3 \* Feferman's system T0 *\
5 \open elements \* [1] 2.1. 2.2. 2.4. *\
7 \decl "rule application" App: { [*Obj] [*Obj] [*Obj] } *Prop
9 \decl "classification predicate" Cl: [*Obj] *Prop
11 \decl "classification membership" Eta: { [*Obj] [*Obj] } *Prop
13 \* we must make an explicit coercion from *Obj to *Term *\
14 \decl "object-to-term-coercion" T: [*Obj] *Term
16 \decl "term application" At: { [*Term] [*Term] } *Term
18 \decl "term-object equivalence" E: { [*Term] [*Obj] } *Prop
22 \open logical_abbreviations \* [1] 2.3. 2.5. *\
24 \def "logical comprehension restricted to classifications"
25 CAll = [q:[*Obj]^1 *Prop] All([x:*Obj]^2 Imp(Cl(x), q(x)))
26 : [[*Obj]^1 *Prop] *Prop
28 \def "logical existence restricted to classifications"
29 CEx = [q:[*Obj]^1 *Prop] Ex([x:*Obj]^2 And(Cl(x), q(x)))
30 : [[*Obj]^1 *Prop] *Prop
32 \def "logical comprehension restricted to a classification"
33 EAll = { [a:*Obj] [q:[*Obj]^1 *Prop] } All([x:*Obj]^2 Imp(Eta(x, a), q(x)))
34 : { [*Obj] [[*Obj]^1 *Prop] } *Prop
36 \def "logical existence restricted to a classification"
37 EEx = { [a:*Obj] [q:[*Obj]^1 *Prop] } Ex([x:*Obj]^2 And(Eta(x, a), q(x)))
38 : { [*Obj] [[*Obj]^1 *Prop] } *Prop
42 \open non_logical_abbreviations \* [1] 2.4. 2.7 *\
44 \def "object application"
45 OAt = { [f:*Obj] [x:*Obj] } At(T(f), T(x)) : { [*Obj] [*Obj] } *Term
47 \def "convergence of a term to an object"
48 Conv = [t:*Term] EX([y:*Obj]^2 E(t, y)) : [*Term] *Prop
50 \def "term-term equivalence"
51 Eq = { [t1:*Term] [t2:*Term] } All([y:*Obj]^2 Iff(E(t1, y), E(t2, y)))
52 : { [*Term] [*Term] } *Prop
54 \def "classification membership of a term"
55 TEta = { [t:*Term] [a:*Obj] } EEx(a, [y:*Obj]^2 E(t, y))
56 : { [*Term] [*Obj] } *Prop
58 \def "operation (rule with inhabited domain)"
59 Op = [f:*Obj] Ex([x:*Obj]^2 Conv(OAt(f, x))) : [*Obj] *Prop
61 \def "classification inclusion"
62 ESub = { [a1:*Obj] [a2:*Obj] } EAll(a1, [x:*Obj]^2 Eta(x, a2))
63 : { [*Obj] [*Obj] } *Prop
65 \def "classification morphism"
66 ETo = { [f:*Obj] [a:*Obj] [ b:*Obj] } EAll(a, [x:*Obj]^2 TEta(OAt(f, x), b))
67 : { [*Obj] [*Obj] [*Obj] } *Prop
71 \open non_logical_axioms \* [1] 2.4. 3.2 *\
73 \* we axiomatize E because *Term is not inductively generated *\
74 \ax e_refl: [y:*Obj] E(T(y), y)
76 \ax e_at_in: [t1:*Term][t2:*Term][f:*Obj][x:*Obj][y:*Obj]
77 [E(t1, f)] [E(t2, x)] [App(f, x, y)] E(At(t1, t2), y)
79 \ax e_at_out: [f:*Obj][x:*Obj][y:*Obj] [E(At(T(f), T(x)), y)] App(f, x, y)
81 \ax "I (i)" id_dec: [x:*Obj][y:*Obj] Or(Id(x, y), NId(x, y))
83 \ax "I (ii)" at_mono: [f:*Obj][x:*Obj][y1:*Obj][y2:*Obj]
84 [E(OAt(f, x), y1)] [E(OAt(f, x), y2)] Id(y1, y2)
86 \ax "I (iii)" eta_cl: [x:*Obj][a:*Obj] [Eta(x, a)] Cl(a)