- •Sense, Denotation and Semantics
- •Sense and denotation in logic
- •The algebraic tradition
- •The syntactic tradition
- •The two semantic traditions
- •Tarski
- •Heyting
- •Natural Deduction
- •The calculus
- •The rules
- •Interpretation of the rules
- •The Curry-Howard Isomorphism
- •Lambda Calculus
- •Types
- •Terms
- •Operational significance
- •Conversion
- •Description of the isomorphism
- •Relevance of the isomorphism
- •The Normalisation Theorem
- •The weak normalisation theorem
- •Proof of the weak normalisation theorem
- •Degree and substitution
- •Degree and conversion
- •Conversion of maximal degree
- •Proof of the theorem
- •The strong normalisation theorem
- •Sequent Calculus
- •The calculus
- •Sequents
- •Structural rules
- •The intuitionistic case
- •Logical rules
- •Some properties of the system without cut
- •The last rule
- •Subformula property
- •Asymmetrical interpretation
- •Sequent Calculus and Natural Deduction
- •Properties of the translation
- •Strong Normalisation Theorem
- •Reducibility
- •Properties of reducibility
- •Atomic types
- •Product type
- •Arrow type
- •Reducibility theorem
- •Pairing
- •Abstraction
- •The theorem
- •The calculus
- •Types
- •Terms
- •Intended meaning
- •Conversions
- •Normalisation theorem
- •Expressive power: examples
- •Booleans
- •Integers
- •Expressive power: results
- •Canonical forms
- •Representable functions
- •Coherence Spaces
- •General ideas
- •Coherence Spaces
- •The web of a coherence space
- •Interpretation
- •Stable functions
- •Parallel Or
- •Direct product of two coherence spaces
- •The Function-Space
- •The trace of a stable function
- •Representation of the function space
- •The Berry order
- •Partial functions
- •Denotational Semantics of T
- •Simple typed calculus
- •Types
- •Terms
- •Properties of the interpretation
- •Booleans
- •Integers
- •Sums in Natural Deduction
- •Defects of the system
- •Standard conversions
- •The need for extra conversions
- •Subformula Property
- •Extension to the full fragment
- •Commuting conversions
- •Properties of conversion
- •The associated functional calculus
- •Empty type
- •Sum type
- •Additional conversions
- •System F
- •The calculus
- •Comments
- •Representation of simple types
- •Booleans
- •Product of types
- •Empty type
- •Sum type
- •Existential type
- •Representation of a free structure
- •Free structure
- •Representation of the constructors
- •Induction
- •Representation of inductive types
- •Integers
- •Lists
- •Binary trees
- •Trees of branching type U
- •The Curry-Howard Isomorphism
- •Coherence Semantics of the Sum
- •Direct sum
- •Lifted sum
- •dI-domains
- •Linearity
- •Characterisation in terms of preservation
- •Linear implication
- •Linearisation
- •Linearised sum
- •Tensor product and units
- •Cut Elimination (Hauptsatz)
- •The key cases
- •The principal lemma
- •The Hauptsatz
- •Resolution
- •Strong Normalisation for F
- •Idea of the proof
- •Reducibility candidates
- •Remarks
- •Reducibility with parameters
- •Substitution
- •Universal abstraction
- •Universal application
- •Reducibility theorem
- •Representation Theorem
- •Representable functions
- •Numerals
- •Total recursive functions
- •Provably total functions
- •Proofs into programs
- •Formulation of HA2
- •Translation of HA2 into F
- •Representation of provably total functions
- •Semantics of System F
- •What is Linear Logic?
48 |
• |
CHAPTER 7. GODEL'S SYSTEM T |
7.2Normalisation theorem
In T, all the reduction sequences are nite and lead to the same normal form.
Proof Part of the result is the extension of Church-Rosser; it is not di cult to extend the proof for the simple system to this more complex case. The other part is a strong normalisation result, for which reducibility is well adapted (it was for T that Tait invented the notion).
First, the notion of neutrality is extended: a term is called neutral if it is not of the form hu; vi, x: v, O, S t, T or F. Then, without changing anything, we show successively:
1. O, T and F are reducible | they are normal terms of atomic type.
2. If t of type Int is reducible (i.e. strongly normalisable), then S t is reducible
| that comes from (S t) = (t).
3.If u, v, t are reducible, then D u v t is reducible | u, v, t are strongly normalisable by (CR 1), and so one can reason by induction on the number(u) + (v) + (t). The neutral term D u v t converts to one of the following terms:
|
D u0 v0 t0 |
with u, v, t reduced respectively to u0, v0, |
t0. |
In |
this case, |
|
|||||
|
we have |
(u0) + (v0) + (t0) < (u) + (v) + (t), |
and |
by |
induction |
|
hypothesis, the term is reducible. |
|
|
|
|
u or v if t is T or F; these two terms are reducible. |
|
|
|
We conclude by (CR 3) that D u v t is reducible.
4. If u, v, t are reducible, then R u v t is reducible | here also we reason by induction, but on (u) + (v) + (t) + `(t), where `(t) is the number of symbols of the normal form of t. In one step, R u v t converts to:
R u0 v0 t0 with etc. | reducible by induction.
u (if t = O) | reducible.
v (R u v w) w, where S w = t; since (w) = (t) and `(w) < `(t), the
induction hypothesis tells us that R u v w is reducible. |
As v and w are, |
v (R u v w) w is reducible by the de nition for U!V . |
|
The use of the induction hypothesis in the nal case is really essential: it is the only occasion, in all the uses so far made of reducibility, where we truly use an induction on reducibility. For the other cases, the cognoscenti will see that we really have no need for induction on a complex predicate, by reformulating (CR 3) in an appropriate way.