Programming and Proving with Higher Inductive Types

Transcription

1 Programming and Proving with Higher Inductive Types Dan Licata Wesleyan University Department of Mathematics and Computer Science

2 Constructive Type Theory Three senses of constructivity: [Martin-Löf] 2

3 Constructive Type Theory [Martin-Löf] Three senses of constructivity: Non-affirmation of certain classical principles provides axiomatic freedom 2

4 Synthetic geometry Euclid s postulates 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any center and distance. 4. That all right angles are equal to one another. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than to right angles,the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the to right angles. 3

5 Synthetic geometry Euclid s postulates 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any center and distance. 4. That all right angles are equal to one another. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than to right angles,the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the to right angles. Cartesian 3

6 Synthetic geometry Euclid s postulates 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any center and distance. 4. That all right angles are equal to one another. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than to right angles,the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the to right angles. models Cartesian 3

7 Synthetic geometry Euclid s postulates 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any center and distance. 4. That all right angles are equal to one another. models Cartesian 3

8 Synthetic geometry Euclid s postulates 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any center and distance. 4. That all right angles are equal to one another. models Cartesian Spherical 3

9 Synthetic geometry Euclid s postulates 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any center and distance. 4. That all right angles are equal to one another. 5. Two distinct lines meet at two antipodal points. models Cartesian Spherical 3

10 Synthetic mathematics Type theory 1.τ ::= b τ 1 τ 2 2.e ::= x e 1 e 2 λx.e 3.(λx.e)e 2 = e[e 2 /x] 4

11 Synthetic mathematics Type theory 1.τ ::= b τ 1 τ 2 2.e ::= x e 1 e 2 λx.e 3.(λx.e)e 2 = e[e 2 /x] Set-theoretic functions 4

12 Synthetic mathematics Type theory 1.τ ::= b τ 1 τ 2 2.e ::= x e 1 e 2 λx.e 3.(λx.e)e 2 = e[e 2 /x] Set-theoretic functions Domain-theoretic functions 4

13 Synthetic mathematics Type theory Set-theoretic functions 1.τ ::= b τ 1 τ 2 2.e ::= x e 1 e 2 λx.e 3.(λx.e)e 2 = e[e 2 /x] 4.Y(f) = f(y(f)) Domain-theoretic functions 4

14 Constructive Type Theory Three senses of constructivity: 5

15 Constructive Type Theory Three senses of constructivity: Non-affirmation of certain classical principles provides axiomatic freedom 5

16 Constructive Type Theory Three senses of constructivity: Non-affirmation of certain classical principles provides axiomatic freedom Computational interpretation supports software verification and proof automation 5

17 Computational Interpretation There is an algorithm that, given a closed program e : bool, computes either an equality e = true, or an equality e = false. 6

18 Computational Interpretation There is an algorithm that, given a closed program e : bool, computes either an equality e = true, or an equality e = false. Requires functions with arbitrary domain/ range to be computable, but stating theorem for bool offers some flexibility 6

19 Computational Interpretation There is an algorithm that, given a closed program e : bool, computes either an equality e = true, or an equality e = false. Requires functions with arbitrary domain/ range to be computable, but stating theorem for bool offers some flexibility Basis for software verification and proof automation 6

20 Constructive Type Theory Three senses of constructivity: Non-affirmation of certain classical principles provides axiomatic freedom Computational interpretation supports software verification and proof automation 7

21 Constructive Type Theory Three senses of constructivity: Non-affirmation of certain classical principles provides axiomatic freedom Computational interpretation supports software verification and proof automation Props-as-types allows proof-relevant mathematics 7

22 Proof relevance x : A 8

23 Proof relevance x : A x =A y equality type 8

24 Proof relevance x : A p : x =A y equality type 8

25 Proof relevance x : A p : x =A y equality type Any structure or property C can be transported along an equality 8

26 Proof relevance x : A p : x =A y equality type Any structure or property C can be transported along an equality transport C (p) : C(x)! C(y) 8

27 Proof relevance x : A p : x =A y equality type Any structure or property C can be transported along an equality transport C (p) : C(x)! C(y) Leibniz s indiscernability of identicals 8

28 Proof relevance x : A p : x =A y equality type Any structure or property C can be transported along an equality transport C (p) : C(x)! C(y) Leibniz s indiscernability of identicals by a function: can it do real work? 8

29 Proof relevance x : A p : x =A y equality type 9

30 Proof relevance x : A p : x =A y equality type p1 = x=y p2 9

31 Proof relevance x : A p : x =A y equality type q : p1 = x=y p2 9

32 Proof relevance x : A p : x =A y equality type q : p1 = x=y p2 q1 = p1=p2 q2 9

33 Proof relevance x : A p : x =A y equality type q : p1 = x=y p2 r : q1 = p1=p2 q2 9

34 ... Proof relevance x : A p : x =A y equality type q : p1 = x=y p2 r : q1 = p1=p2 q2 higher equalities radically expand the kind of math that can be done synthetically 9

35 Homotopy Type Theory type theory category theory homotopy theory [Hofmann,Streicher,Awodey,Warren,Voevodsky Lumsdaine,Gambino,Garner,van den Berg] 10

36 Types as spaces α M N 11

37 Types as spaces type A is a space α M N 11

38 Types as spaces type A is a space α M N programs M:A are points 11

39 Types as spaces type A is a space α M N programs M:A are points proofs of equality α : M =A N are paths 11

40 Types as spaces type A is a space α path operations M N programs M:A are points proofs of equality α : M =A N are paths 11

41 Types as spaces type A is a space id α N M path operations id : M = M (refl) programs M:A are points proofs of equality α : M =A N are paths 11

42 Types as spaces type A is a space id α α -1 N M path operations id : M = M (refl) α -1 : N = M (sym) programs M:A are points proofs of equality α : M =A N are paths 11

43 Types as spaces type A is a space id α α -1 N M β P path operations id : M = M (refl) α -1 : N = M (sym) β o α : M = P (trans) programs M:A are points proofs of equality α : M =A N are paths 11

44 Homotopy Deformation of one path into another α β 12

45 Homotopy Deformation of one path into another α β 12

46 Homotopy Deformation of one path into another α = 2-dimensional path between paths β 12

47 Homotopy Deformation of one path into another α β = 2-dimensional path between paths δ : α =x=y β 12

48 Homotopy Deformation of one path into another α β = 2-dimensional path between paths δ : α =x=y β Then homotopies between homotopies. 12

49 Types as spaces type A is a space id α α -1 N M β path operations id : M = M (refl) α -1 : N = M (sym) β o α : M = P (trans) programs M:A are points P proofs of equality α : M =A N are paths homotopies ul : id o α = M=N α il : α -1 o α = M=M id asc : γ o (β o α) =M=P (γ o β) o α 13

50 Homotopy Type Theory type theory category theory homotopy theory [Hofmann,Streicher,Awodey,Warren,Voevodsky Lumsdaine,Gambino,Garner,van den Berg] 14

51 Types as -groupoids type A is an -groupoid infinite-dimensional algebraic structure, with morphisms, morphisms between morphisms,... each level has a groupoid structure, and they interact morphisms id : M = M (refl) α -1 : N = M (sym) β o α : M = P (trans) morphisms between morphisms ul : id o α = M=N α il : α -1 o α = M=M id asc : γ o (β o α) =M=P (γ o β) o α 15

52 Path induction Type of paths from a to somewhere y2 p2 y1 y3 p1 p3 a is inductively generated by id a 16

53 Path induction Type of paths from a to somewhere y2 p2 y1 y3 p1 p3 a is inductively generated by id a Fix a type A with element a:a. For a family of types C(y:A, p:a=y), to give an element of C(y,p) for all y and p:a=y, suffices to give an element of C(a,id) 16

54 Type theory is a synthetic theory of spaces/ -groupoids 17

55 Homotopy Type Theory type theory new programs and types category theory homotopy theory 18

56 Univalence [Voevodsky] 19

57 Univalence [Voevodsky] Equivalence of types is a generalization to spaces of bijection of sets 19

58 Univalence [Voevodsky] Equivalence of types is a generalization to spaces of bijection of sets Univalence axiom: equality of types (A =Type B) is (equivalent to) equivalence of types (Equiv A B) 19

59 Univalence [Voevodsky] Equivalence of types is a generalization to spaces of bijection of sets Univalence axiom: equality of types (A =Type B) is (equivalent to) equivalence of types (Equiv A B) all structures/properties respect equivalence 19

60 Univalence [Voevodsky] Equivalence of types is a generalization to spaces of bijection of sets Univalence axiom: equality of types (A =Type B) is (equivalent to) equivalence of types (Equiv A B) all structures/properties respect equivalence Not by collapsing equivalence, but by exploiting proof-relevant equality: transport does real work 19

61 Higher inductive types [Bauer,Lumsdaine,Shulman,Warren] New way of forming types: Inductive type specified by generators not only for points (elements), but also for paths 20

62 Constructivity Non-affirmation of classical principles Computational interpretation? Proof-relevant mathematics 21

63 Homotopy Type Theory type theory new programs and types new possibilities for computerchecked proofs category theory homotopy theory 22

64 Outline 1.Certified homotopy theory 2.Certified software 23

65 Outline 1.Certified homotopy theory 2.Certified software 24

66 Homotopy Theory A branch of topology, the study of spaces and continuous deformations 25 [image from wikipedia]

67 Homotopy in HoTT π1(s 1 ) = Z πk<n(s n ) = 0 Hopf fibration π2(s 2 ) = Z π3(s 2 ) = Z James Construction Freudenthal πn(s n ) = Z K(G,n) Cohomology axioms Blakers-Massey Van Kampen Covering spaces Whitehead for n-types π4(s 3 ) = Z? [Brunerie, Finster, Hou, Licata, Lumsdaine, Shulman] 26

68 Homotopy in HoTT π1(s 1 ) = Z πk<n(s n ) = 0 Hopf fibration π2(s 2 ) = Z π3(s 2 ) = Z James Construction Freudenthal πn(s n ) = Z K(G,n) Cohomology axioms Blakers-Massey Van Kampen Covering spaces Whitehead for n-types π4(s 3 ) = Z? [Brunerie, Finster, Hou, Licata, Lumsdaine, Shulman] 26

69 Homotopy Groups Homotopy groups of a space X: π1(x) is fundamental group (group of loops) π2(x) is group of homotopies (2-dimensional loops) π3(x) is group of 3-dimensional loops 27

70 Telling spaces apart = 28

71 Telling spaces apart = fundamental group is non-trivial (Z Z) fundamental group is trivial 28

72 The Circle Circle S 1 is a higher inductive type generated by loop base 29

73 The Circle Circle S 1 is a higher inductive type generated by base : S 1 loop : base = base base loop 29

74 The Circle Circle S 1 is a higher inductive type generated by point base : S 1 loop : base = base base loop 29

75 The Circle Circle S 1 is a higher inductive type generated by point path base : S 1 loop : base = base base loop 29

76 The Circle Circle S 1 is a higher inductive type generated by point path base : S 1 loop : base = base loop -1 id base loop Free type: equipped with structure id loop -1 inv : loop o loop -1 = id... loop o loop 29

77 The Circle Circle recursion: function S 1! X determined by base loop base : X loop : base = base base loop 30

78 The Circle Circle recursion: function S 1! X determined by base loop base : X loop : base = base base loop Circle induction: To prove a predicate P for all points on the circle, suffices to prove P(base), continuously in the loop 30

79 Fundamental group of circle How many different loops are there on the circle, up to homotopy? base loop 31

80 Fundamental group of circle How many different loops are there on the circle, up to homotopy? id base loop 31

81 Fundamental group of circle How many different loops are there on the circle, up to homotopy? id loop base loop 31

82 Fundamental group of circle How many different loops are there on the circle, up to homotopy? id loop loop -1 base loop 31

83 Fundamental group of circle How many different loops are there on the circle, up to homotopy? id loop loop -1 loop o loop base loop 31

84 Fundamental group of circle How many different loops are there on the circle, up to homotopy? id loop loop -1 loop o loop loop -1 o loop -1 base loop 31

85 Fundamental group of circle How many different loops are there on the circle, up to homotopy? id loop loop -1 loop o loop loop -1 o loop -1 loop o loop -1 base loop 31

86 Fundamental group of circle How many different loops are there on the circle, up to homotopy? id loop loop -1 loop o loop loop -1 o loop -1 base loop loop o loop -1 = id 31

87 Fundamental group of circle How many different loops are there on the circle, up to homotopy? base loop id 0 loop loop -1 loop o loop loop -1 o loop -1 loop o loop -1 = id 31

88 Fundamental group of circle How many different loops are there on the circle, up to homotopy? base loop id loop 0 1 loop -1 loop o loop loop -1 o loop -1 loop o loop -1 = id 31

89 Fundamental group of circle How many different loops are there on the circle, up to homotopy? base loop id loop loop loop o loop loop -1 o loop -1 loop o loop -1 = id 31

90 Fundamental group of circle How many different loops are there on the circle, up to homotopy? base loop id loop loop -1 loop o loop loop -1 o loop -1 loop o loop -1 = id 31

91 Fundamental group of circle How many different loops are there on the circle, up to homotopy? base loop id loop loop -1 loop o loop loop -1 o loop loop o loop -1 = id 31

92 Fundamental group of circle How many different loops are there on the circle, up to homotopy? base loop id loop loop -1 loop o loop loop -1 o loop -1 loop o loop -1 = id

93 Fundamental group of circle How many different loops are there on the circle, up to homotopy? base loop id loop loop -1 loop o loop loop -1 o loop integers are codes for paths on the circle loop o loop -1 = id 0 31

94 Fundamental group of circle Definition. Ω(S 1 ) is the type of loops at base i.e. the type (base =S1 base) 32

95 Fundamental group of circle Definition. Ω(S 1 ) is the type of loops at base i.e. the type (base =S1 base) Theorem. Ω(S 1 ) is equivalent to Z, by a map that sends o to + 32

96 Fundamental group of circle Definition. Ω(S 1 ) is the type of loops at base i.e. the type (base =S1 base) Theorem. Ω(S 1 ) is equivalent to Z, by a map that sends o to + Corollary: Fundamental group of the circle is isomorphic to Z 32

97 Fundamental group of circle Definition. Ω(S 1 ) is the type of loops at base i.e. the type (base =S1 base) Theorem. Ω(S 1 ) is equivalent to Z, by a map that sends o to + Corollary: Fundamental group of the circle is isomorphic to Z 0-truncation (set of connected components) of Ω(S 1 ) 32

98 Fundamental group of circle Theorem. Ω(S 1 ) is equivalent to Z Proof (Shulman, L.): two mutually inverse functions wind : Ω(S 1 )! Z loop - : Z! Ω(S 1 ) 33

99 Fundamental group of circle Theorem. Ω(S 1 ) is equivalent to Z Proof (Shulman, L.): two mutually inverse functions wind : Ω(S 1 )! Z loop - : Z! Ω(S 1 ) loop 0 = id loop +n = loop o loop o loop (n times) loop -n = loop -1 o loop -1 o loop -1 (n times) 33

100 Universal Cover wind : Ω(S 1 )! Z defined by lifting a loop to the cover, and giving the other endpoint of 0 34

101 Universal Cover wind : Ω(S 1 )! Z defined by lifting a loop to the cover, and giving the other endpoint of 0 lifting is functorial 34

102 Universal Cover wind : Ω(S 1 )! Z defined by lifting a loop to the cover, and giving the other endpoint of 0 lifting is functorial lifting loop adds 1 34

103 Universal Cover wind : Ω(S 1 )! Z defined by lifting a loop to the cover, and giving the other endpoint of 0 lifting is functorial lifting loop adds 1 lifting loop -1 subtracts 1 34

104 Universal Cover wind : Ω(S 1 )! Z defined by lifting a loop to the cover, and giving the other endpoint of 0 lifting is functorial lifting loop adds 1 lifting loop -1 subtracts 1 Example: wind(loop o loop -1 ) = = 0 34

105 Universal Cover 35

106 Universal Cover Cover : S 1! Type Cover(base) := Z Cover1(loop) := ua(successor) : Z = Z 35

107 Universal Cover defined by circle recursion Cover : S 1! Type Cover(base) := Z Cover1(loop) := ua(successor) : Z = Z 35

108 Universal Cover defined by circle recursion Cover : S 1! Type Cover(base) := Z Cover1(loop) := ua(successor) : Z = Z interpret loop as add 1 bijection 35

109 Universal Cover defined by circle recursion Cover : S 1! Type Cover(base) := Z Cover1(loop) := ua(successor) : Z = Z univalence interpret loop as add 1 bijection 35

110 Winding number wind : Ω(S 1 )! Z wind(p) = transportcover(p,0) lift p to cover, starting at 0 36

111 Winding number wind : Ω(S 1 )! Z wind(p) = transportcover(p,0) lift p to cover, starting at 0 wind(loop -1 o loop) 36

112 Winding number wind : Ω(S 1 )! Z wind(p) = transportcover(p,0) lift p to cover, starting at 0 wind(loop -1 o loop) = transport Cover (loop -1 o loop, 0) 36

113 Winding number wind : Ω(S 1 )! Z wind(p) = transportcover(p,0) lift p to cover, starting at 0 wind(loop -1 o loop) = transport Cover (loop -1 o loop, 0) = transport Cover (loop -1, transport Cover (loop,0)) 36

114 Winding number wind : Ω(S 1 )! Z wind(p) = transportcover(p,0) lift p to cover, starting at 0 wind(loop -1 o loop) = transport Cover (loop -1 o loop, 0) = transport Cover (loop -1, transport Cover (loop,0)) = transport Cover (loop -1, 1) 36

115 Winding number wind : Ω(S 1 )! Z wind(p) = transportcover(p,0) lift p to cover, starting at 0 wind(loop -1 o loop) = transport Cover (loop -1 o loop, 0) = transport Cover (loop -1, transport Cover (loop,0)) = transport Cover (loop -1, 1) = 0 36

116 Fundamental group of the circle The HoTT book Computer-checked 37

117 πn(s n ) in HoTT k th homotopy group n-dimensional sphere [image from wikipedia] 38

118 πn(s n ) in HoTT k th homotopy group n-dimensional sphere [image from wikipedia] 38

119 πn(s n ) = Z for n 1 Proof: Induction on n Base case: π1(s 1 ) = Z Inductive step: πn+1(s n+1 ) = πn(s n ) 39

120 πn(s n ) = Z for n 1 Proof: Induction on n Base case: π1(s 1 ) = Z Inductive step: πn+1(s n+1 ) = πn(s n ) Key lemma: S n n = Ω(S n+1 ) n 39

121 πn(s n ) = Z for n 1 Proof: Induction on n Base case: π1(s 1 ) = Z Inductive step: πn+1(s n+1 ) = πn(s n ) Key lemma: S n n = Ω(S n+1 ) n n-truncation: best approximation of a type such that all (n+1)-paths are equal 39

122 πn(s n ) = Z for n 1 Proof: Induction on n Base case: π1(s 1 ) = Z Inductive step: πn+1(s n+1 ) = πn(s n ) Key lemma: S n n = Ω(S n+1 ) n n-truncation: best approximation of a type such that all (n+1)-paths are equal higher inductive type generated by basen : S n loopn : Ω n (S n ) 39

123 S n n = Ω(S n+1 ) n n-truncation of S n is the type of codes for loops on S n+1

124 S n n = Ω(S n+1 ) n n-truncation of S n is the type of codes for loops on S n+1 Decode: promote n-dimensional loop on S n to n+1-dimensional loop on S n+1

125 S n n = Ω(S n+1 ) n n-truncation of S n is the type of codes for loops on S n+1 Decode: promote n-dimensional loop on S n to n+1-dimensional loop on S n+1 Encode: define fibration Code(x:S n+1 ) with Code(basen+1) := S n n Code(loopn+1) := equivalence S n n S n n rotating by loopn

126 π2(s 2 ): Hopf fibration

127 Synthetic homotopy theory Gap between informal and formal proofs is small Proofs are constructive*: can run them Results apply in a variety of settings, from simplicial sets (hence topological spaces) to Quillen model categories and -topoi* New type-theoretic proofs/methods 42 *work in progress

128 Outline 1.Certified homotopy theory 2.Certified software 43

129 Patches Patch a a 2c2 diff b c d c = < b --- > d Version control Collaborative editing 44

130 a b c id a b c

131 a b c id a b c a p a q a b d d c c e

132 a b c id a b c q o p a p a q a b d d c c e

133 a b c id a b c a b c p a d c q o p a p a q a b d d c c e

134 a b c id a b c a b c p a d c!p q o p a p a q a b d d c c e

135 a b c id a b c a b c p a d c!p q o p undo/rollback a p a q a b d d c c e

136 Patches are paths a b c id a b c a b c p a d c!p q o p undo/rollback a p a q a b d d c c e

137 Merging a d c p a b c q a b e 46

138 Merging a d c p a b c q a b e q a p d e 46

139 Merging p=b d at 1 p a b q q=c e at 2 c a d c a b e q a p d e 46

140 Merging p=b d at 1 p a b q q=c e at 2 c a d c a b e q a p d e p =p q =q 46

141 Merging p=b d at 1 p a b q q=c e at 2 c a d c = a b e q a p d e p =p q =q 46

142 Merging merge : (p q : Patch)! Σq,p :Patch. Maybe(q o p = p o q) 47

143 Merging merge : (p q : Patch)! Σq,p :Patch. Maybe(q o p = p o q) Equational theory of patches = paths between paths 47

144 Basic Patches f i b r a t i o n a 2 f i b f i a a 2 48

145 Basic Patches Repository is a char vector of length n f i b r a t i o n Basic patch is a i where i<n a 2 f i b f i a a 2 48

146 Patches as a HIT Repos:Type 49

147 Patches as a HIT Repos:Type doc[n] points describe repository contents 49

148 Patches as a HIT Repos:Type a b@i doc[n] points describe repository contents paths are patches 49

149 Patches as a HIT Repos:Type compressed a b@i doc[n] points describe repository contents paths are patches 49

150 Patches as a HIT Repos:Type compressed gzip a b@i doc[n] points describe repository contents paths are patches 49

151 Patches as a HIT Repos:Type compressed gzip a b@i doc[n] points describe repository contents paths are patches 49

152 Patches as a HIT Repos:Type compressed gzip a b@i doc[n] points describe repository contents paths are patches doc[n] doc[n] doc[n] doc[n] 49

153 Patches as a HIT Repos:Type compressed gzip a b@i doc[n] points describe repository contents paths are patches doc[n] a b@i c d@j doc[n] doc[n] c d@j a b@i doc[n] 49

154 Patches as a HIT Repos:Type compressed gzip a b@i doc[n] points describe repository contents paths are patches paths between paths are equations between patches doc[n] a b@i c d@j doc[n] doc[n] c d@j a b@i doc[n] 49

155 Patches as a HIT Repos:Type compressed gzip a b@i doc[n] points describe repository contents paths are patches paths between paths are equations between patches doc[n] a b@i c d@j doc[n] doc[n] c d@j a b@i doc[n] i j 49

156 Generators for HIT 50

157 Generators for HIT Repos : Type 50

158 Generators for HIT Repos : Type doc[n] : Repos compressed : Repos 50

159 Generators for HIT Repos : Type doc[n] : Repos compressed : Repos (a b@i) : doc[n] = doc[n] if a,b:char, i<n gzip : doc[n] = compressed 50

160 Generators for HIT Repos : Type doc[n] : Repos compressed : Repos (a b@i) : doc[n] = doc[n] if a,b:char, i<n gzip : doc[n] = compressed commute: (a b at i)o(c d at j) if i j =(c d at j)o(a b at i) 50

161 Type: Patch Type: Repos Elements: Points: Paths: doc[n] a b@i Equality: (a b at i)o(c d at j)= (c d at j)o(a b at i)... id o p = p = p o id po(qor) = (poq)or Paths between paths: commute : (a b at i)o(c d at j)= (c d at j)o(a b at i)!p o p = id = p o!p p=p p=q if q=p p=r if p=q and q=r!p =!p if p = p p o q = p o q if p = p and q = q 51

162 Repos recursion To define a function it suffices to Repos! A 52

163 Repos recursion To define a function it suffices to Repos! A map the element generators of Repos to elements of A 52

164 Repos recursion To define a function it suffices to Repos! A map the element generators of Repos to elements of A map the equality generators of Repos to equalities between the corresponding elements of A 52

165 Repos recursion To define a function it suffices to Repos! A map the element generators of Repos to elements of A map the equality generators of Repos to equalities between the corresponding elements of A map the equality-between-equality generators to equalities between the corresponding equalities in A 52

166 Repos recursion To define a function it suffices to Repos! A map the element generators of Repos to elements of A map the equality generators of Repos to equalities between the corresponding elements of A map the equality-between-equality generators to equalities between the corresponding equalities in A All functions on Repos respect patches All functions on patches respect patch equality 52

167 Interpreter Goal is to define: interp : doc[n] = doc[n]! Bijection (Vec Char n) (Vec Char n) 53

168 Interpreter Goal is to define: interp : doc[n] = doc[n]! Bijection (Vec Char n) (Vec Char n) interp(id) = (λx.x, ) interp(q o p) = (interp q) ob (interp p) interp(!p) =!b (interp p) interp(a b@i) = swapat a b i 53

169 Interpreter Goal is to define: interp : doc[n] = doc[n]! Bijection (Vec Char n) (Vec Char n) interp(id) = (λx.x, ) interp(q o p) = (interp q) ob (interp p) interp(!p) =!b (interp p) interp(a b@i) = swapat a b i But only tool available is RepoDesc recursion: no direct recursion over paths 53

170 interp : doc[n]=doc[n]! Bijection (Vec Char n) (Vec Char n) interp(a b at i) = swapat a b i Need to pick A and define I(doc[n]) := : A I 1 (a b@i) := : I(doc[n]) = I(doc[n]) I2(compose) := 54

171 interp : doc[n]=doc[n]! Bijection (Vec Char n) (Vec Char n) interp(a b at i) = swapat a b i Key idea: pick A = Type and define I(doc[n]) := : Type I 1 (a b@i) := : I(doc[n]) = I(doc[n]) I2(compose) := 55

172 interp : doc[n]=doc[n]! Bijection (Vec Char n) (Vec Char n) interp(a b at i) = swapat a b i Key idea: pick A = Type and define I(doc[n]) := Vec Char n : Type I 1 (a b@i) := : I(doc[n]) = I(doc[n]) I2(compose) := 56

173 interp : doc[n]=doc[n]! Bijection (Vec Char n) (Vec Char n) interp(a b at i) = swapat a b i Key idea: pick A = Type and define I(doc[n]) := Vec Char n : Type I 1 (a b@i) := : Vec Char n = Vec Char n I2(compose) := 57

174 interp : doc[n]=doc[n]! Bijection (Vec Char n) (Vec Char n) interp(a b at i) = swapat a b i Key idea: pick A = Type and define I(doc[n]) := Vec Char n : Type I 1 (a b@i) := ua(swapat a b i) : Vec Char n = Vec Char n I2(compose) := 58

175 interp : doc[n]=doc[n]! Bijection (Vec Char n) (Vec Char n) interp(a b at i) = swapat a b i Key idea: pick A = Type and define I(doc[n]) := Vec Char n : Type I 1 (a b@i) := ua(swapat a b i) : Vec Char n = Vec Char n I2(compose) := univalence 58

176 interp : doc[n]=doc[n]! Bijection (Vec Char n) (Vec Char n) interp(a b at i) = swapat a b i Key idea: pick A = Type and define I(doc[n]) := Vec Char n : Type I 1 (a b@i) := ua(swapat a b i) : Vec Char n = Vec Char n I2(compose) := <proof about swapat> 59

177 interp : doc[n]=doc[n]! Bijection (Vec Char n) (Vec Char n) interp(p) = ua -1 (I1(p)) Key idea: pick A = Type and define I(doc[n]) := Vec Char n : Type I 1 (a b@i) := ua(swapat a b i) : Vec Char n = Vec Char n I2(compose) := <proof about swapat> 60

178 interp : doc[n]=doc[n]! Bijection (Vec Char n) (Vec Char n) interp(p) = ua -1 (I1(p)) Satisfies the desired equations (as propositional equalities): interp(id) = (λx.x, ) interp(q o p) = (interp q) ob (interp p) interp(!p) =!b (interp p) interp(a b@i) = swapat a b i 61

179 Summary 62

180 Summary I : Repos! Type interprets Repos as Types, patches as bijections, satisfying patch equalities 62

181 Summary I : Repos! Type interprets Repos as Types, patches as bijections, satisfying patch equalities Higher inductive elim. defines functions that respect equality: you specify what happens on the generators; homomorphically extended to id,o,!,... 62

182 Summary I : Repos! Type interprets Repos as Types, patches as bijections, satisfying patch equalities Higher inductive elim. defines functions that respect equality: you specify what happens on the generators; homomorphically extended to id,o,!,... Univalence lets you give a computational model of equality proofs (here, patches); guaranteed to satisfy laws 62

183 Summary I : Repos! Type interprets Repos as Types, patches as bijections, satisfying patch equalities Higher inductive elim. defines functions that respect equality: you specify what happens on the generators; homomorphically extended to id,o,!,... Univalence lets you give a computational model of equality proofs (here, patches); guaranteed to satisfy laws Shorter definition and code: 1 basic patch & 4 basic axioms of equality, instead of 4 patches & 14 equations 62

184 Operational semantics Can t run these programs yet Some special cases known, some recent progress: Licata&Harper, POPL 12 Coquand&Barras, 13 Shulman, 13 Bezem&Coquand&Huber, 13 Would support proof automation and programming applications 63

185 Outline 1.Certified homotopy theory 2.Certified software 64

186 Homotopy Type Theory type theory new programs and types category theory new certified programs and proofs homotopy theory 65