9- Generalization and Leibniz Rules Ref: G. Tourlakis, Mathematical - - PowerPoint PPT Presentation

SC/MATH 1090

9- Generalization and Leibniz Rules

Ref: G. Tourlakis, Mathematical Logic, John Wiley & Sons, 2008.

York University

Department of Computer Science and Engineering

York University- MATH 1090

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09-Gen_Leib

Overview

• Insert/Remove the universal quantifier

– Weak generalization Metatheorem – Specialization rule

• The Leibniz Rules

– Boolean Leib (BL) – Weak Leib (unconditional) – Strong Leib (conditional)

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Weak Generalization

• Metatheorem. (Weak Generalization) If ⊢A and

moreover x does not occur free in any formula in the set , then  ⊢(x) A.

– Note x can be free in A. – Proof by induction on length of a -proof of A in Logic(2).

• Corollary. If ⊢A and moreover x does not occur free in

any formula used in the proof, then  ⊢(x) A.

• Corollary. If ⊢A, then ⊢(x)A.

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NEVER Strong Generalization!

• NEVER Correct:

Strong generalization: A ⊢(x)A

• To be able to generalize A, we must have a proof for A.
• If A is an assumption, and not proven, we are not allowed

to generalize.

• Strong generalization is SO WRONG that to show a

formula X is not provable, it is sufficient to show that X can be used to prove strong generalization (Chapter 8).

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Specialization Rule

• Metatheorem. (Specialization Rule) (x)A ⊢A[x:=t]

– Note: The above rule is applicable ONLY if the substitution is defined

• Corollary. (x)A ⊢A

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Remove/ Insert (x)

• A simple template for some Hilbert proofs in 1st order

Logic:

• 1. Use spec to remove the universal quantifiers in the

assumptions.

• 2. Use simple Boolean Logic techniques we learned in

previous chapters.

• 3. Use (weak) gen to insert back the universal quantifiers

provided gen CONDITION.

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Examples

• (x)(y)A  (y)(x)A
• (x)(A  B)  (x)A  (x)B

Distributivity of  over 

• Metatheorem. (-monotonicity) Provided x dnof in any

formula in , if  ⊢A  B then ⊢(x)A  (x)B.

– Corollary. If ⊢A  B then ⊢(x)A  (x)B. – Corollary. If  ⊢A  B then ⊢(x)A  (x)B, provided x dnof in any formula in . – Corollary. If ⊢A  B then ⊢(x)A  (x)B.

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Weak Leibniz (WL)

• Metatheorem. (Weak Leibniz)

If ⊢A  B, then ⊢C[p\A]  C[p\B].

– Proof by induction on the complexity of C. – Also called "Weak Leib with unconditional substitution“ – Note we DONOT have Strong Leib with unconditional substitution:

A  B ⊢C[p\A]  C[p\B] (WRONG!)

• Corollary. (A more generous WL)

If  ⊢A  B and none of the bound variables of C occur free in formulae of , then  ⊢C[p\A]  C[p\B].

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Strong Leibniz (SL)

• Metatheorem. (Strong Leibniz) A  B ⊢C[p:=A]  C[p:=B]

– Proof by induction on the complexity of C. – Also called “Strong Leib with conditional substitution“

• Example: D  (A  B) ⊢ D(C[p:=A]  C[p:=B])
• Exercise: D ◦ (A  B) ⊢ D ◦ (C[p:=A]  C[p:=B])

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Important Tools

• ⊢(x)(A  B)  A (x)B,

provided x dnof in A.

• ⊢(x)(A  B)  A  (x)B,

provided x dnof in A.

• ⊢ (x)(A  B)  A  (x)B,

provided x dnof in A.

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Examples

1) (x)(  A)  ┬ Empty range 2) ⊢ (x) (x=t A)  A[x:=t] provided x not free in t One point rule 3) ⊢ (x) (x=t  A)  A[x:=t] provided x not free in t One point rule-  -version

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Examples (2)

4) ⊢(x)(AB) (x)(AC)  (x)(A  BC)

• Or

⊢(x)AB  (x)AC  (x)A(BC)

• - version (dual of above):

5) ⊢(x)(AB)  (x)(AC)  (x)(A  (BC))

• Or

⊢(x)AB  (x)AC  (x)A(BC)

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Examples (3)

6) ⊢ (x)A  BC  (x)AC  (x)BC Range Split 7) ⊢(x)A(y)BC  (y)B(x)AC

provided y not free in A and x not free in B

Interchange of dummies 8) ⊢(x)A(y)BC  (y)B(x)AC

provided y not free in A and x not free in B

Dual of above 9) (x) (y)A  BC  (x)A (y)BC provided y not free in A Nesting 10) (x) (y)A  BC  (x)A (y)BC provided y not free in A Dual of above

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Dummy Renaming

• Theorem. (Dummy Renaming for ) If z does not occur

in A (free nor bound), then ⊢(x)A  (z)A[x:=z].

• Theorem. (Dummy Renaming for ) If z does not occur in

A (free nor bound), then ⊢(x)A  (z)A[x:=z].

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