SC/MATH 1090 5- Equational Proof Ref: G. Tourlakis, Mathematical Logic , John Wiley & Sons, 2008. York University Department of Computer Science and Engineering 1 York University- MATH 1090 05-Equational
Overview • Equational Proof • Some examples • Using assumptions in equational proofs York University- MATH 1090 05-Equational 2
Equational Proof • An equational-style proof is a proof of the form: A 2 (1) A 1 <annotation> A 3 (2) A 2 <annotation> ... (n-1) A n-1 A n <annotation> A n+1 (n) A n <annotation> A 1 A 2 , A 2 A 3 , ..., A n A n+1 ⊢ A 1 A n+1 • Metatheorem . Corollary . In an equational proof from assumptions , we • have ⊢ A 1 A n+1 . Corollary . In an equational proof from assumptions , we • have ⊢ A 1 iff ⊢ A n+1 . York University- MATH 1090 05-Equational 3
Equational Proof Layout • An equational-style proof is a proof of the form: We write: Instead of A 2 A 1 (1) A 1 <annotation> <annotation> A 3 (2) A 2 <annotation> A 2 ... (n-1) A n-1 A n <annotation> <annotation> A n+1 ... (n) A n <annotation> A n <annotation> A n+1 York University- MATH 1090 05-Equational 4
Equational Proof- framework • To Prove ⊢ A B : Template 1 Template 2 Template 3 We write: Or, we write: Or, we write: an axiom or a A B A proven theorem <annotation> <annotation> <annotation> ... ... ... <annotation> <annotation> <annotation> A B B axiom or proven theorem York University- MATH 1090 05-Equational 5
Equational Proof- framework • To Prove ⊢ A : Template 2 Template 3 Or, we write: Or, we write: an axiom or a proven theorem A <annotation> <annotation> ... ... <annotation> <annotation> axiom or proven A theorem York University- MATH 1090 05-Equational 6
Equational Proof- framework • To Prove ⊢ A : Template 2 Template 3 Or, we write: Or, we write: an axiom or a A hypothesis or <annotation> proven theorem ... <annotation> ... <annotation> axiom or a <annotation> hypothesis or proven theorem A York University- MATH 1090 05-Equational 7
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 York University- MATH 1090 05-Equational 8
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 York University- MATH 1090 05-Equational 9
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 York University- MATH 1090 05-Equational 10
De Morgan theorems • De Morgan 1 ⊢ A B ( A B) or ⊢ (A B) A B • De Morgan 2 ⊢ A B ( A B) or ⊢ (A B) A B York University- MATH 1090 05-Equational 11
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) York University- MATH 1090 05-Equational 12
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) York University- MATH 1090 05-Equational 13
Using Hypotheses (special axioms) in Equational Proofs • A ⊢ A ┬ • Therefore using Leibniz, one can replace occurrences of hypothesis A by ┬ • Conversely, any occurrence of ┬ can be replaced by A. York University- MATH 1090 05-Equational 14
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. York University- MATH 1090 05-Equational 15
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 York University- MATH 1090 05-Equational 16
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