Galois

Galois: 

A Language for Proofs Using Galois connections and Fork Algebras 

Paulo Silva 1 Joost Visser 2 José Oliveira 1 

1 CCTC 

University of Minho 

Braga, Portugal 

2 Software Improvement Group 

The Netherlands 

PLMMS’09 

August 21, 2009 

Munich, Germany 

Paulo Silva (UMinho) Galois PLMMS’09 1 / 28

Outline 

Outline 

1 Introduction 

Motivation 

Objectives 

2 Theoretical background 

Indirect equality 

Galois connections 

Fork algebras 

Point-free transform 

3 Galois and Galculator 

Galois 

Galculator 

4 Summary 

Summary 

Future work 


Outline 

Introduction 


Motivation 

Objectives 




Fork algebras 




Galculator 

4 Summary 

Summary 

Future work 


Introduction 

Motivation 

Whole division 

Prove 

(a ÷ b) ÷ c = a ÷ (c × b) 

for b and c ≠ 0. 

Easy if ÷ is the real number division 

Also valid in natural numbers but the proof is not so straightforward 


Introduction 

Whole division specification 

Motivation 

Implicit definition 

Explicit definition 

c = x ÷ y ⇔ 〈∃ r : 0 r < y : x = c × y + r〉 

x ÷ y = 〈 ∨ z :: z × y x〉 

Galois connection 

z × y x ⇔ z x ÷ y (y > 0) 


Introduction 


Motivation 



c = x ÷ y ⇔ 〈∃ r : 0 r < y : x = c × y + r〉 

x ÷ y = 〈 ∨ z :: z × y x〉 


z × y x ⇔ z x ÷ y (y > 0) 


Introduction 


Motivation 



c = x ÷ y ⇔ 〈∃ r : 0 r < y : x = c × y + r〉 

x ÷ y = 〈 ∨ z :: z × y x〉 


z × y x ⇔ z x ÷ y (y > 0) 


Introduction 

Motivation 

Proof. 

n (a ÷ b) ÷ c 

⇔ { z × y x ⇔ z x ÷ y } 

n × c a ÷ b 

⇔ { z × y x ⇔ z x ÷ y } 

(n × c) × b a 

⇔ { multiplication is associative } 

n × (c × b) a 

⇔ { z × y x ⇔ z x ÷ y } 

n a ÷ (c × b) 


Introduction 

Motivation 

Proof. 

n (a ÷ b) ÷ c 

⇔ { z × y x ⇔ z x ÷ y } 

n × c a ÷ b 

⇔ { z × y x ⇔ z x ÷ y } 

(n × c) × b a 


n × (c × b) a 

⇔ { z × y x ⇔ z x ÷ y } 

n a ÷ (c × b) 


Introduction 

Motivation 

Proof. 

n (a ÷ b) ÷ c 

⇔ { z × y x ⇔ z x ÷ y } 

n × c a ÷ b 

⇔ { z × y x ⇔ z x ÷ y } 

(n × c) × b a 


n × (c × b) a 

⇔ { z × y x ⇔ z x ÷ y } 

n a ÷ (c × b) 


Introduction 

Motivation 

Proof. 

n (a ÷ b) ÷ c 

⇔ { z × y x ⇔ z x ÷ y } 

n × c a ÷ b 

⇔ { z × y x ⇔ z x ÷ y } 

(n × c) × b a 


n × (c × b) a 

⇔ { z × y x ⇔ z x ÷ y } 

n a ÷ (c × b) 


Introduction 

Motivation 

Proof. 

n (a ÷ b) ÷ c 

⇔ { z × y x ⇔ z x ÷ y } 

n × c a ÷ b 

⇔ { z × y x ⇔ z x ÷ y } 

(n × c) × b a 


n × (c × b) a 

⇔ { z × y x ⇔ z x ÷ y } 

n a ÷ (c × b) 


Introduction 

Objectives 

Objectives 

Galculator = Galois connection + calculator 


Build a proof assistant based on Galois connections, their algebra 

and associated tactics 

Language for mathematical reasoning 

Equivalent to first-order logic 

Typed language 

Front-end for the Galculator 


Outline 

Theoretical background 


Motivation 

Objectives 




Fork algebras 




Galculator 

4 Summary 

Summary 

Future work 




Indirect inequality 

Definition (Indirect inequality) 

a ⊑ b ⇔ 〈∀ x :: x ⊑ a ⇒ x ⊑ b〉 

a ⊑ b ⇔ 〈∀ x :: b ⊑ x ⇒ a ⊑ x〉 




Proof. 

a = b 

⇔ { Anti-symmetry } 

a ⊑ b ∧ b ⊑ a 

⇔ { Indirect inequality } 

〈∀ x :: x ⊑ a ⇒ x ⊑ b〉 ∧ 〈∀ x :: x ⊑ b ⇒ x ⊑ a〉 

⇔ { Rearranging quantifiers } 

〈∀ x :: x ⊑ a ⇒ x ⊑ b ∧ x ⊑ b ⇒ x ⊑ a〉 

⇔ { Mutual implication } 

〈∀ x :: x ⊑ a ⇔ x ⊑ b〉 




Proof. 

a = b 


a ⊑ b ∧ b ⊑ a 


〈∀ x :: x ⊑ a ⇒ x ⊑ b〉 ∧ 〈∀ x :: x ⊑ b ⇒ x ⊑ a〉 


〈∀ x :: x ⊑ a ⇒ x ⊑ b ∧ x ⊑ b ⇒ x ⊑ a〉 


〈∀ x :: x ⊑ a ⇔ x ⊑ b〉 




Proof. 

a = b 


a ⊑ b ∧ b ⊑ a 


〈∀ x :: x ⊑ a ⇒ x ⊑ b〉 ∧ 〈∀ x :: x ⊑ b ⇒ x ⊑ a〉 


〈∀ x :: x ⊑ a ⇒ x ⊑ b ∧ x ⊑ b ⇒ x ⊑ a〉 


〈∀ x :: x ⊑ a ⇔ x ⊑ b〉 




Proof. 

a = b 


a ⊑ b ∧ b ⊑ a 


〈∀ x :: x ⊑ a ⇒ x ⊑ b〉 ∧ 〈∀ x :: x ⊑ b ⇒ x ⊑ a〉 


〈∀ x :: x ⊑ a ⇒ x ⊑ b ∧ x ⊑ b ⇒ x ⊑ a〉 


〈∀ x :: x ⊑ a ⇔ x ⊑ b〉 




Proof. 

a = b 


a ⊑ b ∧ b ⊑ a 


〈∀ x :: x ⊑ a ⇒ x ⊑ b〉 ∧ 〈∀ x :: x ⊑ b ⇒ x ⊑ a〉 


〈∀ x :: x ⊑ a ⇒ x ⊑ b ∧ x ⊑ b ⇒ x ⊑ a〉 


〈∀ x :: x ⊑ a ⇔ x ⊑ b〉 





Definition (Indirect equality) 

a = b ⇔ 〈∀ x :: x ⊑ a ⇔ x ⊑ b〉 

a = b ⇔ 〈∀ x :: a ⊑ x ⇔ b ⊑ x〉 





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 ) 


Properties 



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) 



Galois connections — Algebra 


Identity connection 

(id,id) 

(A, ⊑ A ) (A, ⊑ A ) 

Composition 

(f ,g) 

(h,k) 

(h◦f ,g◦k) 

if (A, ⊑) (B, ≼) and (B, ≼) (C, ) then (A, ⊑) (C, ) 

Composition is associative and the identity is its unit. 

Galois connections form a category. 



Galois connections — Algebra 


Converse 

(f ,g) 

(g,f ) 

if (A, ⊑) (B, ≼) then (B, ≽) (A, ⊒) 

Relator 

For every relator F 

(f ,g) 

(Ff ,Fg) 

if (A, ⊑) (B, ≼) then (FA, F ⊑) (FB, F ≼) 


Logic vs. algebra 


Fork algebras 

Logic Algebra 

Propositional logic Boolean algebra 

Intuitionistic propositional logic Heyting algebra 

Predicate logic ?? 



Fork algebras 

Relation algebras 

Extension of Boolean algebras 

Original work of De Morgan, Peirce and Schröder 

Further developed by Tarski in his attempt to formalize set theory 

without variables 

Amenable for syntactic manipulation 

Only one inference rule is needed: substitution of equals by 

equals 

Equational reasoning 



Fork algebras 

Relation algebras 

Extension of Boolean algebras 

Original work of De Morgan, Peirce and Schröder 

Further developed by Tarski in his attempt to formalize set theory 

without variables 

Amenable for syntactic manipulation 

Only one inference rule is needed: substitution of equals by 

equals 

Equational reasoning 



Fork algebras 

Fork algebras 

Limitation of relation algebras 

Relations algebras can express first-order predicates with at most 

three variables 

Fork algebras 

Extend relation algebras with a pairing operator 

Equivalent in expressive and deductive power to first-order logic 



Fork algebras 

Fork algebras 

Limitation of relation algebras 

Relations algebras can express first-order predicates with at most 

three variables 

Fork algebras 

Extend relation algebras with a pairing operator 

Equivalent in expressive and deductive power to first-order logic 



Point-free transform summary 


Pointwise 

Pointfree 

¬(bRa) 

b(¬R)a 

bRa ∧ bSa 

b(R ∩ S)a 

bSa ∨ bSa 

b(R ∪ S)a 

True 

b ⊤ a 

False 

b ⊥ a 

b = a 

b id a 

aRb 

bR ◦ a 

〈∃ c :: bRc ∧ cSa〉 b(R ◦ S)a 

〈∀ x :: xRb ⇒ xSa〉 b(R \ S)a 

〈∀ x :: aRx ⇒ bSx〉 b(S/R)a 

bRa ∧ cSa (b, c)(R ∇ S)a 

bRa ∧ dSc (b, d)(R × S)(a, c) 

〈∀ a, b :: bRa ⇒ bSa〉 R ⊆ S 

〈∀ a, b :: bRa ⇔ bSa〉 R = S 




Point-free definitions 

Definition (Galois connection) 

Definition (Indirect equality) 

f ◦ ◦ ⊑ B = ⊑ A ◦ g 

f = g ⇔ ≼ ◦ f = ≼ ◦ g 

f = g ⇔ f ◦ ◦ ≼ = g ◦ ◦ ≼ 


Outline 

Galois and Galculator 


Motivation 

Objectives 




Fork algebras 




Galculator 

4 Summary 

Summary 

Future work 



Sub-languages of Galois 


Module 

GC 

Definition Axiom Theorem Strategy 

Definition 

Type 

Fork 

Formula 

Term 

Proof Step 

Rewriting 

Combinator 

Derivation 

Order 

Function 


connection 



Architecture of Galculator 

Galculator 

Combine 

GC 

Derive 

Laws 

Relation 

algebra 

Properties 

Theory 

domain 

Derive 

Derive 

Derive 

Rules 

TRS 

Strategies 

Combine 


Outline 

Summary 


Motivation 

Objectives 




Fork algebras 




Galculator 

4 Summary 

Summary 

Future work 


Summary 

Summary 

Summary 

Fork algebras 

Equivalent to first-order logic (same expressive and deductive 

power) 

Single inference rule: substitution of equals for equals 

Equational 

No variables 

Integrates Galois connections and indirect equality 


Provide structure 

Introduce semantic information in syntactic reasoning 


Summary 

Summary 

Summary 


Follows the mathematical concepts 

Alternative to first-order languages 

Typed approach 

Galculator 

Proof assistance prototype based on Galois connections 

Innovative approach 

Uses a point-free equational approach 


Summary 

Future work 

Future work 

Mechanization of point-free transform 

Automated proofs 

Extension of the type system 

Free-theorems 

Evaluation of the language 

Integration with host theorem provers (e.g., Coq) 


The End 

Download 

Source code and documentation available from 

www.di.uminho.pt/research/galculator 

Contact 

Questions to paufil@di.uminho.pt

Galois

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