SLIDE 1
The rewrite tactic: day one
k : N n : N p : P k e : k ≡N n ? : P n
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Lifting proof-relevant unification to higher dimensions Jesper - - PowerPoint PPT Presentation
Lifting proof-relevant unification to higher dimensions Jesper - - PowerPoint PPT Presentation
Lifting proof-relevant unification to higher dimensions Jesper Cockx Dominique Devriese 17 January 2017 The rewrite tactic: day one k : N n : N p : P k e : k N n ? : P n 1 / 22 The rewrite tactic: day one k : N k : N n : N n : N rewrite e
SLIDE 2
SLIDE 3
The rewrite tactic: day one
k : N n : N p : P k e : k ≡N n ? : P n
rewrite e
= = = = ⇒ k : N n : N p : P n e : k ≡N n ? : P n
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SLIDE 4
The unify tactic: day one
k : N n : N p : P k e : k ≡N n ? : P n
unify e
= = = ⇒ n : N p : P n ? : P n
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SLIDE 5
The rewrite tactic: day two
k : N n : N p : P k e : suc k ≡N suc n ? : P n
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SLIDE 6
The rewrite tactic: day two
k : N n : N p : P k e : suc k ≡N suc n ? : P n
rewrite e
= = = = ⇒ k : N n : N p : P k e : suc k ≡N suc n ? : P n
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SLIDE 7
The unify tactic: day two
k : N n : N p : P k e : suc k ≡N suc n ? : P n
unify e
= = = ⇒ n : N p : P n ? : P n
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SLIDE 8
The rewrite tactic: day three
k : N n : N xs : Vec A (suc k) p : P k xs e : suc k ≡N suc n ? : P n (subst (Vec A) e xs)
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SLIDE 9
The rewrite tactic: day three
k : N n : N xs : Vec A (suc k) p : P k xs e : suc k ≡N suc n ? : P n (subst (Vec A) e xs)
rewrite e
= = = = ⇒ error
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SLIDE 10
The unify tactic: day three
k : N n : N xs : Vec A (suc k) p : P k xs e : suc k ≡N suc n ? : P n (subst (Vec A) e xs)
unify e
= = = ⇒ n : N xs : Vec A (suc n) p : P n xs ? : P n xs
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SLIDE 11
Proof-relevant unification Unification of indexed data Lifting unifiers to higher dimensions
SLIDE 12
Proof-relevant unification Unification of indexed data Lifting unifiers to higher dimensions
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Proof-relevant unification
- Represent unification rules internally
- Rules get a computational interpretation
- Core of dependent pattern matching
See Unifiers as Equivalences (ICFP ’16)
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SLIDE 14
Proof-relevant unification: example
(k n : N)(e : suc k ≡N suc n)
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SLIDE 15
Proof-relevant unification: example
(k n : N)(e : suc k ≡N suc n) ≃ (k n : N)(e : k ≡N n)
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SLIDE 16
Proof-relevant unification: example
(k n : N)(e : suc k ≡N suc n) ≃ (k n : N)(e : k ≡N n) ≃ (k : N)
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SLIDE 17
Proof-relevant unification: example
(k n : N)(e : suc k ≡N suc n) ≃ (k n : N)(e : k ≡N n) ≃ (k : N)
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SLIDE 18
Unifiers as equivalences
Goal: given some equations ¯ u ≡∆ ¯ v with free variables in Γ, find an equivalence f of type Γ(¯ e : ¯ u ≡∆ ¯ v) ≃ Γ′
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SLIDE 19
Unification rules
(x : A)(e : x ≡A t) ≃ ⊤ (solution) (suc x ≡N suc y) ≃ (x ≡N y) (injectivity) (left x ≡A⊎B right y) ≃ ⊥ (conflict) (n ≡N suc n) ≃ ⊥ (cycle)
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SLIDE 20
Telescopic equality
The type of an equation can depend on previous equations: (u : Vec A k)(v : Vec A n) (e1 : k ≡N n)(e2 : u ≡Vec A e1 v) This allows us to keep track of dependencies between equations.
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SLIDE 21
Proof-relevant unification Unification of indexed data Lifting unifiers to higher dimensions
SLIDE 22
Injectivity for indexed data
Idea: simplify equations between indices together with equation between constructors: (e1 : i ≡I j)(e2 : c u ≡D e1 c v) ≃ (e : u ≡A v)
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SLIDE 23
Injectivity for indexed data
Idea: simplify equations between indices together with equation between constructors: (e1 : i ≡I j)(e2 : c u ≡D e1 c v) ≃ (e : u ≡A v) Indices of D must be fully general: must be distinct equation variables.
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SLIDE 24
Injectivity for indexed data: example
cons : (n : N)(x : A)(xs : Vec A n) → Vec A (suc n)
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SLIDE 25
Injectivity for indexed data: example
cons : (n : N)(x : A)(xs : Vec A n) → Vec A (suc n) (e1 : suc k ≡N suc n) (e2 : cons k x xs ≡Vec A e1 cons n y ys)
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SLIDE 26
Injectivity for indexed data: example
cons : (n : N)(x : A)(xs : Vec A n) → Vec A (suc n) (e1 : suc k ≡N suc n) (e2 : cons k x xs ≡Vec A e1 cons n y ys) ≃ (e′
1 : k ≡N n)(e′ 2 : x ≡A y)
(e′
3 : xs ≡Vec A e1 ys)
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SLIDE 27
Injectivity for indexed data: example
cons : (n : N)(x : A)(xs : Vec A n) → Vec A (suc n) (e1 : suc k ≡N suc n) (e2 : cons k x xs ≡Vec A e1 cons n y ys) ≃ (e′
1 : k ≡N n)(e′ 2 : x ≡A y)
(e′
3 : xs ≡Vec A e1 ys)
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SLIDE 28
What if the indices are not fully general?
(e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ ???
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SLIDE 29
Solution: generalizing the indices
(e : cons n x xs ≡Vec A (suc n) cons n y ys)
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SLIDE 30
Solution: generalizing the indices
(e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (e1 : suc n ≡N suc n) (e2 : cons n x xs ≡Vec A e1 cons n y ys) (p : e1 ≡suc n≡Nsuc n refl)
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SLIDE 31
Solution: generalizing the indices
(e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (e1 : suc n ≡N suc n) (e2 : cons n x xs ≡Vec A e1 cons n y ys) (p : e1 ≡suc n≡Nsuc n refl) ≃ (e′
1 : n ≡N n)(e′ 2 : x ≡A y)(e′ 3 : xs ≡Vec A e′
1 ys)
(p : cong suc e′
1 ≡suc n≡Nsuc n refl)
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SLIDE 32
Solution: generalizing the indices
(e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (e1 : suc n ≡N suc n) (e2 : cons n x xs ≡Vec A e1 cons n y ys) (p : e1 ≡suc n≡Nsuc n refl) ≃ (e′
1 : n ≡N n)(e′ 2 : x ≡A y)(e′ 3 : xs ≡Vec A e′
1 ys)
(p : cong suc e′
1 ≡suc n≡Nsuc n refl)
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SLIDE 33
Higher-dimensional unification
(e′
1 : n ≡N n)(e′ 2 : x ≡A y)(e′ 3 : xs ≡Vec A e′
1 ys)
(p : cong suc e′
1 ≡suc n≡Nsuc n refl)
Now we have to solve equations between equality proofs!
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SLIDE 34
Proof-relevant unification Unification of indexed data Lifting unifiers to higher dimensions
SLIDE 35
How to solve higher-dimensional equations?
Existing unification rules do not apply. . .
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SLIDE 36
How to solve higher-dimensional equations?
Existing unification rules do not apply. . . We solve the problem in three steps:
- 1. lower the dimension of equations
- 2. solve lower-dimensional equations
- 3. lift unifier to higher dimension
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SLIDE 37
Step 1: lower the dimension of equations
We replace all equation variables by regular variables: instead of (e1 : n ≡N n)(e2 : x ≡A y)(e3 : xs ≡Vec A e1 ys) (p : cong suc e1 ≡suc n≡Nsuc n refl) let’s first consider (k : N)(u : A)(us : Vec A k) (e : suc k ≡N suc n)
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Step 2: solve lower-dimensional equations
This gives us an equivalence f of type (k : N)(u : A)(us : Vec A k) (e : suc k ≡N suc n) ≃ (u : A)(us : Vec A n)
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SLIDE 39
Step 3: lift unifier to higher dimension
We lift f to an equivalence f ↑ of type (e1 : n ≡N n)(e2 : x ≡A y) (e3 : xs ≡Vec A e1 ys) (p : cong suc e1 ≡suc n≡Nsuc n refl) ≃ (e2 : x ≡A y)(e3 : xs ≡Vec A n ys)
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SLIDE 40
Lifting equivalences: (mostly) general case
- Theorem. If we have an equivalence f of type
(x : A)(e : b1 x ≡B x b2 x) ≃ C we can construct f ↑ of type (e : u ≡A v)(p : cong b1 e ≡r≡B es cong b2 e) ≃ (e′ : f u r ≡C f v s)
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SLIDE 41
Conclusion
Proof-relevant unification is useful to deal with many equality constraints.
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SLIDE 42
Conclusion
Proof-relevant unification is useful to deal with many equality constraints. To make it work on indexed datatypes, we need to solve higher-dimensional equations.
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SLIDE 43
Conclusion
Proof-relevant unification is useful to deal with many equality constraints. To make it work on indexed datatypes, we need to solve higher-dimensional equations. We can reuse existing unification rules by lifting them to higher dimensions.
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