5- Equational Proof Ref: G. Tourlakis, Mathematical Logic , John - - PowerPoint PPT Presentation

SC/MATH 1090

5- Equational Proof

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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05-Equational

Overview

• Equational Proof
• Some examples
• Using assumptions in equational proofs

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Equational Proof

• An equational-style proof is a proof of the form:

(1) A1  A2 <annotation> (2) A2  A3 <annotation> ... (n-1) An-1  An <annotation> (n) An  An+1 <annotation>

• Metatheorem.

A1  A2 , A2  A3 , ..., An  An+1 ⊢ A1  An+1

• Corollary. In an equational proof from assumptions , we

have  ⊢ A1  An+1.

• Corollary. In an equational proof from assumptions , we

have  ⊢ A1 iff  ⊢ An+1.

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Equational Proof Layout

• An equational-style proof is a proof of the form:

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Instead of (1) A1  A2 <annotation> (2) A2  A3 <annotation> ... (n-1) An-1  An <annotation> (n) An  An+1 <annotation> We write: A1  <annotation> A2  <annotation> ... An  <annotation> An+1

Equational Proof- framework

• To Prove ⊢ A  B :

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Template 1 We write: A  <annotation> ...  <annotation> B Template 2 Or, we write: an axiom or a proven theorem  <annotation> ...  <annotation> A  B Template 3 Or, we write: A  B  <annotation> ...  <annotation> axiom or proven theorem

Equational Proof- framework

• To Prove ⊢ A :

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Template 2 Or, we write: an axiom or a proven theorem  <annotation> ...  <annotation> A Template 3 Or, we write: A  <annotation> ...  <annotation> axiom or proven theorem

Equational Proof- framework

• To Prove  ⊢ A :

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Template 2 Or, we write: an axiom or a hypothesis or proven theorem  <annotation> ...  <annotation> A Template 3 Or, we write: A  <annotation> ...  <annotation> axiom or a hypothesis or proven theorem

Useful tools: , ┬, and 

• Some properties of 

⊢ (A  B)   A  B ⊢ (A  B)  A  B ⊢  A  A Double Negation

• Some properties of ┬ and 

⊢ ┬    ⊢    ┬ ⊢ A  ┬ ⊢ A    A

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Useful tools: 

• Some properties of 

⊢ A  B  B  A Axiom 6: Symmetry of  ⊢ (A  B)  C  A  (B  C) Axiom 5 : Associativity of  ⊢ A  ( B  C)  (A  B)  C By above theorem, together with axiom 5 and 6, we can prove that in a chain of two ‘’s, we can put the brackets around any subchain and we can move items around (similar to ‘’s). The

general case of any number of ‘’s also holds.

⊢ (A  B)  (C  D)  A  C  B  C  A  D  B  D

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Useful tools:  and not 

• Some properties of 

⊢ A  B  A  B Corollary: ⊢ A  B  A  B  B ⊢ A  (B  C)  A  B  A  C

• Definition:
• Property of

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De Morgan theorems

• De Morgan 1

⊢ A  B  ( A   B)

• r

⊢ (A  B)   A   B

• De Morgan 2

⊢ A  B  ( A   B)

• r

⊢ (A  B)   A   B

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Useful tools: 

• ⊢ A  A  A
• ⊢ A  ┬  A
• ⊢ A    
• Distributivity of  over 

⊢ A  (B  C)  (A  B)  (A  C)

• Distributivity of  over 

⊢ A  (B  C)  (A  B)  (A  C)

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Some more theorems!

• ⊢ (A  B)  C  (A  C)  (B  C)
• ⊢ A (B  C)  (A  B)  (A  C)

Ping-Pong Theorem: ⊢ A  B  (A  B)  (B  A)

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Using Hypotheses (special axioms) in Equational Proofs

• A ⊢ A  ┬
• Therefore using Leibniz, one can replace
• ccurrences of hypothesis A by ┬
• Conversely, any occurrence of ┬ can be

replaced by A.

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Examples- important!

• A, B ⊢ A B
• A A ⊢ A
• A ⊢A  B
• A B ⊢ A
• Metatheorem (Splitting/ Merging Hypotheses)

For any formulae A, B, C and set , we have  {A,B} ⊢ C iff  {A B} ⊢ C.

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



Very important tools!

• A, AB ⊢ B

Modus Ponens

• AB, AC ⊢ BC

Cut Rule

• AB, AB ⊢ B
• AB, A ⊢ B
• A, A ⊢
• AB, BC ⊢ AC

Transitivity of 

• AC, BD ⊢ ABCD
• AC, BC ⊢ ABC

Proof by Cases

• AC, AC ⊢ C

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