statement stringlengths 5 595 | proof stringlengths 0 691 | type stringclasses 2
values | symbolic_name stringlengths 1 27 | library stringclasses 1
value | filename stringclasses 30
values | imports listlengths 0 2 | deps listlengths 0 8 | docstring stringclasses 32
values | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
hcom-type (i : 𝕀) : type | :=
hcom type 0 1 {∂ i} {j => [j=0 => v-test i nat | ∂ i => nat]} | def | hcom-type | test | test/hcom-type.cooltt | [] | [
"v-test"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
hcom-box (i : 𝕀) : hcom-type i | :=
[?asdf, [?,?]] | def | hcom-box | test | test/hcom-type.cooltt | [] | [
"hcom-type"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
has-hlevel : nat → type → type | :=
let aux : nat → type → type :=
elim [
| zero => A => (a a' : A) → path A a a'
| suc {l => ih} =>
A => (a : A) (a' : A) → ih {path A a a'}
]
in
elim [
| zero => A => sig [pt : A, path : (pt' : A) → path A pt pt']
| suc l => aux l
] | def | has-hlevel | test | test/hlevel.cooltt | [
"prelude"
] | [
"path"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
htype : sig [lvl : nat, tp : type] → type | := a => open a in has-hlevel lvl tp | def | htype | test | test/hlevel.cooltt | [
"prelude"
] | [
"has-hlevel"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
contr : type | := htype # [lvl := 0] | def | contr | test | test/hlevel.cooltt | [
"prelude"
] | [
"htype"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
prop : type | := htype # [lvl := 1] | def | prop | test | test/hlevel.cooltt | [
"prelude"
] | [
"htype"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
set : type | := htype # [lvl := 2] | def | set | test | test/hlevel.cooltt | [
"prelude"
] | [
"htype"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
groupoid : type | := htype # [lvl := 3] | def | groupoid | test | test/hlevel.cooltt | [
"prelude"
] | [
"htype"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
contr-prop (A : contr) : prop # [tp := A.tp] | :=
unfold has-hlevel in pt pt' => trans {A.tp} {symm {A.tp} {A.fib.path pt}} {A.fib.path pt'} | def | contr-prop | test | test/hlevel.cooltt | [
"prelude"
] | [
"contr",
"has-hlevel",
"path",
"prop",
"symm",
"trans"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
prop-set (A : prop) : set # [tp := A.tp] | :=
unfold has-hlevel in a b p q i j =>
hcom {A.tp} 0 1 {∂ i ∨ ∂ j} {k =>
[ k=0 ∨ ∂ j ∨ i=0 => A a {p j} k
| i=1 => A a {q j} k
]
} | def | prop-set | test | test/hlevel.cooltt | [
"prelude"
] | [
"has-hlevel",
"prop",
"set"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
raise-hlevel (A : htype) : htype # [lvl := suc {A.lvl}, tp := A.tp] | :=
let aux : (m : nat) (B : htype # [lvl := suc m]) → htype # [lvl := suc {suc m}, tp := B.tp] :=
elim [
| zero => prop-set
| suc {l => ih} => B =>
unfold has-hlevel in
b b' => {ih {struct [tp := path {B.tp} b b', fib := B b b']}}
]
in
let aux2 : (m : nat) (B : htype # [lvl := m]) → ... | def | raise-hlevel | test | test/hlevel.cooltt | [
"prelude"
] | [
"contr-prop",
"has-hlevel",
"htype",
"path",
"prop-set"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
prop-hlevel : (l : nat) (A : prop) → htype # [lvl := suc l, tp := A.tp] | :=
elim [
| zero => A => A
| suc {l => ih} => A => raise-hlevel {ih A}
] | def | prop-hlevel | test | test/hlevel.cooltt | [
"prelude"
] | [
"htype",
"prop",
"raise-hlevel"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
path-hlevel : (l : nat) (A : htype # [lvl := suc l]) (a a' : A.tp) → htype # [lvl := l, tp := path {A.tp} a a'] | :=
elim [
| zero => A a a' =>
unfold has-hlevel in
struct [pt := A a a', path := {prop-set A} a a' {A a a'}]
| suc l =>
unfold has-hlevel in
A => A
] | def | path-hlevel | test | test/hlevel.cooltt | [
"prelude"
] | [
"has-hlevel",
"htype",
"path",
"prop-set"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
path-based-contr (A : type) (a : A) : contr # [tp := (x : A) × path A a x] | :=
unfold has-hlevel in
struct
def pt := [a, i => a]
def path := x i =>
let aux : 𝕀 → A := j =>
hcom A 0 j {∂ i} {k =>
[ k=0 ∨ i=0 => a
| i=1 => {snd x} k
]
}
in
[aux 1, aux]
end | def | path-based-contr | test | test/hlevel.cooltt | [
"prelude"
] | [
"contr",
"has-hlevel",
"path"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
incomplete-trans₁ (A : type) (p : 𝕀 → A) (q : (i : 𝕀) → sub A {i=0} {p 1}) : path A {p 0} {q 1} | :=
i => {! !} | def | incomplete-trans₁ | test | test/holes.cooltt | [
"prelude"
] | [
"path"
] | Test that the cone of silence tactic works | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
incomplete-trans₂ (A : type) (p : 𝕀 → A) (q : (i : 𝕀) → sub A {i=0} {p 1}) : path A {p 0} {q 1} | :=
i => {! trans/filler A ? ? 1 i !} | def | incomplete-trans₂ | test | test/holes.cooltt | [
"prelude"
] | [
"path",
"trans/filler"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
incomplete-trans₃ (A : type) (p : 𝕀 → A) (q : (i : 𝕀) → sub A {i=0} {p 1}) : path A {p 0} {q 1} | :=
i => {! trans/filler A p q 0 i !} | def | incomplete-trans₃ | test | test/holes.cooltt | [
"prelude"
] | [
"path",
"trans/filler"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
incomplete-trans₄ (A : type) (p : 𝕀 → A) (q : (i : 𝕀) → sub A {i=0} {p 1}) : path A {p 0} {q 1} | :=
i => {! trans/filler A p q 1 i !} | def | incomplete-trans₄ | test | test/holes.cooltt | [
"prelude"
] | [
"path",
"trans/filler"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
incomplete-trans₅ (A : type) (p : 𝕀 → A) (q : (i : 𝕀) → sub A {i=0} {p 1}) : path A {p 0} {q 1} | :=
i => {! trans/filler A ?_h1 ?h2 1 i !} | def | incomplete-trans₅ | test | test/holes.cooltt | [
"prelude"
] | [
"path",
"trans/filler"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
incomplete-trans₆ (A : type) (p : 𝕀 → A) (q : (i : 𝕀) → sub A {i=0} {p 1}) : path A {p 0} {q 1} | :=
i => {! trans/filler A ?h3 ?_h4 1 i !} | def | incomplete-trans₆ | test | test/holes.cooltt | [
"prelude"
] | [
"path",
"trans/filler"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
cool-tt : p1::u | := p2::q::t | def | cool-tt | test | test/import.cooltt | [
"prelude"
] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
i : 𝕀 | axiom | i | test | test/inequality.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | ||
j : 𝕀 | axiom | j | test | test/inequality.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | ||
k : 𝕀 | axiom | k | test | test/inequality.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | ||
ij : [i <= j] | axiom | ij | test | test/inequality.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | ||
jk : [j <= k] | axiom | jk | test | test/inequality.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | ||
foo : ext => nat with [i <= k => zero ] | axiom | foo | test | test/inequality.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | ||
bar : ext => nat with [0=0 => zero] | := foo | def | bar | test | test/inequality.cooltt | [] | [
"foo"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
iso : type | :=
sig
def A : type
def B : type
def fwd : A → B
def bwd : B → A
def fwd-bwd : (b : B) → path B {fwd {bwd b}} b
def bwd-fwd : (a : A) → path A {bwd {fwd a}} a
end | def | iso | test | test/isos.cooltt | [
"prelude"
] | [
"path"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
iso/refl (A : type) : iso # [A := A, B := A] | :=
struct
def fwd := x => x
def bwd := x => x
def fwd-bwd := b i => b
def bwd-fwd := a i => a
end | def | iso/refl | test | test/isos.cooltt | [
"prelude"
] | [
"iso"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
iso/symm (I : iso) : iso # [A := I.B, B := I.A] | :=
open I in
struct
def fwd := bwd
def bwd := fwd
def fwd-bwd := bwd-fwd
def bwd-fwd := fwd-bwd
end | def | iso/symm | test | test/isos.cooltt | [
"prelude"
] | [
"iso"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
iso/trans (I1 : iso) (I2 : iso # [ A := I1.B ]) : iso # [A := I1.A, B := I2.B] | :=
struct
def fwd := a => I2.fwd {I1.fwd a}
def bwd := b => I1.bwd {I2.bwd b}
def fwd-bwd := b => trans {I2.B} {j => I2.fwd {I1.fwd-bwd {I2.bwd b} j}} {I2.fwd-bwd b}
def bwd-fwd := a => trans {I1.A} {j => I1.bwd {I2.bwd-fwd {I1.fwd a} j}} {I1.bwd-fwd a}
end | def | iso/trans | test | test/isos.cooltt | [
"prelude"
] | [
"iso",
"trans"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
iso/pair/comm (X Y : type) : iso # [A := X × Y, B := Y × X] | :=
struct
def fwd := ab => [snd ab, fst ab]
def bwd := ba => [snd ba, fst ba]
def fwd-bwd := ba i => ba
def bwd-fwd := ab i => ab
end | def | iso/pair/comm | test | test/isos.cooltt | [
"prelude"
] | [
"iso"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
iso/pair/assoc (X Y Z : type) : iso # [A := {X × Y × Z}, B := {{X × Y} × Z}] | :=
struct
def fwd := a_bc => [[fst a_bc, fst {snd a_bc}], snd {snd a_bc}]
def bwd := ab_c => [fst {fst ab_c}, [snd {fst ab_c}, snd ab_c]]
def fwd-bwd := ab_c i => ab_c
def bwd-fwd := a_bc i => a_bc
end | def | iso/pair/assoc | test | test/isos.cooltt | [
"prelude"
] | [
"iso"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
iso/pair/unit (X : type) : iso # [A := X × unit, B := X] | :=
struct
def fwd := au => fst au
def bwd := a => [a, ⋆]
def fwd-bwd := a i => a
def bwd-fwd := au i => au
end | def | iso/pair/unit | test | test/isos.cooltt | [
"prelude"
] | [
"iso",
"unit",
"⋆"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
curry (A B C : type) : {{A × B} → C} → {A → B → C} | :=
f a b => f [a , b] | def | curry | test | test/isos.cooltt | [
"prelude"
] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
uncurry (A B C : type) : {A → B → C} → {{A × B} → C} | :=
f ab => f {fst ab} {snd ab} | def | uncurry | test | test/isos.cooltt | [
"prelude"
] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
iso/curry (X Y Z : type) : iso # [A := X → Y → Z, B := {X × Y} → Z] | :=
struct
def fwd := uncurry X Y Z
def bwd := curry X Y Z
def fwd-bwd := ab_c i => ab_c
def bwd-fwd := a_b_c i => a_b_c
end | def | iso/curry | test | test/isos.cooltt | [
"prelude"
] | [
"curry",
"iso",
"uncurry"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
iso/lhs (C : type) (I : iso) : iso # [A := {I.A} → C, B := {I.B} → C] | :=
open I in
struct
def fwd := ac b => ac {bwd b}
def bwd := bc a => bc {fwd a}
def fwd-bwd := bc i b => bc {fwd-bwd b i}
def bwd-fwd := ac i a => ac {bwd-fwd a i}
end | def | iso/lhs | test | test/isos.cooltt | [
"prelude"
] | [
"iso"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
iso/rhs (C : type) (I : iso) : iso # [A := C → {I.A}, B := C → {I.B} ] | :=
open I in
struct
def fwd := ca c => fwd {ca c}
def bwd := cb c => bwd {cb c}
def fwd-bwd := cb i c => fwd-bwd {cb c} i
def bwd-fwd := ca i c => bwd-fwd {ca c} i
end | def | iso/rhs | test | test/isos.cooltt | [
"prelude"
] | [
"iso"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
iso/flip (X Y Z : type) : iso # [A := X → Y → Z, B := Y → X → Z] | :=
struct
def fwd := abc b a => abc a b
def bwd := bac a b => bac b a
def fwd-bwd := bac i => bac
def bwd-fwd := abc i => abc
end | def | iso/flip | test | test/isos.cooltt | [
"prelude"
] | [
"iso"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
x::y : nat | := 0 | def | x::y | test | test/names.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
x (_ : nat) : nat | := 0 | def | x | test | test/names.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
y : nat | := x x::y | def | y | test | test/names.cooltt | [] | [
"x::y"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
x::y::z (u::v : nat) : nat | := u::v | def | x::y::z | test | test/names.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
evan-test (A : type) (φ : 𝔽) (a : A)
(p : sub {path A a a} φ {_ => a})
: (j : 𝕀) → A | :=
p | def | evan-test | test | test/nat-path.cooltt | [
"nat",
"prelude"
] | [
"path"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
J (A : type) (p : 𝕀 → A) (C : {(i : 𝕀) → sub A {i=0} {p 0}} → type) (d : C {_ => p 0}) : C p | :=
coe {i =>
C {hfill A 0 {∂ i} {k => [k=0 ∨ i=0 => p 0 | i=1 => p k]}}
} 0 1 d | def | J | test | test/nat-path.cooltt | [
"nat",
"prelude"
] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
J/eq (A : type) (p : 𝕀 → A) (C : {(i : 𝕀) → sub A {i=0} {p 0}} → type) (d : C {_ => p 0}) : path {C {_ => p 0}} {J A {_ => p 0} C d} d | :=
let square : 𝕀 → 𝕀 → A := i => hfill A 0 {∂ i} {_ => p 0} in
k =>
let mot : 𝕀 → type :=
i => C {hfill A 0 {∂ k ∨ ∂ i} {j => [k=0 => square i j | j=0 ∨ k=1 ∨ ∂ i => p 0]}}
in
unfold J in
com mot 0 1 {∂ k} {i => [k=0 => coe {j => C {square j}} 0 i d | k=1 ∨ i=0 => d]} | def | J/eq | test | test/nat-path.cooltt | [
"nat",
"prelude"
] | [
"path",
"square"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
trans-left-unit (A : type) (p : 𝕀 → A) : path {path A {p 0} {p 1}} p {trans A {_ => p 0} p} | :=
k i =>
unfold trans in
hcom A 0 1 {k=0 ∨ ∂ i} {j =>
[ j=0 ∨ i=0 => p 0
| i=1 => p j
| k=0 =>
hcom A 0 1 {∂ i ∨ ∂ j} {l =>
let filler : 𝕀 → A := k => trans/filler A {_ => p 0} p k l in
[ l=0 ∨ i=0 ∨ j=1 => filler i
| i=1 ∨ j=0 => filler j
]
}
]
} | def | trans-left-unit | test | test/nat-path.cooltt | [
"nat",
"prelude"
] | [
"path",
"trans",
"trans/filler"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
trans-right-unit (A : type) (p : 𝕀 → A) : path {path A {p 0} {p 1}} p {trans A p {_ => p 1}} | :=
unfold trans in
trans/filler A p {_ => p 1} | def | trans-right-unit | test | test/nat-path.cooltt | [
"nat",
"prelude"
] | [
"path",
"trans",
"trans/filler"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
trans-symm-refl (A : type) (p : 𝕀 → A) : path {path A {p 0} {p 0}} {_ => p 0} {trans A p {symm A p}} | :=
k i =>
unfold trans symm in
hcom A 0 1 {k=0 ∨ ∂ i} {j =>
symm/filler A p j i
} | def | trans-symm-refl | test | test/nat-path.cooltt | [
"nat",
"prelude"
] | [
"path",
"symm",
"symm/filler",
"trans"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
test (p : 𝕀 → nat) : (i : 𝕀) → nat | :=
let fun : nat → nat :=
elim [
| zero => zero
| suc _ => zero
]
in
i => fun {symm nat p i} | def | test | test | test/nat-path.cooltt | [
"nat",
"prelude"
] | [
"symm"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
test2 : (i : 𝕀) → nat | :=
let fun : nat → nat :=
elim [
| zero => zero
| suc _ => zero
]
in
i =>
fun {symm nat {_ => zero} i} | def | test2 | test | test/nat-path.cooltt | [
"nat",
"prelude"
] | [
"symm"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
test2' : sub {𝕀 → nat} ⊤ {i => hcom nat 0 1 {∂ i} {_ => 0}} | := unfold symm in test2 | def | test2' | test | test/nat-path.cooltt | [
"nat",
"prelude"
] | [
"symm",
"test2"
] | The following illustrates that although cooltt's quasi-normal forms aren't as fully reduced as they could be, it is of no consequence for definitional equivalence. That is, we don't bother pushing eliminators through all the branches of a disjunction split, but our equational theory acts as if we do. | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
+ : nat → nat → nat | :=
elim [
| zero => n => n
| suc {_ => ih} => n => suc {ih n}
] | def | + | test | test/nat.cooltt | [
"prelude"
] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
pred : nat → nat | :=
elim [
| zero => zero
| suc {n => _} => n
] | def | pred | test | test/nat.cooltt | [
"prelude"
] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
+-left-unit (x : nat) : path nat {+ 0 x} x | :=
unfold + in
_ => x | def | +-left-unit | test | test/nat.cooltt | [
"prelude"
] | [
"+",
"path"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
+-suc-l (x y : nat) : path nat {+ {suc x} y} {suc {+ x y}} | :=
unfold + in
_ => suc {+ x y} | def | +-suc-l | test | test/nat.cooltt | [
"prelude"
] | [
"+",
"path"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
+-right-unit : (x : nat) → path nat {+ x 0} x | :=
elim [
| zero =>
+-left-unit 0
| suc {y => ih} =>
equation nat begin
+ {suc y} 0 =[ +-suc-l y 0 ]
suc {+ y 0} =[ i => suc {ih i} ]
suc y
end
] | def | +-right-unit | test | test/nat.cooltt | [
"prelude"
] | [
"+",
"+-left-unit",
"+-suc-l",
"path"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
+-suc-r : (x y : nat) → path nat {+ x {suc y}} {suc {+ x y}} | :=
elim [
| zero => x =>
equation nat begin
+ 0 {suc x} =[ +-left-unit {suc x} ]
suc x =[ i => suc {symm nat {+-left-unit x} i} ]
suc {+ 0 x}
end
| suc {x => ih} => y =>
equation nat begin
+ {suc x} {suc y} =[ +-suc-l x {suc y} ]
suc {+ x {suc y}} =[ i => suc {ih y i} ]
... | def | +-suc-r | test | test/nat.cooltt | [
"prelude"
] | [
"+",
"+-left-unit",
"+-suc-l",
"path",
"symm"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
+-comm : (x y : nat) → path nat {+ y x} {+ x y} | :=
elim [
| zero =>
y =>
equation nat begin
+ y 0 =[ +-right-unit y ]
y =[ symm nat {+-left-unit y} ]
+ 0 y
end
| suc {y => ih} =>
z =>
equation nat begin
+ z {suc y} =[ +-suc-r z y ]
suc {+ z y} =[ i => suc {ih z i} ]
suc {+ y z} =[ symm nat {+-suc-l y z} ]... | def | +-comm | test | test/nat.cooltt | [
"prelude"
] | [
"+",
"+-left-unit",
"+-right-unit",
"+-suc-l",
"+-suc-r",
"path",
"symm"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
+-assoc : (x y z : nat) → path nat {+ {+ x y} z} {+ x {+ y z}} | :=
elim [
| zero => y z =>
equation nat begin
+ {+ 0 y} z =[ i => + {+-left-unit y i} z ]
+ y z =[ symm nat {+-left-unit {+ y z}} ]
+ 0 {+ y z}
end
| suc {x => ih} => y z =>
equation nat begin
+ {+ {suc x} y} z =[ i => + {+-suc-l x y i} z ]
+ {suc {+ x y}} z =[ +-suc-l {+... | def | +-assoc | test | test/nat.cooltt | [
"prelude"
] | [
"+",
"+-left-unit",
"+-suc-l",
"path",
"symm"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
el : type | :=
sig
def A : type
def a : A
end | def | el | test | test/patch.cooltt | [
"prelude"
] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
el-patch : type | := el # [A := nat, a := 4] | def | el-patch | test | test/patch.cooltt | [
"prelude"
] | [
"el"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
el-patch-partial : type | := el # [ A := nat ] | def | el-patch-partial | test | test/patch.cooltt | [
"prelude"
] | [
"el"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
patch/inhabit : el-patch | :=
struct
def A := nat
def a := 4
end | def | patch/inhabit | test | test/patch.cooltt | [
"prelude"
] | [
"el-patch"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
patch-partial/inhabit/infer : el-patch-partial | := struct def a := 4 end | def | patch-partial/inhabit/infer | test | test/patch.cooltt | [
"prelude"
] | [
"el-patch-partial"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
patch/inhabit/infer : el-patch | := struct end | def | patch/inhabit/infer | test | test/patch.cooltt | [
"prelude"
] | [
"el-patch"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
patch/inhabit/hole : el-patch | := struct def A := ? def a := ? end | def | patch/inhabit/hole | test | test/patch.cooltt | [
"prelude"
] | [
"el-patch"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
patch-depends : type | := {sig [A : type, B : type]} # [A := nat, B := A] | def | patch-depends | test | test/patch.cooltt | [
"prelude"
] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
patch-depends/inhabit : patch-depends | := struct def A := nat def B := nat end | def | patch-depends/inhabit | test | test/patch.cooltt | [
"prelude"
] | [
"patch-depends"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
testing (A Z : type) (B : A → type) (p : Z → sig [x : A, bx : B x]) (z : Z) : sig [x : A, bx : B x] # [x := p z.x, bx := p z.bx] | :=
p z | def | testing | test | test/patch.cooltt | [
"prelude"
] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
category : type | :=
sig
def ob : type
def hom : sig [s : ob, t : ob] → type
def idn : (x : ob) → hom # [s := x, t := x]
def seq : (f : hom) → (g : hom # [s := f.t]) → hom # [s := f.s, t := g.t]
def seqL :
abstract seqL∷tp ←
(f : hom) → path {hom # [s := f.s, t := f.t]} {seq {idn {f.s}} f} f
def s... | def | category | test | test/patch.cooltt | [
"prelude"
] | [
"path"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
types : category | :=
unfold category in
struct
def ob := type
def hom := args => {args.s} → {args.t}
def idn := _ x => x
def seq := f g x => g {f x}
def seqL := unfold seqL∷tp in f _ => f
def seqR := unfold seqR∷tp in f _ => f
def seqA := unfold seqA∷tp in f g h _ x => h {g {f x}}
end | def | types | test | test/patch.cooltt | [
"prelude"
] | [
"category"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
test-auto (fam : sig [x : nat] → type) : type | := fam | def | test-auto | test | test/patch.cooltt | [
"prelude"
] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
test-auto-patch (fam : sig [x : nat] → type) : type | := fam # [x := 0] | def | test-auto-patch | test | test/patch.cooltt | [
"prelude"
] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
U : type | := type | def | U | test | test/patch.cooltt | [
"prelude"
] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
test-unfold-total (fam : sig [x : nat] → U) : U | := fam | def | test-unfold-total | test | test/patch.cooltt | [
"prelude"
] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
test-extra-field : sub el true {struct [A := nat, a := 0]} | :=
struct [B := 0, A := nat, x := 0, a := 0, y := 0] | def | test-extra-field | test | test/patch.cooltt | [
"prelude"
] | [
"el"
] | Extra fields are ignored and elaborated away | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
test-extra-field-patch : sub {el # [A := nat]} true {struct [A := nat, a := 0]} | :=
struct [B := 0, x := 0, a := 0, y := 0] | def | test-extra-field-patch | test | test/patch.cooltt | [
"prelude"
] | [
"el"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
test-extra-field-include : sub el true {struct [A := nat, a := 0]} | :=
unfold category in
struct
include types
include test-extra-field
include types
end | def | test-extra-field-include | test | test/patch.cooltt | [
"prelude"
] | [
"category",
"el",
"test-extra-field",
"types"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
formation : {
(A : 𝕀 → type) (a : A 0) (b : A 1) → type
} | := {
A a b =>
ext i => A i with [i=0 => a | i=1 => b]
} | def | formation | test | test/path-types.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
path (A : type) (a : A) (b : A) : type | :=
ext i => A with [i=0 => a | i=1 => b] | def | path | test | test/path-types.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
myrefl : {
(A : type) (a : A) → path A a a
} | := {
A a i => a
} | def | myrefl | test | test/path-types.cooltt | [] | [
"path"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
funext : {
(A B : type) (f : (x : A) → B) (g : (x : A) → B)
(h : (x : A) → path B {f x} {g x})
→ path {(x : A) → B} f g
} | := {
A B f g h i x =>
h x i
} | def | funext | test | test/path-types.cooltt | [] | [
"path"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
funextdep : {
(A : 𝕀 → type) (B : (i : 𝕀) → A i → type)
(f : (i : 𝕀) → [∂ i] → (x : A i) → B i x)
(h : (p : (i : 𝕀) → A i) → ext i => B i {p i} with [∂ i => f i {p i}])
→ ext i => (x : A i) → B i x with [∂ i => f i]
} | := {
A B f h i x =>
h {j => coe A i j x} i
} | def | funextdep | test | test/path-types.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
pairext : {
(A B : type) (p : (x : A) × B) (q : (x : A) × B)
(h : (x : path A {fst p} {fst q}) × path B {snd p} {snd q})
→ path {(x : A) × B} p q
} | := {
A B p q h i =>
[ {fst h} i
, {snd h} i
]
} | def | pairext | test | test/path-types.cooltt | [] | [
"path"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
path (A : type) (x y : A) : type | :=
ext i => A with [i=0 => x | i=1 => y] | def | path | test | test/prelude.cooltt | [] | [] | Path Types, and some common operations | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
path-p (A : 𝕀 → type) (x : A 0) (y : A 1) : type | :=
ext i => A i with [i=0 => x | i=1 => y] | def | path-p | test | test/prelude.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
refl (A : type) (x : A) : path A x x | :=
i => x | def | refl | test | test/prelude.cooltt | [] | [
"path"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
symm/filler (A : type) (p : 𝕀 → A) (i : 𝕀) : (j : 𝕀) → sub A {∂ i} [ i=1 => p 0 | i=0 => p j ] | :=
hfill 0 {j =>
[ j=0 ∨ i=1 => p 0
| i=0 => p j
]
} | def | symm/filler | test | test/prelude.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
symm (A : type) (p : 𝕀 → A) : path A {p 1} {p 0} | :=
i => symm/filler A p i 1 | def | symm | test | test/prelude.cooltt | [] | [
"path",
"symm/filler"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
trans/filler (A : type) (p : 𝕀 → A) (q : (i : 𝕀) → sub A {i=0} {p 1}) (j : 𝕀) (i : 𝕀) : A | :=
hcom A 0 j {∂ i} {j =>
[ j=0 ∨ i=0 => p i
| i=1 => q j
]
} | def | trans/filler | test | test/prelude.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
trans (A : type) (p : 𝕀 → A) (q : (i : 𝕀) → sub A {i=0} {p 1}) : path A {p 0} {q 1} | :=
trans/filler A p q 1 | def | trans | test | test/prelude.cooltt | [] | [
"path",
"trans/filler"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
cong (A : type) (B : A → type) (f : (a : A) → B a) (x y : A) (p : path A x y) : path-p {i => B {p i}} {f x} {f y} | :=
i => f {p i} | def | cong | test | test/prelude.cooltt | [] | [
"path",
"path-p"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
funext (A : type) (B : type) (f : (x : A) → B) (g : (x : A) → B) (h : (x : A) → path B {f x} {g x}) : path {(x : A) → B} f g | :=
i x => h x i | def | funext | test | test/prelude.cooltt | [] | [
"path"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
unit : type | := ext => nat with [⊤ => 0] | def | unit | test | test/prelude.cooltt | [] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
⋆ : unit | := 0 | def | ⋆ | test | test/prelude.cooltt | [] | [
"unit"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
unit/eta (x : unit) : path unit ⋆ x | :=
_ => x | def | unit/eta | test | test/prelude.cooltt | [] | [
"path",
"unit",
"⋆"
] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 | |
test : type | :=
sig
def x : type
def y : type
def fn : x -> y
end | def | test | test | test/record.cooltt | [
"prelude"
] | [] | https://github.com/RedPRL/cooltt | b39bf29900451cb43ae6fbd9af5aa33d59e18935 |
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