Category-Partial Orders Prologue and Category- partial Proof - - PowerPoint PPT Presentation

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Category-Partial Orders Prologue and Category- partial Proof - - PowerPoint PPT Presentation

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C-POs and Proof Principles Ralf Hinze Category-Partial Orders Prologue and Category- partial Proof Principles orders Split c-pos Monic c-pos Ralf Hinze Initial objects Computing Laboratory, University of Oxford Proof Wolfson

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C-POs and Proof Principles Ralf Hinze Prologue Category- partial

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Category-Partial Orders and Proof Principles

Ralf Hinze

Computing Laboratory, University of Oxford Wolfson Building, Parks Road, Oxford, OX1 3QD, England ralf.hinze@comlab.ox.ac.uk http://www.comlab.ox.ac.uk/ralf.hinze/

June 2008

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data List = [ ] | Nat : List append :: (List, List) → List append ([ ], bs) = bs append (a : as, bs) = a : append (as, bs)

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∀x : List . P(x) P(x) ⇐ ⇒ append (x, [ ]) = x

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Case P([ ]): append ([ ], [ ]) = { definition of append } [ ] Case P(a : as): append (a : as, [ ]) = { definition of append } a : append (as, [ ]) = { ex hypothesi } a : as

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append :: (List, List) → List append (as, bs) = foldr (:) bs as

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append (as, [ ]) = { definition of append } foldr (:) [ ] as = { reflection: foldr (:) [ ] = id } as

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Cat + ⊆ ?

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Ordering objects

A category-partial order is a pair C, ⊑ where

◮ C is a category and ◮ ⊑ is a subcategory of C that is a partial order on the

  • bjects of C.

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Ordering morphisms

Let f : A → B and g : C → D, then f ⊑ g iff A ⊑ C, B ⊑ D and the following diagram commutes. A f ≻ B C g ≻ D ⇐ ⇒ ⊑B,D · f = g · ⊑A,C

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Properties

◮ ⊑ on morphisms is a partial order. ◮ Let f, g : A → B, then

f ⊑ g ⇐ ⇒ f = g

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Examples

◮ Set, ⊆: f ⊑ g iff f is the restriction of g to A. ◮ Functor categories: DC is a c-po if D is one.

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In Set: B A ⊆ ≻ f ≻ C ⊆ ≻ ⇐ ⇒ f = ⊆

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Split transitivity

Let f : A → B, then ⊑B,C · f = ⊑A,C ⇐ ⇒ f = ⊑A,B A c-po that satisfies this property is called a split c-po.

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An equivalent formulation

Let f : A → B, then f ⊑ idC = ⇒ f = ⊑A,B A f ≻ B C idC ≻ C

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In Set the inclusion morphisms are monos. ⊆ · f1 = ⊆ · f2 ⇐ ⇒ f1 = f2

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Monic c-pos

A c-po is called a monic c-po if the arrows ⊑A,B are monos: Let f1, f2 : A → B in C, then ⊑B,C · f1 = ⊑B,C · f2 ⇐ ⇒ f1 = f2

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Properties

In a monic c-po the lower arrow uniquely determines the upper arrow. A f ≻ B C g ≻ D

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Contracts

Let f : A → B, then f ∈ R → S :⇐ ⇒ ∃g : R → S . g ⊑ f Think of R → S as a contract with precondition R and postcondition S. But note that the postcondition can’t be weaker than B. R ≻ S A f ≻ B

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An aside: The category of contracts

◮ Identity:

R ⊑ A ⇐ ⇒ idA ∈ R → R

◮ Composition: Let f : A → B and g : B → C, then

f ∈ R → S ∧ g ∈ S → T = ⇒ g · f ∈ R → T R ≻ R A idA ≻ A R ≻ S ≻ T A f ≻ B g ≻ C

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Initial objects

¡A ≻ A

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Universal property

¡A = h ⇐ ⇒ h : 0 → A

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Reflection

Set A = 0 and h = id0. We obtain ¡0 = id0

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Fusion

Let f : B → A and set h = f · ¡B : 0 → A. We obtain ¡A = f · ¡B ⇐ = f : B → A B ¡B

  • f

≻ A ¡

A

≻

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The mother of all proof principles

f : A → R g : B → R f · ¡A = g · ¡B A ≺ ¡

A

B ¡

B

≻ R ¡R

  • ≺

g f ≻

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A special case: Set B = R and g = idR. We obtain f : A → R f · ¡A = ¡R So, the proof principle implies fusion.

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A more special case: Set f = ⊑A,R. We obtain A ⊑ R ¡A ⊑ ¡R A ⊑ R ¡R ∈ 0 → A ¡A ≻ A

  • ¡R

≻ R

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An even more special case: Set R = 0. We obtain A ⊑ 0 ¡A ⊑ ¡0 A ⊑ 0 ¡0 ∈ 0 → A Recall that ¡0 = id0. So, in a split c-po this implies: A ⊑ 0 A = 0

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An even more special case: Set R = 0. We obtain A ⊑ 0 ¡A ⊑ ¡0 A ⊑ 0 ¡0 ∈ 0 → A Recall that ¡0 = id0. So, in a split c-po this implies: A ⊑ 0 A = 0

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Cospans

Given: objects A and B. A cospan is an object C with two morphisms f : A → C and g : A → C. A f ≻ C ≺ g B

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The category of cospans

Cospans are the objects of the category Cospan(A, B). Morphisms in Cospan(A, B) are morphisms in the underlying category that make the following diagram commute. A f ≻ C ≺ g B A idA

  • f′

≻ C′ h

  • ≺

g′ B idB

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Coproducts

The initial object in Cospan(A, B) is the coproduct of A and B. Notation: A inl≻ A + B ≺inr B A idA

  • f

≻ C f ▽ g

  • ≺

g B idB

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Ordering cospans

C, f, g ⊑ C′, f′, g′ :⇐ ⇒ C ⊑ C′ ∧ f ⊑ f′ ∧ g ⊑ g′

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Proof principle: case analysis

R ⊑ C ¡C ∈ 0 → R R ⊑ C f ∈ A → R g ∈ B → R f ▽ g ∈ A + B → R

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A special case: C = 0. R ⊑ 0 R = 0 R ⊑ A + B inl ∈ A → R inr ∈ B → R R = A + B

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F-algebras

Given: a functor F : C → C. An F-algebra is an object T with a morphism f : FT → T. FT f ≻ T

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The category of F-algebras

F-algebras are the objects of the category Alg(F). Morphisms in Alg(F) are morphisms in the underlying category that make the following diagram commute. FT f ≻ T FT ′ Fh

  • f′

≻ T ′ h

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Initial algebras

The initial object in Alg(F) is the least fixed point of F. Notation: F(µT) in ≻ µT FT Ff

  • f

≻ T f

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Ordering F-algebras

T, f ⊑ T ′, f′ :⇐ ⇒ T ⊑ T ′ ∧ f ⊑ f′

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Proof principle: induction

R ⊑ C ¡C ∈ 0 → R R ⊑ C f ∈ FR → R f ∈ µF → R

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A special case: C = 0. R ⊑ 0 R = 0 R ⊑ µF in ∈ FR → R R = µF

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data Base A = 1 + Nat × A type List = µBase P = { x : List | append (x, [ ]) = x } nil ∈ 1 → P cons ∈ Nat × P → P nil ▽ cons ∈ Base P → P in ∈ Base P → P P = List

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Correctness of insertion sort

Ord = { x : List | ordered x } nil ∈ 1 → Ord insert ∈ Nat × Ord → Ord nil ▽ insert ∈ Base Ord → Ord nil ▽ insert ∈ List → Ord

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Epilogue

◮ Simple and general framework for studying proof

principles.

◮ Nicely links proof principles to contracts. ◮ The development dualises: proof principles for terminal

  • bjects (products, final coalgebras).

◮ But, the requirements also dualise: epic inclusion,

cosplit transitivity which doesn’t hold in Set. {0} ∅ ⊆ ≻ ⊆ ≻ {1} {

  • →

1 } ≻

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