Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 103 items • Updated • 3
fact stringlengths 4 23.6k | statement stringlengths 1 3.21k | proof stringlengths 1 23.6k | type stringclasses 3
values | symbolic_name stringlengths 1 31 | library stringclasses 3
values | filename stringclasses 77
values | imports listlengths 0 4 | deps listlengths 0 37 | docstring stringclasses 301
values | line_start int64 4 2.12k | line_end int64 4 2.12k | has_proof bool 1
class | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
isAssociative (M : U) (op : M -> M -> M) : U
= (a b c : M) -> Path M (op a (op b c)) (op (op a b) c) | isAssociative (M : U) (op : M -> M -> M) : U | = (a b c : M) -> Path M (op a (op b c)) (op (op a b) c) | def | isAssociative | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"op"
] | null | 14 | 15 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
propisAssociative (M : U) (sM : set M) (op : M -> M -> M)
: prop (isAssociative M op)
= let
B (a b c : M) : U
= Path M (op a (op b c)) (op (op a b) c)
h (a b c : M) : prop (B a b c)
= sM (op a (op b c)) (op (op a b) c)
in propPi3 M B h | propisAssociative (M : U) (sM : set M) (op : M -> M -> M)
: prop (isAssociative M op) | = let
B (a b c : M) : U
= Path M (op a (op b c)) (op (op a b) c)
h (a b c : M) : prop (B a b c)
= sM (op a (op b c)) (op (op a b) c)
in propPi3 M B h | def | propisAssociative | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"isAssociative",
"op",
"prop",
"propPi3",
"set"
] | null | 17 | 24 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
hasLeftIdentity (M : U) (op : M -> M -> M) (id : M) : U
= (x : M) -> Path M (op id x) x | hasLeftIdentity (M : U) (op : M -> M -> M) (id : M) : U | = (x : M) -> Path M (op id x) x | def | hasLeftIdentity | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"id",
"op"
] | null | 28 | 29 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
prophasLeftIdentity (M : U) (sM : set M) (op : M -> M -> M) (id : M)
: prop (hasLeftIdentity M op id)
= let
B (x : M) : U
= Path M (op id x) x
h (x : M) : prop (B x)
= sM (op id x) x
in propPi M B h | prophasLeftIdentity (M : U) (sM : set M) (op : M -> M -> M) (id : M)
: prop (hasLeftIdentity M op id) | = let
B (x : M) : U
= Path M (op id x) x
h (x : M) : prop (B x)
= sM (op id x) x
in propPi M B h | def | prophasLeftIdentity | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"hasLeftIdentity",
"id",
"op",
"prop",
"propPi",
"set"
] | null | 31 | 38 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
hasRightIdentity (M : U) (op : M -> M -> M) (id : M) : U
= (x : M) -> Path M (op x id) x | hasRightIdentity (M : U) (op : M -> M -> M) (id : M) : U | = (x : M) -> Path M (op x id) x | def | hasRightIdentity | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"id",
"op"
] | null | 40 | 41 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
prophasRightIdentity (M : U) (sM : set M) (op : M -> M -> M) (id : M)
: prop (hasRightIdentity M op id)
= let
B (x : M) : U
= Path M (op x id) x
h (x : M) : prop (B x)
= sM (op x id) x
in propPi M B h | prophasRightIdentity (M : U) (sM : set M) (op : M -> M -> M) (id : M)
: prop (hasRightIdentity M op id) | = let
B (x : M) : U
= Path M (op x id) x
h (x : M) : prop (B x)
= sM (op x id) x
in propPi M B h | def | prophasRightIdentity | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"hasRightIdentity",
"id",
"op",
"prop",
"propPi",
"set"
] | null | 43 | 50 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
hasIdentity (M : U) (op : M -> M -> M) (id : M) : U
= (_ : hasLeftIdentity M op id)
* (hasRightIdentity M op id) | hasIdentity (M : U) (op : M -> M -> M) (id : M) : U | = (_ : hasLeftIdentity M op id)
* (hasRightIdentity M op id) | def | hasIdentity | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"hasLeftIdentity",
"hasRightIdentity",
"id",
"op"
] | null | 52 | 54 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
prophasIdentity (M : U) (sM : set M) (op : M -> M -> M) (id : M)
: prop (hasIdentity M op id)
= propAnd (hasLeftIdentity M op id) (hasRightIdentity M op id)
(prophasLeftIdentity M sM op id) (prophasRightIdentity M sM op id) | prophasIdentity (M : U) (sM : set M) (op : M -> M -> M) (id : M)
: prop (hasIdentity M op id) | = propAnd (hasLeftIdentity M op id) (hasRightIdentity M op id)
(prophasLeftIdentity M sM op id) (prophasRightIdentity M sM op id) | def | prophasIdentity | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"hasIdentity",
"hasLeftIdentity",
"hasRightIdentity",
"id",
"op",
"prop",
"propAnd",
"prophasLeftIdentity",
"prophasRightIdentity",
"set"
] | null | 56 | 59 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
isCommutative (M : U) (op : M -> M -> M) : U
= (x y : M) -> Path M (op x y) (op y x) | isCommutative (M : U) (op : M -> M -> M) : U | = (x y : M) -> Path M (op x y) (op y x) | def | isCommutative | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"op"
] | null | 63 | 64 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
propisCommutative (M : U) (sM : set M) (op : M -> M -> M)
: prop (isCommutative M op)
= let
B (x y : M) : U
= Path M (op x y) (op y x)
h (x y : M) : prop (B x y)
= sM (op x y) (op y x)
in propPi2 M B h | propisCommutative (M : U) (sM : set M) (op : M -> M -> M)
: prop (isCommutative M op) | = let
B (x y : M) : U
= Path M (op x y) (op y x)
h (x y : M) : prop (B x y)
= sM (op x y) (op y x)
in propPi2 M B h | def | propisCommutative | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"isCommutative",
"op",
"prop",
"propPi2",
"set"
] | null | 66 | 73 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
isLeftCancellative (M : U) (op : M -> M -> M) : U
= (c x y : M) -> Path M (op c x) (op c y) -> Path M x y | isLeftCancellative (M : U) (op : M -> M -> M) : U | = (c x y : M) -> Path M (op c x) (op c y) -> Path M x y | def | isLeftCancellative | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"op"
] | null | 77 | 78 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
propisLeftCancellative (M : U) (sM : set M) (op : M -> M -> M)
: prop (isLeftCancellative M op)
= let
B (c x y : M) : U
= Path M (op c x) (op c y) -> Path M x y
h (c x y : M) : prop (B c x y)
= let
B0 (p : Path M (op c x) (op c y)) : U
= Path M x y
h0 (p : Path M (op c ... | propisLeftCancellative (M : U) (sM : set M) (op : M -> M -> M)
: prop (isLeftCancellative M op) | = let
B (c x y : M) : U
= Path M (op c x) (op c y) -> Path M x y
h (c x y : M) : prop (B c x y)
= let
B0 (p : Path M (op c x) (op c y)) : U
= Path M x y
h0 (p : Path M (op c x) (op c y)) : prop (B0 p)
= sM x y
in propPi (Path M (op c x) (op c y)) B0 h0
in ... | def | propisLeftCancellative | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"isLeftCancellative",
"op",
"prop",
"propPi",
"propPi3",
"set"
] | null | 80 | 92 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
isRightCancellative (M : U) (op : M -> M -> M) : U
= (c x y : M) -> Path M (op x c) (op y c) -> Path M x y | isRightCancellative (M : U) (op : M -> M -> M) : U | = (c x y : M) -> Path M (op x c) (op y c) -> Path M x y | def | isRightCancellative | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"op"
] | null | 94 | 95 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
propisRightCancellative (M : U) (sM : set M) (op : M -> M -> M)
: prop (isRightCancellative M op)
= let
B (c x y : M) : U
= Path M (op x c) (op y c) -> Path M x y
h (c x y : M) : prop (B c x y)
= let
B0 (p : Path M (op x c) (op y c)) : U
= Path M x y
h0 (p : Path M (op ... | propisRightCancellative (M : U) (sM : set M) (op : M -> M -> M)
: prop (isRightCancellative M op) | = let
B (c x y : M) : U
= Path M (op x c) (op y c) -> Path M x y
h (c x y : M) : prop (B c x y)
= let
B0 (p : Path M (op x c) (op y c)) : U
= Path M x y
h0 (p : Path M (op x c) (op y c)) : prop (B0 p)
= sM x y
in propPi (Path M (op x c) (op y c)) B0 h0
in ... | def | propisRightCancellative | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"isRightCancellative",
"op",
"prop",
"propPi",
"propPi3",
"set"
] | null | 97 | 109 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
isCancellative (M : U) (op : M -> M -> M) : U
= (_ : isLeftCancellative M op)
* (isRightCancellative M op) | isCancellative (M : U) (op : M -> M -> M) : U | = (_ : isLeftCancellative M op)
* (isRightCancellative M op) | def | isCancellative | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"isLeftCancellative",
"isRightCancellative",
"op"
] | null | 111 | 113 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
propisCancellative (M : U) (sM : set M) (op : M -> M -> M)
: prop (isCancellative M op)
= propAnd (isLeftCancellative M op) (isRightCancellative M op)
(propisLeftCancellative M sM op) (propisRightCancellative M sM op) | propisCancellative (M : U) (sM : set M) (op : M -> M -> M)
: prop (isCancellative M op) | = propAnd (isLeftCancellative M op) (isRightCancellative M op)
(propisLeftCancellative M sM op) (propisRightCancellative M sM op) | def | propisCancellative | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"isCancellative",
"isLeftCancellative",
"isRightCancellative",
"op",
"prop",
"propAnd",
"propisLeftCancellative",
"propisRightCancellative",
"set"
] | null | 115 | 118 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
hasLeftInverse (G : U) (op : G -> G -> G) (id : G) (inv : G -> G) : U
= (x : G) -> Path G (op (inv x) x) id | hasLeftInverse (G : U) (op : G -> G -> G) (id : G) (inv : G -> G) : U | = (x : G) -> Path G (op (inv x) x) id | def | hasLeftInverse | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"id",
"inv",
"op"
] | null | 122 | 123 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
prophasLeftInverse (G : U) (sG : set G) (op : G -> G -> G) (id : G) (inv : G -> G)
: prop (hasLeftInverse G op id inv)
= let
B (x : G) : U
= Path G (op (inv x) x) id
h (x : G) : prop (B x)
= sG (op (inv x) x) id
in propPi G B h | prophasLeftInverse (G : U) (sG : set G) (op : G -> G -> G) (id : G) (inv : G -> G)
: prop (hasLeftInverse G op id inv) | = let
B (x : G) : U
= Path G (op (inv x) x) id
h (x : G) : prop (B x)
= sG (op (inv x) x) id
in propPi G B h | def | prophasLeftInverse | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"hasLeftInverse",
"id",
"inv",
"op",
"prop",
"propPi",
"set"
] | null | 125 | 132 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
hasRightInverse (G : U) (op : G -> G -> G) (id : G) (inv : G -> G) : U
= (x : G) -> Path G (op x (inv x)) id | hasRightInverse (G : U) (op : G -> G -> G) (id : G) (inv : G -> G) : U | = (x : G) -> Path G (op x (inv x)) id | def | hasRightInverse | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"id",
"inv",
"op"
] | null | 134 | 135 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
prophasRightInverse (G : U) (sG : set G) (op : G -> G -> G) (id : G) (inv : G -> G)
: prop (hasRightInverse G op id inv)
= let
B (x : G) : U
= Path G (op x (inv x)) id
h (x : G) : prop (B x)
= sG (op x (inv x)) id
in propPi G B h | prophasRightInverse (G : U) (sG : set G) (op : G -> G -> G) (id : G) (inv : G -> G)
: prop (hasRightInverse G op id inv) | = let
B (x : G) : U
= Path G (op x (inv x)) id
h (x : G) : prop (B x)
= sG (op x (inv x)) id
in propPi G B h | def | prophasRightInverse | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"hasRightInverse",
"id",
"inv",
"op",
"prop",
"propPi",
"set"
] | null | 137 | 144 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
hasInverse (G : U) (op : G -> G -> G) (id : G) (inv : G -> G) : U
= (_ : hasLeftInverse G op id inv)
* (hasRightInverse G op id inv) | hasInverse (G : U) (op : G -> G -> G) (id : G) (inv : G -> G) : U | = (_ : hasLeftInverse G op id inv)
* (hasRightInverse G op id inv) | def | hasInverse | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"hasLeftInverse",
"hasRightInverse",
"id",
"inv",
"op"
] | null | 146 | 148 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
prophasInverse (G : U) (sG : set G) (op : G -> G -> G) (id : G) (inv : G -> G)
: prop (hasInverse G op id inv)
= propAnd (hasLeftInverse G op id inv) (hasRightInverse G op id inv)
(prophasLeftInverse G sG op id inv) (prophasRightInverse G sG op id inv) | prophasInverse (G : U) (sG : set G) (op : G -> G -> G) (id : G) (inv : G -> G)
: prop (hasInverse G op id inv) | = propAnd (hasLeftInverse G op id inv) (hasRightInverse G op id inv)
(prophasLeftInverse G sG op id inv) (prophasRightInverse G sG op id inv) | def | prophasInverse | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"hasInverse",
"hasLeftInverse",
"hasRightInverse",
"id",
"inv",
"op",
"prop",
"propAnd",
"prophasLeftInverse",
"prophasRightInverse",
"set"
] | null | 150 | 153 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
isLeftDistributive (R : U) (add : R -> R -> R) (mul : R -> R -> R) : U
= (a b c : R) -> Path R (mul a (add b c)) (add (mul a b) (mul a c)) | isLeftDistributive (R : U) (add : R -> R -> R) (mul : R -> R -> R) : U | = (a b c : R) -> Path R (mul a (add b c)) (add (mul a b) (mul a c)) | def | isLeftDistributive | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"add"
] | null | 157 | 158 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
propisLeftDistributive (R : U) (sR : set R) (add : R -> R -> R) (mul : R -> R -> R)
: prop (isLeftDistributive R add mul)
= let
B (a b c : R) : U
= Path R (mul a (add b c)) (add (mul a b) (mul a c))
h (a b c : R) : prop (B a b c)
= sR (mul a (add b c)) (add (mul a b) (mul a c))
in propPi3 R B ... | propisLeftDistributive (R : U) (sR : set R) (add : R -> R -> R) (mul : R -> R -> R)
: prop (isLeftDistributive R add mul) | = let
B (a b c : R) : U
= Path R (mul a (add b c)) (add (mul a b) (mul a c))
h (a b c : R) : prop (B a b c)
= sR (mul a (add b c)) (add (mul a b) (mul a c))
in propPi3 R B h | def | propisLeftDistributive | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"add",
"isLeftDistributive",
"prop",
"propPi3",
"set"
] | null | 160 | 167 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
isRightDistributive (R : U) (add : R -> R -> R) (mul : R -> R -> R) : U
= (a b c : R) -> Path R (mul (add b c) a) (add (mul b a) (mul c a)) | isRightDistributive (R : U) (add : R -> R -> R) (mul : R -> R -> R) : U | = (a b c : R) -> Path R (mul (add b c) a) (add (mul b a) (mul c a)) | def | isRightDistributive | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"add"
] | null | 169 | 170 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
propisRightDistributive (R : U) (sR : set R) (add : R -> R -> R) (mul : R -> R -> R)
: prop (isRightDistributive R add mul)
= let
B (a b c : R) : U
= Path R (mul (add b c) a) (add (mul b a) (mul c a))
h (a b c : R) : prop (B a b c)
= sR (mul (add b c) a) (add (mul b a) (mul c a))
in propPi3 R ... | propisRightDistributive (R : U) (sR : set R) (add : R -> R -> R) (mul : R -> R -> R)
: prop (isRightDistributive R add mul) | = let
B (a b c : R) : U
= Path R (mul (add b c) a) (add (mul b a) (mul c a))
h (a b c : R) : prop (B a b c)
= sR (mul (add b c) a) (add (mul b a) (mul c a))
in propPi3 R B h | def | propisRightDistributive | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"add",
"isRightDistributive",
"prop",
"propPi3",
"set"
] | null | 172 | 179 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
isDistributive (R : U) (add : R -> R -> R) (mul : R -> R -> R) : U
= (_ : isLeftDistributive R add mul)
* (isRightDistributive R add mul) | isDistributive (R : U) (add : R -> R -> R) (mul : R -> R -> R) : U | = (_ : isLeftDistributive R add mul)
* (isRightDistributive R add mul) | def | isDistributive | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"add",
"isLeftDistributive",
"isRightDistributive"
] | null | 181 | 183 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
propisDistributive (R : U) (sR : set R) (add : R -> R -> R) (mul : R -> R -> R)
: prop (isDistributive R add mul)
= propAnd (isLeftDistributive R add mul) (isRightDistributive R add mul)
(propisLeftDistributive R sR add mul) (propisRightDistributive R sR add mul) | propisDistributive (R : U) (sR : set R) (add : R -> R -> R) (mul : R -> R -> R)
: prop (isDistributive R add mul) | = propAnd (isLeftDistributive R add mul) (isRightDistributive R add mul)
(propisLeftDistributive R sR add mul) (propisRightDistributive R sR add mul) | def | propisDistributive | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"add",
"isDistributive",
"isLeftDistributive",
"isRightDistributive",
"prop",
"propAnd",
"propisLeftDistributive",
"propisRightDistributive",
"set"
] | null | 185 | 188 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
preservesOp (A B : U) (oA : A -> A -> A) (oB : B -> B -> B) (f : A -> B) : U
= (a0 a1 : A) -> Path B (f (oA a0 a1)) (oB (f a0) (f a1)) | preservesOp (A B : U) (oA : A -> A -> A) (oB : B -> B -> B) (f : A -> B) : U | = (a0 a1 : A) -> Path B (f (oA a0 a1)) (oB (f a0) (f a1)) | def | preservesOp | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path"
] | null | 192 | 193 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
proppreservesOp (A B : U) (sB : set B) (oA : A -> A -> A) (oB : B -> B -> B)
(f : A -> B) : prop (preservesOp A B oA oB f)
= propPi2 A (\ (a0 a1 : A) -> Path B (f (oA a0 a1)) (oB (f a0) (f a1)))
(\ (a0 a1 : A) -> sB (f (oA a0 a1)) (oB (f a0) (f a1))) | proppreservesOp (A B : U) (sB : set B) (oA : A -> A -> A) (oB : B -> B -> B)
(f : A -> B) : prop (preservesOp A B oA oB f) | = propPi2 A (\ (a0 a1 : A) -> Path B (f (oA a0 a1)) (oB (f a0) (f a1)))
(\ (a0 a1 : A) -> sB (f (oA a0 a1)) (oB (f a0) (f a1))) | def | proppreservesOp | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"preservesOp",
"prop",
"propPi2",
"set"
] | null | 195 | 198 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
preservesId (A B : U) (iA : A) (iB : B) (f : A -> B) : U
= Path B (f iA) iB | preservesId (A B : U) (iA : A) (iB : B) (f : A -> B) : U | = Path B (f iA) iB | def | preservesId | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path"
] | null | 200 | 201 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
proppreservesId (A B : U) (sB : set B) (iA : A) (iB : B) (f : A -> B)
: prop (preservesId A B iA iB f)
= sB (f iA) iB | proppreservesId (A B : U) (sB : set B) (iA : A) (iB : B) (f : A -> B)
: prop (preservesId A B iA iB f) | = sB (f iA) iB | def | proppreservesId | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"preservesId",
"prop",
"set"
] | null | 203 | 205 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
ismonoid (M : SET) : U
= (op : M.1 -> M.1 -> M.1)
* (_ : isAssociative M.1 op)
* (id : M.1)
* (hasIdentity M.1 op id) | ismonoid (M : SET) : U | = (op : M.1 -> M.1 -> M.1)
* (_ : isAssociative M.1 op)
* (id : M.1)
* (hasIdentity M.1 op id) | def | ismonoid | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"SET",
"hasIdentity",
"id",
"isAssociative",
"op"
] | null | 216 | 220 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
monoid : U
= (X : SET) * ismonoid X | monoid : U | = (X : SET) * ismonoid X | def | monoid | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"SET",
"ismonoid"
] | null | 222 | 223 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
isgroup (G : SET) : U
= (m : ismonoid G)
* (inv : G.1 -> G.1)
* (hasInverse G.1 m.1 m.2.2.1 inv) | isgroup (G : SET) : U | = (m : ismonoid G)
* (inv : G.1 -> G.1)
* (hasInverse G.1 m.1 m.2.2.1 inv) | def | isgroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"SET",
"hasInverse",
"inv",
"ismonoid"
] | Group | 226 | 229 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
group : U
= (X : SET) * isgroup X | group : U | = (X : SET) * isgroup X | def | group | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"SET",
"isgroup"
] | null | 231 | 232 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
iscmonoid (M : SET) : U
= (m : ismonoid M)
* (isCommutative M.1 m.1) | iscmonoid (M : SET) : U | = (m : ismonoid M)
* (isCommutative M.1 m.1) | def | iscmonoid | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"SET",
"isCommutative",
"ismonoid"
] | null | 239 | 241 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
cmonoid : U
= (X : SET) * iscmonoid X | cmonoid : U | = (X : SET) * iscmonoid X | def | cmonoid | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"SET",
"iscmonoid"
] | null | 243 | 244 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
iscgroup (G : SET) : U
= (g : isgroup G)
* (isCommutative G.1 g.1.1) | iscgroup (G : SET) : U | = (g : isgroup G)
* (isCommutative G.1 g.1.1) | def | iscgroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"SET",
"isCommutative",
"isgroup"
] | null | 248 | 250 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
cgroup : U
= (X : SET) * iscgroup X | cgroup : U | = (X : SET) * iscgroup X | def | cgroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"SET",
"iscgroup"
] | null | 252 | 253 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
isring (R : SET) : U
= (mul : ismonoid R)
* (add : iscgroup R)
* (isDistributive R.1 add.1.1.1 mul.1) | isring (R : SET) : U | = (mul : ismonoid R)
* (add : iscgroup R)
* (isDistributive R.1 add.1.1.1 mul.1) | def | isring | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"SET",
"add",
"isDistributive",
"iscgroup",
"ismonoid"
] | null | 260 | 263 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
ring : U
= (X : SET) * isring X | ring : U | = (X : SET) * isring X | def | ring | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"SET",
"isring"
] | null | 265 | 266 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
iscring (R : SET) : U
= (mul : iscmonoid R)
* (add : iscgroup R)
* (isDistributive R.1 add.1.1.1 mul.1.1) | iscring (R : SET) : U | = (mul : iscmonoid R)
* (add : iscgroup R)
* (isDistributive R.1 add.1.1.1 mul.1.1) | def | iscring | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"SET",
"add",
"isDistributive",
"iscgroup",
"iscmonoid"
] | null | 273 | 276 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
cring : U
= (X : SET) * iscring X | cring : U | = (X : SET) * iscring X | def | cring | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"SET",
"iscring"
] | null | 278 | 279 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
opMonoid (m : monoid) : m.1.1 -> m.1.1 -> m.1.1
= m.2.1 | opMonoid (m : monoid) : m.1.1 -> m.1.1 -> m.1.1 | = m.2.1 | def | opMonoid | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"monoid"
] | null | 286 | 287 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
isAssocMonoid (m : monoid) : isAssociative m.1.1 (opMonoid m)
= m.2.2.1 | isAssocMonoid (m : monoid) : isAssociative m.1.1 (opMonoid m) | = m.2.2.1 | def | isAssocMonoid | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"isAssociative",
"monoid",
"opMonoid"
] | null | 289 | 290 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
idMonoid (m : monoid) : m.1.1
= m.2.2.2.1 | idMonoid (m : monoid) : m.1.1 | = m.2.2.2.1 | def | idMonoid | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"monoid"
] | null | 292 | 293 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
hasIdMonoid (m : monoid) : hasIdentity m.1.1 (opMonoid m) (idMonoid m)
= m.2.2.2.2 | hasIdMonoid (m : monoid) : hasIdentity m.1.1 (opMonoid m) (idMonoid m) | = m.2.2.2.2 | def | hasIdMonoid | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"hasIdentity",
"idMonoid",
"monoid",
"opMonoid"
] | null | 295 | 296 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
opCMonoid (m : cmonoid) : m.1.1 -> m.1.1 -> m.1.1
= m.2.1.1 | opCMonoid (m : cmonoid) : m.1.1 -> m.1.1 -> m.1.1 | = m.2.1.1 | def | opCMonoid | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cmonoid"
] | null | 300 | 301 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
isAssocCMonoid (m : cmonoid) : isAssociative m.1.1 (opCMonoid m)
= m.2.1.2.1 | isAssocCMonoid (m : cmonoid) : isAssociative m.1.1 (opCMonoid m) | = m.2.1.2.1 | def | isAssocCMonoid | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cmonoid",
"isAssociative",
"opCMonoid"
] | null | 303 | 304 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
idCMonoid (m : cmonoid) : m.1.1
= m.2.1.2.2.1 | idCMonoid (m : cmonoid) : m.1.1 | = m.2.1.2.2.1 | def | idCMonoid | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cmonoid"
] | null | 306 | 307 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
hasIdCMonoid (m : cmonoid) : hasIdentity m.1.1 (opCMonoid m) (idCMonoid m)
= m.2.1.2.2.2 | hasIdCMonoid (m : cmonoid) : hasIdentity m.1.1 (opCMonoid m) (idCMonoid m) | = m.2.1.2.2.2 | def | hasIdCMonoid | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cmonoid",
"hasIdentity",
"idCMonoid",
"opCMonoid"
] | null | 309 | 310 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
isCommCMonoid (m : cmonoid) : isCommutative m.1.1 (opCMonoid m)
= m.2.2 | isCommCMonoid (m : cmonoid) : isCommutative m.1.1 (opCMonoid m) | = m.2.2 | def | isCommCMonoid | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cmonoid",
"isCommutative",
"opCMonoid"
] | null | 312 | 313 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
opGroup (g : group) : g.1.1 -> g.1.1 -> g.1.1
= g.2.1.1 | opGroup (g : group) : g.1.1 -> g.1.1 -> g.1.1 | = g.2.1.1 | def | opGroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"group"
] | null | 317 | 318 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
isAssocGroup (g : group) : isAssociative g.1.1 (opGroup g)
= g.2.1.2.1 | isAssocGroup (g : group) : isAssociative g.1.1 (opGroup g) | = g.2.1.2.1 | def | isAssocGroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"group",
"isAssociative",
"opGroup"
] | null | 320 | 321 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
idGroup (g : group) : g.1.1
= g.2.1.2.2.1 | idGroup (g : group) : g.1.1 | = g.2.1.2.2.1 | def | idGroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"group"
] | null | 323 | 324 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
hasIdGroup (g : group) : hasIdentity g.1.1 (opGroup g) (idGroup g)
= g.2.1.2.2.2 | hasIdGroup (g : group) : hasIdentity g.1.1 (opGroup g) (idGroup g) | = g.2.1.2.2.2 | def | hasIdGroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"group",
"hasIdentity",
"idGroup",
"opGroup"
] | null | 326 | 327 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
invGroup (g : group) : g.1.1 -> g.1.1
= g.2.2.1 | invGroup (g : group) : g.1.1 -> g.1.1 | = g.2.2.1 | def | invGroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"group"
] | null | 329 | 330 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
hasInvGroup (g : group) : hasInverse g.1.1 (opGroup g) (idGroup g) (invGroup g)
= g.2.2.2 | hasInvGroup (g : group) : hasInverse g.1.1 (opGroup g) (idGroup g) (invGroup g) | = g.2.2.2 | def | hasInvGroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"group",
"hasInverse",
"idGroup",
"invGroup",
"opGroup"
] | null | 332 | 333 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
opCGroup (g : cgroup) : g.1.1 -> g.1.1 -> g.1.1
= g.2.1.1.1 | opCGroup (g : cgroup) : g.1.1 -> g.1.1 -> g.1.1 | = g.2.1.1.1 | def | opCGroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cgroup"
] | null | 337 | 338 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
isAssocCGroup (g : cgroup) : isAssociative g.1.1 (opCGroup g)
= g.2.1.1.2.1 | isAssocCGroup (g : cgroup) : isAssociative g.1.1 (opCGroup g) | = g.2.1.1.2.1 | def | isAssocCGroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cgroup",
"isAssociative",
"opCGroup"
] | null | 340 | 341 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
idCGroup (g : cgroup) : g.1.1
= g.2.1.1.2.2.1 | idCGroup (g : cgroup) : g.1.1 | = g.2.1.1.2.2.1 | def | idCGroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cgroup"
] | null | 343 | 344 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
hasIdCGroup (g : cgroup) : hasIdentity g.1.1 (opCGroup g) (idCGroup g)
= g.2.1.1.2.2.2 | hasIdCGroup (g : cgroup) : hasIdentity g.1.1 (opCGroup g) (idCGroup g) | = g.2.1.1.2.2.2 | def | hasIdCGroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cgroup",
"hasIdentity",
"idCGroup",
"opCGroup"
] | null | 346 | 347 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
isCommCGroup (g : cgroup) : isCommutative g.1.1 (opCGroup g)
= g.2.2 | isCommCGroup (g : cgroup) : isCommutative g.1.1 (opCGroup g) | = g.2.2 | def | isCommCGroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cgroup",
"isCommutative",
"opCGroup"
] | null | 349 | 350 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
invCGroup (g : cgroup) : g.1.1 -> g.1.1
= g.2.1.2.1 | invCGroup (g : cgroup) : g.1.1 -> g.1.1 | = g.2.1.2.1 | def | invCGroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cgroup"
] | null | 352 | 353 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
hasInvCGroup (g : cgroup) : hasInverse g.1.1 (opCGroup g) (idCGroup g) (invCGroup g)
= g.2.1.2.2 | hasInvCGroup (g : cgroup) : hasInverse g.1.1 (opCGroup g) (idCGroup g) (invCGroup g) | = g.2.1.2.2 | def | hasInvCGroup | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cgroup",
"hasInverse",
"idCGroup",
"invCGroup",
"opCGroup"
] | null | 355 | 356 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
lem_group_lcancellative (g : group) (c x y : g.1.1) (p : Path g.1.1 (opGroup g c x) (opGroup g c y))
: Path g.1.1 x y
= <i> comp (<_> g.1.1) (opGroup g (invGroup g c) (p @ i))
[ (i = 0) -> <j> comp (<_> g.1.1) (opGroup g ((hasInvGroup g).1 c @ j) x)
[ (j = 0) -> <k> isAssocGroup g (invGroup g c) c x @ -... | lem_group_lcancellative (g : group) (c x y : g.1.1) (p : Path g.1.1 (opGroup g c x) (opGroup g c y))
: Path g.1.1 x y | = <i> comp (<_> g.1.1) (opGroup g (invGroup g c) (p @ i))
[ (i = 0) -> <j> comp (<_> g.1.1) (opGroup g ((hasInvGroup g).1 c @ j) x)
[ (j = 0) -> <k> isAssocGroup g (invGroup g c) c x @ -k
, (j = 1) -> (hasIdGroup g).1 x ]
, (i = 1) -> <j> comp (<_> g.1.1) (opGroup g ((hasInvGroup g).1 c @ j) y)
... | def | lem_group_lcancellative | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"group",
"hasIdGroup",
"hasInvGroup",
"invGroup",
"isAssocGroup",
"opGroup"
] | null | 364 | 372 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
lem_group_rcancellative (g : group) (c x y : g.1.1) (p : Path g.1.1 (opGroup g x c) (opGroup g y c))
: Path g.1.1 x y
= <i> comp (<_> g.1.1) (opGroup g (p @ i) (invGroup g c))
[ (i = 0) -> <j> comp (<_> g.1.1) (opGroup g x ((hasInvGroup g).2 c @ j))
[ (j = 0) -> isAssocGroup g x c (invGroup g c)
... | lem_group_rcancellative (g : group) (c x y : g.1.1) (p : Path g.1.1 (opGroup g x c) (opGroup g y c))
: Path g.1.1 x y | = <i> comp (<_> g.1.1) (opGroup g (p @ i) (invGroup g c))
[ (i = 0) -> <j> comp (<_> g.1.1) (opGroup g x ((hasInvGroup g).2 c @ j))
[ (j = 0) -> isAssocGroup g x c (invGroup g c)
, (j = 1) -> (hasIdGroup g).2 x ]
, (i = 1) -> <j> comp (<_> g.1.1) (opGroup g y ((hasInvGroup g).2 c @ j))
[... | def | lem_group_rcancellative | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"group",
"hasIdGroup",
"hasInvGroup",
"invGroup",
"isAssocGroup",
"opGroup"
] | null | 374 | 382 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
lem_group_cancellative (g : group) : isCancellative g.1.1 (opGroup g)
= (lem_group_lcancellative g, lem_group_rcancellative g) | lem_group_cancellative (g : group) : isCancellative g.1.1 (opGroup g) | = (lem_group_lcancellative g, lem_group_rcancellative g) | def | lem_group_cancellative | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"group",
"isCancellative",
"lem_group_lcancellative",
"lem_group_rcancellative",
"opGroup"
] | null | 384 | 385 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
lem_cgroup_inv_dist (g : cgroup) (a b : g.1.1)
: Path g.1.1 (opCGroup g (invCGroup g a) (invCGroup g b)) (invCGroup g (opCGroup g a b))
= let
a' : g.1.1
= invCGroup g a
b' : g.1.1
= invCGroup g b
x : g.1.1
= opCGroup g a b
x' : g.1.1
= invCGroup g (opCGroup g a b)
p0 : Pa... | lem_cgroup_inv_dist (g : cgroup) (a b : g.1.1)
: Path g.1.1 (opCGroup g (invCGroup g a) (invCGroup g b)) (invCGroup g (opCGroup g a b)) | = let
a' : g.1.1
= invCGroup g a
b' : g.1.1
= invCGroup g b
x : g.1.1
= opCGroup g a b
x' : g.1.1
= invCGroup g (opCGroup g a b)
p0 : Path g.1.1 (opCGroup g x' a) b'
= <i> comp (<_> g.1.1) (opCGroup g ((hasInvCGroup g).1 (opCGroup g a b) @ i) b')
[ (i = 1) -> (h... | def | lem_cgroup_inv_dist | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"cgroup",
"hasIdCGroup",
"hasInvCGroup",
"invCGroup",
"isAssocCGroup",
"isCommCGroup",
"opCGroup",
"p0",
"p1"
] | (a · b)⁻¹ ≡ a⁻¹ · b⁻¹ | 388 | 419 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
lemma_cgroup_inv_id (g : cgroup) : Path g.1.1 (invCGroup g (idCGroup g)) (idCGroup g)
= <i> comp (<_> g.1.1) ((hasIdCGroup g).2 (invCGroup g (idCGroup g)) @ -i)
[ (i = 0) -> <_> invCGroup g (idCGroup g)
, (i = 1) -> (hasInvCGroup g).1 (idCGroup g) ] | lemma_cgroup_inv_id (g : cgroup) : Path g.1.1 (invCGroup g (idCGroup g)) (idCGroup g) | = <i> comp (<_> g.1.1) ((hasIdCGroup g).2 (invCGroup g (idCGroup g)) @ -i)
[ (i = 0) -> <_> invCGroup g (idCGroup g)
, (i = 1) -> (hasInvCGroup g).1 (idCGroup g) ] | def | lemma_cgroup_inv_id | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"cgroup",
"hasIdCGroup",
"hasInvCGroup",
"idCGroup",
"invCGroup"
] | e⁻¹ ≡ e | 422 | 425 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
ismonoidhom (a b : monoid) (f : a.1.1 -> b.1.1) : U
= (_ : preservesOp a.1.1 b.1.1 (opMonoid a) (opMonoid b) f)
* (preservesId a.1.1 b.1.1 (idMonoid a) (idMonoid b) f) | ismonoidhom (a b : monoid) (f : a.1.1 -> b.1.1) : U | = (_ : preservesOp a.1.1 b.1.1 (opMonoid a) (opMonoid b) f)
* (preservesId a.1.1 b.1.1 (idMonoid a) (idMonoid b) f) | def | ismonoidhom | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"idMonoid",
"monoid",
"opMonoid",
"preservesId",
"preservesOp"
] | The monoid homomorphism preserves both the structure and the identity element | 432 | 434 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
propismonoidhom (a b : monoid) (f : a.1.1 -> b.1.1) : prop (ismonoidhom a b f)
= propAnd (preservesOp a.1.1 b.1.1 (opMonoid a) (opMonoid b) f)
(preservesId a.1.1 b.1.1 (idMonoid a) (idMonoid b) f)
(proppreservesOp a.1.1 b.1.1 b.1.2 (opMonoid a) (opMonoid b) f)
(proppreservesId a.1.1 b.1.1 b.1.2 (idM... | propismonoidhom (a b : monoid) (f : a.1.1 -> b.1.1) : prop (ismonoidhom a b f) | = propAnd (preservesOp a.1.1 b.1.1 (opMonoid a) (opMonoid b) f)
(preservesId a.1.1 b.1.1 (idMonoid a) (idMonoid b) f)
(proppreservesOp a.1.1 b.1.1 b.1.2 (opMonoid a) (opMonoid b) f)
(proppreservesId a.1.1 b.1.1 b.1.2 (idMonoid a) (idMonoid b) f) | def | propismonoidhom | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"idMonoid",
"ismonoidhom",
"monoid",
"opMonoid",
"preservesId",
"preservesOp",
"prop",
"propAnd",
"proppreservesId",
"proppreservesOp"
] | null | 436 | 440 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
monoidhom (a b : monoid) : U
= (f : a.1.1 -> b.1.1)
* (ismonoidhom a b f) | monoidhom (a b : monoid) : U | = (f : a.1.1 -> b.1.1)
* (ismonoidhom a b f) | def | monoidhom | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"ismonoidhom",
"monoid"
] | null | 442 | 444 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
setmonoidhom (a b : monoid) : set (monoidhom a b)
= let
setf : set (a.1.1 -> b.1.1)
= setPi a.1.1 (\ (_ : a.1.1) -> b.1.1) (\ (_ : a.1.1) -> b.1.2)
setm (f : a.1.1 -> b.1.1) : set (ismonoidhom a b f)
= propSet (ismonoidhom a b f) (propismonoidhom a b f)
in setSig (a.1.1 -> b.1.1) (ismonoidhom a... | setmonoidhom (a b : monoid) : set (monoidhom a b) | = let
setf : set (a.1.1 -> b.1.1)
= setPi a.1.1 (\ (_ : a.1.1) -> b.1.1) (\ (_ : a.1.1) -> b.1.2)
setm (f : a.1.1 -> b.1.1) : set (ismonoidhom a b f)
= propSet (ismonoidhom a b f) (propismonoidhom a b f)
in setSig (a.1.1 -> b.1.1) (ismonoidhom a b) setf setm | def | setmonoidhom | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"ismonoidhom",
"monoid",
"monoidhom",
"propSet",
"propismonoidhom",
"set",
"setPi",
"setSig"
] | null | 446 | 452 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
cmonoidhom (a b : cmonoid) : U
= monoidhom (a.1, a.2.1) (b.1, b.2.1) | cmonoidhom (a b : cmonoid) : U | = monoidhom (a.1, a.2.1) (b.1, b.2.1) | def | cmonoidhom | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cmonoid",
"monoidhom"
] | null | 456 | 457 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
grouphom (a b : group) : U
= monoidhom (a.1, a.2.1) (b.1, b.2.1) | grouphom (a b : group) : U | = monoidhom (a.1, a.2.1) (b.1, b.2.1) | def | grouphom | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"group",
"monoidhom"
] | null | 459 | 460 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
cgrouphom (a b : cgroup) : U
= monoidhom (a.1, a.2.1.1) (b.1, b.2.1.1) | cgrouphom (a b : cgroup) : U | = monoidhom (a.1, a.2.1.1) (b.1, b.2.1.1) | def | cgrouphom | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cgroup",
"monoidhom"
] | null | 462 | 463 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
cmoncgrouphom (a : cmonoid) (b : cgroup) : U
= monoidhom (a.1, a.2.1) (b.1, b.2.1.1) | cmoncgrouphom (a : cmonoid) (b : cgroup) : U | = monoidhom (a.1, a.2.1) (b.1, b.2.1.1) | def | cmoncgrouphom | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cgroup",
"cmonoid",
"monoidhom"
] | null | 465 | 466 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
grouphom' (a b : group) (f : a.1.1 -> b.1.1)
(pO : preservesOp a.1.1 b.1.1 (opGroup a) (opGroup b) f)
: monoidhom (a.1, a.2.1) (b.1, b.2.1)
= let
p : Path b.1.1 (opGroup b (f (idGroup a)) (f (idGroup a))) (opGroup b (f (idGroup a)) (idGroup b))
= <i> comp (<_> b.1.1) (f ((hasIdGroup a).1 (idGroup a) @... | grouphom' (a b : group) (f : a.1.1 -> b.1.1)
(pO : preservesOp a.1.1 b.1.1 (opGroup a) (opGroup b) f)
: monoidhom (a.1, a.2.1) (b.1, b.2.1) | = let
p : Path b.1.1 (opGroup b (f (idGroup a)) (f (idGroup a))) (opGroup b (f (idGroup a)) (idGroup b))
= <i> comp (<_> b.1.1) (f ((hasIdGroup a).1 (idGroup a) @ i))
[ (i = 0) -> pO (idGroup a) (idGroup a)
, (i = 1) -> <j> (hasIdGroup b).2 (f (idGroup a)) @ -j ]
in (f, pO, (lem_group_cancel... | def | grouphom' | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"group",
"hasIdGroup",
"idGroup",
"lem_group_cancellative",
"monoidhom",
"opGroup",
"preservesOp"
] | null | 470 | 478 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
cgrouphom' (a b : cgroup) (f : a.1.1 -> b.1.1)
(pO : preservesOp a.1.1 b.1.1 (opCGroup a) (opCGroup b) f)
: monoidhom (a.1, a.2.1.1) (b.1, b.2.1.1)
= grouphom' (a.1, a.2.1) (b.1, b.2.1) f pO | cgrouphom' (a b : cgroup) (f : a.1.1 -> b.1.1)
(pO : preservesOp a.1.1 b.1.1 (opCGroup a) (opCGroup b) f)
: monoidhom (a.1, a.2.1.1) (b.1, b.2.1.1) | = grouphom' (a.1, a.2.1) (b.1, b.2.1) f pO | def | cgrouphom' | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"cgroup",
"grouphom'",
"monoidhom",
"opCGroup",
"preservesOp"
] | null | 480 | 483 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
monoidhomcomp (a b c : monoid)
(f : monoidhom a b) (g : monoidhom b c)
: monoidhom a c
= let
h (x : a.1.1) : c.1.1
= g.1 (f.1 x)
pO (x0 x1 : a.1.1) : Path c.1.1 (h (a.2.1 x0 x1)) (c.2.1 (h x0) (h x1))
= <i> comp (<_> c.1.1) (g.1 (f.2.1 x0 x1 @ i))
[ (i = 0) -> <_> h (a.2.1 x0 x1)
... | monoidhomcomp (a b c : monoid)
(f : monoidhom a b) (g : monoidhom b c)
: monoidhom a c | = let
h (x : a.1.1) : c.1.1
= g.1 (f.1 x)
pO (x0 x1 : a.1.1) : Path c.1.1 (h (a.2.1 x0 x1)) (c.2.1 (h x0) (h x1))
= <i> comp (<_> c.1.1) (g.1 (f.2.1 x0 x1 @ i))
[ (i = 0) -> <_> h (a.2.1 x0 x1)
, (i = 1) -> g.2.1 (f.1 x0) (f.1 x1) ]
pI : Path c.1.1 (h (idMonoid a)) (idMonoid c)
... | def | monoidhomcomp | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"idMonoid",
"monoid",
"monoidhom"
] | g ∘ f | 486 | 500 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
lemma_monoidcomp0 (a b c d : monoid) (f : monoidhom a b)
(g : monoidhom b c) (h : monoidhom c d)
: Path (monoidhom a d) (monoidhomcomp a c d (monoidhomcomp a b c f g) h)
(monoidhomcomp a b d f (monoidhomcomp b c d g h))
= let
f0 : monoidhom a d
= monoidhomcomp a c d (monoidhom... | lemma_monoidcomp0 (a b c d : monoid) (f : monoidhom a b)
(g : monoidhom b c) (h : monoidhom c d)
: Path (monoidhom a d) (monoidhomcomp a c d (monoidhomcomp a b c f g) h)
(monoidhomcomp a b d f (monoidhomcomp b c d g h)) | = let
f0 : monoidhom a d
= monoidhomcomp a c d (monoidhomcomp a b c f g) h
f1 : monoidhom a d
= monoidhomcomp a b d f (monoidhomcomp b c d g h)
in lemSig (a.1.1 -> d.1.1) (ismonoidhom a d) (propismonoidhom a d) f0 f1 (<_> f0.1) | def | lemma_monoidcomp0 | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"f0",
"f1",
"ismonoidhom",
"lemSig",
"monoid",
"monoidhom",
"monoidhomcomp",
"propismonoidhom"
] | h ∘ (g ∘ f) ≡ (h ∘ g) ∘ f | 503 | 512 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
idmonoidhom (a : monoid) : monoidhom a a
= (\ (x : a.1.1) -> x, \ (a0 a1 : a.1.1) -> <_> opMonoid a a0 a1, <_> idMonoid a) | idmonoidhom (a : monoid) : monoidhom a a | = (\ (x : a.1.1) -> x, \ (a0 a1 : a.1.1) -> <_> opMonoid a a0 a1, <_> idMonoid a) | def | idmonoidhom | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"idMonoid",
"monoid",
"monoidhom",
"opMonoid"
] | 1_a | 515 | 516 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
lemma_idmonoidhom0 (a b : monoid) (f : monoidhom a b)
: Path (monoidhom a b) (monoidhomcomp a a b (idmonoidhom a) f) f
= let
h : monoidhom a b
= monoidhomcomp a a b (idmonoidhom a) f
in lemSig (a.1.1 -> b.1.1) (ismonoidhom a b) (propismonoidhom a b) h f (<_> f.1) | lemma_idmonoidhom0 (a b : monoid) (f : monoidhom a b)
: Path (monoidhom a b) (monoidhomcomp a a b (idmonoidhom a) f) f | = let
h : monoidhom a b
= monoidhomcomp a a b (idmonoidhom a) f
in lemSig (a.1.1 -> b.1.1) (ismonoidhom a b) (propismonoidhom a b) h f (<_> f.1) | def | lemma_idmonoidhom0 | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"idmonoidhom",
"ismonoidhom",
"lemSig",
"monoid",
"monoidhom",
"monoidhomcomp",
"propismonoidhom"
] | f ∘ 1_a ≡ f | 519 | 524 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
lemma_idmonoidhom1 (a b : monoid) (f : monoidhom a b)
: Path (monoidhom a b) (monoidhomcomp a b b f (idmonoidhom b)) f
= let
h : monoidhom a b
= monoidhomcomp a b b f (idmonoidhom b)
in lemSig (a.1.1 -> b.1.1) (ismonoidhom a b) (propismonoidhom a b) h f (<_> f.1) | lemma_idmonoidhom1 (a b : monoid) (f : monoidhom a b)
: Path (monoidhom a b) (monoidhomcomp a b b f (idmonoidhom b)) f | = let
h : monoidhom a b
= monoidhomcomp a b b f (idmonoidhom b)
in lemSig (a.1.1 -> b.1.1) (ismonoidhom a b) (propismonoidhom a b) h f (<_> f.1) | def | lemma_idmonoidhom1 | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"idmonoidhom",
"ismonoidhom",
"lemSig",
"monoid",
"monoidhom",
"monoidhomcomp",
"propismonoidhom"
] | 1_b ∘ f ≡ f | 527 | 532 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
lem_grouphom_inv (g h : group) (f : grouphom g h) (x : g.1.1)
: Path h.1.1 (f.1 (invGroup g x)) (invGroup h (f.1 x))
= let
x' : g.1.1
= invGroup g x
y : h.1.1
= f.1 x
y' : h.1.1
= invGroup h y
p0 : Path h.1.1 (opGroup h y (f.1 x')) (idGroup h)
= <i> comp (<_> h.1.1) (f.1 ((g.... | lem_grouphom_inv (g h : group) (f : grouphom g h) (x : g.1.1)
: Path h.1.1 (f.1 (invGroup g x)) (invGroup h (f.1 x)) | = let
x' : g.1.1
= invGroup g x
y : h.1.1
= f.1 x
y' : h.1.1
= invGroup h y
p0 : Path h.1.1 (opGroup h y (f.1 x')) (idGroup h)
= <i> comp (<_> h.1.1) (f.1 ((g.2.2.2).2 x @ i))
[ (i = 0) -> f.2.1 x x'
, (i = 1) -> f.2.2 ]
p1 : Path h.1.1 (opGroup h (opGroup h y... | def | lem_grouphom_inv | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"Path",
"group",
"grouphom",
"hasIdGroup",
"idGroup",
"invGroup",
"isAssocGroup",
"opGroup",
"p0",
"p1"
] | f(x⁻¹) ≡ f(x)⁻¹ | 535 | 558 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
AscCom : U
= (A : U)
* (op : A -> A -> A)
* (_ : isAssociative A op)
* (isCommutative A op) | AscCom : U | = (A : U)
* (op : A -> A -> A)
* (_ : isAssociative A op)
* (isCommutative A op) | def | AscCom | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"isAssociative",
"isCommutative",
"op"
] | null | 564 | 568 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
swp0 (A : AscCom) (a b c : A.1)
: Path A.1 (A.2.1 a (A.2.1 b c)) (A.2.1 b (A.2.1 a c))
= let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in <i> comp (<_> A.1) (as... | swp0 (A : AscCom) (a b c : A.1)
: Path A.1 (A.2.1 a (A.2.1 b c)) (A.2.1 b (A.2.1 a c)) | = let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in <i> comp (<_> A.1) (asc a b c @ i)
[ (i = 0) -> <_> op a (op b c)
, (i = 1) -> <j> comp (<_> A.1) (op (cm... | def | swp0 | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"AscCom",
"Path",
"asc",
"cm",
"op"
] | null | 570 | 583 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
swp1 (A : AscCom) (a b c : A.1)
: Path A.1 (A.2.1 a (A.2.1 b c)) (A.2.1 c (A.2.1 b a))
= let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in <i> comp (<_> A.1) (op... | swp1 (A : AscCom) (a b c : A.1)
: Path A.1 (A.2.1 a (A.2.1 b c)) (A.2.1 c (A.2.1 b a)) | = let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in <i> comp (<_> A.1) (op a (cm b c @ i))
[ (i = 0) -> <_> op a (op b c)
, (i = 1) -> <j> comp (<_> A.1) (sw... | def | swp1 | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"AscCom",
"Path",
"asc",
"cm",
"op",
"swp0"
] | null | 585 | 598 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
swp2 (A : AscCom) (a b c : A.1)
: Path A.1 (A.2.1 (A.2.1 a b) c) (A.2.1 (A.2.1 c b) a)
= let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in <i> comp (<_> A.1) (cm... | swp2 (A : AscCom) (a b c : A.1)
: Path A.1 (A.2.1 (A.2.1 a b) c) (A.2.1 (A.2.1 c b) a) | = let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in <i> comp (<_> A.1) (cm (op a b) c @ i)
[ (i = 0) -> <_> op (op a b) c
, (i = 1) -> <j> comp (<_> A.1) (sw... | def | swp2 | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"AscCom",
"Path",
"asc",
"cm",
"op",
"swp0"
] | null | 600 | 613 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
swp3 (A : AscCom) (a b c : A.1)
: Path A.1 (A.2.1 (A.2.1 a b) c) (A.2.1 (A.2.1 a c) b)
= let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in <i> comp (<_> A.1) (cm... | swp3 (A : AscCom) (a b c : A.1)
: Path A.1 (A.2.1 (A.2.1 a b) c) (A.2.1 (A.2.1 a c) b) | = let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in <i> comp (<_> A.1) (cm (op a b) c @ i)
[ (i = 0) -> <_> op (op a b) c
, (i = 1) -> <j> comp (<_> A.1) (sw... | def | swp3 | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"AscCom",
"Path",
"asc",
"cm",
"op",
"swp1"
] | null | 615 | 628 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
swp4 (A : AscCom) (a b c d : A.1)
: Path A.1 (A.2.1 a (A.2.1 b (A.2.1 c d))) (A.2.1 c (A.2.1 b (A.2.1 a d)))
= let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in ... | swp4 (A : AscCom) (a b c d : A.1)
: Path A.1 (A.2.1 a (A.2.1 b (A.2.1 c d))) (A.2.1 c (A.2.1 b (A.2.1 a d))) | = let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in <i> comp (<_> A.1) (swp0 A a b (op c d) @ i)
[ (i = 0) -> <_> op a (op b (op c d))
, (i = 1) -> <j> comp ... | def | swp4 | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"AscCom",
"Path",
"asc",
"cm",
"op",
"swp0"
] | null | 630 | 643 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
swp5 (A : AscCom) (a b c d : A.1)
: Path A.1 (A.2.1 a (A.2.1 b (A.2.1 c d))) (A.2.1 d (A.2.1 b (A.2.1 c a)))
= let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in ... | swp5 (A : AscCom) (a b c d : A.1)
: Path A.1 (A.2.1 a (A.2.1 b (A.2.1 c d))) (A.2.1 d (A.2.1 b (A.2.1 c a))) | = let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in <i> comp (<_> A.1) (swp0 A a b (op c d) @ i)
[ (i = 0) -> <_> op a (op b (op c d))
, (i = 1) -> <j> comp ... | def | swp5 | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"AscCom",
"Path",
"asc",
"cm",
"op",
"swp0",
"swp1"
] | null | 645 | 658 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
swp6 (A : AscCom) (a b c d : A.1)
: Path A.1 (A.2.1 (A.2.1 a b) (A.2.1 c d)) (A.2.1 (A.2.1 c b) (A.2.1 a d))
= let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in ... | swp6 (A : AscCom) (a b c d : A.1)
: Path A.1 (A.2.1 (A.2.1 a b) (A.2.1 c d)) (A.2.1 (A.2.1 c b) (A.2.1 a d)) | = let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in <i> comp (<_> A.1) (asc a b (op c d) @ -i)
[ (i = 0) -> <_> op (op a b) (op c d)
, (i = 1) -> <j> comp (<... | def | swp6 | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"AscCom",
"Path",
"asc",
"cm",
"op",
"swp4"
] | null | 660 | 673 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
swp7 (A : AscCom) (a b c d : A.1)
: Path A.1 (A.2.1 (A.2.1 a b) (A.2.1 c d)) (A.2.1 (A.2.1 d b) (A.2.1 c a))
= let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in ... | swp7 (A : AscCom) (a b c d : A.1)
: Path A.1 (A.2.1 (A.2.1 a b) (A.2.1 c d)) (A.2.1 (A.2.1 d b) (A.2.1 c a)) | = let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in <i> comp (<_> A.1) (asc a b (op c d) @ -i)
[ (i = 0) -> <_> op (op a b) (op c d)
, (i = 1) -> <j> comp (<... | def | swp7 | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"AscCom",
"Path",
"asc",
"cm",
"op",
"swp5"
] | null | 675 | 688 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
swp8 (A : AscCom) (a b c d : A.1)
: Path A.1 (A.2.1 (A.2.1 a b) (A.2.1 c d)) (A.2.1 (A.2.1 a c) (A.2.1 b d))
= let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in ... | swp8 (A : AscCom) (a b c d : A.1)
: Path A.1 (A.2.1 (A.2.1 a b) (A.2.1 c d)) (A.2.1 (A.2.1 a c) (A.2.1 b d)) | = let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in <i> comp (<_> A.1) (asc a b (op c d) @ -i)
[ (i = 0) -> <_> op (op a b) (op c d)
, (i = 1) -> <j> comp (<... | def | swp8 | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"AscCom",
"Path",
"asc",
"cm",
"op",
"swp0"
] | null | 690 | 703 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
swp9 (A : AscCom) (a b c d : A.1)
: Path A.1 (A.2.1 (A.2.1 a b) (A.2.1 c d)) (A.2.1 (A.2.1 a d) (A.2.1 c b))
= let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in ... | swp9 (A : AscCom) (a b c d : A.1)
: Path A.1 (A.2.1 (A.2.1 a b) (A.2.1 c d)) (A.2.1 (A.2.1 a d) (A.2.1 c b)) | = let
op : A.1 -> A.1 -> A.1
= A.2.1
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
in <i> comp (<_> A.1) (asc a b (op c d) @ -i)
[ (i = 0) -> <_> op (op a b) (op c d)
, (i = 1) -> <j> comp (<... | def | swp9 | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"AscCom",
"Path",
"asc",
"cm",
"op",
"swp1"
] | null | 705 | 718 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
swptrans (A : AscCom) (k l : A.1) (a b c : and A.1 A.1)
(p0 : Path A.1 (A.2.1 k (A.2.1 a.1 b.2)) (A.2.1 k (A.2.1 b.1 a.2)))
(p1 : Path A.1 (A.2.1 l (A.2.1 b.1 c.2)) (A.2.1 l (A.2.1 c.1 b.2)))
: Path A.1 (A.2.1 (A.2.1 k (A.2.1 l b.2)) (A.2.1 a.1 c.2))
(A.2.1 (A.2.1 k (A.2.1 l b.2)) (A.2.1 c.1 a.2))
... | swptrans (A : AscCom) (k l : A.1) (a b c : and A.1 A.1)
(p0 : Path A.1 (A.2.1 k (A.2.1 a.1 b.2)) (A.2.1 k (A.2.1 b.1 a.2)))
(p1 : Path A.1 (A.2.1 l (A.2.1 b.1 c.2)) (A.2.1 l (A.2.1 c.1 b.2)))
: Path A.1 (A.2.1 (A.2.1 k (A.2.1 l b.2)) (A.2.1 a.1 c.2))
(A.2.1 (A.2.1 k (A.2.1 l b.2)) (A.2.1 c.1 a.2)) | = let
op : A.1 -> A.1 -> A.1
= A.2.1
op3 (x y z : A.1) : A.1
= op x (op y z)
asc (x y z : A.1) : Path A.1 (op x (op y z)) (op (op x y) z)
= A.2.2.1 x y z
cm (x y : A.1) : Path A.1 (op x y) (op y x)
= A.2.2.2 x y
p2 : Path A.1 (op (op l c.2) (op3 k a.1 b.2)) (op (op3 k l b.2) ... | def | swptrans | examples | examples/algstruct.ctt | [
"pi",
"prelude",
"sigma"
] | [
"AscCom",
"Path",
"and",
"asc",
"cm",
"op",
"op3",
"p0",
"p1",
"swp0",
"swp1",
"swp6",
"swp9"
] | null | 720 | 748 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
pos = pos1
| x0 (p : pos)
| x1 (p : pos) | pos | = pos1
| x0 (p : pos)
| x1 (p : pos) | data | pos | examples | examples/binnat.ctt | [
"nat"
] | [] | Positive binary numbers like in Coq | 18 | 20 | true | https://github.com/mortberg/cubicaltt | 9baa6f2491cc61dbd4fd81d58323c04100381451 |
End of preview. Expand in Data Studio
cubicaltt
Declarations from cubicaltt, an experimental cubical type theory.
Source
- Repository: https://github.com/mortberg/cubicaltt
- Commit:
9baa6f2491cc61dbd4fd81d58323c04100381451 - Files: 78
- 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: 1,986
- With proof: 1,986 (100.0%)
- With docstring: 335 (16.9%)
- Libraries: 3
By type
| Type | Count |
|---|---|
| def | 1,934 |
| data | 51 |
| hdata | 1 |
Example
isAssociative (M : U) (op : M -> M -> M) : U
= (a b c : M) -> Path M (op a (op b c)) (op (op a b) c)
- type: def | symbolic_name:
isAssociative| examples/algstruct.ctt:14
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{cubicaltt_dataset,
title = {cubicaltt},
author = {Norton, Charles},
year = {2026},
note = {Extracted from https://github.com/mortberg/cubicaltt, commit 9baa6f2491cc},
url = {https://huggingface.co/datasets/phanerozoic/cubicaltt}
}
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