\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]^1 *Prop] All([x:*Obj]^2 Imp(Cl(x), q(x)))      
           : [[*Obj]^1 *Prop] *Prop

   \def "logical existence restricted to classifications"
      CEx = [q:[*Obj]^1 *Prop] Ex([x:*Obj]^2 And(Cl(x), q(x)))      
          : [[*Obj]^1 *Prop] *Prop

   \def "logical comprehension restricted to a classification"
      EAll = { [a:*Obj] [q:[*Obj]^1 *Prop] } All([x:*Obj]^2 Imp(Eta(x, a), q(x)))      
           : { [*Obj] [[*Obj]^1 *Prop] } *Prop

   \def "logical existence restricted to a classification"
      EEx = { [a:*Obj] [q:[*Obj]^1 *Prop] } Ex([x:*Obj]^2 And(Eta(x, a), q(x)))
          : { [*Obj] [[*Obj]^1 *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]^2 E(t, y)) : [*Term] *Prop

   \def "term-term equivalence"
      Eq = { [t1:*Term] [t2:*Term] } All([y:*Obj]^2 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]^2 E(t, y))
           : { [*Term] [*Obj] } *Prop

   \def "operation (rule with inhabited domain)"
      Op = [f:*Obj] Ex([x:*Obj]^2 Conv(OAt(f, x))) : [*Obj] *Prop

   \def "classification inclusion"
      ESub = { [a1:*Obj] [a2:*Obj] } EAll(a1, [x:*Obj]^2 Eta(x, a2))
           : { [*Obj] [*Obj] } *Prop

    \def "classification morphism"
      ETo = { [f:*Obj] [a:*Obj] [ b:*Obj] } EAll(a, [x:*Obj]^2 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