- •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?
76 |
CHAPTER 10. SUMS IN NATURAL DEDUCTION |
10.3.2Extension to the full fragment
For the full calculus, we come against an enormous di culty: it is no longer true that the conclusion of an elimination is a subformula of its principal premise: the \C" of the three elimination rules has nothing to do with the eliminated formula. So we are led to restricting the notion of principal branch to those eliminations which are well-behaved (^1E, ^2E, )E and 8E) and we can try to extend our theorem. Of course it will be necessary to restrict part (ii) to the case where ends in a \good" elimination.
The theorem is proved as before in the case of introductions, but the case of eliminations is more complex:
If ends in a good elimination, look at its principal premise A: we shall be embarrassed in the case where A is the conclusion of a bad elimination. Otherwise we conclude the existence of a principal branch.
If ends in a bad elimination, look again at its principal premise A: it is not the conclusion of an introduction. If A is a hypothesis or the conclusion of a good elimination, it is a subformula of a hypothesis, and the result follows easily. There still remains the case where A comes from a bad elimination.
To sum up, it would be necessary to get rid of con gurations formed from a succession of two rules: a bad elimination of which the conclusion C is the principal premise of an elimination, good or bad. Once we have done this, we can recover the Subformula Property. A quick calculation shows that the number of con gurations is 3 7 = 21 and there is no question of considering them one by one. In any case, the removal of these con gurations is certainly necessary, as the
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which does not satisfy the Subformula Property.
10.4Commuting conversions
In what follows, C ... r denotes an elimination of the principal premise C, the
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10.4. COMMUTING CONVERSIONS |
77 |
1. commutation of ?E:
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3. commutation of 9E:
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Example The most complicated situation is:
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CHAPTER 10. |
SUMS IN NATURAL DEDUCTION |
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We see in particular that the general case (with an unspeci ed elimination r) is more intelligible than its 21 specialisations.
10.5Properties of conversion
First of all, the normal form, if it exists, is unique: that follows again from a Church-Rosser property. The result remains true in this case, since the con icts of the kind
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are easily resolved, because the second deduction converts to the rst.
It is possible to extend the results obtained for the (^; ); 8) fragment to the full calculus, at the price of boring complications. [Prawitz] gives all the technical details for doing this. The abstract properties of reducibility for this case are in [Gir72], and there are no real problems when we extend this to existential quanti cation over types.