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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