Depending on equations
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1 Depending on equations A proof-relevant framework for unification in dependent type theory Jesper Cockx DistriNet KU Leuven 3 September 2017
2 Unification for dependent types Unification is used for many purposes: logic programming, type inference, term rewriting, automated theorem proving, natural language processing,... This talk: checking definitions by dependent pattern matching 1 / 52
3 Disclaimer My work is on dependently typed languages, I know little about unification. 2 / 52
4 Disclaimer My work is on dependently typed languages, I know little about unification. This talk is about first-order unification: (suc x = suc y) = (x = y) x:=y == OK 2 / 52
5 Disclaimer My work is on dependently typed languages, I know little about unification. This talk is about first-order unification: (suc x = suc y) = (x = y) x:=y == OK (suc x = zero) = 2 / 52
6 Disclaimer My work is on dependently typed languages, I know little about unification. This talk is about first-order unification: (suc x = suc y) = (x = y) x:=y == OK (suc x = zero) =... but there will be types everywhere! 2 / 52
7 Dependent types: the big five During this presentation, we ll spot: Dependent functions: (x : A) B x 3 / 52
8 Dependent types: the big five During this presentation, we ll spot: Dependent functions: (x : A) B x Indexed datatypes: Vec A n,... 3 / 52
9 Dependent types: the big five During this presentation, we ll spot: Dependent functions: (x : A) B x Indexed datatypes: Vec A n,... Identity types: x A y 3 / 52
10 Dependent types: the big five During this presentation, we ll spot: Dependent functions: (x : A) B x Indexed datatypes: Vec A n,... Identity types: x A y Universes: Type i 3 / 52
11 Dependent types: the big five During this presentation, we ll spot: Dependent functions: (x : A) B x Indexed datatypes: Vec A n,... Identity types: x A y Universes: Type i Univalence: (A B) (A B) 3 / 52
12 Dependent types: the big five During this presentation, we ll spot: Dependent functions: (x : A) B x Indexed datatypes: Vec A n,... Identity types: x A y Universes: Type i Univalence: (A B) (A B) and see how they interact with unification! 3 / 52
13 Depending on equations Checking dependently typed programs Unification in dependent type theory Unification of dependently typed terms
14 Depending on equations Checking dependently typed programs Unification in dependent type theory Unification of dependently typed terms
15 Why use dependent types? With dependent types, you can... 4 / 52
16 Why use dependent types? With dependent types, you can guarantee that a program matches its specification 4 / 52
17 Why use dependent types? With dependent types, you can guarantee that a program matches its specification... use the same language for writing programs and proofs 4 / 52
18 Why use dependent types? With dependent types, you can guarantee that a program matches its specification... use the same language for writing programs and proofs... develop programs and proofs interactively 4 / 52
19 Dependent types A dependent type is a family of types, depending on a term of a base type. Per Martin-Löf 5 / 52
20 Dependent types A dependent type is a family of types, depending on a term of a base type. Per Martin-Löf e.g. Vec A n is the type of vectors of length n. 5 / 52
21 The Agda language Agda is a purely functional language 6 / 52
22 The Agda language Agda is a purely functional language... with a strong, static type system 6 / 52
23 The Agda language Agda is a purely functional language... with a strong, static type system... for writing programs and proofs 6 / 52
24 The Agda language Agda is a purely functional language... with a strong, static type system... for writing programs and proofs... with datatypes and pattern matching 6 / 52
25 The Agda language Agda is a purely functional language... with a strong, static type system... for writing programs and proofs... with datatypes and pattern matching... with first-class dependent types 6 / 52
26 The Agda language Agda is a purely functional language... with a strong, static type system... for writing programs and proofs... with datatypes and pattern matching... with first-class dependent types... with support for interactive development 6 / 52
27 The Agda language Agda is a purely functional language... with a strong, static type system... for writing programs and proofs... with datatypes and pattern matching... with first-class dependent types... with support for interactive development All examples are (mostly) valid Agda code! 6 / 52
28 Using dependent types With dependent types, we can give more precise types to our programs: replicate : (n : N) A Vec A n 7 / 52
29 Using dependent types With dependent types, we can give more precise types to our programs: replicate : (n : N) A Vec A n replicate 10 a : Vec Char 10 7 / 52
30 Using dependent types With dependent types, we can give more precise types to our programs: replicate : (n : N) A Vec A n tail : (n : N) Vec A (suc n) Vec A n 7 / 52
31 Using dependent types With dependent types, we can give more precise types to our programs: replicate : (n : N) A Vec A n tail : (n : N) Vec A (suc n) Vec A n append : (m n : N) Vec A m Vec A n Vec A (m + n) 7 / 52
32 Simple pattern matching data N : Type where zero : N suc : N N 8 / 52
33 Simple pattern matching data N : Type where zero : N suc : N N minimum : N N N minimum x y = { } 8 / 52
34 Simple pattern matching data N : Type where zero : N suc : N N minimum : N N N minimum zero y = { } minimum (suc x) y = { } 8 / 52
35 Simple pattern matching data N : Type where zero : N suc : N N minimum : N N N minimum zero y = zero minimum (suc x) y = { } 8 / 52
36 Simple pattern matching data N : Type where zero : N suc : N N minimum : N N N minimum zero y = zero minimum (suc x) zero = { } minimum (suc x) (suc y) = { } 8 / 52
37 Simple pattern matching data N : Type where zero : N suc : N N minimum : N N N minimum zero y = zero minimum (suc x) zero = zero minimum (suc x) (suc y) = { } 8 / 52
38 Simple pattern matching data N : Type where zero : N suc : N N minimum : N N N minimum zero y = zero minimum (suc x) zero = zero minimum (suc x) (suc y) = suc (minimum x y) 8 / 52
39 Dependent pattern matching data Vec (A : Type) : N Type where nil : Vec A zero cons : (n : N) A Vec A n Vec A (suc n) 9 / 52
40 Dependent pattern matching data Vec (A : Type) : N Type where nil : Vec A zero cons : (n : N) A Vec A n Vec A (suc n) tail : (k : N) Vec A (suc k) Vec A k tail k xs = { } 9 / 52
41 Dependent pattern matching data Vec (A : Type) : N Type where nil : Vec A zero cons : (n : N) A Vec A n Vec A (suc n) tail : (k : N) Vec A (suc k) Vec A k tail k nil = { } -- suc k = zero tail k (cons n x xs) = { } -- suc k = suc n 9 / 52
42 Dependent pattern matching data Vec (A : Type) : N Type where nil : Vec A zero cons : (n : N) A Vec A n Vec A (suc n) tail : (k : N) Vec A (suc k) Vec A k tail k nil = { } -- impossible tail k (cons n x xs) = { } -- suc k = suc n 9 / 52
43 Dependent pattern matching data Vec (A : Type) : N Type where nil : Vec A zero cons : (n : N) A Vec A n Vec A (suc n) tail : (k : N) Vec A (suc k) Vec A k tail k (cons n x xs) = { } -- suc k = suc n 9 / 52
44 Dependent pattern matching data Vec (A : Type) : N Type where nil : Vec A zero cons : (n : N) A Vec A n Vec A (suc n) tail : (k : N) Vec A (suc k) Vec A k tail k (cons n x xs) = { } -- k = n 9 / 52
45 Dependent pattern matching data Vec (A : Type) : N Type where nil : Vec A zero cons : (n : N) A Vec A n Vec A (suc n) tail : (k : N) Vec A (suc k) Vec A k tail.n (cons n x xs) = { } 9 / 52
46 Dependent pattern matching data Vec (A : Type) : N Type where nil : Vec A zero cons : (n : N) A Vec A n Vec A (suc n) tail : (k : N) Vec A (suc k) Vec A k tail.n (cons n x xs) = xs 9 / 52
47 Specialization by unification Agda uses unification to: eliminate impossible cases specialize the result type 10 / 52
48 Specialization by unification Agda uses unification to: eliminate impossible cases specialize the result type The output of unification can change Agda s notion of equality! 10 / 52
49 Specialization by unification Agda uses unification to: eliminate impossible cases specialize the result type The output of unification can change Agda s notion of equality! Main question: How to make sure the output of unification is correct? 10 / 52
50 Depending on equations Checking dependently typed programs Unification in dependent type theory Unification of dependently typed terms
51 Q: What is the fastest way to start a fight between type theorists? 11 / 52
52 Q: What is the fastest way to start a fight between type theorists? A: Mention the topic of equality. 11 / 52
53 The identity type x A y... a dependent type depending on x, y : A. 12 / 52
54 The identity type x A y... a dependent type depending on x, y : A.... type theory s built-in notion of equality. 12 / 52
55 The identity type x A y... a dependent type depending on x, y : A.... type theory s built-in notion of equality.... the type of proofs that x = y. 12 / 52
56 Operations on the identity type refl : x A x 13 / 52
57 Operations on the identity type refl : x A x sym : x A y y A x 13 / 52
58 Operations on the identity type refl : x A x sym : x A y y A x trans : x A y y A z x A z 13 / 52
59 Operations on the identity type refl : x A x sym : x A y y A x trans : x A y y A z x A z cong f : x A y f x B f y 13 / 52
60 Operations on the identity type refl : x A x sym : x A y y A x trans : x A y y A z x A z cong f : x A y f x B f y subst P : x A y P x P y 13 / 52
61 Unification problems as telescopes A unification problem consists of 1. Flexible variables x 1 : A 1, x 2 : A 2, Equations u 1 = v 1 : B 1, / 52
62 Unification problems as telescopes A unification problem consists of 1. Flexible variables x 1 : A 1, x 2 : A 2, Equations u 1 = v 1 : B 1,... This can be represented as a telescope: (x 1 : A 1 )(x 2 : A 2 )... (e 1 : u 1 B1 v 1 )(e 2 : u 2 B2 v 2 )... e.g. (k : N)(n : N)(e : suc k N suc n) 14 / 52
63 Unification problems as telescopes A unification problem consists of 1. Flexible variables Γ 2. Equations u 1 = v 1 : B 1,... This can be represented as a telescope: Γ (e 1 : u 1 B1 v 1 )(e 2 : u 2 B2 v 2 )... e.g. (k : N)(n : N)(e : suc k N suc n) 14 / 52
64 Unification problems as telescopes A unification problem consists of 1. Flexible variables Γ 2. Equations ū = v : This can be represented as a telescope: Γ(ē : ū v) e.g. (k : N)(n : N)(e : suc k N suc n) 14 / 52
65 Unifiers as telescope maps A unifier of ū and v is a substitution σ : Γ Γ such that ūσ = vσ. 15 / 52
66 Unifiers as telescope maps A unifier of ū and v is a substitution σ : Γ Γ such that ūσ = vσ. This can be represented as a telescope map: f : Γ Γ(ē : ū v) e.g. f : () (n : N)(e : n N zero) f () = zero; refl 15 / 52
67 Evidence of unification A map f : () (n : N)(e : n N zero) gives us two things: 16 / 52
68 Evidence of unification A map f : () (n : N)(e : n N zero) gives us two things: 1. A value for n such that n N zero 16 / 52
69 Evidence of unification A map f : () (n : N)(e : n N zero) gives us two things: 1. A value for n such that n N zero 2. Explicit evidence e of n N zero 16 / 52
70 Evidence of unification A map f : () (n : N)(e : n N zero) gives us two things: 1. A value for n such that n N zero 2. Explicit evidence e of n N zero = Unification is guaranteed to respect! 16 / 52
71 Three valid unifiers f 1 : (k : N) (k n : N)(e : k N n) f 1 k = k; k; refl f 2 : () (k n : N)(e : k N n) f 2 () = zero; zero; refl f 3 : (k n : N) (k n : N)(e : k N n) f 3 k n = k; k; refl 17 / 52
72 Most general unifiers A most general unifier of ū and v is a unifier σ such that for any σ with ūσ = vσ, there is a ν such that σ = σ ν. 18 / 52
73 Most general unifiers A most general unifier of ū and v is a unifier σ such that for any σ with ūσ = vσ, there is a ν such that σ = σ ν. This is quite difficult to translate to type theory directly / 52
74 Most general unifiers A most general unifier of ū and v is a unifier σ such that for any σ with ūσ = vσ, there is a ν such that σ = σ ν. This is quite difficult to translate to type theory directly... Intuition: if f : Γ Γ(ē : ū v) is MGU, we can go back from Γ(ē : ū v) to Γ without losing any information. 18 / 52
75 Equivalences A function f : A B is an equivalence if it has both a left and a right inverse: islinv : (x : A) g 1 (f x) A x isrinv : (y : B) f (g 2 y) B y In this case, we write f : A B. 19 / 52
76 Most general unifiers are equivalences! f : Γ(ē : ū v) Γ 20 / 52
77 Example of unification (k n : N)(e : suc k N suc n) 21 / 52
78 Example of unification (k n : N)(e : suc k N suc n) (k n : N)(e : k N n) 21 / 52
79 Example of unification (k n : N)(e : suc k N suc n) (k n : N)(e : k N n) (k : N) 21 / 52
80 Example of unification (k n : N)(e : suc k N suc n) (k n : N)(e : k N n) (k : N) f : (k : N) (k n : N)(e : suc k N suc n) f k = k; k; refl 21 / 52
81 The solution rule solution : (x : A)(e : x A t) () 22 / 52
82 The deletion rule deletion : (e : t A t) () 23 / 52
83 The injectivity rule injectivity suc : (e : suc x N suc y) (e : x N y) 24 / 52
84 Negative unification rules A negative unification rule applies to impossible equations, e.g. suc x = zero. 25 / 52
85 Negative unification rules A negative unification rule applies to impossible equations, e.g. suc x = zero. This can be represented by an equivalence: (e : suc x N zero) where is the empty type. 25 / 52
86 The conflict rule conflict suc,zero : (e : suc x N zero) 26 / 52
87 The cycle rule cycle n,suc n : (e : n N suc n) 27 / 52
88 Unifiers as equivalences By requiring unifiers to be equivalences: we exclude bad unification rules we can safely introduce new rules 28 / 52
89 Unifiers as equivalences By requiring unifiers to be equivalences: we exclude bad unification rules we can safely introduce new rules Next, we ll explore how this idea can help us. Any questions so far? 28 / 52
90 Depending on equations Checking dependently typed programs Unification in dependent type theory Unification of dependently typed terms
91 Time for the interesting bits! Equations between types Heterogeneous equations Equations on indexed datatypes Equations between equations 29 / 52
92 Equations between types Types are first-class terms of type Type: Bool : Type, N : Type, N N : Type, / 52
93 Equations between types Types are first-class terms of type Type: Bool : Type, N : Type, N N : Type,... We can form equations between types, e.g. Bool Type Bool. 30 / 52
94 Equations between types Types are first-class terms of type Type: Bool : Type, N : Type, N N : Type,... We can form equations between types, e.g. Bool Type Bool. Q: Can we apply the deletion rule? 30 / 52
95 Equations between types Types are first-class terms of type Type: Bool : Type, N : Type, N N : Type,... We can form equations between types, e.g. Bool Type Bool. Q: Can we apply the deletion rule? A: Depends on which type theory we use! 30 / 52
96 The univalence axiom (2009) Vladimir Voevodsky 31 / 52
97 The univalence axiom (2009) Isomorphic types can be identified. Vladimir Voevodsky 31 / 52
98 The univalence axiom (2009) Isomorphic types can be identified. Vladimir Voevodsky (A B) (A B) 31 / 52
99 The univalence axiom (2009) Bool is equal to Bool in two ways: Bool true false 32 / 52
100 The univalence axiom (2009) Bool is equal to Bool in two ways: Bool true false true Bool false 32 / 52
101 The univalence axiom (2009) Bool is equal to Bool in two ways: Bool true false true Bool false 32 / 52
102 The univalence axiom (2009) Bool is equal to Bool in two ways: Bool true false true Bool false 32 / 52
103 Limiting the deletion rule The deletion rule does not always hold: there might be multiple proofs of x A x. E.g. Bool Type Bool has two elements. 33 / 52
104 Limiting the deletion rule The deletion rule does not always hold: there might be multiple proofs of x A x. E.g. Bool Type Bool has two elements. We cannot use deletion in this case! 33 / 52
105 Heterogeneous equations Σ n:n Vec A n is the type of pairs (n, xs) where n : N and xs : Vec A n. 34 / 52
106 Heterogeneous equations Σ n:n Vec A n is the type of pairs (n, xs) where n : N and xs : Vec A n. (e : (0, nil) Σn:N Vec A n (1, cons 0 x xs)) (e 1 : 0 N 1)(e 2 : nil Vec A??? cons 0 x xs) 34 / 52
107 Heterogeneous equations Σ n:n Vec A n is the type of pairs (n, xs) where n : N and xs : Vec A n. (e : (0, nil) Σn:N Vec A n (1, cons 0 x xs)) (e 1 : 0 N 1)(e 2 : nil Vec A??? cons 0 x xs) What is the type of e 2? 34 / 52
108 Heterogeneous equations Solution: use equation variables as placeholders for their solutions: (e : (0, nil) Σn:N Vec A n (1, cons 0 x xs)) (e 1 : 0 N 1)(e 2 : nil Vec A e1 cons 0 x xs) 35 / 52
109 Heterogeneous equations Solution: use equation variables as placeholders for their solutions: (e : (0, nil) Σn:N Vec A n (1, cons 0 x xs)) (e 1 : 0 N 1)(e 2 : nil Vec A e1 cons 0 x xs) This is called a telescopic equality. 35 / 52
110 Be careful with heterogeneous equations! (e : (Bool, true) ΣA:Type A (Bool, false)) 36 / 52
111 Be careful with heterogeneous equations! (e : (Bool, true) ΣA:Type A (Bool, false)) (e 1 : Bool Type Bool)(e 2 : true e1 false) 36 / 52
112 Be careful with heterogeneous equations! (e : (Bool, true) ΣA:Type A (Bool, false)) (e 1 : Bool Type Bool)(e 2 : true e1 false) 36 / 52
113 Be careful with heterogeneous equations! (e : (Bool, true) ΣA:Type A (Bool, false)) (e 1 : Bool Type Bool)(e 2 : true e1 false) The conflict rule does not apply! 36 / 52
114 Be careful with heterogeneous equations! (e : (Bool, true) ΣA:Type Bool (Bool, false)) 37 / 52
115 Be careful with heterogeneous equations! (e : (Bool, true) ΣA:Type Bool (Bool, false)) (e 1 : Bool Type Bool)(e 2 : true Bool false) 37 / 52
116 Be careful with heterogeneous equations! (e : (Bool, true) ΣA:Type Bool (Bool, false)) (e 1 : Bool Type Bool)(e 2 : true Bool false) Whether a unification rule can be applied depends on the type of the equation! 37 / 52
117 Injectivity for indexed data Do standard unification rules apply to constructors of indexed datatypes? (e : cons n x xs Vec A (suc n) cons n y ys)??? 38 / 52
118 Injectivity for indexed data Idea: simplify equations between indices together with equation between constructors: (e 1 : suc k N suc n) (e 2 : cons k x xs Vec A e1 cons n y ys) 39 / 52
119 Injectivity for indexed data Idea: simplify equations between indices together with equation between constructors: (e 1 : suc k N suc n) (e 2 : 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 e 1 ys) 39 / 52
120 Injectivity for indexed data Idea: simplify equations between indices together with equation between constructors: (e 1 : suc k N suc n) (e 2 : 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 e 1 ys) Length of the Vec must be fully general: must be an equation variable. 39 / 52
121 The image datatype The type Im f y consists of elements image x such that f x = y: data Im (f : A B) : B Type where image : (x : A) Im f (f x) 40 / 52
122 Solving unsolvable equations (x 1 x 2 : A)(e 1 : f x 1 B f x 2 ) (e 2 : image x 1 Im f e1 image x 2 ) 41 / 52
123 Solving unsolvable equations (x 1 x 2 : A)(e 1 : f x 1 B f x 2 ) (e 2 : image x 1 Im f e1 image x 2 ) (x 1 x 2 : A)(e : x 1 A x 2 ) 41 / 52
124 Solving unsolvable equations (x 1 x 2 : A)(e 1 : f x 1 B f x 2 ) (e 2 : image x 1 Im f e1 image x 2 ) (x 1 x 2 : A)(e : x 1 A x 2 ) (x 1 : A) 41 / 52
125 What if the indices are not fully general? (e : cons n x xs Vec A (suc n) cons n y ys) 42 / 52
126 What if the indices are not fully general? (e : cons n x xs Vec A (suc n) cons n y ys) (e 1 : suc n N suc n) (e 2 : cons n x xs Vec A e1 cons n y ys) (p : e 1 suc n N suc n refl) 42 / 52
127 What if the indices are not fully general? (e : cons n x xs Vec A (suc n) cons n y ys) (e 1 : suc n N suc n) (e 2 : cons n x xs Vec A e1 cons n y ys) (p : e 1 suc n N suc 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 N suc n refl) 42 / 52
128 What if the indices are not fully general? (e : cons n x xs Vec A (suc n) cons n y ys) (e 1 : suc n N suc n) (e 2 : cons n x xs Vec A e1 cons n y ys) (p : e 1 suc n N suc 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 N suc n refl) 42 / 52
129 Higher-dimensional equations (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 N suc n refl) We call an equation between equality proofs (e.g. p) a higher-dimensional equation. 43 / 52
130 How to solve higher-dimensional equations? Existing unification rules do not apply / 52
131 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 44 / 52
132 Step 1: lower the dimension of equations We replace all equation variables by regular variables: instead of (e 1 : n N n)(e 2 : x A y)(e 3 : xs Vec A e1 ys) (p : cong suc e 1 suc n N suc n refl) let s first consider (k : N)(u : A)(us : Vec A k) (e : suc k N suc n) 45 / 52
133 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) 46 / 52
134 Step 3: lift unifier to higher dimension We lift f to an equivalence f of type (e 1 : n N n)(e 2 : x A y) (e 3 : xs Vec A e1 ys) (p : cong suc e 1 suc n N suc n refl) (e 2 : x A y)(e 3 : xs Vec A n ys) 47 / 52
135 Final result of steps 1-3 (e : cons n x xs Vec A (suc n) cons n y ys) (e 2 : x A y)(e 3 : xs Vec A n ys) 48 / 52
136 Final result of steps 1-3 (e : cons n x xs Vec A (suc n) cons n y ys) (e 2 : x A y)(e 3 : xs Vec A n ys) This is the forcing rule for cons. 48 / 52
137 Lifting equivalences: (mostly) general case Theorem. If we have an equivalence f of type (x : A)(e : b 1 x B x b 2 x) C we can construct f of type (e : u A v)(p : cong b 1 e r B e s cong b 2 e) (e : f u r C f v s) 49 / 52
138 Implementation in Agda This is all used by Agda to check definitions by dependent pattern matching. More general than before Fixed many bugs Implementation matches theory You can try it for yourself: wiki.portal.chalmers.se/agda 50 / 52
139 Conclusion Unification rules should return evidence of their correctness. 51 / 52
140 Conclusion Unification rules should return evidence of their correctness. A most general unifier can be represented internally as an equivalence. 51 / 52
141 Conclusion Unification rules should return evidence of their correctness. A most general unifier can be represented internally as an equivalence. Unification cannot ignore the types! 51 / 52
142 Questions? If you want to know more, you can: Try out Agda: wiki.portal.chalmers.se/agda Look at the source: github.com/agda/agda Read my thesis: Dependent pattern matching and proof-relevant unification (2017) 52 / 52
143 Two applications of unification Filling in implicit arguments Checking definitions by pattern matching 52 / 52
144 Two applications of unification Filling in implicit arguments Higher order Checking definitions by pattern matching First order 52 / 52
145 Two applications of unification Filling in implicit arguments Higher order Syntactic Checking definitions by pattern matching First order Semantic 52 / 52
146 Two applications of unification Filling in implicit arguments Higher order Syntactic MGU optional Checking definitions by pattern matching First order Semantic MGU required 52 / 52
147 Two applications of unification Filling in implicit arguments Higher order Syntactic MGU optional Checking definitions by pattern matching First order Semantic MGU required Focus of this talk 52 / 52
148 Two notions of equality Definitional equality x = y : A Propositional equality e : x A y Weaker Stronger 52 / 52
149 Two notions of equality Definitional equality x = y : A Propositional equality e : x A y Weaker Decidable Stronger Undecidable 52 / 52
150 Two notions of equality Definitional equality x = y : A Propositional equality e : x A y Weaker Decidable Meta-theoretic Stronger Undecidable Internal to theory 52 / 52
151 Two notions of equality Definitional equality x = y : A Propositional equality e : x A y Weaker Decidable Meta-theoretic Implicit Stronger Undecidable Internal to theory Explicit 52 / 52
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