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phanerozoic
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cubicaltt

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 288 values	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)	def	isAssociative	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "op" ]		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	def	propisAssociative	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "isAssociative", "op", "prop", "propPi3", "set" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		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	def	prophasLeftIdentity	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "hasLeftIdentity", "id", "op", "prop", "propPi", "set" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		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	def	prophasRightIdentity	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "hasRightIdentity", "id", "op", "prop", "propPi", "set" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		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)	def	prophasIdentity	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "hasIdentity", "hasLeftIdentity", "hasRightIdentity", "id", "op", "prop", "propAnd", "prophasLeftIdentity", "prophasRightIdentity", "set" ]		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)	def	isCommutative	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "op" ]		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	def	propisCommutative	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "isCommutative", "op", "prop", "propPi2", "set" ]		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	def	isLeftCancellative	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "op" ]		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 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" ]		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	def	isRightCancellative	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "op" ]		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 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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		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)	def	propisCancellative	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "isCancellative", "isLeftCancellative", "isRightCancellative", "op", "prop", "propAnd", "propisLeftCancellative", "propisRightCancellative", "set" ]		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	def	hasLeftInverse	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "id", "inv", "op" ]		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	def	prophasLeftInverse	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "hasLeftInverse", "id", "inv", "op", "prop", "propPi", "set" ]		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	def	hasRightInverse	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "id", "inv", "op" ]		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	def	prophasRightInverse	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "hasRightInverse", "id", "inv", "op", "prop", "propPi", "set" ]		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)	def	hasInverse	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "hasLeftInverse", "hasRightInverse", "id", "inv", "op" ]		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)	def	prophasInverse	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "hasInverse", "hasLeftInverse", "hasRightInverse", "id", "inv", "op", "prop", "propAnd", "prophasLeftInverse", "prophasRightInverse", "set" ]		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))	def	isLeftDistributive	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "add" ]		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 h	def	propisLeftDistributive	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "add", "isLeftDistributive", "prop", "propPi3", "set" ]		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))	def	isRightDistributive	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "add" ]		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 B h	def	propisRightDistributive	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "add", "isRightDistributive", "prop", "propPi3", "set" ]		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)	def	isDistributive	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "add", "isLeftDistributive", "isRightDistributive" ]		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)	def	propisDistributive	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "add", "isDistributive", "isLeftDistributive", "isRightDistributive", "prop", "propAnd", "propisLeftDistributive", "propisRightDistributive", "set" ]		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))	def	preservesOp	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path" ]		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)))	def	proppreservesOp	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "preservesOp", "prop", "propPi2", "set" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		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	def	proppreservesId	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "preservesId", "prop", "set" ]		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)	def	ismonoid	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "SET", "hasIdentity", "id", "isAssociative", "op" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
monoid : U	= (X : SET) * ismonoid X	def	monoid	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "SET", "ismonoid" ]		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)	def	isgroup	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "SET", "hasInverse", "inv", "ismonoid" ]	Group	https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
group : U	= (X : SET) * isgroup X	def	group	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "SET", "isgroup" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
iscmonoid (M : SET) : U	= (m : ismonoid M) * (isCommutative M.1 m.1)	def	iscmonoid	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "SET", "isCommutative", "ismonoid" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
cmonoid : U	= (X : SET) * iscmonoid X	def	cmonoid	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "SET", "iscmonoid" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
cgroup : U	= (X : SET) * iscgroup X	def	cgroup	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "SET", "iscgroup" ]		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)	def	isring	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "SET", "add", "isDistributive", "iscgroup", "ismonoid" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
ring : U	= (X : SET) * isring X	def	ring	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "SET", "isring" ]		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)	def	iscring	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "SET", "add", "isDistributive", "iscgroup", "iscmonoid" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
cring : U	= (X : SET) * iscring X	def	cring	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "SET", "iscring" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
idMonoid (m : monoid) : m.1.1	= m.2.2.2.1	def	idMonoid	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "monoid" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
idCMonoid (m : cmonoid) : m.1.1	= m.2.1.2.2.1	def	idCMonoid	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "cmonoid" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
isCommCMonoid (m : cmonoid) : isCommutative m.1.1 (opCMonoid m)	= m.2.2	def	isCommCMonoid	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "cmonoid", "isCommutative", "opCMonoid" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
idGroup (g : group) : g.1.1	= g.2.1.2.2.1	def	idGroup	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "group" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
invGroup (g : group) : g.1.1 -> g.1.1	= g.2.2.1	def	invGroup	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "group" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
idCGroup (g : cgroup) : g.1.1	= g.2.1.1.2.2.1	def	idCGroup	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "cgroup" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
isCommCGroup (g : cgroup) : isCommutative g.1.1 (opCGroup g)	= g.2.2	def	isCommCGroup	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "cgroup", "isCommutative", "opCGroup" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
invCGroup (g : cgroup) : g.1.1 -> g.1.1	= g.2.1.2.1	def	invCGroup	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "cgroup" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		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 @ -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" ]		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) , (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" ]		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)	def	lem_group_cancellative	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "group", "isCancellative", "lem_group_lcancellative", "lem_group_rcancellative", "opGroup" ]		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 : 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	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) ]	def	lemma_cgroup_inv_id	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "cgroup", "hasIdCGroup", "hasInvCGroup", "idCGroup", "invCGroup" ]	e⁻¹ ≡ e	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)	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	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 (idMonoid a) (idMonoid b) f)	def	propismonoidhom	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "idMonoid", "ismonoidhom", "monoid", "opMonoid", "preservesId", "preservesOp", "prop", "propAnd", "proppreservesId", "proppreservesOp" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		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 b) setf setm	def	setmonoidhom	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "ismonoidhom", "monoid", "monoidhom", "propSet", "propismonoidhom", "set", "setPi", "setSig" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		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) @ 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" ]		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	def	cgrouphom'	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "cgroup", "grouphom'", "monoidhom", "opCGroup", "preservesOp" ]		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) , (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	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 (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	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)	def	idmonoidhom	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "idMonoid", "monoid", "monoidhom", "opMonoid" ]	1_a	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)	def	lemma_idmonoidhom0	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "idmonoidhom", "ismonoidhom", "lemSig", "monoid", "monoidhom", "monoidhomcomp", "propismonoidhom" ]	f ∘ 1_a ≡ f	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)	def	lemma_idmonoidhom1	examples	examples/algstruct.ctt	[ "pi", "prelude", "sigma" ]	[ "Path", "idmonoidhom", "ismonoidhom", "lemSig", "monoid", "monoidhom", "monoidhomcomp", "propismonoidhom" ]	1_b ∘ f ≡ f	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.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	https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
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" ]		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) (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" ]		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 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" ]		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 (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" ]		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 (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" ]		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 <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" ]		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 <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" ]		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 <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" ]		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 <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" ]		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 <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" ]		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 <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" ]		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))	= 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" ]		https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451
pos	= pos1 \| x0 (p : pos) \| x1 (p : pos)	data	pos	examples	examples/binnat.ctt	[ "nat" ]	[]	Positive binary numbers like in Coq	https://github.com/mortberg/cubicaltt	9baa6f2491cc61dbd4fd81d58323c04100381451

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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
statement	string	Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof
proof	string	Verbatim proof/body, empty if the declaration has 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 `Require`/`Import` modules
deps	list[string]	Intra-corpus identifiers referenced
docstring	string	Preceding documentation comment, empty if absent
source_url	string	Upstream repository
commit	string	Upstream commit extracted

Statistics

Entries: 1,874
With proof: 1,874 (100.0%)
With docstring: 316 (16.9%)
Libraries: 3

By type

Type	Count
def	1,829
data	44
hdata	1

Example

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

type: def | symbolic_name: propisAssociative | examples/algstruct.ctt

Use

Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint and together form the complete declaration, for proof 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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