19 - Scala. Eliminators into dependent types (induction)
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This document describes eliminators (also called induction principles) for dependent types in dependent type theory, including: - Empty, unit, sum, product, and function types - Dependent pairs and dependent functions - Boolean, natural numbers, lists, and vectors - Identity types For each type, it provides the eliminator signature and definition, describing how to eliminate values of that type into a dependent type.
![Empty type
Eliminator into type C
Γ C : ∗
Γ, x : 0 abortC x : C
Eliminator into dependent type C
Γ, u : 0 C : ∗
Γ, x : 0 abortC x : C[u ← x]
ind0 ≡ abort :
C:0→∗ x:0
C(x)
abort(C) ≡ C
Dmytro Mitin Eliminators into dependent types](https://image.slidesharecdn.com/19-eliminatorsintodependenttypesinduction-190722073406/85/19-Scala-Eliminators-into-dependent-types-induction-2-320.jpg)
![Unit type
Eliminator into type C
Γ c : C
Γ, x : 1 case1(c)(x) : C
x match { case () => c }
Eliminator into dependent type C
Γ, u : 1 C : ∗ Γ c : C[u ← 1]
Γ, x : 1 case1(λu.C)(c)(x) : C[u ← x]
ind1 ≡ case1 :
C:1→∗
C(1) →
x:1
C(x)
case1(C, c, 1) ≡ c
Dmytro Mitin Eliminators into dependent types](https://image.slidesharecdn.com/19-eliminatorsintodependenttypesinduction-190722073406/85/19-Scala-Eliminators-into-dependent-types-induction-3-320.jpg)
![Sum type
Eliminator into type C
Γ C : ∗ Γ, a : A c1 : C Γ, b : B c2 : C
Γ, s : A + B case+(λa.c1, λb.c2)(s) : C
s match { case Inl(a) => c1(a); case Inr(b) => c2(b) }
Eliminator into dependent type C
Γ, u : A + B C : ∗
Γ, a : A c1 : C[u ← inl a] Γ, b : B c2 : C[u ← inr b]
Γ, s : A + B case+(λu.C)(λa.c1, λb.c2)(s) : C[u ← s]
ind+ ≡ case+ :
C:A+B→∗ a:A
C(inl a) ×
b:B
C(inr b) →
s:A+B
C(s)
case+(C, f , g, inl a) ≡ f (a)
case+(C, f , g, inr b) ≡ g(b)
Dmytro Mitin Eliminators into dependent types](https://image.slidesharecdn.com/19-eliminatorsintodependenttypesinduction-190722073406/85/19-Scala-Eliminators-into-dependent-types-induction-4-320.jpg)
![Product type
Eliminators into types A and B
Γ p : A × B
Γ fst p : A
Γ p : A × B
Γ snd p : B
Eliminator into dependent type C
Γ, u : A × B C : ∗ Γ, a : A, b : B c : C[u ← (a, b)]
Γ, p : A × B c[a ← fst p, b ← snd p]
≡ case×(λu.C)(λab.c)(p)
: C[u ← p]
ind× ≡ case× :
C:A×B→∗ a:A b:B
C(a, b) →
p:A×B
C(p)
case×(C, f , (a, b)) ≡ f (a)(b)
Dmytro Mitin Eliminators into dependent types](https://image.slidesharecdn.com/19-eliminatorsintodependenttypesinduction-190722073406/85/19-Scala-Eliminators-into-dependent-types-induction-5-320.jpg)
![Function type
Eliminators into type B
Γ f : A → B Γ a : A
Γ f a : B
Higher-order eliminator[1] [2] [3] into dependent type C
Γ, u : A → B C : ∗
(Γ, x : A b : B) c : C[u ← λx.b]
Γ, g : A → B c : C[u ← λ(g)]
Γ, f : A → B caseΠ(λu.C) λg.c (f ) : C[u ← f ]
funsplit ≡ ind→ ≡ case→ :
C:(A→B)→∗ g:A→B
C(λx.b) →
f :A→B
C(f )
case→(C, h, λx.b) ≡ h(λx.b)
Dmytro Mitin Eliminators into dependent types](https://image.slidesharecdn.com/19-eliminatorsintodependenttypesinduction-190722073406/85/19-Scala-Eliminators-into-dependent-types-induction-6-320.jpg)
![Dependent pair type (Σ-type)
Eliminators into types A and B
Γ p :
x:A
B
Γ fst p : A
Γ p :
x:A
B
Γ snd p : B[x ← fst p]
Eliminator into dependent type C
Γ, u :
x:A
B C : ∗ Γ, a : A, b : B[u ← a] c : C[u ← (a, b)]
Γ, p :
x:A
B c[a ← fst p, b ← snd p]
≡ caseΣ(λu.C)(λab.c)(p)
: C[u ← p]
split ≡ indΣ ≡ caseΣ :
C:
x:A
B(x)→∗ a:A b:B(a)
C(a, b) →
p:
x:A
B(x)
C(p)
caseΣ(C, f , (a, b)) ≡ f (a)(b)
Dmytro Mitin Eliminators into dependent types](https://image.slidesharecdn.com/19-eliminatorsintodependenttypesinduction-190722073406/85/19-Scala-Eliminators-into-dependent-types-induction-7-320.jpg)
![Dependent function type (Π-type)
Eliminators into type B
Γ f :
x:A
B Γ a : A
Γ f a : B[x ← a]
Higher-order eliminator[1] [2] [3] into dependent type C
Γ, u :
x:A
B C : ∗ (Γ, x : A b : B) c : C[u ← λx.b]
Γ, g :
x:A
B c : C[u ← λ(g)]
Γ, f :
x:A
B caseΠ(λu.C) λg.c (f ) : C[u ← f ]
funsplit ≡ indΠ ≡ caseΠ :
C:
x:A
B(x)→∗ g:
x:A
B(x)
C(λx.b) →
f :
x:A
B(x)
C(f )
caseΠ(C, h, λx.b) ≡ h(λx.b)
Dmytro Mitin Eliminators into dependent types](https://image.slidesharecdn.com/19-eliminatorsintodependenttypesinduction-190722073406/85/19-Scala-Eliminators-into-dependent-types-induction-8-320.jpg)
![Boolean type
Eliminator into type C
Γ c1 : C Γ c2 : C
Γ, x : Bool if(c1, c2)(x) : C
x match { case True => c1; case False => c2 }
Eliminator into dependent type C
Γ, u : Bool C : ∗ Γ c1 : C[u ← true] Γ c2 : C[u ← false]
Γ, x : Bool if(c1, c2)(x)
caseBool(λu.C)(c1,c2)(x)
: C[u ← x]
indBool ≡ caseBool :
C:Bool→∗
C(true) × C(false) →
x:Bool
C(x)
caseBool(C, c1, c2, true) ≡ c1
caseBool(C, c1, c2, false) ≡ c2
Dmytro Mitin Eliminators into dependent types](https://image.slidesharecdn.com/19-eliminatorsintodependenttypesinduction-190722073406/85/19-Scala-Eliminators-into-dependent-types-induction-9-320.jpg)
![Type of natural numbers
Eliminator into type C
Γ c1 : C Γ, n : Nat c2 : C
Γ, x : Nat caseNat(c1, λn.c2)(x) : C
x match { case Zero => c1; case Suc(n) => c2(n) }
Eliminator into dependent type C
Γ, u : Nat C : ∗ Γ c1 : C[u ← zero]
Γ, n : Nat, c : C[u ← n] c2 : C[u ← suc n]
Γ, x : Nat caseNat(λu.C)(c1, λnc.c2)(x) : C[u ← x]
indNat ≡ caseNat :
C:Nat→∗
C(zero) ×
n:Nat
C(n) → C(suc n) →
x:Nat
C(x)
caseNat(C, c1, f , zero) ≡ c1
caseNat(C, c1, f , suc n) ≡ f (n)(caseNat(C, c1, f , n))
Dmytro Mitin Eliminators into dependent types](https://image.slidesharecdn.com/19-eliminatorsintodependenttypesinduction-190722073406/85/19-Scala-Eliminators-into-dependent-types-induction-10-320.jpg)
![List type
Eliminator into type C
Γ c1 : C Γ, h : A, t : List A c2 : C
Γ, x : List A caseList(x)(c1, λht.c2) : C
x match { case Nil => c1; case Cons(h, t) => c2(h, t) }
Eliminator into dependent type C
Γ, u : List A C : ∗ Γ c1 : C[u ← nil]
Γ, h : A, t : List A, c : C[u ← t] c2 : C[u ← cons h t]
Γ, x : List A caseList(λu.C)(c1, λhtc.c2)(x) : C[u ← x]
indList ≡ caseList :
C:List A→∗
C(nil) ×
h:A t:List A
C(t) → C(cons h t) →
x:List A
C(x)
caseList(C, c1, f , nil) ≡ c1
caseList(C, c1, f , cons h t) ≡ f (h, t)(caseList(C, c1, f , t))
Dmytro Mitin Eliminators into dependent types](https://image.slidesharecdn.com/19-eliminatorsintodependenttypesinduction-190722073406/85/19-Scala-Eliminators-into-dependent-types-induction-11-320.jpg)
![Type of fixed-length vectors
Eliminator into dependent type C
Γ, n : Nat, u : Vect n A C : ∗ Γ c1 : C[n ← zero, u ← nil]
Γ, h : A, m : Nat, t : Vect m A, c : C[n ← m, u ← t]
c2 : C[n ← suc m, u ← cons h t]
Γ, k : Nat, x : Vect k A
caseVect(λnu.C)(c1, λhmtc.c2)(k, x) : C[n ← k, u ← x]
indVect ≡ caseVect :
C:
n:Nat
(Vect n A→∗)
C(zero, nil) ×
h:A m:Nat t:Vect m A
C(m, t)
→ C(suc m, cons h t) →
x:
k:Nat
Vect k A
C(x)
caseVect C, c1, f , (zero, nil) ≡ c1
caseVect C, c1, f , (suc m, cons h t) ≡ f (h, m, t)(caseVect(C, c1, f , (m, t)))
Dmytro Mitin Eliminators into dependent types](https://image.slidesharecdn.com/19-eliminatorsintodependenttypesinduction-190722073406/85/19-Scala-Eliminators-into-dependent-types-induction-12-320.jpg)
![Identity type
Eliminator into dependent type C
Γ, a1 : A, a2 : A, u : a1 =A a2 C : ∗
Γ, a : A c : C[a1 ← a, a2 ← a, u ← refl a]
Γ, a1 : A, a2 : A, x : a1 =A a2 caseId(λa1a2u.C)(λa.c)(λa1a2.x)
: C[a1 ← a1, a2 ← a2, u ← x]
J ≡ indId ≡ caseId :
C:
a1:A a2:A
(a1=Aa2→∗) a:A
C(a, a, refl a) →
a1:A a2:A x:a1=Aa2
C(a1, a2, x)
caseId(C, f , a, a, refl a) = f (a)
Dmytro Mitin Eliminators into dependent types](https://image.slidesharecdn.com/19-eliminatorsintodependenttypesinduction-190722073406/85/19-Scala-Eliminators-into-dependent-types-induction-13-320.jpg)
19 - Scala. Eliminators into dependent types (induction)
- 1. Eliminators into dependent types Dmytro Mitin https://stepik.org/course/Introduction-to-programming- with-dependent-types-in-Scala-2294/ March 2017 Dmytro Mitin Eliminators into dependent types
- 2. Empty type Eliminator into type C Γ C : ∗ Γ, x : 0 abortC x : C Eliminator into dependent type C Γ, u : 0 C : ∗ Γ, x : 0 abortC x : C[u ← x] ind0 ≡ abort : C:0→∗ x:0 C(x) abort(C) ≡ C Dmytro Mitin Eliminators into dependent types
- 3. Unit type Eliminator into type C Γ c : C Γ, x : 1 case1(c)(x) : C x match { case () => c } Eliminator into dependent type C Γ, u : 1 C : ∗ Γ c : C[u ← 1] Γ, x : 1 case1(λu.C)(c)(x) : C[u ← x] ind1 ≡ case1 : C:1→∗ C(1) → x:1 C(x) case1(C, c, 1) ≡ c Dmytro Mitin Eliminators into dependent types
- 4. Sum type Eliminator into type C Γ C : ∗ Γ, a : A c1 : C Γ, b : B c2 : C Γ, s : A + B case+(λa.c1, λb.c2)(s) : C s match { case Inl(a) => c1(a); case Inr(b) => c2(b) } Eliminator into dependent type C Γ, u : A + B C : ∗ Γ, a : A c1 : C[u ← inl a] Γ, b : B c2 : C[u ← inr b] Γ, s : A + B case+(λu.C)(λa.c1, λb.c2)(s) : C[u ← s] ind+ ≡ case+ : C:A+B→∗ a:A C(inl a) × b:B C(inr b) → s:A+B C(s) case+(C, f , g, inl a) ≡ f (a) case+(C, f , g, inr b) ≡ g(b) Dmytro Mitin Eliminators into dependent types
- 5. Product type Eliminators into types A and B Γ p : A × B Γ fst p : A Γ p : A × B Γ snd p : B Eliminator into dependent type C Γ, u : A × B C : ∗ Γ, a : A, b : B c : C[u ← (a, b)] Γ, p : A × B c[a ← fst p, b ← snd p] ≡ case×(λu.C)(λab.c)(p) : C[u ← p] ind× ≡ case× : C:A×B→∗ a:A b:B C(a, b) → p:A×B C(p) case×(C, f , (a, b)) ≡ f (a)(b) Dmytro Mitin Eliminators into dependent types
- 6. Function type Eliminators into type B Γ f : A → B Γ a : A Γ f a : B Higher-order eliminator[1] [2] [3] into dependent type C Γ, u : A → B C : ∗ (Γ, x : A b : B) c : C[u ← λx.b] Γ, g : A → B c : C[u ← λ(g)] Γ, f : A → B caseΠ(λu.C) λg.c (f ) : C[u ← f ] funsplit ≡ ind→ ≡ case→ : C:(A→B)→∗ g:A→B C(λx.b) → f :A→B C(f ) case→(C, h, λx.b) ≡ h(λx.b) Dmytro Mitin Eliminators into dependent types
- 7. Dependent pair type (Σ-type) Eliminators into types A and B Γ p : x:A B Γ fst p : A Γ p : x:A B Γ snd p : B[x ← fst p] Eliminator into dependent type C Γ, u : x:A B C : ∗ Γ, a : A, b : B[u ← a] c : C[u ← (a, b)] Γ, p : x:A B c[a ← fst p, b ← snd p] ≡ caseΣ(λu.C)(λab.c)(p) : C[u ← p] split ≡ indΣ ≡ caseΣ : C: x:A B(x)→∗ a:A b:B(a) C(a, b) → p: x:A B(x) C(p) caseΣ(C, f , (a, b)) ≡ f (a)(b) Dmytro Mitin Eliminators into dependent types
- 8. Dependent function type (Π-type) Eliminators into type B Γ f : x:A B Γ a : A Γ f a : B[x ← a] Higher-order eliminator[1] [2] [3] into dependent type C Γ, u : x:A B C : ∗ (Γ, x : A b : B) c : C[u ← λx.b] Γ, g : x:A B c : C[u ← λ(g)] Γ, f : x:A B caseΠ(λu.C) λg.c (f ) : C[u ← f ] funsplit ≡ indΠ ≡ caseΠ : C: x:A B(x)→∗ g: x:A B(x) C(λx.b) → f : x:A B(x) C(f ) caseΠ(C, h, λx.b) ≡ h(λx.b) Dmytro Mitin Eliminators into dependent types
- 9. Boolean type Eliminator into type C Γ c1 : C Γ c2 : C Γ, x : Bool if(c1, c2)(x) : C x match { case True => c1; case False => c2 } Eliminator into dependent type C Γ, u : Bool C : ∗ Γ c1 : C[u ← true] Γ c2 : C[u ← false] Γ, x : Bool if(c1, c2)(x) caseBool(λu.C)(c1,c2)(x) : C[u ← x] indBool ≡ caseBool : C:Bool→∗ C(true) × C(false) → x:Bool C(x) caseBool(C, c1, c2, true) ≡ c1 caseBool(C, c1, c2, false) ≡ c2 Dmytro Mitin Eliminators into dependent types
- 10. Type of natural numbers Eliminator into type C Γ c1 : C Γ, n : Nat c2 : C Γ, x : Nat caseNat(c1, λn.c2)(x) : C x match { case Zero => c1; case Suc(n) => c2(n) } Eliminator into dependent type C Γ, u : Nat C : ∗ Γ c1 : C[u ← zero] Γ, n : Nat, c : C[u ← n] c2 : C[u ← suc n] Γ, x : Nat caseNat(λu.C)(c1, λnc.c2)(x) : C[u ← x] indNat ≡ caseNat : C:Nat→∗ C(zero) × n:Nat C(n) → C(suc n) → x:Nat C(x) caseNat(C, c1, f , zero) ≡ c1 caseNat(C, c1, f , suc n) ≡ f (n)(caseNat(C, c1, f , n)) Dmytro Mitin Eliminators into dependent types
- 11. List type Eliminator into type C Γ c1 : C Γ, h : A, t : List A c2 : C Γ, x : List A caseList(x)(c1, λht.c2) : C x match { case Nil => c1; case Cons(h, t) => c2(h, t) } Eliminator into dependent type C Γ, u : List A C : ∗ Γ c1 : C[u ← nil] Γ, h : A, t : List A, c : C[u ← t] c2 : C[u ← cons h t] Γ, x : List A caseList(λu.C)(c1, λhtc.c2)(x) : C[u ← x] indList ≡ caseList : C:List A→∗ C(nil) × h:A t:List A C(t) → C(cons h t) → x:List A C(x) caseList(C, c1, f , nil) ≡ c1 caseList(C, c1, f , cons h t) ≡ f (h, t)(caseList(C, c1, f , t)) Dmytro Mitin Eliminators into dependent types
- 12. Type of fixed-length vectors Eliminator into dependent type C Γ, n : Nat, u : Vect n A C : ∗ Γ c1 : C[n ← zero, u ← nil] Γ, h : A, m : Nat, t : Vect m A, c : C[n ← m, u ← t] c2 : C[n ← suc m, u ← cons h t] Γ, k : Nat, x : Vect k A caseVect(λnu.C)(c1, λhmtc.c2)(k, x) : C[n ← k, u ← x] indVect ≡ caseVect : C: n:Nat (Vect n A→∗) C(zero, nil) × h:A m:Nat t:Vect m A C(m, t) → C(suc m, cons h t) → x: k:Nat Vect k A C(x) caseVect C, c1, f , (zero, nil) ≡ c1 caseVect C, c1, f , (suc m, cons h t) ≡ f (h, m, t)(caseVect(C, c1, f , (m, t))) Dmytro Mitin Eliminators into dependent types
- 13. Identity type Eliminator into dependent type C Γ, a1 : A, a2 : A, u : a1 =A a2 C : ∗ Γ, a : A c : C[a1 ← a, a2 ← a, u ← refl a] Γ, a1 : A, a2 : A, x : a1 =A a2 caseId(λa1a2u.C)(λa.c)(λa1a2.x) : C[a1 ← a1, a2 ← a2, u ← x] J ≡ indId ≡ caseId : C: a1:A a2:A (a1=Aa2→∗) a:A C(a, a, refl a) → a1:A a2:A x:a1=Aa2 C(a1, a2, x) caseId(C, f , a, a, refl a) = f (a) Dmytro Mitin Eliminators into dependent types