Obj => *Obj -> *Prop

\close

\open non_logical_axioms \* [1] 2.4. 3.2 *\

\* we axiomatize E because *Term is not inductively generated *\
   \ax e_refl: [y:*Obj] E(T(y), y)

   \ax e_at_in: [t1:*Term][t2:*Term][f:*Obj][x:*Obj][y:*Obj] 
                E(t1, f) -> E(t2, x) -> App(f, x, y) -> E(At(t1, t2), y) 
\*
   \ax e_at_out: [f:*Obj][x:*Obj][y:*Obj] E(At(T(f), T(x)), y) -> App(f,x,y) 
*\
   \ax "I (i)" id_dec: [x:*Obj][y:*Obj] Or(Id(x, y), NId(x, y))
   
   \ax "I (ii)" at_mono: [f:*Obj][x:*Obj][y1:*Obj][y2:*Obj]
                         E(OAt(f, x), y1) -> E(OAt(f, x), y2) -> Id(y1, y2)

   \ax "I (iii)" eta_cl: [x:*Obj][a:*Obj] Eta(x, a) -> Cl(a)

\close