Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 103 items • Updated • 3
fact stringlengths 10 2.87k | statement stringlengths 2 731 | proof stringlengths 5 2.72k | type stringclasses 2
values | symbolic_name stringlengths 1 31 | library stringclasses 6
values | filename stringclasses 65
values | imports listlengths 0 10 | deps listlengths 0 15 | docstring stringclasses 19
values | line_start int64 1 266 | line_end int64 3 272 | has_proof bool 1
class | source_url stringclasses 1
value | commit stringclasses 1
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is-biinv-equiv (A B : type) (f : A → B) : type =
section A B f × retraction A B f | is-biinv-equiv (A B : type) (f : A → B) : type | =
section A B f × retraction A B f | def | is-biinv-equiv | library/basics | library/basics/biinv-equiv.red | [
"basics.isotoequiv",
"basics.retract",
"prelude"
] | [
"retraction",
"section"
] | null | 7 | 8 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
biinv-equiv (A B : type) : type = (f : A → B) × is-biinv-equiv A B f | biinv-equiv (A B : type) : type | = (f : A → B) × is-biinv-equiv A B f | def | biinv-equiv | library/basics | library/basics/biinv-equiv.red | [
"basics.isotoequiv",
"basics.retract",
"prelude"
] | [
"is-biinv-equiv"
] | null | 10 | 10 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
biinv-equiv→iso (A B : type) : biinv-equiv A B → iso A B =
λ (f,(g,α),h,β) →
let β' (a : A) : path _ (g (f a)) a =
λ i →
comp 0 1 (h (α (f a) i)) [
| i=0 j → β (g (f a)) j
| i=1 j → β a j
]
in
(f,g,α,β') | biinv-equiv→iso (A B : type) : biinv-equiv A B → iso A B | =
λ (f,(g,α),h,β) →
let β' (a : A) : path _ (g (f a)) a =
λ i →
comp 0 1 (h (α (f a) i)) [
| i=0 j → β (g (f a)) j
| i=1 j → β a j
]
in
(f,g,α,β') | def | biinv-equiv→iso | library/basics | library/basics/biinv-equiv.red | [
"basics.isotoequiv",
"basics.retract",
"prelude"
] | [
"biinv-equiv",
"iso",
"path"
] | null | 12 | 21 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
lcoh (A B : type) (f : A → B) (g : B → A) (f-g : (b : _) → path _ (f (g b)) b) : type =
(g-f : (a : _) → path _ (g (f a)) a)
× (a : A) → path (path _ (f (g (f a))) (f a)) (λ i → f (g-f a i)) (f-g (f a)) | lcoh (A B : type) (f : A → B) (g : B → A) (f-g : (b : _) → path _ (f (g b)) b) : type | =
(g-f : (a : _) → path _ (g (f a)) a)
× (a : A) → path (path _ (f (g (f a))) (f a)) (λ i → f (g-f a i)) (f-g (f a)) | def | lcoh | library/basics | library/basics/ha-equiv.red | [
"basics.retract",
"prelude"
] | [
"path"
] | null | 6 | 8 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
is-ha-equiv (A B : type) (f : A → B) : type =
(g : B → A)
× (f-g : (b : _) → path _ (f (g b)) b)
× lcoh A B f g f-g | is-ha-equiv (A B : type) (f : A → B) : type | =
(g : B → A)
× (f-g : (b : _) → path _ (f (g b)) b)
× lcoh A B f g f-g | def | is-ha-equiv | library/basics | library/basics/ha-equiv.red | [
"basics.retract",
"prelude"
] | [
"lcoh",
"path"
] | null | 10 | 13 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
ha-equiv (A B : type) : type = (f : A → B) × is-ha-equiv A B f | ha-equiv (A B : type) : type | = (f : A → B) × is-ha-equiv A B f | def | ha-equiv | library/basics | library/basics/ha-equiv.red | [
"basics.retract",
"prelude"
] | [
"is-ha-equiv"
] | null | 15 | 15 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
ha-equiv/symm (A B : type) (e : ha-equiv A B) : ha-equiv B A =
let (f, g, f-g, g-f, adj) = e in
let adj' (b : B) : path (path _ (g (f (g b))) (g b)) (λ i → g (f-g b i)) (g-f (g b)) =
λ j i →
let cap0 : A =
comp 1 0 (g (f-g (f-g b i) j)) [
| i=0 k → g (adj (g b) k j)
| i=1 | ∂[j] → refl
... | ha-equiv/symm (A B : type) (e : ha-equiv A B) : ha-equiv B A | =
let (f, g, f-g, g-f, adj) = e in
let adj' (b : B) : path (path _ (g (f (g b))) (g b)) (λ i → g (f-g b i)) (g-f (g b)) =
λ j i →
let cap0 : A =
comp 1 0 (g (f-g (f-g b i) j)) [
| i=0 k → g (adj (g b) k j)
| i=1 | ∂[j] → refl
]
in
let filler (x k : 𝕀) : A =
comp 0 x (g... | def | ha-equiv/symm | library/basics | library/basics/ha-equiv.red | [
"basics.retract",
"prelude"
] | [
"ha-equiv",
"path",
"weak-connection/and",
"weak-connection/or"
] | this symmetry function is exactly involutive on all but the highest coherence | 18 | 49 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
equiv→ha-equiv (A B : type) (e : equiv A B) : ha-equiv A B =
let (f, c) = e in
let g (b : B) = c b .fst .fst in
let f-g (b : B) = c b .fst .snd in
let p (a : A) = symm (fiber A B f (f a)) (c (f a) .snd (a, refl)) in
( f
, g
, f-g
, λ a i → p a i .fst
, λ a j i →
comp 1 0 (p a i .snd j) [
| i=0... | equiv→ha-equiv (A B : type) (e : equiv A B) : ha-equiv A B | =
let (f, c) = e in
let g (b : B) = c b .fst .fst in
let f-g (b : B) = c b .fst .snd in
let p (a : A) = symm (fiber A B f (f a)) (c (f a) .snd (a, refl)) in
( f
, g
, f-g
, λ a i → p a i .fst
, λ a j i →
comp 1 0 (p a i .snd j) [
| i=0 k → weak-connection/and B (f-g (f a)) j k
| i=1 → refl... | def | equiv→ha-equiv | library/basics | library/basics/ha-equiv.red | [
"basics.retract",
"prelude"
] | [
"equiv",
"fiber",
"ha-equiv",
"symm",
"weak-connection/and",
"weak-connection/or"
] | null | 51 | 67 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
stable (A : type) : type =
neg (neg A) → A | stable (A : type) : type | =
neg (neg A) → A | def | stable | library/basics | library/basics/hedberg.red | [
"basics.retract",
"data.or",
"data.void",
"prelude"
] | [
"neg"
] | null | 6 | 7 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
dec (A : type) : type =
or A (neg A) | dec (A : type) : type | =
or A (neg A) | def | dec | library/basics | library/basics/hedberg.red | [
"basics.retract",
"data.or",
"data.void",
"prelude"
] | [
"neg",
"or"
] | null | 9 | 10 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
discrete (A : type) : type =
(x y : A) → dec (path A x y) | discrete (A : type) : type | =
(x y : A) → dec (path A x y) | def | discrete | library/basics | library/basics/hedberg.red | [
"basics.retract",
"data.or",
"data.void",
"prelude"
] | [
"dec",
"path"
] | null | 12 | 13 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
dec→stable (A : type) : dec A → stable A =
elim [
| inl a → λ _ → a
| inr f → λ g → elim (g f) []
] | dec→stable (A : type) : dec A → stable A | =
elim [
| inl a → λ _ → a
| inr f → λ g → elim (g f) []
] | def | dec→stable | library/basics | library/basics/hedberg.red | [
"basics.retract",
"data.or",
"data.void",
"prelude"
] | [
"dec",
"stable"
] | null | 15 | 19 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
neg/is-prop-over (A : 𝕀 → type)
: is-prop-over (λ i → neg (A i))
= prop→prop-over (λ i → neg (A i)) (neg/prop (A 1)) | neg/is-prop-over (A : 𝕀 → type)
: is-prop-over (λ i → neg (A i)) | = prop→prop-over (λ i → neg (A i)) (neg/prop (A 1)) | def | neg/is-prop-over | library/basics | library/basics/hedberg.red | [
"basics.retract",
"data.or",
"data.void",
"prelude"
] | [
"is-prop-over",
"neg",
"neg/prop",
"prop→prop-over"
] | null | 21 | 23 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
paths-stable→set (A : type) (st : (x y : A) → stable (path A x y)) : is-set A =
λ a b p q i j →
let square (k m : 𝕀) : A =
comp 0 k a [
| m=0 → p
| m=1 → q
]
in
let mycap (k m : 𝕀) = st (p k) (q k) (λ c → c (square k)) m in
comp 0 1 (mycap j i) [
| i=0 k →
st (p j) (p j)
(neg/is... | paths-stable→set (A : type) (st : (x y : A) → stable (path A x y)) : is-set A | =
λ a b p q i j →
let square (k m : 𝕀) : A =
comp 0 k a [
| m=0 → p
| m=1 → q
]
in
let mycap (k m : 𝕀) = st (p k) (q k) (λ c → c (square k)) m in
comp 0 1 (mycap j i) [
| i=0 k →
st (p j) (p j)
(neg/is-prop-over (λ j → neg (path A (p j) (p j)))
(λ c → c (square 0))
... | def | paths-stable→set | library/basics | library/basics/hedberg.red | [
"basics.retract",
"data.or",
"data.void",
"prelude"
] | [
"is-set",
"neg",
"neg/is-prop-over",
"path",
"square",
"stable",
"weak-connection/or"
] | Hedberg's theorem for stable path types | 26 | 45 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
discrete→set (A : type) (d : discrete A) : is-set A =
paths-stable→set A (λ x y → dec→stable (path A x y) (d x y)) | discrete→set (A : type) (d : discrete A) : is-set A | =
paths-stable→set A (λ x y → dec→stable (path A x y) (d x y)) | def | discrete→set | library/basics | library/basics/hedberg.red | [
"basics.retract",
"data.or",
"data.void",
"prelude"
] | [
"dec→stable",
"discrete",
"is-set",
"path",
"paths-stable→set"
] | Hedberg's theorem for decidable path types | 48 | 49 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
hrel/set-equiv
(A : type) (R : A → A → type)
(R/prop : (x y : A) → is-prop (R x y))
(R/refl : (x : A) → R x x)
(R/id : (x y : A) → R x y → path A x y)
: (is-set A) × ((x y : A) → equiv (R x y) (path A x y))
=
let eq = path-retract/equiv A R (λ a b →
( R/id a b
, λ p → coe 0 1 (R/refl a) in λ... | hrel/set-equiv
(A : type) (R : A → A → type)
(R/prop : (x y : A) → is-prop (R x y))
(R/refl : (x : A) → R x x)
(R/id : (x y : A) → R x y → path A x y)
: (is-set A) × ((x y : A) → equiv (R x y) (path A x y)) | =
let eq = path-retract/equiv A R (λ a b →
( R/id a b
, λ p → coe 0 1 (R/refl a) in λ j → R a (p j)
, λ rab → R/prop a b (coe 0 1 (R/refl a) in λ j → R a (R/id a b rab j)) rab
)) in
( λ x y → coe 0 1 (R/prop x y) in λ j → is-prop (ua _ _ (eq x y) j)
, eq
) | def | hrel/set-equiv | library/basics | library/basics/hedberg.red | [
"basics.retract",
"data.or",
"data.void",
"prelude"
] | [
"equiv",
"is-prop",
"is-set",
"path",
"path-retract/equiv",
"ua"
] | null | 51 | 65 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
paths-stable→set/alt (A : type) (st : (x y : A) → stable (path A x y)) : is-set A =
(hrel/set-equiv A (λ x y → neg (neg (path A x y)))
(λ x y → neg/prop (neg (path A x y)))
(λ _ np → np refl)
st
).fst | paths-stable→set/alt (A : type) (st : (x y : A) → stable (path A x y)) : is-set A | =
(hrel/set-equiv A (λ x y → neg (neg (path A x y)))
(λ x y → neg/prop (neg (path A x y)))
(λ _ np → np refl)
st
).fst | def | paths-stable→set/alt | library/basics | library/basics/hedberg.red | [
"basics.retract",
"data.or",
"data.void",
"prelude"
] | [
"hrel/set-equiv",
"is-set",
"neg",
"neg/prop",
"path",
"stable"
] | Hedberg's theorem is a corollary of above | 68 | 73 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
iso (A B : type) : type =
(f : A → B)
× (g : B → A)
× ((b : _) → path _ (f (g b)) b)
× (a : _) → path _ (g (f a)) a | iso (A B : type) : type | =
(f : A → B)
× (g : B → A)
× ((b : _) → path _ (f (g b)) b)
× (a : _) → path _ (g (f a)) a | def | iso | library/basics | library/basics/isotoequiv.red | [
"prelude"
] | [
"path"
] | null | 7 | 11 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
iso/refl (A : type) : iso A A =
( λ f → f
, λ g → g
, λ _ → refl
, λ _ → refl
) | iso/refl (A : type) : iso A A | =
( λ f → f
, λ g → g
, λ _ → refl
, λ _ → refl
) | def | iso/refl | library/basics | library/basics/isotoequiv.red | [
"prelude"
] | [
"iso"
] | null | 13 | 18 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
iso/symm (A B : type) (I : iso A B) : iso B A =
let (f,g,α,β) = I in (g,f,β,α) | iso/symm (A B : type) (I : iso A B) : iso B A | =
let (f,g,α,β) = I in (g,f,β,α) | def | iso/symm | library/basics | library/basics/isotoequiv.red | [
"prelude"
] | [
"iso"
] | null | 20 | 21 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
iso/trans (A B C : type) (I1 : iso A B) (I2 : iso B C) : iso A C =
let (f1,g1,α1,β1) = I1 in
let (f2,g2,α2,β2) = I2 in
( λ a → f2 (f1 a)
, λ c → g1 (g2 c)
, λ c → trans _ (λ j → f2 (α1 (g2 c) j)) (α2 c)
, λ a → trans _ (λ j → g1 (β2 (f1 a) j)) (β1 a)
) | iso/trans (A B C : type) (I1 : iso A B) (I2 : iso B C) : iso A C | =
let (f1,g1,α1,β1) = I1 in
let (f2,g2,α2,β2) = I2 in
( λ a → f2 (f1 a)
, λ c → g1 (g2 c)
, λ c → trans _ (λ j → f2 (α1 (g2 c) j)) (α2 c)
, λ a → trans _ (λ j → g1 (β2 (f1 a) j)) (β1 a)
) | def | iso/trans | library/basics | library/basics/isotoequiv.red | [
"prelude"
] | [
"iso",
"trans"
] | null | 23 | 30 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
iso/fiber/prop-over
(A B : type)
(I : iso A B) (b : 𝕀 → B)
: is-prop-over (λ i → fiber _ _ (I.fst) (b i))
=
let (f, g, α, β) = I in
let sq (b : B) (fib : fiber _ _ f b) (j k : 𝕀) : A =
comp k j (β (fib.fst) k) [
| k=1 → refl
| k=0 j → g (fib.snd j)
]
in
λ fib0 fib1 →
let sq2 (i k : �... | iso/fiber/prop-over
(A B : type)
(I : iso A B) (b : 𝕀 → B)
: is-prop-over (λ i → fiber _ _ (I.fst) (b i)) | =
let (f, g, α, β) = I in
let sq (b : B) (fib : fiber _ _ f b) (j k : 𝕀) : A =
comp k j (β (fib.fst) k) [
| k=1 → refl
| k=0 j → g (fib.snd j)
]
in
λ fib0 fib1 →
let sq2 (i k : 𝕀) : A =
comp 0 k (g (b i)) [
| i=0 → sq (b 0) fib0 1
| i=1 → sq (b 1) fib1 1
]
in
λ i →
( re... | def | iso/fiber/prop-over | library/basics | library/basics/isotoequiv.red | [
"prelude"
] | [
"fiber",
"is-prop-over",
"iso"
] | null | 32 | 67 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
iso→equiv (A B : type) (I : iso A B) : equiv A B =
let (f, g, α, β) = I in
(f , λ b → ((g b, α b), λ fib → iso/fiber/prop-over _ _ I (λ _ → b) fib (g b, α b))) | iso→equiv (A B : type) (I : iso A B) : equiv A B | =
let (f, g, α, β) = I in
(f , λ b → ((g b, α b), λ fib → iso/fiber/prop-over _ _ I (λ _ → b) fib (g b, α b))) | def | iso→equiv | library/basics | library/basics/isotoequiv.red | [
"prelude"
] | [
"equiv",
"iso",
"iso/fiber/prop-over"
] | null | 69 | 71 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
equiv→iso (A B : type) (e : equiv A B) : iso A B =
( e .fst
, λ b → e .snd b .fst .fst
, λ b → e .snd b .fst .snd
, λ a i → symm (fiber A B (e .fst) (e .fst a)) (e .snd (e .fst a) .snd (a, refl)) i .fst
) | equiv→iso (A B : type) (e : equiv A B) : iso A B | =
( e .fst
, λ b → e .snd b .fst .fst
, λ b → e .snd b .fst .snd
, λ a i → symm (fiber A B (e .fst) (e .fst a)) (e .snd (e .fst a) .snd (a, refl)) i .fst
) | def | equiv→iso | library/basics | library/basics/isotoequiv.red | [
"prelude"
] | [
"equiv",
"fiber",
"iso",
"symm"
] | null | 79 | 84 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
retraction (A B : type) (f : A → B) : type =
(g : B → A) × (a : A) → path A (g (f a)) a | retraction (A B : type) (f : A → B) : type | =
(g : B → A) × (a : A) → path A (g (f a)) a | def | retraction | library/basics | library/basics/retract.red | [
"basics.isotoequiv",
"prelude"
] | [
"path"
] | null | 4 | 5 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
section (A B : type) (f : A → B) : type =
(g : B → A) × (b : B) → path B (f (g b)) b | section (A B : type) (f : A → B) : type | =
(g : B → A) × (b : B) → path B (f (g b)) b | def | section | library/basics | library/basics/retract.red | [
"basics.isotoequiv",
"prelude"
] | [
"path"
] | null | 7 | 8 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
retract (A B : type) : type =
(f : A → B) × retraction A B f | retract (A B : type) : type | =
(f : A → B) × retraction A B f | def | retract | library/basics | library/basics/retract.red | [
"basics.isotoequiv",
"prelude"
] | [
"retraction"
] | null | 10 | 11 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
retract/path-action (A B : type)
(f : A → B) (retr : retraction A B f) (a a' : A)
: retract (path _ a a') (path B (f a) (f a'))
=
let (g,α) = retr in
( λ p i → f (p i)
, λ q i → comp 0 1 (g (q i)) [i=0 → α a | i=1 → α a']
, λ p j i → comp j 1 (α (p i) j) [i=0 → α a | i=1 → α a']
) | retract/path-action (A B : type)
(f : A → B) (retr : retraction A B f) (a a' : A)
: retract (path _ a a') (path B (f a) (f a')) | =
let (g,α) = retr in
( λ p i → f (p i)
, λ q i → comp 0 1 (g (q i)) [i=0 → α a | i=1 → α a']
, λ p j i → comp j 1 (α (p i) j) [i=0 → α a | i=1 → α a']
) | def | retract/path-action | library/basics | library/basics/retract.red | [
"basics.isotoequiv",
"prelude"
] | [
"path",
"retract",
"retraction"
] | null | 13 | 21 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
retract/hlevel : (l : hlevel) (A B : type)
→ retract A B → has-hlevel l B → has-hlevel l A
=
elim [
| contr → λ A B (f,g,α) B/contr →
( g (B/contr .fst)
, λ a i →
comp 0 1 (g (B/contr .snd (f a) i)) [
| i=0 → α a
| i=1 → refl
]
)
| hsuc l →
elim l [
| contr → λ A B ... | retract/hlevel : (l : hlevel) (A B : type)
→ retract A B → has-hlevel l B → has-hlevel l A | =
elim [
| contr → λ A B (f,g,α) B/contr →
( g (B/contr .fst)
, λ a i →
comp 0 1 (g (B/contr .snd (f a) i)) [
| i=0 → α a
| i=1 → refl
]
)
| hsuc l →
elim l [
| contr → λ A B (f,g,α) B/prop a a' i →
comp 0 1 (g (B/prop (f a) (f a') i)) [
| i=0 → α a
| ... | def | retract/hlevel | library/basics | library/basics/retract.red | [
"basics.isotoequiv",
"prelude"
] | [
"has-hlevel",
"hlevel",
"path",
"retract",
"retract/path-action"
] | null | 23 | 47 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
path-retract/preserves-refl (A : type) (R : A → A → type)
(ret : (x y : A) → retract (R x y) (path A x y)) (x : A)
: path _ (ret x x .fst (ret x x .snd .fst refl)) refl
=
let s (x y : A) : R x y → path A x y = ret x y .fst in
let r (x y : A) : path A x y → R x y = ret x y .snd .fst in
let q = s x x (r x x r... | path-retract/preserves-refl (A : type) (R : A → A → type)
(ret : (x y : A) → retract (R x y) (path A x y)) (x : A)
: path _ (ret x x .fst (ret x x .snd .fst refl)) refl | =
let s (x y : A) : R x y → path A x y = ret x y .fst in
let r (x y : A) : path A x y → R x y = ret x y .snd .fst in
let q = s x x (r x x refl) in
let cap1 : [i j] A [
| j=0 → x
| j=1 → q i
| i=0 → q j
| i=1 → s x x (r x x q) j
]
=
λ i j →
s x (q i) (r x (q i) (λ k → weak-connect... | def | path-retract/preserves-refl | library/basics | library/basics/retract.red | [
"basics.isotoequiv",
"prelude"
] | [
"path",
"retract",
"weak-connection/and"
] | Adapted from https://github.com/HoTT/book/issues/718 Any family of retracts of the path family preserves refl through the other round-trip | 51 | 91 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
path-retract/equiv (A : type) (R : A → A → type)
(ret : (x y : A) → retract (R x y) (path A x y)) (a b : A)
: equiv (R a b) (path A a b)
=
let preserves-refl = path-retract/preserves-refl A R ret a in
iso→equiv (R a b) (path A a b)
( ret a b .fst
, ret a b .snd .fst
, λ p → J A p (λ q → path _ (re... | path-retract/equiv (A : type) (R : A → A → type)
(ret : (x y : A) → retract (R x y) (path A x y)) (a b : A)
: equiv (R a b) (path A a b) | =
let preserves-refl = path-retract/preserves-refl A R ret a in
iso→equiv (R a b) (path A a b)
( ret a b .fst
, ret a b .snd .fst
, λ p → J A p (λ q → path _ (ret a (q 1) .fst (ret a (q 1) .snd .fst q)) q) preserves-refl
, ret a b .snd .snd
) | def | path-retract/equiv | library/basics | library/basics/retract.red | [
"basics.isotoequiv",
"prelude"
] | [
"equiv",
"iso→equiv",
"path",
"path-retract/preserves-refl",
"retract"
] | null | 95 | 105 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
equiv-section/prop (A B : type) (f : A → B) (c : is-equiv A B f)
: is-prop (section A B f) =
λ (g0,p0) (g1,p1) i →
let α (b : B) : path (fiber A B f b) (g0 b, p0 b) (g1 b, p1 b) =
contr→prop (fiber A B f b) (c b) (g0 b, p0 b) (g1 b, p1 b)
in
(λ b → α b i .fst, λ b → α b i .snd) | equiv-section/prop (A B : type) (f : A → B) (c : is-equiv A B f)
: is-prop (section A B f) | =
λ (g0,p0) (g1,p1) i →
let α (b : B) : path (fiber A B f b) (g0 b, p0 b) (g1 b, p1 b) =
contr→prop (fiber A B f b) (c b) (g0 b, p0 b) (g1 b, p1 b)
in
(λ b → α b i .fst, λ b → α b i .snd) | def | equiv-section/prop | library/basics | library/basics/retract.red | [
"basics.isotoequiv",
"prelude"
] | [
"contr→prop",
"fiber",
"is-equiv",
"is-prop",
"path",
"section"
] | null | 107 | 113 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
precompose-equiv (A B C : type) (e : equiv A B) : equiv (B → C) (A → C) =
let (f,g,α,β) = equiv→iso _ _ e in
iso→equiv (B → C) (A → C)
( λ h a → h (f a)
, λ k b → k (g b)
, λ k i a → k (β a i)
, λ h i b → h (α b i)
) | precompose-equiv (A B C : type) (e : equiv A B) : equiv (B → C) (A → C) | =
let (f,g,α,β) = equiv→iso _ _ e in
iso→equiv (B → C) (A → C)
( λ h a → h (f a)
, λ k b → k (g b)
, λ k i a → k (β a i)
, λ h i b → h (α b i)
) | def | precompose-equiv | library/basics | library/basics/retract.red | [
"basics.isotoequiv",
"prelude"
] | [
"equiv",
"equiv→iso",
"iso→equiv"
] | TODO this does not really belong in this file | 116 | 123 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
equiv-retraction/prop (A B : type) (f : A → B) (c : is-equiv A B f)
: is-prop (retraction A B f) =
λ (g0,q0) (g1,q1) i →
let p =
contr→prop _ (precompose-equiv A B A (f,c) .snd (λ a → a))
(g0, λ j b → q0 b j) (g1, λ j b → q1 b j)
in
(p i .fst, λ b j → p i .snd j b) | equiv-retraction/prop (A B : type) (f : A → B) (c : is-equiv A B f)
: is-prop (retraction A B f) | =
λ (g0,q0) (g1,q1) i →
let p =
contr→prop _ (precompose-equiv A B A (f,c) .snd (λ a → a))
(g0, λ j b → q0 b j) (g1, λ j b → q1 b j)
in
(p i .fst, λ b j → p i .snd j b) | def | equiv-retraction/prop | library/basics | library/basics/retract.red | [
"basics.isotoequiv",
"prelude"
] | [
"contr→prop",
"is-equiv",
"is-prop",
"precompose-equiv",
"retraction"
] | null | 125 | 132 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
biinv-int where
| zero
| suc (z : biinv-int)
| predl (z : biinv-int)
| predr (z : biinv-int)
| predl-suc (z : biinv-int) (i : 𝕀) [i=0 → predl (suc z) | i=1 → z]
| suc-predr (z : biinv-int) (i : 𝕀) [i=0 → suc (predr z) | i=1 → z] | biinv-int | where
| zero
| suc (z : biinv-int)
| predl (z : biinv-int)
| predr (z : biinv-int)
| predl-suc (z : biinv-int) (i : 𝕀) [i=0 → predl (suc z) | i=1 → z]
| suc-predr (z : biinv-int) (i : 𝕀) [i=0 → suc (predr z) | i=1 → z] | data | biinv-int | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [] | null | 8 | 14 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
predl/ha-equiv : ha-equiv biinv-int biinv-int =
equiv→ha-equiv _ _
(iso→equiv biinv-int biinv-int
( λ z → predl z
, λ z → suc z
, λ z i → predl-suc z i
, λ z i →
comp 0 1 (suc (predl-suc (predr z) i)) [
| i=0 j → suc (predl (suc-predr z j))
| i=1 j → suc-predr z j
... | predl/ha-equiv : ha-equiv biinv-int biinv-int | =
equiv→ha-equiv _ _
(iso→equiv biinv-int biinv-int
( λ z → predl z
, λ z → suc z
, λ z i → predl-suc z i
, λ z i →
comp 0 1 (suc (predl-suc (predr z) i)) [
| i=0 j → suc (predl (suc-predr z j))
| i=1 j → suc-predr z j
]
)) | def | predl/ha-equiv | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"equiv→ha-equiv",
"ha-equiv",
"iso→equiv"
] | null | 16 | 27 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
suc-predl = predl/ha-equiv .snd .snd .snd .fst | suc-predl | = predl/ha-equiv .snd .snd .snd .fst | def | suc-predl | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"predl/ha-equiv"
] | null | 29 | 29 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
predl-adj = predl/ha-equiv .snd .snd .snd .snd | predl-adj | = predl/ha-equiv .snd .snd .snd .snd | def | predl-adj | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"predl/ha-equiv"
] | null | 30 | 30 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
suc-adj = ha-equiv/symm _ _ predl/ha-equiv .snd .snd .snd .snd | suc-adj | = ha-equiv/symm _ _ predl/ha-equiv .snd .snd .snd .snd | def | suc-adj | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"ha-equiv/symm",
"predl/ha-equiv"
] | null | 31 | 31 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
suc/equiv : equiv biinv-int biinv-int =
iso→equiv biinv-int biinv-int
( λ z → suc z
, λ z → predl z
, suc-predl
, λ z i → predl-suc z i
) | suc/equiv : equiv biinv-int biinv-int | =
iso→equiv biinv-int biinv-int
( λ z → suc z
, λ z → predl z
, suc-predl
, λ z i → predl-suc z i
) | def | suc/equiv | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"equiv",
"iso→equiv",
"suc-predl"
] | null | 33 | 39 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
predl~predr (z : biinv-int) : path biinv-int (predl z) (predr z) =
λ j →
equiv-section/prop biinv-int biinv-int (λ z → suc z) (suc/equiv .snd)
(λ z → predl z, suc-predl)
(λ z → predr z, λ z i → suc-predr z i)
j .fst z | predl~predr (z : biinv-int) : path biinv-int (predl z) (predr z) | =
λ j →
equiv-section/prop biinv-int biinv-int (λ z → suc z) (suc/equiv .snd)
(λ z → predl z, suc-predl)
(λ z → predr z, λ z i → suc-predr z i)
j .fst z | def | predl~predr | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"equiv-section/prop",
"path",
"suc-predl",
"suc/equiv"
] | null | 41 | 46 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
suc-predl~suc-predr (z : biinv-int) : [j i] biinv-int [
| i=0 → suc (predl~predr z j)
| i=1 → z
| j=0 → suc-predl z i
| j=1 → suc-predr z i
]
=
λ j i →
equiv-section/prop biinv-int biinv-int (λ z → suc z) (suc/equiv .snd)
(λ z → predl z, suc-predl)
(λ z → predr z, λ z i → suc-predr z i)
j .s... | suc-predl~suc-predr (z : biinv-int) : [j i] biinv-int [
| i=0 → suc (predl~predr z j)
| i=1 → z
| j=0 → suc-predl z i
| j=1 → suc-predr z i
] | =
λ j i →
equiv-section/prop biinv-int biinv-int (λ z → suc z) (suc/equiv .snd)
(λ z → predl z, suc-predl)
(λ z → predr z, λ z i → suc-predr z i)
j .snd z i | def | suc-predl~suc-predr | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"equiv-section/prop",
"predl~predr",
"suc-predl",
"suc/equiv"
] | null | 48 | 59 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
fwd/pos : nat → biinv-int = elim [zero → zero | suc (_ → ih/n) → suc ih/n] | fwd/pos : nat → biinv-int | = elim [zero → zero | suc (_ → ih/n) → suc ih/n] | def | fwd/pos | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"nat"
] | null | 63 | 63 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
fwd/negsuc : nat → biinv-int = elim [zero → predl zero | suc (_ → ih/n) → predl ih/n] | fwd/negsuc : nat → biinv-int | = elim [zero → predl zero | suc (_ → ih/n) → predl ih/n] | def | fwd/negsuc | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"nat"
] | null | 65 | 65 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
fwd : int → biinv-int = elim [pos n → fwd/pos n | negsuc n → fwd/negsuc n] | fwd : int → biinv-int | = elim [pos n → fwd/pos n | negsuc n → fwd/negsuc n] | def | fwd | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"fwd/negsuc",
"fwd/pos",
"int"
] | null | 67 | 67 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
bwd : biinv-int → int =
elim [
| zero → pos zero
| suc (z → ih/z) → isuc ih/z
| predl (z → ih/z) → pred ih/z
| predr (z → ih/z) → pred ih/z
| predl-suc (z → ih/z) i → pred-isuc ih/z i
| suc-predr (z → ih/z) i → isuc-pred ih/z i
] | bwd : biinv-int → int | =
elim [
| zero → pos zero
| suc (z → ih/z) → isuc ih/z
| predl (z → ih/z) → pred ih/z
| predr (z → ih/z) → pred ih/z
| predl-suc (z → ih/z) i → pred-isuc ih/z i
| suc-predr (z → ih/z) i → isuc-pred ih/z i
] | def | bwd | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"int",
"isuc",
"isuc-pred",
"pred",
"pred-isuc"
] | null | 69 | 77 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
bwd-fwd/pos : (n : nat) → path _ (bwd (fwd/pos n)) (pos n) =
elim [
| zero → refl
| suc (_ → n/ih) → λ k → isuc (n/ih k)
] | bwd-fwd/pos : (n : nat) → path _ (bwd (fwd/pos n)) (pos n) | =
elim [
| zero → refl
| suc (_ → n/ih) → λ k → isuc (n/ih k)
] | def | bwd-fwd/pos | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"bwd",
"fwd/pos",
"isuc",
"nat",
"path"
] | null | 79 | 83 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
bwd-fwd/negsuc : (n : nat) → path _ (bwd (fwd/negsuc n)) (negsuc n) =
elim [
| zero → refl
| suc (_ → n/ih) → λ k → pred (n/ih k)
] | bwd-fwd/negsuc : (n : nat) → path _ (bwd (fwd/negsuc n)) (negsuc n) | =
elim [
| zero → refl
| suc (_ → n/ih) → λ k → pred (n/ih k)
] | def | bwd-fwd/negsuc | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"bwd",
"fwd/negsuc",
"nat",
"path",
"pred"
] | null | 85 | 89 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
bwd-fwd : (n : int) → path _ (bwd (fwd n)) n =
elim [pos n → bwd-fwd/pos n | negsuc n → bwd-fwd/negsuc n] | bwd-fwd : (n : int) → path _ (bwd (fwd n)) n | =
elim [pos n → bwd-fwd/pos n | negsuc n → bwd-fwd/negsuc n] | def | bwd-fwd | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"bwd",
"bwd-fwd/negsuc",
"bwd-fwd/pos",
"fwd",
"int",
"path"
] | null | 91 | 92 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
fwd/isuc/negsuc : (n : nat)
→ path _ (fwd (isuc (negsuc n))) (suc (fwd (negsuc n)))
=
elim [
| zero → λ k → symm' biinv-int (λ j → suc-predl zero j) k
| suc n → λ k → symm' biinv-int (λ j → suc-predl (fwd/negsuc n) j) k
] | fwd/isuc/negsuc : (n : nat)
→ path _ (fwd (isuc (negsuc n))) (suc (fwd (negsuc n))) | =
elim [
| zero → λ k → symm' biinv-int (λ j → suc-predl zero j) k
| suc n → λ k → symm' biinv-int (λ j → suc-predl (fwd/negsuc n) j) k
] | def | fwd/isuc/negsuc | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"fwd",
"fwd/negsuc",
"isuc",
"nat",
"path",
"suc-predl",
"symm'"
] | null | 94 | 100 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
fwd/isuc : (n : int) → path _ (fwd (isuc n)) (suc (fwd n)) =
elim [pos n → refl | negsuc n → fwd/isuc/negsuc n] | fwd/isuc : (n : int) → path _ (fwd (isuc n)) (suc (fwd n)) | =
elim [pos n → refl | negsuc n → fwd/isuc/negsuc n] | def | fwd/isuc | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"fwd",
"fwd/isuc/negsuc",
"int",
"isuc",
"path"
] | null | 102 | 103 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
fwd/pred/pos : (n : nat)
→ path _ (fwd (pred (pos n))) (predl (fwd/pos n))
=
elim [
| zero → refl
| suc n → λ k → symm' biinv-int (λ i → predl-suc (fwd/pos n) i) k
] | fwd/pred/pos : (n : nat)
→ path _ (fwd (pred (pos n))) (predl (fwd/pos n)) | =
elim [
| zero → refl
| suc n → λ k → symm' biinv-int (λ i → predl-suc (fwd/pos n) i) k
] | def | fwd/pred/pos | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"fwd",
"fwd/pos",
"nat",
"path",
"pred",
"symm'"
] | null | 105 | 111 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
fwd/pred : (n : int) → path _ (fwd (pred n)) (predl (fwd n)) =
elim [pos n → fwd/pred/pos n | negsuc n → refl] | fwd/pred : (n : int) → path _ (fwd (pred n)) (predl (fwd n)) | =
elim [pos n → fwd/pred/pos n | negsuc n → refl] | def | fwd/pred | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"fwd",
"fwd/pred/pos",
"int",
"path",
"pred"
] | null | 113 | 114 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
fwd/pred-isuc/negsuc : (n : nat) → [i k] biinv-int [
| i=0 → trans biinv-int (fwd/pred (isuc (negsuc n))) (λ k → predl (fwd/isuc/negsuc n k)) k
| i=1 → fwd/negsuc n
| k=0 → fwd (pred-isuc (negsuc n) i)
| k=1 → predl-suc (fwd/negsuc n) i
]
=
elim [
| zero → λ i k →
comp 0 1 (predl (symm'/filler biinv... | fwd/pred-isuc/negsuc : (n : nat) → [i k] biinv-int [
| i=0 → trans biinv-int (fwd/pred (isuc (negsuc n))) (λ k → predl (fwd/isuc/negsuc n k)) k
| i=1 → fwd/negsuc n
| k=0 → fwd (pred-isuc (negsuc n) i)
| k=1 → predl-suc (fwd/negsuc n) i
] | =
elim [
| zero → λ i k →
comp 0 1 (predl (symm'/filler biinv-int (λ i → suc-predl zero i) i k)) [
| i=0 m → trans/unit/l biinv-int (λ k → predl (fwd/isuc/negsuc zero k)) m k
| i=1 | k=0 → refl
| k=1 m → predl-adj zero m i
]
| suc n → λ i k →
comp 0 1 (predl (symm'/filler biinv-int (λ i → ... | def | fwd/pred-isuc/negsuc | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"fwd",
"fwd/isuc/negsuc",
"fwd/negsuc",
"fwd/pred",
"isuc",
"nat",
"pred-isuc",
"predl-adj",
"suc-predl",
"symm'/filler",
"trans",
"trans/unit/l"
] | null | 117 | 137 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
fwd/pred-isuc : (n : int) → [i k] biinv-int [
| i=0 → trans biinv-int (fwd/pred (isuc n)) (λ k → predl (fwd/isuc n k)) k
| i=1 | k=0 → fwd (pred-isuc n i)
| k=1 → predl-suc (fwd n) i
]
=
elim [
| pos n → λ i k →
comp 0 1 (symm'/filler biinv-int (λ i → predl-suc (fwd/pos n) i) i k) [
| i=0 m → tran... | fwd/pred-isuc : (n : int) → [i k] biinv-int [
| i=0 → trans biinv-int (fwd/pred (isuc n)) (λ k → predl (fwd/isuc n k)) k
| i=1 | k=0 → fwd (pred-isuc n i)
| k=1 → predl-suc (fwd n) i
] | =
elim [
| pos n → λ i k →
comp 0 1 (symm'/filler biinv-int (λ i → predl-suc (fwd/pos n) i) i k) [
| i=0 m → trans/unit/r biinv-int (fwd/pred/pos (suc n)) m k
| i=1 | ∂[k] → refl
]
| negsuc n → fwd/pred-isuc/negsuc n
] | def | fwd/pred-isuc | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"fwd",
"fwd/isuc",
"fwd/pos",
"fwd/pred",
"fwd/pred-isuc/negsuc",
"fwd/pred/pos",
"int",
"isuc",
"pred-isuc",
"symm'/filler",
"trans",
"trans/unit/r"
] | null | 139 | 152 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
fwd/isuc-pred/pos : (n : nat) → [i k] biinv-int [
| i=0 → trans biinv-int (fwd/isuc (pred (pos n))) (λ k → suc (fwd/pred/pos n k)) k
| i=1 → fwd/pos n
| k=0 → fwd (isuc-pred (pos n) i)
| k=1 → suc-predl (fwd/pos n) i
]
=
elim [
| zero → λ i k →
comp 0 1 (symm'/filler biinv-int (λ i → suc-predl zero ... | fwd/isuc-pred/pos : (n : nat) → [i k] biinv-int [
| i=0 → trans biinv-int (fwd/isuc (pred (pos n))) (λ k → suc (fwd/pred/pos n k)) k
| i=1 → fwd/pos n
| k=0 → fwd (isuc-pred (pos n) i)
| k=1 → suc-predl (fwd/pos n) i
] | =
elim [
| zero → λ i k →
comp 0 1 (symm'/filler biinv-int (λ i → suc-predl zero i) i k) [
| i=0 m → trans/unit/r biinv-int (fwd/isuc/negsuc zero) m k
| i=1 | ∂[k] → refl
]
| suc n → λ i k →
comp 0 1 (suc (symm'/filler biinv-int (λ i → predl-suc (fwd/pos n) i) i k)) [
| i=0 m → trans/unit/... | def | fwd/isuc-pred/pos | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"fwd",
"fwd/isuc",
"fwd/isuc/negsuc",
"fwd/pos",
"fwd/pred/pos",
"isuc-pred",
"nat",
"pred",
"suc-adj",
"suc-predl",
"symm'/filler",
"trans",
"trans/unit/l",
"trans/unit/r"
] | null | 154 | 173 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
fwd/isuc-pred : (n : int) → [i k] biinv-int [
| i=0 → trans biinv-int (fwd/isuc (pred n)) (λ k → suc (fwd/pred n k)) k
| i=1 → fwd n
| k=0 → fwd (isuc-pred n i)
| k=1 → suc-predl (fwd n) i
]
=
elim [
| pos n → fwd/isuc-pred/pos n
| negsuc n → λ i k →
comp 0 1 (symm'/filler biinv-int (λ i → suc-pre... | fwd/isuc-pred : (n : int) → [i k] biinv-int [
| i=0 → trans biinv-int (fwd/isuc (pred n)) (λ k → suc (fwd/pred n k)) k
| i=1 → fwd n
| k=0 → fwd (isuc-pred n i)
| k=1 → suc-predl (fwd n) i
] | =
elim [
| pos n → fwd/isuc-pred/pos n
| negsuc n → λ i k →
comp 0 1 (symm'/filler biinv-int (λ i → suc-predl (fwd/negsuc n) i) i k) [
| i=0 m → trans/unit/r biinv-int (fwd/isuc/negsuc (suc n)) m k
| i=1 | ∂[k] → refl
]
] | def | fwd/isuc-pred | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"fwd",
"fwd/isuc",
"fwd/isuc-pred/pos",
"fwd/isuc/negsuc",
"fwd/negsuc",
"fwd/pred",
"int",
"isuc-pred",
"pred",
"suc-predl",
"symm'/filler",
"trans",
"trans/unit/r"
] | null | 176 | 190 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
fwd-bwd : (z : biinv-int) → path _ (fwd (bwd z)) z =
elim [
| zero → refl
| suc (z → z/ih) → trans biinv-int (fwd/isuc (bwd z)) (λ k → suc (z/ih k))
| predl (z → z/ih) → trans biinv-int (fwd/pred (bwd z)) (λ k → predl (z/ih k))
| predr (z → z/ih) → trans biinv-int (fwd/pred (bwd z)) (λ k → predl~predr (z/ih k... | fwd-bwd : (z : biinv-int) → path _ (fwd (bwd z)) z | =
elim [
| zero → refl
| suc (z → z/ih) → trans biinv-int (fwd/isuc (bwd z)) (λ k → suc (z/ih k))
| predl (z → z/ih) → trans biinv-int (fwd/pred (bwd z)) (λ k → predl (z/ih k))
| predr (z → z/ih) → trans biinv-int (fwd/pred (bwd z)) (λ k → predl~predr (z/ih k) k)
| predl-suc (z → z/ih) i → λ k →
comp 0 ... | def | fwd-bwd | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"bwd",
"fwd",
"fwd/isuc",
"fwd/isuc-pred",
"fwd/pred",
"fwd/pred-isuc",
"isuc",
"path",
"pred",
"predl~predr",
"suc-predl~suc-predr",
"trans",
"trans/filler",
"weak-connection/and"
] | null | 192 | 216 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
int-equiv-biinv-int : equiv int biinv-int =
iso→equiv _ _ (fwd,bwd,fwd-bwd,bwd-fwd) | int-equiv-biinv-int : equiv int biinv-int | =
iso→equiv _ _ (fwd,bwd,fwd-bwd,bwd-fwd) | def | int-equiv-biinv-int | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"bwd",
"bwd-fwd",
"equiv",
"fwd",
"fwd-bwd",
"int",
"iso→equiv"
] | null | 218 | 219 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
suc'/pair : (z : biinv-int) → (y : biinv-int) × path biinv-int (suc z) y =
elim [
| zero → (suc zero, refl)
| suc z → (suc (suc z), refl)
| predl z → (z, suc-predl z)
| predr z → (z, λ j → suc-predr z j)
| predl-suc (z → z/ih) i →
let filler (i j : 𝕀) : biinv-int =
comp 0 j (suc (predl-suc z i)) ... | suc'/pair : (z : biinv-int) → (y : biinv-int) × path biinv-int (suc z) y | =
elim [
| zero → (suc zero, refl)
| suc z → (suc (suc z), refl)
| predl z → (z, suc-predl z)
| predr z → (z, λ j → suc-predr z j)
| predl-suc (z → z/ih) i →
let filler (i j : 𝕀) : biinv-int =
comp 0 j (suc (predl-suc z i)) [i=0 → suc-predl (suc z) | i=1 → z/ih .snd]
in
(filler i 1, fille... | def | suc'/pair | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"path",
"suc-predl"
] | null | 223 | 239 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
suc' (z : biinv-int) : biinv-int = suc'/pair z .fst | suc' (z : biinv-int) : biinv-int | = suc'/pair z .fst | def | suc' | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"suc'/pair"
] | null | 241 | 241 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
suc'-predl (z : biinv-int) : path _ (suc' (predl z)) z = refl | suc'-predl (z : biinv-int) : path _ (suc' (predl z)) z | = refl | def | suc'-predl | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"path",
"suc'"
] | null | 243 | 243 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
suc'-predr (z : biinv-int) : path _ (suc' (predr z)) z = refl | suc'-predr (z : biinv-int) : path _ (suc' (predr z)) z | = refl | def | suc'-predr | library/cool | library/cool/biinv-int.red | [
"basics.ha-equiv",
"basics.isotoequiv",
"basics.retract",
"data.int",
"data.nat",
"prelude"
] | [
"biinv-int",
"path",
"suc'"
] | null | 244 | 244 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
le : nat → nat → type =
elim [
| zero → λ _ → unit
| suc (m → f) →
elim [
| zero → void
| suc n → f n
]
] | le : nat → nat → type | =
elim [
| zero → λ _ → unit
| suc (m → f) →
elim [
| zero → void
| suc n → f n
]
] | def | le | library/cool | library/cool/complete-induction.red | [
"data.nat",
"data.or",
"data.unit",
"data.void",
"prelude"
] | [
"nat",
"unit",
"void"
] | null | 7 | 15 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
le/suc/right : (n m : nat) → le n m → le n (suc m) =
elim [
| zero → λ _ _ → ★
| suc (n' → f) →
elim [
| zero → elim []
| suc m' → λ l → f m' l
]
] | le/suc/right : (n m : nat) → le n m → le n (suc m) | =
elim [
| zero → λ _ _ → ★
| suc (n' → f) →
elim [
| zero → elim []
| suc m' → λ l → f m' l
]
] | def | le/suc/right | library/cool | library/cool/complete-induction.red | [
"data.nat",
"data.or",
"data.unit",
"data.void",
"prelude"
] | [
"le",
"nat"
] | null | 17 | 25 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
le/suc : (n m : nat) → le n m → le (suc n) (suc m) =
elim [
| zero → λ _ _ → ★
| suc _ → λ _ l → l
] | le/suc : (n m : nat) → le n m → le (suc n) (suc m) | =
elim [
| zero → λ _ _ → ★
| suc _ → λ _ l → l
] | def | le/suc | library/cool | library/cool/complete-induction.red | [
"data.nat",
"data.or",
"data.unit",
"data.void",
"prelude"
] | [
"le",
"nat"
] | null | 27 | 31 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
le/refl : (n : nat) → le n n =
elim [
| zero → ★
| suc (_ → f) → f
] | le/refl : (n : nat) → le n n | =
elim [
| zero → ★
| suc (_ → f) → f
] | def | le/refl | library/cool | library/cool/complete-induction.red | [
"data.nat",
"data.or",
"data.unit",
"data.void",
"prelude"
] | [
"le",
"nat"
] | null | 33 | 37 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
le/zero/implies/zero : (n : nat) → (le n zero) → path nat zero n =
elim [
| zero → λ _ → refl
| suc n' → elim []
] | le/zero/implies/zero : (n : nat) → (le n zero) → path nat zero n | =
elim [
| zero → λ _ → refl
| suc n' → elim []
] | def | le/zero/implies/zero | library/cool | library/cool/complete-induction.red | [
"data.nat",
"data.or",
"data.unit",
"data.void",
"prelude"
] | [
"le",
"nat",
"path"
] | null | 39 | 43 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
le/case : (m n : nat) → (le n (suc m)) → or (path nat n (suc m)) (le n m) =
elim [
| zero →
elim [
| zero → λ _ → inr ★
| suc n' →
elim n' [
| zero → λ _ → inl refl
| suc _ → λ p → inr p
]
]
| suc (m' → c) →
elim [
| zero → λ _ → inr ★
| suc n' → λ p →
eli... | le/case : (m n : nat) → (le n (suc m)) → or (path nat n (suc m)) (le n m) | =
elim [
| zero →
elim [
| zero → λ _ → inr ★
| suc n' →
elim n' [
| zero → λ _ → inl refl
| suc _ → λ p → inr p
]
]
| suc (m' → c) →
elim [
| zero → λ _ → inr ★
| suc n' → λ p →
elim (c n' p) [
| inl p → inl (λ i → suc (p i))
| inr l → inr (le... | def | le/case | library/cool | library/cool/complete-induction.red | [
"data.nat",
"data.or",
"data.unit",
"data.void",
"prelude"
] | [
"le",
"le/suc",
"nat",
"or",
"path"
] | null | 45 | 65 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
weak/induction (P : nat → type) : type =
P zero
→ ((n : nat) → P n → P (suc n))
→ (n : nat)
→ P n | weak/induction (P : nat → type) : type | =
P zero
→ ((n : nat) → P n → P (suc n))
→ (n : nat)
→ P n | def | weak/induction | library/cool | library/cool/complete-induction.red | [
"data.nat",
"data.or",
"data.unit",
"data.void",
"prelude"
] | [
"nat"
] | null | 67 | 71 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
realize/weak/induction (P : nat → type) : weak/induction P =
λ p0 ps →
elim [
| zero → p0
| suc (n' → pn') → ps n' pn'
] | realize/weak/induction (P : nat → type) : weak/induction P | =
λ p0 ps →
elim [
| zero → p0
| suc (n' → pn') → ps n' pn'
] | def | realize/weak/induction | library/cool | library/cool/complete-induction.red | [
"data.nat",
"data.or",
"data.unit",
"data.void",
"prelude"
] | [
"nat",
"weak/induction"
] | null | 73 | 78 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
complete/induction (P : nat → type) : type =
P zero
→ ((n : nat) → ((k : nat) → (le k n) → P k) → P (suc n))
→ (n : nat)
→ P n | complete/induction (P : nat → type) : type | =
P zero
→ ((n : nat) → ((k : nat) → (le k n) → P k) → P (suc n))
→ (n : nat)
→ P n | def | complete/induction | library/cool | library/cool/complete-induction.red | [
"data.nat",
"data.or",
"data.unit",
"data.void",
"prelude"
] | [
"le",
"nat"
] | null | 80 | 84 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
complete/implies/weak
(P : nat → type)
(complete : complete/induction P)
: weak/induction P
=
λ p0 ps →
complete p0 (λ n f → ps n (f n (le/refl n))) | complete/implies/weak
(P : nat → type)
(complete : complete/induction P)
: weak/induction P | =
λ p0 ps →
complete p0 (λ n f → ps n (f n (le/refl n))) | def | complete/implies/weak | library/cool | library/cool/complete-induction.red | [
"data.nat",
"data.or",
"data.unit",
"data.void",
"prelude"
] | [
"complete/induction",
"le/refl",
"nat",
"weak/induction"
] | null | 86 | 92 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
weak/implies/complete
(P : nat → type)
(weak : (P' : nat → type) → weak/induction P')
: complete/induction P
=
λ p0 ps →
let P' (n : nat) : type = (k : nat) → (le k n) → P k in
let P'0 : P' zero =
λ k k/le/0 →
coe 0 1 p0 in λ i →
P (le/zero/implies/zero k k/le/0 i)
in
let f (n : nat) (p'n ... | weak/implies/complete
(P : nat → type)
(weak : (P' : nat → type) → weak/induction P')
: complete/induction P | =
λ p0 ps →
let P' (n : nat) : type = (k : nat) → (le k n) → P k in
let P'0 : P' zero =
λ k k/le/0 →
coe 0 1 p0 in λ i →
P (le/zero/implies/zero k k/le/0 i)
in
let f (n : nat) (p'n : P' n) : (P' (suc n)) =
λ k k/le/sn →
elim (le/case n k k/le/sn) [
| inl p → coe 1 0 (ps n p'n) in λ i →... | def | weak/implies/complete | library/cool | library/cool/complete-induction.red | [
"data.nat",
"data.or",
"data.unit",
"data.void",
"prelude"
] | [
"complete/induction",
"le",
"le/case",
"le/refl",
"le/zero/implies/zero",
"nat",
"weak/induction"
] | null | 94 | 114 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
(A : type) ⊢ F where
| η (a : A)
| ☆ (s t : F)
| ε
| idn/r (s : F) (i : 𝕀) [
| i=0 → ☆ s ε
| i=1 → s
]
| idn/l (s : F) (i : 𝕀) [
| i=0 → ☆ ε s
| i=1 → s
]
| ass (s t u : F) (i : 𝕀) [
| i=0 → ☆ s (☆ t u)
| i=1 → ☆ (☆ s t) u
] | (A : type) ⊢ F | where
| η (a : A)
| ☆ (s t : F)
| ε
| idn/r (s : F) (i : 𝕀) [
| i=0 → ☆ s ε
| i=1 → s
]
| idn/l (s : F) (i : 𝕀) [
| i=0 → ☆ ε s
| i=1 → s
]
| ass (s t u : F) (i : 𝕀) [
| i=0 → ☆ s (☆ t u)
| i=1 → ☆ (☆ s t) u
] | data | F | library/cool | library/cool/free-monoid.red | [
"data.list"
] | [] | Probably need to truncate this to get the right type | 7 | 22 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
quote (A : type) : list A → F A =
elim [
| nil → ε
| cons x (xs → ih) →
☆ (η x) ih
] | quote (A : type) : list A → F A | =
elim [
| nil → ε
| cons x (xs → ih) →
☆ (η x) ih
] | def | quote | library/cool | library/cool/free-monoid.red | [
"data.list"
] | [
"list"
] | null | 24 | 29 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
eval (A : type) : F A → list A =
elim [
| η a →
cons a nil
| ☆ (s → ih/s) (t → ih/t) →
append A ih/s ih/t
| ε →
nil
| idn/l s i →
refl
| idn/r (s → ih/s) i →
append/idn/r A ih/s i
| ass (s → ih/s) (t → ih/t) (u → ih/u) i →
append/ass A ih/s ih/t ih/u i
] | eval (A : type) : F A → list A | =
elim [
| η a →
cons a nil
| ☆ (s → ih/s) (t → ih/t) →
append A ih/s ih/t
| ε →
nil
| idn/l s i →
refl
| idn/r (s → ih/s) i →
append/idn/r A ih/s i
| ass (s → ih/s) (t → ih/t) (u → ih/u) i →
append/ass A ih/s ih/t ih/u i
] | def | eval | library/cool | library/cool/free-monoid.red | [
"data.list"
] | [
"append",
"append/ass",
"append/idn/r",
"list"
] | null | 31 | 45 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
nbe (A : type) (s : F A) : F A =
quote A (eval A s) | nbe (A : type) (s : F A) : F A | =
quote A (eval A s) | def | nbe | library/cool | library/cool/free-monoid.red | [
"data.list"
] | [
"eval",
"quote"
] | null | 47 | 48 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
double : nat → nat =
elim [
| zero → zero
| suc (n' → f) → suc (suc f)
] | double : nat → nat | =
elim [
| zero → zero
| suc (n' → f) → suc (suc f)
] | def | double | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"nat"
] | null | 6 | 10 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
sub : nat → nat → nat =
elim [
| zero → λ _ → zero
| suc (m' → sub/m') →
elim [
| zero → suc m'
| suc n' → sub/m' n'
]
] | sub : nat → nat → nat | =
elim [
| zero → λ _ → zero
| suc (m' → sub/m') →
elim [
| zero → suc m'
| suc n' → sub/m' n'
]
] | def | sub | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"nat"
] | null | 12 | 20 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
sub/le : (n m : nat) → le (sub n m) n =
elim [
| zero → λ _ → ★
| suc (n' → f) → λ m → elim m [
| zero → le/refl n'
| suc m' → le/suc/right (sub n' m') n' (f m')
]
] | sub/le : (n m : nat) → le (sub n m) n | =
elim [
| zero → λ _ → ★
| suc (n' → f) → λ m → elim m [
| zero → le/refl n'
| suc m' → le/suc/right (sub n' m') n' (f m')
]
] | def | sub/le | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"le",
"le/refl",
"le/suc/right",
"nat",
"sub"
] | null | 22 | 29 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
mod/prop : nat → type = λ _ → nat → nat | mod/prop : nat → type | = λ _ → nat → nat | def | mod/prop | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"nat"
] | null | 31 | 31 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
mod : (n : nat) → mod/prop n =
let complete = weak/implies/complete mod/prop (λ P → realize/weak/induction P) in
complete
(λ _ → zero)
(λ n f → λ m →
elim m [
| zero → zero
| suc m' →
elim (sub (suc n) m) [
| zero →
elim (sub m (suc n)) [
| zero → zero
... | mod : (n : nat) → mod/prop n | =
let complete = weak/implies/complete mod/prop (λ P → realize/weak/induction P) in
complete
(λ _ → zero)
(λ n f → λ m →
elim m [
| zero → zero
| suc m' →
elim (sub (suc n) m) [
| zero →
elim (sub m (suc n)) [
| zero → zero
| suc _ → suc n
... | def | mod | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"mod/prop",
"nat",
"realize/weak/induction",
"sub",
"sub/le",
"weak/implies/complete"
] | null | 33 | 50 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
id/nat : nat → nat =
elim [
| zero → zero
| suc (_ → f) → suc f
] | id/nat : nat → nat | =
elim [
| zero → zero
| suc (_ → f) → suc f
] | def | id/nat | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"nat"
] | null | 52 | 56 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
eta : (n : nat) → path nat (id/nat n) n =
elim [
| zero → refl
| suc (_ → p) → λ i → suc (p i)
] | eta : (n : nat) → path nat (id/nat n) n | =
elim [
| zero → refl
| suc (_ → p) → λ i → suc (p i)
] | def | eta | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"id/nat",
"nat",
"path"
] | null | 58 | 62 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
sub/plus/path : (m n : nat) → le n m → path nat (plus n (sub m n)) m =
elim [
| zero → λ n p →
let path/n/0 = symm nat (le/zero/implies/zero n p) in
trans nat (eta n) path/n/0
| suc (m' → f) → elim [
| zero → refl
| suc n' → λ p i → suc ((f n' p) i)
]
] | sub/plus/path : (m n : nat) → le n m → path nat (plus n (sub m n)) m | =
elim [
| zero → λ n p →
let path/n/0 = symm nat (le/zero/implies/zero n p) in
trans nat (eta n) path/n/0
| suc (m' → f) → elim [
| zero → refl
| suc n' → λ p i → suc ((f n' p) i)
]
] | def | sub/plus/path | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"eta",
"le",
"le/zero/implies/zero",
"nat",
"path",
"plus",
"sub",
"symm",
"trans"
] | null | 64 | 73 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
plus/le : (m n : nat) → le m (plus m n) =
elim [
| zero → λ _ → ★
| suc (m' → f) → f
] | plus/le : (m n : nat) → le m (plus m n) | =
elim [
| zero → λ _ → ★
| suc (m' → f) → f
] | def | plus/le | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"le",
"nat",
"plus"
] | null | 75 | 79 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
le/trans : (m n l : nat) → le m n → le n l → le m l =
elim [
| zero → λ _ _ _ _ → ★
| suc (m' → f) →
elim [
| zero → λ _ → elim []
| suc n' →
elim [
| zero → λ _ → elim []
| suc l' → f n' l'
]
]
] | le/trans : (m n l : nat) → le m n → le n l → le m l | =
elim [
| zero → λ _ _ _ _ → ★
| suc (m' → f) →
elim [
| zero → λ _ → elim []
| suc n' →
elim [
| zero → λ _ → elim []
| suc l' → f n' l'
]
]
] | def | le/trans | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"le",
"nat"
] | null | 81 | 93 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
sub/le/implies/le : (m n k : nat) → path nat (suc k) (sub m n) → le n m =
elim [
| zero →
elim [
| zero → λ _ _ → ★
| suc n' → λ _ p → coe 1 0 ★ in λ i → le (p i) zero
]
| suc (m' → f) →
elim [
| zero → λ _ _ → ★
| suc n' → f n'
]
] | sub/le/implies/le : (m n k : nat) → path nat (suc k) (sub m n) → le n m | =
elim [
| zero →
elim [
| zero → λ _ _ → ★
| suc n' → λ _ p → coe 1 0 ★ in λ i → le (p i) zero
]
| suc (m' → f) →
elim [
| zero → λ _ _ → ★
| suc n' → f n'
]
] | def | sub/le/implies/le | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"le",
"nat",
"path",
"sub"
] | null | 95 | 107 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
suc/right/path : (m n : nat) → path nat (plus (suc m) n) (plus m (suc n)) =
elim [
| zero → refl
| suc (_ → f) → λ n i → suc (f n i)
] | suc/right/path : (m n : nat) → path nat (plus (suc m) n) (plus m (suc n)) | =
elim [
| zero → refl
| suc (_ → f) → λ n i → suc (f n i)
] | def | suc/right/path | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"nat",
"path",
"plus"
] | null | 109 | 113 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
path/implies/le (p : dim → nat) : le (p 0) (p 1) =
coe 0 1 (le/refl (p 0)) in (λ i → le (p 0) (p i)) | path/implies/le (p : dim → nat) : le (p 0) (p 1) | =
coe 0 1 (le/refl (p 0)) in (λ i → le (p 0) (p i)) | def | path/implies/le | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"le",
"le/refl",
"nat"
] | null | 115 | 116 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
gcd/prop (m : nat) : type =
(x y : nat) → le (plus x y) m → nat | gcd/prop (m : nat) : type | =
(x y : nat) → le (plus x y) m → nat | def | gcd/prop | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"le",
"nat",
"plus"
] | null | 118 | 119 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
gcd' : (m : nat) → gcd/prop m =
let complete = weak/implies/complete gcd/prop realize/weak/induction in
complete
(λ _ _ _ → zero)
(λ m f → λ x y →
elim x [
| zero → λ _ → y
| suc x' →
elim y [
| zero → λ _ → x
| suc y' →
elim (sub y x) [
| zero →... | gcd' : (m : nat) → gcd/prop m | =
let complete = weak/implies/complete gcd/prop realize/weak/induction in
complete
(λ _ _ _ → zero)
(λ m f → λ x y →
elim x [
| zero → λ _ → y
| suc x' →
elim y [
| zero → λ _ → x
| suc y' →
elim (sub y x) [
| zero → λ x+y'/le/m →
let... | def | gcd' | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"gcd/prop",
"le",
"le/trans",
"nat",
"path",
"path/implies/le",
"plus",
"plus/comm",
"plus/le",
"realize/weak/induction",
"sub",
"sub/le/implies/le",
"sub/plus/path",
"suc/right/path",
"weak/implies/complete"
] | null | 121 | 169 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
gcd (m n : nat) : nat = gcd' (plus m n) m n (le/refl (plus m n)) | gcd (m n : nat) : nat | = gcd' (plus m n) m n (le/refl (plus m n)) | def | gcd | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"gcd'",
"le/refl",
"nat",
"plus"
] | null | 171 | 171 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
n1 : nat = suc zero | n1 : nat | = suc zero | def | n1 | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"nat"
] | null | 173 | 173 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
n2 : nat = double n1 | n2 : nat | = double n1 | def | n2 | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"double",
"n1",
"nat"
] | null | 174 | 174 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
n3 : nat = suc n2 | n3 : nat | = suc n2 | def | n3 | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"n2",
"nat"
] | null | 175 | 175 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
n4 : nat = double n2 | n4 : nat | = double n2 | def | n4 | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"double",
"n2",
"nat"
] | null | 176 | 176 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
n5 : nat = suc n4 | n5 : nat | = suc n4 | def | n5 | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"n4",
"nat"
] | null | 177 | 177 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
n6 : nat = plus n2 n4 | n6 : nat | = plus n2 n4 | def | n6 | library/cool | library/cool/gcd.red | [
"cool.complete-induction",
"data.nat",
"data.void",
"prelude"
] | [
"n2",
"n4",
"nat",
"plus"
] | null | 178 | 178 | true | https://github.com/RedPRL/redtt | ae76658873a647eb43d8cf84365a9d68e9a3273c |
End of preview. Expand in Data Studio
redtt
Declarations from redtt, a cubical/computational type theory system.
Source
- Repository: https://github.com/RedPRL/redtt
- Commit:
ae76658873a647eb43d8cf84365a9d68e9a3273c - Files: 66
- 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: 501
- With proof: 501 (100.0%)
- With docstring: 19 (3.8%)
- Libraries: 6
By type
| Type | Count |
|---|---|
| def | 478 |
| data | 23 |
Example
biinv-equiv→iso (A B : type) : biinv-equiv A B → iso A B =
λ (f,(g,α),h,β) →
let β' (a : A) : path _ (g (f a)) a =
λ i →
comp 0 1 (h (α (f a) i)) [
| i=0 j → β (g (f a)) j
| i=1 j → β a j
]
in
(f,g,α,β')
- type: def | symbolic_name:
biinv-equiv→iso| library/basics/biinv-equiv.red:12
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{redtt_dataset,
title = {redtt},
author = {Norton, Charles},
year = {2026},
note = {Extracted from https://github.com/RedPRL/redtt, commit ae76658873a6},
url = {https://huggingface.co/datasets/phanerozoic/redtt}
}
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