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Type Theory
Marene Dimmendaal, Pleun Koldewijn
Overview
- What is ‘proof by reflection’?
- The two main classes:
- Direct computation proofs
- Algebraic computation proofs
- Example Direct proof
- Example Algebraic proof
- Summary
Type Theory Proof by reflection Marene Dimmendaal, Pleun Koldewijn - - PowerPoint PPT Presentation
Type Theory Proof by reflection Marene Dimmendaal, Pleun Koldewijn - - PowerPoint PPT Presentation
Type Theory Proof by reflection Marene Dimmendaal, Pleun Koldewijn Overview - What is proof by reflection? - The two main classes: - Direct computation proofs - Algebraic computation proofs - Example Direct proof - Example
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What is proof by reflection?
- Statements involving computations
- Automated proof development system
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General presentation
- Coq file
- Complex combinations of reasoning steps replaced by few
computation steps
- Two classes of problems:
Direct computation proofs Algebraic computation proofs
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Direct computation proofs
Proof of “C t”: Proof of ‘C t’:
predicate function
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Algebraic computational proofs
The reflection process relies on the following theorem:
Functions and
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Example direct computation proof
Proved by computing remainders Coq proof: A reasonably sized natural number is prime
In this case, C x is the mathematical statement for ‘x is prime’, i.e. there is no integer n (not equal to 1 or x) which divides x.
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Setting up reflection
Only smaller number need to be checked: Existence of a divisor:
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Function for division
To check presence of divisors:
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Primality
To check primality:
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Function check_range
Isn’t it simpler with two arguments?
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Duration of the functions
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Duration of the functions
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Deduced result
This is our f_correct!
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Primality proof
TTTT
This proof takes a few minutes while the naïve procedure could not cope with a number this size.
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Example Algebraic computational proofs
For set A and a binary operation *, we have that Associativity law: ( x * y ) * z = x * ( y * z ) for all x y z in A
Easily use For With
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Example Algebraic computational proofs
x y z t u x y z t u
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Example Algebraic computational proofs
x y z t u x y z t u
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Data Type and functions
Data Type A : Function f :
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Data Type and functions
Data Type A : Function i :
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The required proofs
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Example Algebraic computational proofs
x y z t u x y z t u
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Using the proof
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Ltac: transforming equation to binary tree
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Ltac: automated proof steps
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Ltac: automated proof steps
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Example Algebraic computational proofs
x y z t u x y z t u
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Generic version
For set A and a binary operation *, we have that Associativity law: ( x * y ) * z = x * ( y * z ) for all x y z in A
f x ( f ( f y z ) ( f t u ) ) represented as
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Generic version
f x ( f ( f y z ) ( f t x ) ) x y z t x x y z t x
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Generic version - theorems
Original : Generic :
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Generic version
f x ( f ( f y z ) ( f t x ) ) f x ( f ( f y z ) ( f t x ) ) cons x ( cons y ( cons z ( cons t ( cons x ) ) ) ) x y z t x x y z t x
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Generic version
f x ( f ( f y z ) ( f t x ) ) f x ( f ( f y z ) ( f t x ) ) cons x ( cons y ( cons z ( cons t ( cons x ) ) ) ) 1 2 3 1 2 3
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Generic version
f x ( f ( f y z ) ( f t x ) ) f x ( f ( f y z ) ( f t x ) ) cons x ( cons y ( cons z ( cons t ( cons x ) ) ) ) 1 2 3 1 2 3
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Generic version - theorems
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Generic version
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Generic version - with commutativity
f x ( f ( f y z ) ( f t x ) ) 1 2 3 1 2 3
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Generic version - with commutativity
f x ( f ( f y z ) ( f t x ) ) 1 2 3 1 2 3 1 2 3