Galois

More documents

Recommendations

Info

Galois connections Theoretical background Galois connections Definition (Galois connection) Given two preordered sets (A, ⊑ A ) and (B, ⊑ B ) and two functions B f A and g A B , the pair (f , g) is a Galois connection if and only if, for all a ∈ A and b ∈ B: f a ⊑ B b ⇔ a ⊑ A g b Graphical notation ⊑ A A f g B ⊑ B (f ,g) or (A, ⊑ A ) (B, ⊑ B ) Paulo Silva (UMinho) Galois PLMMS’09 12 / 28
Properties Theoretical background Galois connections Property f a ⊑ B b ⇔ a ⊑ A g b a ⊑ A a ′ ⇒ f a ⊑ B f a ′ b ⊑ B b ′ ⇒ g b ⊑ A g b ′ a ⊑ A g (f a) f (g b) ⊑ B b f (g (f a)) = f a g (f (g b)) = g b g (b ⊓ B b ′ ) = g b ⊓ A g b ′ f (a ⊔ A a ′ ) = f a ⊔ B f a ′ g ⊤ B = ⊤ A f ⊥ A = ⊥ B Description “Shunting rule” Monotonicity (LA) Monotonicity (UA) Lower cancellation Upper cancellation Semi-inverse Semi-inverse Distributivity (UA over meet) Distributivity (LA over join) Top-preservation (UA) Bottom-preservation (LA) Paulo Silva (UMinho) Galois PLMMS’09 13 / 28
Page 1 and 2: Galois: A Language for Proofs Using
Page 3 and 4: Outline Introduction 1 Introduction
Page 5 and 6: Introduction Whole division specifi
Page 7 and 8: Introduction Whole division specifi
Page 9 and 10: Introduction Motivation Proof. n (
Page 11 and 12: Introduction Motivation Proof. n (
Page 13 and 14: Introduction Objectives Objectives
Page 15 and 16: Theoretical background Indirect equ
Page 21: Theoretical background Indirect equ
Page 25 and 26: Theoretical background Galois conne
Page 27 and 28: Theoretical background Fork algebra
Page 29 and 30: Theoretical background Fork algebra
Page 31 and 32: Theoretical background Point-free t
Page 33 and 34: Outline Galois and Galculator 1 Int
Page 35 and 36: Galois and Galculator Architecture
Page 37 and 38: Summary Summary Summary Fork algebr
Page 39 and 40: Summary Future work Future work Mec

Galois

Create successful ePaper yourself

Delete template?

Save as template?