1 Strong normalization - natural deduction. The inference figure shemata: ⢠Implication introduction: [A] .... B. A â B. (â I). ⢠Implication elimination: A A â B. B.
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Strong normalization - natural deduction
The inference figure shemata: • Implication introduction: [A] .. .. B (→ I) A→B • Implication elimination: A
A → B (→ E) B
Symbols written in square brackets have the following meaning: An arbitrary number (possibly zero) of formulae of this form, all formally identical, may be adjoined to the inference figure as assumption formulae. They must then be initial formula of the derivation and occur, moreover, in those paths of the proof to which the particular upper formula of the inference figure belongs. (I.e., that upper formula above which the square bracket occur in the scheme. This formula may itself be an assumption formula.) A formula occurrence in a deduction that is the consequence of an application of introduction rule and major premiss of an application of elimination rule is said to be a maximum formula. Maximum formula can be eliminated from derivation on the following way. The right derivation is a reduction of the derivation to the left. [A] .. .. B A→B B
D A
D A .. .. B
D can be empty.
But [A] denotes set of assumptions. So, it can happen following situations: A
A . .. . A A .. . D B A→B A B
D A
D D A . .. . A .. . B
D A
Derivation D can contain several maximum formulas which can produce new maximum formulas, in the same way. And the number of maximum formulas are increasing. How to show that this procedure will eliminate all maximum formulas in finite number of steps? Uniqueness of normal form follow from: If P reduces to M and P reduces to N then there exists T such that M reduces to T and N reduces to T .
