Proof Assistant Projects
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Digesting proof assistant libraries for AI ingestion. β’ 103 items β’ Updated β’ 3
fact stringlengths 5 876 | statement stringlengths 5 595 | proof stringlengths 0 691 | type stringclasses 2
values | symbolic_name stringlengths 1 27 | library stringclasses 1
value | filename stringclasses 32
values | imports listlengths 0 2 | deps listlengths 0 8 | docstring stringclasses 32
values | line_start int64 1 228 | line_end int64 1 229 | has_proof bool 2
classes | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
+ : nat β nat β nat :=
elim [
| zero => n => n
| suc {_ => ih} => n => suc {ih n}
] | + : nat β nat β nat | :=
elim [
| zero => n => n
| suc {_ => ih} => n => suc {ih n}
] | def | + | test | test/abstract.cooltt | [
"hlevel",
"prelude"
] | [] | null | 6 | 10 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
+0L : (x : nat) β path nat {+ 0 x} x :=
x _ => x | +0L : (x : nat) β path nat {+ 0 x} x | :=
x _ => x | def | +0L | test | test/abstract.cooltt | [
"hlevel",
"prelude"
] | [
"+",
"path"
] | null | 14 | 15 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
+SL : (x y : nat) β path nat {+ {suc x} y} {suc {+ x y}} :=
x y _ => suc {+ x y} | +SL : (x y : nat) β path nat {+ {suc x} y} {suc {+ x y}} | :=
x y _ => suc {+ x y} | def | +SL | test | test/abstract.cooltt | [
"hlevel",
"prelude"
] | [
"+",
"path"
] | null | 19 | 20 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
+0R : (x : nat) β path nat {+ x 0} x :=
elim [
| zero => +0L 0
| suc {x => ih} =>
equation nat begin
+ {suc x} 0 =[ +SL x 0 ]
suc {+ x 0} =[ i => suc {ih i} ]
suc x
end
] | +0R : (x : nat) β path nat {+ x 0} x | :=
elim [
| zero => +0L 0
| suc {x => ih} =>
equation nat begin
+ {suc x} 0 =[ +SL x 0 ]
suc {+ x 0} =[ i => suc {ih i} ]
suc x
end
] | def | +0R | test | test/abstract.cooltt | [
"hlevel",
"prelude"
] | [
"+",
"+0L",
"+SL",
"path"
] | null | 23 | 32 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
+SR : (x y : nat) β path nat {+ x {suc y}} {suc {+ x y}} :=
elim [
| zero =>
y =>
equation nat begin
+ 0 {suc y} =[ +0L {suc y} ]
suc y =[ i => suc {symm nat {+0L y} i} ]
suc {+ 0 y}
end
| suc {x => ih} =>
y =>
equation nat begin
+ {suc x} {suc y} =[ +SL x {suc y} ]
... | +SR : (x y : nat) β path nat {+ x {suc y}} {suc {+ x y}} | :=
elim [
| zero =>
y =>
equation nat begin
+ 0 {suc y} =[ +0L {suc y} ]
suc y =[ i => suc {symm nat {+0L y} i} ]
suc {+ 0 y}
end
| suc {x => ih} =>
y =>
equation nat begin
+ {suc x} {suc y} =[ +SL x {suc y} ]
suc {+ x {suc y}} =[ i => suc {ih y i} ]
suc {su... | def | +SR | test | test/abstract.cooltt | [
"hlevel",
"prelude"
] | [
"+",
"+0L",
"+SL",
"path",
"symm"
] | null | 35 | 52 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
+A : (x y z : nat) β path nat {+ {+ x y} z} {+ x {+ y z}} :=
elim [
| zero =>
y z =>
equation nat begin
+ {+ 0 y} z =[ i => + {+0L y i} z ]
+ y z =[ symm nat {+0L {+ y z}} ]
+ 0 {+ y z}
end
| suc {x => ih} =>
y z =>
equation nat begin
+ {+ {suc x} y} z =[ i => + {+SL x ... | +A : (x y z : nat) β path nat {+ {+ x y} z} {+ x {+ y z}} | :=
elim [
| zero =>
y z =>
equation nat begin
+ {+ 0 y} z =[ i => + {+0L y i} z ]
+ y z =[ symm nat {+0L {+ y z}} ]
+ 0 {+ y z}
end
| suc {x => ih} =>
y z =>
equation nat begin
+ {+ {suc x} y} z =[ i => + {+SL x y i} z ]
+ {suc {+ x y}} z =[ +SL {+ x y} z ]
... | def | +A | test | test/abstract.cooltt | [
"hlevel",
"prelude"
] | [
"+",
"+0L",
"+SL",
"path",
"symm"
] | null | 55 | 73 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
natβ·set : set # [tp := nat] | natβ·set : set # [tp := nat] | axiom | natβ·set | test | test/abstract.cooltt | [
"hlevel",
"prelude"
] | [
"set"
] | null | 79 | 79 | false | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
foo (p : path nat {+ 0 0} 0) : unfold + in path {path nat {+ 0 0} 0} p {_ => 0} :=
unfold + has-hlevel in
natβ·set 0 0 p {_ => 0} | foo (p : path nat {+ 0 0} 0) : unfold + in path {path nat {+ 0 0} 0} p {_ => 0} | :=
unfold + has-hlevel in
natβ·set 0 0 p {_ => 0} | def | foo | test | test/abstract.cooltt | [
"hlevel",
"prelude"
] | [
"+",
"has-hlevel",
"natβ·set",
"path"
] | null | 81 | 83 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
two : nat := + 1 1 | two : nat | := + 1 1 | def | two | test | test/abstract.cooltt | [
"hlevel",
"prelude"
] | [
"+"
] | null | 87 | 87 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
test : path nat two 2 :=
unfold two + in
i => 2 | test : path nat two 2 | :=
unfold two + in
i => 2 | def | test | test | test/abstract.cooltt | [
"hlevel",
"prelude"
] | [
"+",
"path",
"two"
] | null | 89 | 91 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
test2 : path-p {i => path nat two {test i}} {_ => two} test :=
i j =>
unfold two + in
2 | test2 : path-p {i => path nat two {test i}} {_ => two} test | :=
i j =>
unfold two + in
2 | def | test2 | test | test/abstract.cooltt | [
"hlevel",
"prelude"
] | [
"+",
"path",
"path-p",
"test",
"two"
] | null | 94 | 97 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
abs-test : nat :=
suc {abstract abs-test::foo β 41} | abs-test : nat | :=
suc {abstract abs-test::foo β 41} | def | abs-test | test | test/abstract.cooltt | [
"hlevel",
"prelude"
] | [] | null | 101 | 102 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
monoid : type :=
sig
def t : type
def op : t β t β t
def emp : t
def opL : (u : t) β path t {op emp u} u
def opR : (u : t) β path t {op u emp} u
def opA : (u v w : t) β path t {op {op u v} w} {op u {op v w}}
end | monoid : type | :=
sig
def t : type
def op : t β t β t
def emp : t
def opL : (u : t) β path t {op emp u} u
def opR : (u : t) β path t {op u emp} u
def opA : (u v w : t) β path t {op {op u v} w} {op u {op v w}}
end | def | monoid | test | test/algebra.cooltt | [
"nat",
"prelude"
] | [
"path"
] | null | 5 | 13 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
commutative-monoid : type :=
sig
include monoid
def opC : (u v : t) β path t {op u v} {op v u}
end | commutative-monoid : type | :=
sig
include monoid
def opC : (u v : t) β path t {op u v} {op v u}
end | def | commutative-monoid | test | test/algebra.cooltt | [
"nat",
"prelude"
] | [
"monoid",
"path"
] | null | 16 | 20 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
multiplicative-monoid : type :=
sig
include monoid
renaming [op β mul; emp β one; opL β mulL; opR β mulR; opA β mulA]
end | multiplicative-monoid : type | :=
sig
include monoid
renaming [op β mul; emp β one; opL β mulL; opR β mulR; opA β mulA]
end | def | multiplicative-monoid | test | test/algebra.cooltt | [
"nat",
"prelude"
] | [
"monoid"
] | null | 22 | 26 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
additive-monoid : type :=
sig
include monoid
renaming [op β add; emp β zer; opL β addL; opR β addR; opA β addA]
end | additive-monoid : type | :=
sig
include monoid
renaming [op β add; emp β zer; opL β addL; opR β addR; opA β addA]
end | def | additive-monoid | test | test/algebra.cooltt | [
"nat",
"prelude"
] | [
"monoid"
] | null | 28 | 32 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
abelian-group : type :=
sig
include additive-monoid
def neg : t β t
def addC : (u v : t) β path t {add u v} {add v u}
def add-neg : (u : t) β path t {add {neg u} u} zer
end | abelian-group : type | :=
sig
include additive-monoid
def neg : t β t
def addC : (u v : t) β path t {add u v} {add v u}
def add-neg : (u : t) β path t {add {neg u} u} zer
end | def | abelian-group | test | test/algebra.cooltt | [
"nat",
"prelude"
] | [
"additive-monoid",
"path"
] | null | 34 | 40 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
ring : type :=
sig
def t : type
include abelian-group # [ t ::= t ]
include multiplicative-monoid # [ t ::= t ]
end | ring : type | :=
sig
def t : type
include abelian-group # [ t ::= t ]
include multiplicative-monoid # [ t ::= t ]
end | def | ring | test | test/algebra.cooltt | [
"nat",
"prelude"
] | [
"abelian-group",
"multiplicative-monoid"
] | null | 42 | 47 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
monoid/nat : monoid # [t := nat] :=
struct
def op := +
def emp := 0
def opL := +-left-unit
def opR := +-right-unit
def opA := +-assoc
end | monoid/nat : monoid # [t := nat] | :=
struct
def op := +
def emp := 0
def opL := +-left-unit
def opR := +-right-unit
def opA := +-assoc
end | def | monoid/nat | test | test/algebra.cooltt | [
"nat",
"prelude"
] | [
"+",
"+-assoc",
"+-left-unit",
"+-right-unit",
"monoid"
] | null | 49 | 56 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
monoid/nat/+ : monoid # [t := nat, op := +] :=
struct
def emp := 0
def opL := +-left-unit
def opR := +-right-unit
def opA := +-assoc
end | monoid/nat/+ : monoid # [t := nat, op := +] | :=
struct
def emp := 0
def opL := +-left-unit
def opR := +-right-unit
def opA := +-assoc
end | def | monoid/nat/+ | test | test/algebra.cooltt | [
"nat",
"prelude"
] | [
"+",
"+-assoc",
"+-left-unit",
"+-right-unit",
"monoid"
] | null | 60 | 66 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
additive-monoid/nat : additive-monoid # [ t := nat ] :=
struct
include monoid/nat
renaming [op β add; emp β zer; opL β addL; opR β addR; opA β addA]
end | additive-monoid/nat : additive-monoid # [ t := nat ] | :=
struct
include monoid/nat
renaming [op β add; emp β zer; opL β addL; opR β addR; opA β addA]
end | def | additive-monoid/nat | test | test/algebra.cooltt | [
"nat",
"prelude"
] | [
"additive-monoid",
"monoid/nat"
] | null | 68 | 72 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
monoid/unit : monoid # [t := unit] :=
struct
def op := _ _ => 0
def emp := 0
def opL := _ _ => 0
def opR := _ _ => 0
def opA := _ _ _ _ => 0
end | monoid/unit : monoid # [t := unit] | :=
struct
def op := _ _ => 0
def emp := 0
def opL := _ _ => 0
def opR := _ _ => 0
def opA := _ _ _ _ => 0
end | def | monoid/unit | test | test/algebra.cooltt | [
"nat",
"prelude"
] | [
"monoid",
"unit"
] | null | 76 | 83 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
abelian-group/unit : abelian-group # [t := unit] :=
struct
include monoid/unit
renaming [op β add; emp β zer; opL β addL; opR β addR; opA β addA]
def neg := _ => 0
def addC := _ _ _ => 0
def add-neg := _ _ => 0
end | abelian-group/unit : abelian-group # [t := unit] | :=
struct
include monoid/unit
renaming [op β add; emp β zer; opL β addL; opR β addR; opA β addA]
def neg := _ => 0
def addC := _ _ _ => 0
def add-neg := _ _ => 0
end | def | abelian-group/unit | test | test/algebra.cooltt | [
"nat",
"prelude"
] | [
"abelian-group",
"monoid/unit",
"unit"
] | null | 85 | 92 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
ring/unit : ring # [t := unit] :=
struct
include abelian-group/unit
include monoid/unit
renaming [op β mul; emp β one; opL β mulL; opR β mulR; opA β mulA]
end | ring/unit : ring # [t := unit] | :=
struct
include abelian-group/unit
include monoid/unit
renaming [op β mul; emp β one; opL β mulL; opR β mulR; opA β mulA]
end | def | ring/unit | test | test/algebra.cooltt | [
"nat",
"prelude"
] | [
"abelian-group/unit",
"monoid/unit",
"ring",
"unit"
] | null | 94 | 99 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
empty : type := path nat 0 1 | empty : type | := path nat 0 1 | def | empty | test | test/base-types.cooltt | [
"prelude"
] | [
"path"
] | null | 4 | 4 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
abort (P : empty -> type) (e : empty) : P e :=
unfold empty in
let myelim : nat -> type := elim [ zero => unit | suc _ => P e ] in
coe {i => myelim {e i}} 0 1 β | abort (P : empty -> type) (e : empty) : P e | :=
unfold empty in
let myelim : nat -> type := elim [ zero => unit | suc _ => P e ] in
coe {i => myelim {e i}} 0 1 β | def | abort | test | test/base-types.cooltt | [
"prelude"
] | [
"empty",
"unit",
"β"
] | null | 7 | 10 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
sum (A B : type) : type :=
let famβ·shifted : nat -> type := elim [ zero => B | suc _ => empty ] in
let fam : nat -> type := elim [ zero => A | suc n => famβ·shifted n ] in
(n : nat) * fam n | sum (A B : type) : type | :=
let famβ·shifted : nat -> type := elim [ zero => B | suc _ => empty ] in
let fam : nat -> type := elim [ zero => A | suc n => famβ·shifted n ] in
(n : nat) * fam n | def | sum | test | test/base-types.cooltt | [
"prelude"
] | [
"empty"
] | null | 13 | 16 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
inl (A B : type) (x : A) : sum A B :=
[ 0 , x ] | inl (A B : type) (x : A) : sum A B | :=
[ 0 , x ] | def | inl | test | test/base-types.cooltt | [
"prelude"
] | [
"sum"
] | null | 19 | 20 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
inr (A B : type) (y : B) : sum A B :=
[ 1 , y ] | inr (A B : type) (y : B) : sum A B | :=
[ 1 , y ] | def | inr | test | test/base-types.cooltt | [
"prelude"
] | [
"sum"
] | null | 23 | 24 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
case (A B : type) (P : sum A B -> type)
(P/inl : (a : A) -> P {inl A B a})
(P/inr : (b : B) -> P {inr A B b})
(s : sum A B) : P s
:=
let fam/shifted : nat -> type := elim [ zero => B | suc _ => empty ] in
let curried/shifted : (n : nat) (e : fam/shifted n) -> unfold sum in P [ suc n , e ] :=
unfold inr ... | case (A B : type) (P : sum A B -> type)
(P/inl : (a : A) -> P {inl A B a})
(P/inr : (b : B) -> P {inr A B b})
(s : sum A B) : P s | :=
let fam/shifted : nat -> type := elim [ zero => B | suc _ => empty ] in
let curried/shifted : (n : nat) (e : fam/shifted n) -> unfold sum in P [ suc n , e ] :=
unfold inr in
elim [ zero => P/inr | suc n => e => abort {_ => P [ suc {suc n} , e ]} e ]
in
let fam : nat -> type := elim [ zero => A | suc ... | def | case | test | test/base-types.cooltt | [
"prelude"
] | [
"abort",
"empty",
"inl",
"inr",
"sum"
] | null | 26 | 42 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
reflβ‘reflΒ―ΒΉ (A : type) (x : A) : path {path A x x} {refl A x} {symm A {refl A x}} :=
j i =>
unfold symm in
symm/filler A {refl A x} i j | reflβ‘reflΒ―ΒΉ (A : type) (x : A) : path {path A x x} {refl A x} {symm A {refl A x}} | :=
j i =>
unfold symm in
symm/filler A {refl A x} i j | def | reflβ‘reflΒ―ΒΉ | test | test/bruno.cooltt | [
"prelude"
] | [
"path",
"refl",
"symm",
"symm/filler"
] | Lemma 3.2.1 (inversion unit) | 6 | 9 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
reflβ‘reflβrefl (A : type) (x : A) : path {path A x x} {refl A x} {trans A {refl A x} {refl A x}} :=
unfold trans in
trans/filler A {refl A x} {refl A x} | reflβ‘reflβrefl (A : type) (x : A) : path {path A x x} {refl A x} {trans A {refl A x} {refl A x}} | :=
unfold trans in
trans/filler A {refl A x} {refl A x} | def | reflβ‘reflβrefl | test | test/bruno.cooltt | [
"prelude"
] | [
"path",
"refl",
"trans",
"trans/filler"
] | Lemma 3.2.2 (composition unit) | 12 | 14 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
pβ‘pβrefl (A : type) (p : π β A) : path {path A {p 0} {p 1}} p {trans A p {refl A {p 1}}} :=
unfold trans in
trans/filler A p {refl A {p 1}} | pβ‘pβrefl (A : type) (p : π β A) : path {path A {p 0} {p 1}} p {trans A p {refl A {p 1}}} | :=
unfold trans in
trans/filler A p {refl A {p 1}} | def | pβ‘pβrefl | test | test/bruno.cooltt | [
"prelude"
] | [
"path",
"refl",
"trans",
"trans/filler"
] | Lemma 3.2.3 (right unit) | 17 | 19 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
reflβ‘pβpΒ―ΒΉ (A : type) (p : π β A) : path {path A {p 0} {p 0}} {refl A {p 0}} {trans A p {symm A p}} :=
k i =>
unfold trans symm in
hcom 0 1 {j =>
[ j=0 β¨ i=0 => p i
| i=1 β¨ k=0 => symm/filler A p j i
| k=1 => trans/filler A p {symm A p} j i
]
} | reflβ‘pβpΒ―ΒΉ (A : type) (p : π β A) : path {path A {p 0} {p 0}} {refl A {p 0}} {trans A p {symm A p}} | :=
k i =>
unfold trans symm in
hcom 0 1 {j =>
[ j=0 β¨ i=0 => p i
| i=1 β¨ k=0 => symm/filler A p j i
| k=1 => trans/filler A p {symm A p} j i
]
} | def | reflβ‘pβpΒ―ΒΉ | test | test/bruno.cooltt | [
"prelude"
] | [
"path",
"refl",
"symm",
"symm/filler",
"trans",
"trans/filler"
] | Lemma 3.2.4 (right cancellation) | 22 | 30 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
square (A : type) : type :=
sig
def top : ext i => A with []
def bot : ext i => A with []
def left : ext i => A with [i=0 => top 0 | i=1 => bot 0]
def right : ext i => A with [i=0 => top 1 | i=1 => bot 1]
def filler : ext i j => A with [i=0 => top j | i=1 => bot j | j=0 => left i | j=1 => right i]... | square (A : type) : type | :=
sig
def top : ext i => A with []
def bot : ext i => A with []
def left : ext i => A with [i=0 => top 0 | i=1 => bot 0]
def right : ext i => A with [i=0 => top 1 | i=1 => bot 1]
def filler : ext i j => A with [i=0 => top j | i=1 => bot j | j=0 => left i | j=1 => right i]
end | def | square | test | test/bruno.cooltt | [
"prelude"
] | [] | null | 32 | 39 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
swap (A : type) (sq : square A)
: square A #
[top := sq.bot,
bot := sq.top,
left := symm A {sq.left},
right := symm A {sq.right}] :=
unfold symm in
struct
def filler :=
i k => hcom 0 1 {j =>
[ i=0 => sq.filler j k
| i=1 β¨ j=0 => sq.top k
| k=0 => symm/filler A ... | swap (A : type) (sq : square A)
: square A #
[top := sq.bot,
bot := sq.top,
left := symm A {sq.left},
right := symm A {sq.right}] | :=
unfold symm in
struct
def filler :=
i k => hcom 0 1 {j =>
[ i=0 => sq.filler j k
| i=1 β¨ j=0 => sq.top k
| k=0 => symm/filler A {sq.left} i j
| k=1 => symm/filler A {sq.right} i j
]
}
end | def | swap | test | test/bruno.cooltt | [
"prelude"
] | [
"square",
"symm",
"symm/filler"
] | Lemma 3.2.5 (square swap) | 42 | 58 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
symm-invol (A : type) (p : π β A) : path {path A {p 0} {p 1}} p {symm A {symm A p}} :=
let a : A := p 0 in
let b : A := p 1 in
let true-at-refl : path {path A b b} {refl A b} {symm A {symm A {refl A b}}} :=
trans {path A b b} {reflβ‘reflΒ―ΒΉ A b} {i => symm A {reflβ‘reflΒ―ΒΉ A b i}}
in
let back : square A :=
... | symm-invol (A : type) (p : π β A) : path {path A {p 0} {p 1}} p {symm A {symm A p}} | :=
let a : A := p 0 in
let b : A := p 1 in
let true-at-refl : path {path A b b} {refl A b} {symm A {symm A {refl A b}}} :=
trans {path A b b} {reflβ‘reflΒ―ΒΉ A b} {i => symm A {reflβ‘reflΒ―ΒΉ A b i}}
in
let back : square A :=
unfold trans in
struct
def top := symm A p
def bot := trans A {sym... | def | symm-invol | test | test/bruno.cooltt | [
"prelude"
] | [
"path",
"refl",
"reflβ‘reflΒ―ΒΉ",
"square",
"swap",
"symm",
"trans",
"trans/filler"
] | Lemma 3.2.6 (inversability) | 61 | 87 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
opβ (A : type) (p : π β A) : path-p {i => path A {p 0} {symm A p i}} p {refl A {p 0}} :=
k i =>
unfold trans in
hcom 0 1 {j =>
[ i=0 => p 0
| i=1 => symm A p k
| j=0 => trans/filler A p {symm A p} k i
| k=0 => p i
| k=1 => symm {path A {p 0} {p 0}} {reflβ‘pβpΒ―ΒΉ A p} j i
]
} | opβ (A : type) (p : π β A) : path-p {i => path A {p 0} {symm A p i}} p {refl A {p 0}} | :=
k i =>
unfold trans in
hcom 0 1 {j =>
[ i=0 => p 0
| i=1 => symm A p k
| j=0 => trans/filler A p {symm A p} k i
| k=0 => p i
| k=1 => symm {path A {p 0} {p 0}} {reflβ‘pβpΒ―ΒΉ A p} j i
]
} | def | opβ | test | test/bruno.cooltt | [
"prelude"
] | [
"path",
"path-p",
"refl",
"reflβ‘pβpΒ―ΒΉ",
"symm",
"trans",
"trans/filler"
] | Lemma 3.2.7(i) (opposite identification) | 90 | 100 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
reflβ‘pΒ―ΒΉβp (A : type) (p : π β A) : path {path A {p 1} {p 1}} {refl A {p 1}} {trans A {symm A p} p} :=
k i =>
unfold trans in
hcom 0 1 {j =>
[ i=0 => p 1
| i=1 => symm {path A {p 0} {p 1}} {symm-invol A p} k j
| j=0 => symm A p i
| k=0 => opβ A {symm A p} j i
| k=1 => trans/filler A {symm A p... | reflβ‘pΒ―ΒΉβp (A : type) (p : π β A) : path {path A {p 1} {p 1}} {refl A {p 1}} {trans A {symm A p} p} | :=
k i =>
unfold trans in
hcom 0 1 {j =>
[ i=0 => p 1
| i=1 => symm {path A {p 0} {p 1}} {symm-invol A p} k j
| j=0 => symm A p i
| k=0 => opβ A {symm A p} j i
| k=1 => trans/filler A {symm A p} p j i
]
} | def | reflβ‘pΒ―ΒΉβp | test | test/bruno.cooltt | [
"prelude"
] | [
"opβ",
"path",
"refl",
"symm",
"symm-invol",
"trans",
"trans/filler"
] | Lemma 3.2.8 (left cancellation) | 103 | 113 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
opβ (A : type) (p : π β A) : path-p {i => path A {p 1} {p i}} {symm A p} {refl A {p 1}} :=
k i =>
unfold trans in
hcom 0 1 {j =>
[ i=0 => p 1
| i=1 => p k
| j=0 => trans/filler A {symm A p} p k i
| k=0 => symm A p i
| k=1 => symm {path A {p 1} {p 1}} {reflβ‘pΒ―ΒΉβp A p} j i
]
} | opβ (A : type) (p : π β A) : path-p {i => path A {p 1} {p i}} {symm A p} {refl A {p 1}} | :=
k i =>
unfold trans in
hcom 0 1 {j =>
[ i=0 => p 1
| i=1 => p k
| j=0 => trans/filler A {symm A p} p k i
| k=0 => symm A p i
| k=1 => symm {path A {p 1} {p 1}} {reflβ‘pΒ―ΒΉβp A p} j i
]
} | def | opβ | test | test/bruno.cooltt | [
"prelude"
] | [
"path",
"path-p",
"refl",
"reflβ‘pΒ―ΒΉβp",
"symm",
"trans",
"trans/filler"
] | Lemma 3.2.7(ii) (opposite identification) the paper mentions that this can be solved in a "similar argument" to 3.2.7(i), but for the truly symmetric proof (nearly identical to opβ) you need 3.2.8. so this is kind of nonlinear and might not be how the author intended? | 120 | 130 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
pβ‘reflβp (A : type) (p : π β A) : path {path A {p 0} {p 1}} p {trans A {refl A {p 0}} p} :=
k i =>
unfold trans in
hcom 0 1 {j =>
[ i=0 => p 0
| i=1 => opβ A p j k
| j=0 => opβ A p k i
| k=0 => p i
| k=1 => trans/filler A {refl A {p 0}} p j i
]
} | pβ‘reflβp (A : type) (p : π β A) : path {path A {p 0} {p 1}} p {trans A {refl A {p 0}} p} | :=
k i =>
unfold trans in
hcom 0 1 {j =>
[ i=0 => p 0
| i=1 => opβ A p j k
| j=0 => opβ A p k i
| k=0 => p i
| k=1 => trans/filler A {refl A {p 0}} p j i
]
} | def | pβ‘reflβp | test | test/bruno.cooltt | [
"prelude"
] | [
"opβ",
"opβ",
"path",
"refl",
"trans",
"trans/filler"
] | Lemma 3.2.9 (left unit) | 133 | 143 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
bottom-id (A : type) (Ξ± : π β π β A) (Ξ² : (i j : π) β sub A {β j β¨ i=0} {Ξ± i j}) : path {path A {Ξ± 1 0} {Ξ² 1 1}} {Ξ± 1} {Ξ² 1} :=
k i =>
hcom 0 1 {j =>
[ β i β¨ j=0 β¨ k=0 => Ξ± j i
| k=1 => Ξ² j i
]
} | bottom-id (A : type) (Ξ± : π β π β A) (Ξ² : (i j : π) β sub A {β j β¨ i=0} {Ξ± i j}) : path {path A {Ξ± 1 0} {Ξ² 1 1}} {Ξ± 1} {Ξ² 1} | :=
k i =>
hcom 0 1 {j =>
[ β i β¨ j=0 β¨ k=0 => Ξ± j i
| k=1 => Ξ² j i
]
} | def | bottom-id | test | test/bruno.cooltt | [
"prelude"
] | [
"path"
] | Lemma 3.2.10 (bottom identification) we represent squares as just their filler here, not the struct | 147 | 153 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
assoc
(A : type)
(p : π β A)
(q : (i : π) β sub A {i=0} {p 1})
(r : (i : π) β sub A {i=0} {q 1})
: path {path A {p 0} {r 1}} {trans A {trans A p q} r} {trans A p {trans A q r}}
:=
unfold trans in
let Ξ± : square A :=
struct
def top := p
def bot := trans A {trans A p q} r
def left := ... | assoc
(A : type)
(p : π β A)
(q : (i : π) β sub A {i=0} {p 1})
(r : (i : π) β sub A {i=0} {q 1})
: path {path A {p 0} {r 1}} {trans A {trans A p q} r} {trans A p {trans A q r}} | :=
unfold trans in
let Ξ± : square A :=
struct
def top := p
def bot := trans A {trans A p q} r
def left := refl A {p 0}
def right := trans A q r
def filler :=
k i => hcom 0 1 {j =>
[ i=0 => p 0
| i=1 => trans/filler A q r j k
| j=0 => trans/fill... | def | assoc | test | test/bruno.cooltt | [
"prelude"
] | [
"bottom-id",
"path",
"refl",
"square",
"trans",
"trans/filler"
] | Lemma 3.2.11 (associativity) | 156 | 189 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
hsymm (A : π β type) (p : (i : π) β A i) : path-p {symm type A} {p 1} {p 0} :=
i =>
unfold symm in
com {symm/filler type A i} 0 1 {β i} {j =>
[ i=0 => p j
| i=1 β¨ j=0 => p 0
]
} | hsymm (A : π β type) (p : (i : π) β A i) : path-p {symm type A} {p 1} {p 0} | :=
i =>
unfold symm in
com {symm/filler type A i} 0 1 {β i} {j =>
[ i=0 => p j
| i=1 β¨ j=0 => p 0
]
} | def | hsymm | test | test/bruno.cooltt | [
"prelude"
] | [
"path-p",
"symm",
"symm/filler"
] | Lemma 3.3.1 (heterogeneous inversion) | 192 | 199 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
htrans
(A : π β type)
(B : (i : π) β sub type {i=0} {A 1})
(p : (i : π) β A i) (q : (i : π) β sub {B i} {i=0} {p 1})
: path-p {trans type A B} {p 0} {q 1}
:=
i =>
unfold trans in
com {j => trans/filler type A B j i} 0 1 {β i} {j =>
[ j=0 β¨ i=0 => p i
| i=1 => q j
]
} | htrans
(A : π β type)
(B : (i : π) β sub type {i=0} {A 1})
(p : (i : π) β A i) (q : (i : π) β sub {B i} {i=0} {p 1})
: path-p {trans type A B} {p 0} {q 1} | :=
i =>
unfold trans in
com {j => trans/filler type A B j i} 0 1 {β i} {j =>
[ j=0 β¨ i=0 => p i
| i=1 => q j
]
} | def | htrans | test | test/bruno.cooltt | [
"prelude"
] | [
"path-p",
"trans",
"trans/filler"
] | Lemma 3.3.2 (heterogeneous composition) | 202 | 214 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
is-refl (A : type) (p : π β A) : path-p {i => path A {p 0} {p i}} {refl A {p 0}} p :=
k i =>
hcom 0 1 {j =>
[ i=0 => p 0
| i=1 => opβ A p k j
| j=0 => pβ‘pβrefl A p k i
| k=0 => opβ A p j i
| k=1 => symm {path A {p 0} {p 1}} {pβ‘pβrefl A p} j i
]
} | is-refl (A : type) (p : π β A) : path-p {i => path A {p 0} {p i}} {refl A {p 0}} p | :=
k i =>
hcom 0 1 {j =>
[ i=0 => p 0
| i=1 => opβ A p k j
| j=0 => pβ‘pβrefl A p k i
| k=0 => opβ A p j i
| k=1 => symm {path A {p 0} {p 1}} {pβ‘pβrefl A p} j i
]
} | def | is-refl | test | test/bruno.cooltt | [
"prelude"
] | [
"opβ",
"opβ",
"path",
"path-p",
"pβ‘pβrefl",
"refl",
"symm"
] | Theorem 4.1.1 (path induction) | 217 | 226 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
J (A : type) (a : A) (P : (x : A) β {path A a x} β type) (u : P a {refl A a}) (p : (i : π) β sub A {i=0} a) : P {p 1} p :=
coe {i => P {p i} {is-refl A p i}} 0 1 u | J (A : type) (a : A) (P : (x : A) β {path A a x} β type) (u : P a {refl A a}) (p : (i : π) β sub A {i=0} a) : P {p 1} p | :=
coe {i => P {p i} {is-refl A p i}} 0 1 u | def | J | test | test/bruno.cooltt | [
"prelude"
] | [
"is-refl",
"path",
"refl"
] | null | 228 | 229 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
path (A : type) (a b : A) : type :=
ext i => A with [i=0 => a | i=1 => b] | path (A : type) (a b : A) : type | :=
ext i => A with [i=0 => a | i=1 => b] | def | path | test | test/circle.cooltt | [] | [] | null | 1 | 2 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
Ξ©1s1 : type :=
path circle base base | Ξ©1s1 : type | :=
path circle base base | def | Ξ©1s1 | test | test/circle.cooltt | [] | [
"path"
] | null | 4 | 5 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
loopn : nat -> Ξ©1s1 :=
elim [
| zero => _ => base
| suc {n => loopn} =>
i =>
hcom circle 0 1 {β i} {k =>
[ k=0 => loopn i
| i=0 => base
| i=1 => loop k
]
}
] | loopn : nat -> Ξ©1s1 | :=
elim [
| zero => _ => base
| suc {n => loopn} =>
i =>
hcom circle 0 1 {β i} {k =>
[ k=0 => loopn i
| i=0 => base
| i=1 => loop k
]
}
] | def | loopn | test | test/circle.cooltt | [] | [
"Ξ©1s1"
] | null | 7 | 18 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
_ (A : π β type) (src : π) (trg : π) (x : A src) : sub {A trg} {src=trg} x :=
coe A src trg x | _ (A : π β type) (src : π) (trg : π) (x : A src) : sub {A trg} {src=trg} x | :=
coe A src trg x | def | _ | test | test/coercion.cooltt | [
"prelude"
] | [] | This is the Cartesian coercion operator. | 4 | 5 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
coe/fwd (A : π β type) (x : A 0) : A 1 :=
coe A 0 1 x | coe/fwd (A : π β type) (x : A 0) : A 1 | :=
coe A 0 1 x | def | coe/fwd | test | test/coercion.cooltt | [
"prelude"
] | [] | A special case of coercion is that if we have a path of types A0 = A1, we can cast/coerce (x : A0) to an element of A1. | 9 | 10 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
coe/bwd (A : π β type) (x : A 1) : A 0 :=
coe A 1 0 x | coe/bwd (A : π β type) (x : A 1) : A 0 | :=
coe A 1 0 x | def | coe/bwd | test | test/coercion.cooltt | [
"prelude"
] | [] | ...and conversely. | 13 | 14 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
transport/fwd (A : type) (B : A β type) (p : π β A) (x : B {p 0}) : B {p 1} :=
coe/fwd {i => B {p i}} x | transport/fwd (A : type) (B : A β type) (p : π β A) (x : B {p 0}) : B {p 1} | :=
coe/fwd {i => B {p i}} x | def | transport/fwd | test | test/coercion.cooltt | [
"prelude"
] | [
"coe/fwd"
] | By combining coe with ap (the fact that functions respect paths), we can show that if we have a path (a0 = a1 : A) and an A-indexed family B of types, then we can transport (x : B a0) to an element of (B a1). | 19 | 20 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
_ (A : π β type) (x : A 0) : path-p A x {coe/fwd A x} :=
i => coe A 0 i x | _ (A : π β type) (x : A 0) : path-p A x {coe/fwd A x} | :=
i => coe A 0 i x | def | _ | test | test/coercion.cooltt | [
"prelude"
] | [
"coe/fwd",
"path-p"
] | If we coerce (x : A0) to an interval variable, we get a dependent path from x to the coercion of x. That's because of the side condition that coe is the identity function when src=trg. | 25 | 26 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
heterogenize (A : π β type) (p : π β A 0) : path-p A {p 0} {coe/fwd A {p 1}} :=
i => coe A 0 i {p i} | heterogenize (A : π β type) (p : π β A 0) : path-p A {p 0} {coe/fwd A {p 1}} | :=
i => coe A 0 i {p i} | def | heterogenize | test | test/coercion.cooltt | [
"prelude"
] | [
"coe/fwd",
"path-p"
] | Here's another use of coercing to an interval variable. If we have a homogeneous path in A0, we can turn it into a heterogeneous path in A from its left endpoint to the coercion of its right endpoint. | 31 | 32 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
homogenize (A : π β type) (p : (i : π) β A i) : path {A 1} {coe/fwd A {p 0}} {p 1} :=
i => coe A i 1 {p i} | homogenize (A : π β type) (p : (i : π) β A i) : path {A 1} {coe/fwd A {p 0}} {p 1} | :=
i => coe A i 1 {p i} | def | homogenize | test | test/coercion.cooltt | [
"prelude"
] | [
"coe/fwd",
"path"
] | Dually, we can coerce *from* an interval variable to turn a heterogeneous path into a homogeneous one. | 36 | 37 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
mycoe/fun
(A : (i : π) β type) (B : (i: π) β type)
(coe/A : (r : π) (x : A r) (i : π) β sub {A i} {i=r} x)
(coe/B : (r : π) (x : B r) (i : π) β sub {B i} {i=r} x)
(r : π) (f : (_ : A r) β B r) (i : π)
: sub {(_ : A i) β B i} {i=r} f
:=
x =>
coe/B r {f {coe/A i x r}} i | mycoe/fun
(A : (i : π) β type) (B : (i: π) β type)
(coe/A : (r : π) (x : A r) (i : π) β sub {A i} {i=r} x)
(coe/B : (r : π) (x : B r) (i : π) β sub {B i} {i=r} x)
(r : π) (f : (_ : A r) β B r) (i : π)
: sub {(_ : A i) β B i} {i=r} f | :=
x =>
coe/B r {f {coe/A i x r}} i | def | mycoe/fun | test | test/com.cooltt | [] | [] | null | 1 | 9 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
mycom/fun
(A B : π β type)
(com/A : (r : π) (Ο : π½) (p : (i : π) β [i=r β¨ Ο] β A i) (i : π) β sub {A i} {i=r β¨ Ο} {p i})
(com/B : (r : π) (Ο : π½) (p : (i : π) β [i=r β¨ Ο] β B i) (i : π) β sub {B i} {i=r β¨ Ο} {p i})
(r : π) (Ο : π½) (p : (i : π) β [i=r β¨ Ο] β A i β B i) (i : π)
: sub {(_ : A i) β B... | mycom/fun
(A B : π β type)
(com/A : (r : π) (Ο : π½) (p : (i : π) β [i=r β¨ Ο] β A i) (i : π) β sub {A i} {i=r β¨ Ο} {p i})
(com/B : (r : π) (Ο : π½) (p : (i : π) β [i=r β¨ Ο] β B i) (i : π) β sub {B i} {i=r β¨ Ο} {p i})
(r : π) (Ο : π½) (p : (i : π) β [i=r β¨ Ο] β A i β B i) (i : π)
: sub {(_ : A i) β B... | :=
x =>
com/B r Ο {j => p j {com/A i β₯ {_ => x} j}} i | def | mycom/fun | test | test/com.cooltt | [] | [] | null | 11 | 19 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
coe/intro (A : π β type) (r r' : π) (x : A r) : sub {A r'} {r=r'} x :=
coe A r r' x | coe/intro (A : π β type) (r r' : π) (x : A r) : sub {A r'} {r=r'} x | :=
coe A r r' x | def | coe/intro | test | test/com.cooltt | [] | [] | null | 23 | 24 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
coe/pi
(A : π β type) (B : (i : π) β A i β type)
(r r' : π)
(f : (x : A r) β B r x)
: sub {(x : A r') β B r' x} β€ {x => coe {i => B i {coe A r' i x}} r r' {f {coe A r' r x}}}
:=
coe {i => (x : A i) β B i x} r r' f | coe/pi
(A : π β type) (B : (i : π) β A i β type)
(r r' : π)
(f : (x : A r) β B r x)
: sub {(x : A r') β B r' x} β€ {x => coe {i => B i {coe A r' i x}} r r' {f {coe A r' r x}}} | :=
coe {i => (x : A i) β B i x} r r' f | def | coe/pi | test | test/com.cooltt | [] | [] | null | 26 | 32 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
coe/sigma
(A : π β type) (B : (i : π) β A i β type)
(r r' : π)
(p : (x : A r) Γ B r x)
: sub {(x : A r') Γ B r' x} β€ [coe A r r' {fst p}, coe {i => B i {coe A r i {fst p}}} r r' {snd p}]
:=
coe {i => (x : A i) Γ B i x} r r' p | coe/sigma
(A : π β type) (B : (i : π) β A i β type)
(r r' : π)
(p : (x : A r) Γ B r x)
: sub {(x : A r') Γ B r' x} β€ [coe A r r' {fst p}, coe {i => B i {coe A r i {fst p}}} r r' {snd p}] | :=
coe {i => (x : A i) Γ B i x} r r' p | def | coe/sigma | test | test/com.cooltt | [] | [] | null | 36 | 42 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
pathd (A : π β type) (a : A 0) (b : A 1) : type :=
ext i => A i with [i=0 => a | i=1 => b] | pathd (A : π β type) (a : A 0) (b : A 1) : type | :=
ext i => A i with [i=0 => a | i=1 => b] | def | pathd | test | test/com.cooltt | [] | [] | null | 46 | 47 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
coe/pathd
(A : π -> π -> type)
(r r' : π)
(a : (i : π) -> A i 0)
(b : (i : π) -> A i 1)
(m : pathd {A r} {a r} {b r})
: sub {pathd {A r'} {a r'} {b r'}} β€ {j =>
com {i => A i j} r r' {β j} {i =>
[j=0 => a i | j=1 => b i | i=r => m j]
}
}
:=
coe {i => pathd {A i} {a i} {b i}}... | coe/pathd
(A : π -> π -> type)
(r r' : π)
(a : (i : π) -> A i 0)
(b : (i : π) -> A i 1)
(m : pathd {A r} {a r} {b r})
: sub {pathd {A r'} {a r'} {b r'}} β€ {j =>
com {i => A i j} r r' {β j} {i =>
[j=0 => a i | j=1 => b i | i=r => m j]
}
} | :=
coe {i => pathd {A i} {a i} {b i}} r r' m | def | coe/pathd | test | test/com.cooltt | [] | [
"pathd"
] | null | 50 | 62 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
hcom/intro
(A : type) (r r' : π) (Ο : π½)
(p : (i : π) β [i=r β¨ Ο] β A)
: sub A {r=r' β¨ Ο} {p r'}
:=
hcom A r r' Ο p | hcom/intro
(A : type) (r r' : π) (Ο : π½)
(p : (i : π) β [i=r β¨ Ο] β A)
: sub A {r=r' β¨ Ο} {p r'} | :=
hcom A r r' Ο p | def | hcom/intro | test | test/com.cooltt | [] | [] | null | 66 | 71 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
hcom/fun
(A B : type) (r r' : π) (Ο : π½)
(p : (i : π) β [i=r β¨ Ο] β A β B)
: sub {A β B} β€ {x => hcom B r r' Ο {j => p j x}}
:=
hcom {A β B} r r' Ο p | hcom/fun
(A B : type) (r r' : π) (Ο : π½)
(p : (i : π) β [i=r β¨ Ο] β A β B)
: sub {A β B} β€ {x => hcom B r r' Ο {j => p j x}} | :=
hcom {A β B} r r' Ο p | def | hcom/fun | test | test/com.cooltt | [] | [] | null | 73 | 78 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
com/intro
(A : π β type) (r r' : π) (Ο : π½)
(p : (i : π) β [i=r β¨ Ο] β A i)
: sub {A r'} {r=r' β¨ Ο} {p r'}
:=
com A r r' Ο p | com/intro
(A : π β type) (r r' : π) (Ο : π½)
(p : (i : π) β [i=r β¨ Ο] β A i)
: sub {A r'} {r=r' β¨ Ο} {p r'} | :=
com A r r' Ο p | def | com/intro | test | test/com.cooltt | [] | [] | null | 82 | 87 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
com/decomposition
(A : π β type) (r r' : π) (Ο : π½)
(p : (i : π) β [i=r β¨ Ο] β A i)
: sub {A r'} β€ {hcom {A r'} r r' Ο {j => coe A j r' {p j}}}
:=
com A r r' Ο p | com/decomposition
(A : π β type) (r r' : π) (Ο : π½)
(p : (i : π) β [i=r β¨ Ο] β A i)
: sub {A r'} β€ {hcom {A r'} r r' Ο {j => coe A j r' {p j}}} | :=
com A r r' Ο p | def | com/decomposition | test | test/com.cooltt | [] | [] | null | 91 | 96 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
fully-patched (fam : sig [x : nat] -> type) (fib : fam {struct [x := 0]}) : fam # [x := 0] :=
fib | fully-patched (fam : sig [x : nat] -> type) (fib : fam {struct [x := 0]}) : fam # [x := 0] | :=
fib | def | fully-patched | test | test/cool-total-space.cooltt | [] | [] | null | 4 | 5 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
not-fully-patched (fam : sig [x : nat] -> type) (fib : fam {struct [x := 0]}) : fam :=
struct [x := 0, fib := fib] | not-fully-patched (fam : sig [x : nat] -> type) (fib : fam {struct [x := 0]}) : fam | :=
struct [x := 0, fib := fib] | def | not-fully-patched | test | test/cool-total-space.cooltt | [] | [] | null | 8 | 9 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
no-insert-fib (fam : sig [x : nat] -> type) (total : fam) : nat :=
total.x | no-insert-fib (fam : sig [x : nat] -> type) (total : fam) : nat | :=
total.x | def | no-insert-fib | test | test/cool-total-space.cooltt | [] | [] | null | 12 | 13 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
insert-fib-plain (fam : sig [x : nat] -> type) (total : fam) : fam {struct [x := total.x]} :=
total | insert-fib-plain (fam : sig [x : nat] -> type) (total : fam) : fam {struct [x := total.x]} | :=
total | def | insert-fib-plain | test | test/cool-total-space.cooltt | [] | [] | null | 16 | 17 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
insert-fib-pi : sig [fam : sig [x : nat] -> type, test : fam -> nat] :=
struct
def fam := _ => nat -> nat
def test := total => total 0
end | insert-fib-pi : sig [fam : sig [x : nat] -> type, test : fam -> nat] | :=
struct
def fam := _ => nat -> nat
def test := total => total 0
end | def | insert-fib-pi | test | test/cool-total-space.cooltt | [] | [
"test"
] | null | 20 | 24 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
insert-fib-sg : sig [fam : sig [x : nat] -> type, test : fam -> nat] :=
struct
def fam := _ => nat * nat
def test := total => fst total
end | insert-fib-sg : sig [fam : sig [x : nat] -> type, test : fam -> nat] | :=
struct
def fam := _ => nat * nat
def test := total => fst total
end | def | insert-fib-sg | test | test/cool-total-space.cooltt | [] | [
"test"
] | null | 27 | 31 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
no-insert-fib-record : sig [fam : sig [x : nat] -> type, test : fam -> nat] :=
struct
def fam := _ => sig [y : nat]
def test := total => total.fib.y
end | no-insert-fib-record : sig [fam : sig [x : nat] -> type, test : fam -> nat] | :=
struct
def fam := _ => sig [y : nat]
def test := total => total.fib.y
end | def | no-insert-fib-record | test | test/cool-total-space.cooltt | [] | [
"test"
] | null | 34 | 38 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
test-hole (fam : sig [x : nat] -> type) : fam # [x := 0] := ? | test-hole (fam : sig [x : nat] -> type) : fam # [x := 0] | := ? | def | test-hole | test | test/cool-total-space.cooltt | [] | [] | null | 42 | 42 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
boundary-test : (i : π) β [β i] β nat :=
i =>
[ i=1 => 5
| i=0 => 19
] | boundary-test : (i : π) β [β i] β nat | :=
i =>
[ i=1 => 5
| i=0 => 19
] | def | boundary-test | test | test/elab.cooltt | [] | [] | null | 1 | 5 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
reflexivity : (A : type) (a : A) (i : π) β A :=
A a _ => a | reflexivity : (A : type) (a : A) (i : π) β A | :=
A a _ => a | def | reflexivity | test | test/elab.cooltt | [] | [] | null | 9 | 10 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
pi-code-test : type := (x : nat) β nat | pi-code-test : type | := (x : nat) β nat | def | pi-code-test | test | test/elab.cooltt | [] | [] | null | 13 | 13 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
foo : pi-code-test :=
x => x | foo : pi-code-test | :=
x => x | def | foo | test | test/elab.cooltt | [] | [
"pi-code-test"
] | null | 15 | 16 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
simple-let :
(A : type) (a : A) -> A
:=
A a =>
let b : A := a in
b | simple-let :
(A : type) (a : A) -> A | :=
A a =>
let b : A := a in
b | def | simple-let | test | test/elab.cooltt | [] | [] | null | 21 | 26 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
hole-in-type :
(x y z : nat)
β ?tyhole
:=
y z => ?tmhole | hole-in-type :
(x y z : nat)
β ?tyhole | :=
y z => ?tmhole | def | hole-in-type | test | test/elab.cooltt | [] | [] | null | 31 | 35 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
path (A : type) (a b : A) : type :=
ext i => A with [i=0 => a | i=1 => b] | path (A : type) (a b : A) : type | :=
ext i => A with [i=0 => a | i=1 => b] | def | path | test | test/elab.cooltt | [] | [] | null | 37 | 38 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
bar : (x : nat) β (y : nat) Γ path nat x y :=
x =>
[x, ?hole1] | bar : (x : nat) β (y : nat) Γ path nat x y | :=
x =>
[x, ?hole1] | def | bar | test | test/elab.cooltt | [] | [
"path"
] | null | 40 | 42 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
equational/trans (a : type) (x y z : a) (p : path a x y) (q : path a y z) : path a x z :=
equation a begin
x =[ p ]
y =[ q ]
z
end | equational/trans (a : type) (x y z : a) (p : path a x y) (q : path a y z) : path a x z | :=
equation a begin
x =[ p ]
y =[ q ]
z
end | def | equational/trans | test | test/equation.cooltt | [
"prelude"
] | [
"path"
] | null | 3 | 8 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
equational/refl/nat : path nat 4 4 :=
equation nat begin
4 =[]
4
end | equational/refl/nat : path nat 4 4 | :=
equation nat begin
4 =[]
4
end | def | equational/refl/nat | test | test/equation.cooltt | [
"prelude"
] | [
"path"
] | null | 12 | 16 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
refl2 (A : type) (p : (i : π) β A) : sub {path {path A {p 0} {p 1}} p p} β€ {_ => p}
:=
_ => p | refl2 (A : type) (p : (i : π) β A) : sub {path {path A {p 0} {p 1}} p p} β€ {_ => p} | :=
_ => p | def | refl2 | test | test/evan.cooltt | [
"prelude"
] | [
"path"
] | null | 3 | 5 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
test (A : type) (p : (i : π) β A) : (j : π) β path A {p 0} {p 1}
:=
j => refl2 A p j | test (A : type) (p : (i : π) β A) : (j : π) β path A {p 0} {p 1} | :=
j => refl2 A p j | def | test | test | test/evan.cooltt | [
"prelude"
] | [
"path",
"refl2"
] | null | 7 | 9 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
a1 : nat := 0 | a1 : nat | := 0 | def | a1 | test | test/export.cooltt | [] | [] | null | 1 | 1 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
a2 : nat := 10 | a2 : nat | := 10 | def | a2 | test | test/export.cooltt | [] | [] | null | 2 | 2 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
a3 : nat := 20 | a3 : nat | := 20 | def | a3 | test | test/export.cooltt | [] | [] | null | 3 | 3 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
special-j (A : type) (x : A) (B : (Ο : π½) β {(i : π) β sub A {i=0 β¨ Ο} x} β type)
(d : B β€ {_ => x})
(Ο : π½) (p : (i : π) β sub A {i=0 β¨ Ο} x)
: sub {B Ο p} Ο d
:=
let filler : π β π β A :=
j i =>
hcom A 0 i {β j β¨ Ο} {i =>
[ i=0 β¨ j=0 β¨ Ο => p 0
| j=1 => p i
]
}
in
com... | special-j (A : type) (x : A) (B : (Ο : π½) β {(i : π) β sub A {i=0 β¨ Ο} x} β type)
(d : B β€ {_ => x})
(Ο : π½) (p : (i : π) β sub A {i=0 β¨ Ο} x)
: sub {B Ο p} Ο d | :=
let filler : π β π β A :=
j i =>
hcom A 0 i {β j β¨ Ο} {i =>
[ i=0 β¨ j=0 β¨ Ο => p 0
| j=1 => p i
]
}
in
com {j => B {Ο β¨ j=0} {filler j}} 0 1 {Ο} {j => d} | def | special-j | test | test/groupoid-laws.cooltt | [
"prelude"
] | [] | null | 7 | 20 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
trans (A : type) (p : (i : π) β A)
: (Ο : π½) (q : (i : π) β sub A {i=0 β¨ Ο} {p 1})
β sub {path A {p 0} {q 1}} Ο p
:=
special-j A {p 1} {_ q => path A {p 0} {q 1}} p | trans (A : type) (p : (i : π) β A)
: (Ο : π½) (q : (i : π) β sub A {i=0 β¨ Ο} {p 1})
β sub {path A {p 0} {q 1}} Ο p | :=
special-j A {p 1} {_ q => path A {p 0} {q 1}} p | def | trans | test | test/groupoid-laws.cooltt | [
"prelude"
] | [
"path",
"special-j"
] | null | 22 | 26 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
assoc (A : type)
(p : (i : π) β A)
(Ο : π½) (q : (i : π) β sub A {i=0 β¨ Ο} {p 1})
: (Ο : π½) (r : (i : π) β sub A {i=0 β¨ Ο} {q 1})
β sub {path {path A {p 0} {r 1}} {trans A {trans A p Ο q} Ο r} {trans A p {Ο β§ Ο} {trans A q Ο r}}}
Ο {_ => trans A p Ο q}
:=
special-j A {q 1}
{Ο r => path {path A {... | assoc (A : type)
(p : (i : π) β A)
(Ο : π½) (q : (i : π) β sub A {i=0 β¨ Ο} {p 1})
: (Ο : π½) (r : (i : π) β sub A {i=0 β¨ Ο} {q 1})
β sub {path {path A {p 0} {r 1}} {trans A {trans A p Ο q} Ο r} {trans A p {Ο β§ Ο} {trans A q Ο r}}}
Ο {_ => trans A p Ο q} | :=
special-j A {q 1}
{Ο r => path {path A {p 0} {r 1}} {trans A {trans A p Ο q} Ο r} {trans A p {Ο β§ Ο} {trans A q Ο r}}}
{_ => trans A p Ο q} | def | assoc | test | test/groupoid-laws.cooltt | [
"prelude"
] | [
"path",
"special-j",
"trans"
] | null | 29 | 38 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
pentagonType (A : type)
(p : (i : π) β A)
(Ο : π½) (q : (i : π) β sub A {i=0 β¨ Ο} {p 1})
(Ο : π½) (r : (i : π) β sub A {i=0 β¨ Ο} {q 1})
(Ο : π½) (s : (i : π) β sub A {i=0 β¨ Ο} {r 1})
: type
:=
path
{path
{path A {p 0} {s 1}}
{trans A {trans A {trans A p Ο q} Ο r} Ο s}
{trans ... | pentagonType (A : type)
(p : (i : π) β A)
(Ο : π½) (q : (i : π) β sub A {i=0 β¨ Ο} {p 1})
(Ο : π½) (r : (i : π) β sub A {i=0 β¨ Ο} {q 1})
(Ο : π½) (s : (i : π) β sub A {i=0 β¨ Ο} {r 1})
: type | :=
path
{path
{path A {p 0} {s 1}}
{trans A {trans A {trans A p Ο q} Ο r} Ο s}
{trans A p {Ο β§ Ο β§ Ο} {trans A q {Ο β§ Ο} {trans A r Ο s}}}}
{trans {path A {p 0} {s 1}}
{assoc A {trans A p Ο q} Ο r Ο s}
{Ο β§ Ο}
{assoc A p Ο q {Ο β§ Ο} {trans A r Ο s}}}
{trans {path A ... | def | pentagonType | test | test/groupoid-laws.cooltt | [
"prelude"
] | [
"assoc",
"path",
"trans"
] | null | 40 | 62 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
pentagon (A : type)
(p : (i : π) β A)
(Ο : π½) (q : (i : π) β sub A {i=0 β¨ Ο} {p 1})
(Ο : π½) (r : (i : π) β sub A {i=0 β¨ Ο} {q 1})
: (Ο : π½) (s : (i : π) β sub A {i=0 β¨ Ο} {r 1})
β pentagonType A p Ο q Ο r Ο s
:=
special-j A {r 1}
{pentagonType A p Ο q Ο r}
{special-j A {q 1}
{Ο r => p... | pentagon (A : type)
(p : (i : π) β A)
(Ο : π½) (q : (i : π) β sub A {i=0 β¨ Ο} {p 1})
(Ο : π½) (r : (i : π) β sub A {i=0 β¨ Ο} {q 1})
: (Ο : π½) (s : (i : π) β sub A {i=0 β¨ Ο} {r 1})
β pentagonType A p Ο q Ο r Ο s | :=
special-j A {r 1}
{pentagonType A p Ο q Ο r}
{special-j A {q 1}
{Ο r => pentagonType A p Ο q Ο r β€ {_ => r 1}}
{special-j A {p 1}
{Ο q => pentagonType A p Ο q β€ {_ => q 1} β€ {_ => q 1}}
{_ _ => p}
Ο q}
Ο r} | def | pentagon | test | test/groupoid-laws.cooltt | [
"prelude"
] | [
"pentagonType",
"special-j"
] | null | 64 | 79 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
trans' (A : type) (p : (i : π) β A) (q : (i : π) β sub A {i=0} {p 1})
: path A {p 0} {q 1}
:=
trans A p β₯ q | trans' (A : type) (p : (i : π) β A) (q : (i : π) β sub A {i=0} {p 1})
: path A {p 0} {q 1} | :=
trans A p β₯ q | def | trans' | test | test/groupoid-laws.cooltt | [
"prelude"
] | [
"path",
"trans"
] | null | 83 | 86 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
assoc' (A : type)
(p : (i : π) β A)
(q : (i : π) β sub A {i=0} {p 1})
(r : (i : π) β sub A {i=0} {q 1})
: path {path A {p 0} {r 1}} {trans' A {trans' A p q} r} {trans' A p {trans' A q r}}
:=
assoc A p β₯ q β₯ r | assoc' (A : type)
(p : (i : π) β A)
(q : (i : π) β sub A {i=0} {p 1})
(r : (i : π) β sub A {i=0} {q 1})
: path {path A {p 0} {r 1}} {trans' A {trans' A p q} r} {trans' A p {trans' A q r}} | :=
assoc A p β₯ q β₯ r | def | assoc' | test | test/groupoid-laws.cooltt | [
"prelude"
] | [
"assoc",
"path",
"trans'"
] | null | 89 | 95 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
pentagon' (A : type)
(p : (i : π) β A)
(q : (i : π) β sub A {i=0} {p 1})
(r : (i : π) β sub A {i=0} {q 1})
(s : (i : π) β sub A {i=0} {r 1})
: path
{path
{path A {p 0} {s 1}}
{trans' A {trans' A {trans' A p q} r} s}
{trans' A p {trans' A q {trans' A r s}}}}
{trans' {path A {p... | pentagon' (A : type)
(p : (i : π) β A)
(q : (i : π) β sub A {i=0} {p 1})
(r : (i : π) β sub A {i=0} {q 1})
(s : (i : π) β sub A {i=0} {r 1})
: path
{path
{path A {p 0} {s 1}}
{trans' A {trans' A {trans' A p q} r} s}
{trans' A p {trans' A q {trans' A r s}}}}
{trans' {path A {p... | :=
pentagon A p β₯ q β₯ r β₯ s | def | pentagon' | test | test/groupoid-laws.cooltt | [
"prelude"
] | [
"assoc'",
"path",
"pentagon",
"trans'"
] | null | 98 | 117 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
test (A : type)
(p : (i : π) β A)
(q : (i : π) β sub A {i=0} {p 1})
(r : (i : π) β sub A {i=0} {q 1})
(s : (i : π) β sub A {i=0} {r 1})
: (j : π) β path A {p 0} {r 1}
:=
j => assoc A p β₯ q β₯ r j | test (A : type)
(p : (i : π) β A)
(q : (i : π) β sub A {i=0} {p 1})
(r : (i : π) β sub A {i=0} {q 1})
(s : (i : π) β sub A {i=0} {r 1})
: (j : π) β path A {p 0} {r 1} | :=
j => assoc A p β₯ q β₯ r j | def | test | test | test/groupoid-laws.cooltt | [
"prelude"
] | [
"assoc",
"path"
] | null | 119 | 126 | true | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
End of preview. Expand in Data Studio
cooltt
Declarations from cooltt, a cubical/computational type theory system.
Source
- Repository: https://github.com/RedPRL/cooltt
- Commit:
b39bf29900451cb43ae6fbd9af5aa33d59e18935 - Files: 32
- License: other
Schema
| Column | Type | Description |
|---|---|---|
| fact | string | Verbatim declaration with the leading keyword removed: signature and body/proof joined |
| statement | string | Signature with the leading keyword removed (verbatim slice) |
| proof | string | Verbatim proof/body, empty if none |
| type | string | Declaration keyword |
| symbolic_name | string | Declaration identifier |
| library | string | Sub-library |
| filename | string | Repository-relative source path |
| imports | list[string] | File-level import modules |
| deps | list[string] | Intra-corpus identifiers referenced |
| docstring | string | Preceding documentation comment, null if absent |
| line_start | int | First source line |
| line_end | int | Last source line |
| has_proof | bool | Whether a proof block was captured |
| source_url | string | Upstream repository |
| commit | string | Upstream commit extracted |
Statistics
- Entries: 235
- With proof: 228 (97.0%)
- With docstring: 32 (13.6%)
- Libraries: 1
By type
| Type | Count |
|---|---|
| def | 228 |
| axiom | 7 |
Example
+0R : (x : nat) β path nat {+ x 0} x :=
elim [
| zero => +0L 0
| suc {x => ih} =>
equation nat begin
+ {suc x} 0 =[ +SL x 0 ]
suc {+ x 0} =[ i => suc {ih i} ]
suc x
end
]
- type: def | symbolic_name:
+0R| test/abstract.cooltt:23
Use
Statement and proof are available both joined (fact) and split (statement, proof) for
proof-term modeling, autoformalization, retrieval, and dependency analysis via deps.
Citation
@misc{cooltt_dataset,
title = {cooltt},
author = {Norton, Charles},
year = {2026},
note = {Extracted from https://github.com/RedPRL/cooltt, commit b39bf2990045},
url = {https://huggingface.co/datasets/phanerozoic/cooltt}
}
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