number-theory

184T., ticket entailment logic 106TAc 19-20TA2 15-16axioms and rules for 15-16basic facts about 12-29comparison with HS86 version 18TA2--deducible formula 16-deduction 16detailed definition 148-9length of 149translated to a typed term 66-formula 15-typable term 27TA2+p, TA2+pn 52TA2+p[o] (for both TA2+p, TA2+pn) 52TA2+p-typable, TA2}pn-typable 52tail of a type, Tail( ) 115, 155subtail 155tail position 153term (means A-term) 1abstract 1application Iclosed 4composite 1construction-tree of 141-2subterm 2typable (see typable term)typed (see typed term)term models 59term-variable 1terminates, said of a reduction 5theorem (= provable formula)in a Natural Deduction logic 79in an axiom-based logic 89ticket entailment logic (T.) 106TMM#, TMMpn (term models) 59TMpln] (for both TMp, TMpn) 59TNS(I-) 119-20trivial component of a substitution 35TT(r) 64typable basis 159typable termin TA2 27in TA2+p, TA2+p,r 52type-assignment (expression) 14type-assignment theory (see TA2)Type-checking algorithm (see PT algo.)type-context 14for nf-schemes 119minimum, Con() 65type-erasure 65type-invariance, strong 158type-substitutions (see subs. into types)type-theories in general xChurch-style and Curry-style 13-14type-variables 12interpretation in a A-model 58typednf-scheme 120substitution 68term 63-73, esp. 64encoding a TA2-deduction 66variable 64a-conversion 69Indexf3-, q-redex, reduction 69/3-, q-, $n-normal form 69structure of /3-nf 110types 12atomic 12composite 12construction-tree of 152condensed 153-4inhabitants, tables of 50, 177-8principal (also see PT) 32with intersection 18Types( ) 26Typesno eq( ) 56Typesp( ), Typespn( ) 55unextendable argument-branch 130unifiable 40unification 40most general (= m.g.u.) 41Unification algorithm 42-3historical note on 43use in PT algorithm 34, 45-6Unification theorem 42unifier 40most general (= m.g.u.) 41union of substitutions (U) 35uniqueness of TA2-deductionsholds for nf's and Al-terms 22vacuous discharging 16, 76, 78valid in a A-model 58variablebound 2,68changing (see a-conversion)free 2, 68new 119propositional 75term- 1type- 12interpreting in a A-model 58typed 64variable-domain of a substitution 35variable-range of a substitution 35Vars( ) 13, 31, 149W (combinator) 4principal type of 18deduction for 161(W) (axiom) 81, 90weak normalization (see WN)Weakening lemma 18WN (= weak normalization) 27-8WN theoremfor TA2-typable terms 28for TA2+p and TA2+pn 55for typed terms 71historical note 72Y (fixed-point combinator) 4untypability of 28a-conversion 3of typed terms 69a-invariance lemma for TA2 23