- •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?
118 |
CHAPTER 14. STRONG NORMALISATION FOR F |
14.3Reducibility theorem
A term t of type T is said reducible if it is in REDT [SN=X], where X1; : : : ; Xm are the free type variables of T , and SNi is the set of strongly normalisable terms of type Xi.
As in chapter 6 we can prove the
Theorem All terms of F are reducible.
and hence, by (CR 1), the
Corollary All terms of F are strongly normalisable.
We need a more general result, which uses substitution twice (once for types, and again for terms) and from which the theorem follows by putting Ri = SNi and uj = xj:
Proposition |
Let t be |
a term of type T . |
Suppose all the free variables of |
t are among |
x1; : : : ; xn |
of types U1; : : : ; Un, |
and all the free type variables of |
T; U1; : : : ; Un are among X1; : : : ; Xm. If R1; : : : ; Rm are reducibility candidates of types V1; : : : ; Vm and u1; : : : ; un are terms of types U1[V =X]; : : : ; Un[V =X] which are in REDU1 [R=X]; : : : ; REDUn [R=X] then t[V =X][u=x] 2 REDT [R=X].
The proof is similar to 6.3.3. The new cases are handled using 14.2.2 and 14.2.3.