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core
Empty ∈ Set
univ ∈ (↓A ∈ Set; Empty) A
Not ∈ (Set) Set
not ∈ (↓A ∈ Set; (A) Empty) Not(A)
Unit ∈ Set
unit ∈ Unit
Bool ∈ Set
tt ∈ Bool
ff ∈ Bool
Σ ∈ (A ∈ Set; B ∈ (A) Set) Set
pair ∈ (↓A ∈ Set; ↓B ∈ (A) Set; a ∈ A; b ∈ B(a)) Σ(A, B)
Prod ∈ (Set; Set) Set
prod ∈ (↓A, ↓B ∈ Set; a ∈ A; b ∈ B) Prod(A, B)
Plus ∈ (A, B ∈ Set) Set
inl ∈ (↓A, ↓B ∈ Set; a ∈ A) Plus(A, B)
inr ∈ (↓A, ↓B ∈ Set; b ∈ B) Plus(A, B)
Lift ∈ (Set) Set
bot ∈ (A ∈ Set) Lift(A)
in ∈ (↓A ∈ Set; A) Lift(A)
Π ∈ (A ∈ Set; B ∈ (A) Set) Set
λ ∈ (↓A ∈ Set; B ∈ (A) Set; f ∈ (a ∈ A) B(a)) Π(A, B)
Fun ∈ (A, B ∈ Set) Set
fun ∈ (↓A, ↓B ∈ Set; f ∈ (A) B) Fun(A, B)
Id ∈ (↓A ∈ Set; a, b ∈ A) Set
id ∈ (↓A ∈ Set; x ∈ A) Id(x, x)
I ∈ (↓A ∈ Set; A) A
o ∈ (↓A, ↓B, ↓C ∈ Set; (B) C; (A) B; A) C
Ý
rel
Refl ∈ (↓A ∈ Set; R ∈ (A; A) Set) Set
refl ∈ (↓A ∈ Set; R ∈ (A; A) Set; (a ∈ A) R(a, a)) Refl(R)
Sym ∈ (↓A ∈ Set; R ∈ (A; A) Set) Set
sym ∈ (↓A ∈ Set;
↓R ∈ (A; A) Set;
(a, a’ ∈ A; R(a, a’)) R(a’, a)) Sym(R)
Trans ∈ (↓A ∈ Set; R ∈ (A; A) Set) Set
trans ∈ (↓A ∈ Set;
↓R ∈ (A; A) Set;
(a, a’, a’’ ∈ A; R(a, a’); R(a’, a’’)) R(a, a’’)) Trans(R)
Dec ∈ (↓A ∈ Set; P ∈ (A) Set) Set
dec ∈ (↓A ∈ Set; ↓P ∈ (A) Set; (a ∈ A) Plus(P(a), Not(P(a)))) Dec(P)
DecR ∈ (↓A ∈ Set; R ∈ (A; A) Set) Set
decR ∈ (↓A ∈ Set;
↓R ∈ (A; A) Set;
(a, a’ ∈ A) Plus(R(a, a’), Not(R(a, a’)))) DecR(R)
Tot ∈ (↓A ∈ Set; R ∈ (A; A) Set) Set
tot ∈ (↓A ∈ Set;
↓R ∈ (A; A) Set;
(a, a’ ∈ A) Plus(R(a, a’), R(a’, a))) Tot(R)
Acc ∈ (↓A ∈ Set; (A; A) Set; A) Set
acc ∈ (↓A ∈ Set;
R ∈ (A; A) Set;
a ∈ A;
(b ∈ A; R(b, a)) Acc(R, b)) Acc(R, a)
id
substId ∈ (↓A ∈ Set; ↓a, ↓b ∈ A; Id(a, b); C ∈ (A) Set; C(a)) C(b)
respId ∈ (↓A, ↓B ∈ Set; ↓a, ↓a’ ∈ A; f ∈ (A) B; Id(a, a’)) Id(f(a), f(a’))
reflId ∈ (A ∈ Set) Refl(Id())
symId1 ∈ (A ∈ Set; a, b ∈ A; Id(a, b)) Id(b, a)
symId ∈ (A ∈ Set) Sym(Id())
transId1 ∈ (A ∈ Set; a, b, c ∈ A; Id(a, b); Id(b, c)) Id(a, c)
transId ∈ (A ∈ Set) Trans(Id())
Ý
algebra
Assoc ∈ (A ∈ Set; op ∈ (A; A) A) Set
assoc ∈ (A ∈ Set;
op ∈ (A; A) A;
(x, y, z ∈ A) Id(op(op(x, y), z), op(x, op(y, z)))) Assoc(A,
op)
Comm ∈ (A ∈ Set; op ∈ (A; A) A) Set
comm ∈ (A ∈ Set;
op ∈ (A; A) A;
(x, y ∈ A) Id(op(x, y), op(y, x))) Comm(A, op)
WithUnitL ∈ (A ∈ Set; op ∈ (A; A) A) Set
withUnitL ∈ (A ∈ Set;
op ∈ (A; A) A;
e ∈ A;
(x ∈ A) Id(op(e, x), x)) WithUnitL(A, op)
Semigroup ∈ (A ∈ Set; op ∈ (A; A) A) Set
semigroup ∈ (A ∈ Set;
op ∈ (A; A) A;
Assoc(A, op)) Semigroup(A, op)
WithUnitR ∈ (A ∈ Set; op ∈ (A; A) A) Set
withUnitR ∈ (A ∈ Set;
op ∈ (A; A) A;
e ∈ A;
(x ∈ A) Id(op(x, e), x)) WithUnitR(A, op)
Monoid ∈ (A ∈ Set; op ∈ (A; A) A) Set
monoid ∈ (A ∈ Set;
op ∈ (A; A) A;
Assoc(A, op);
WithUnitL(A, op);
WithUnitR(A, op)) Monoid(A, op)
CommMonoid ∈ (A ∈ Set; op ∈ (A; A) A) Set
commMonoid ∈ (A ∈ Set;
op ∈ (A; A) A;
Assoc(A, op);
Comm(A, op);
WithUnitL(A, op);
WithUnitR(A, op)) CommMonoid(A, op)
Ý
nat
Nat ∈ Set
0 ∈ Nat
s ∈ (Nat) Nat
Lt ∈ (Nat; Nat) Set
lt0 ∈ (i ∈ Nat) Lt(0, s(i))
lt1 ∈ (↓i, ↓j ∈ Nat; Lt(i, j)) Lt(s(i), s(j))
add ∈ (Nat; Nat) Nat
injS ∈ (↓i, ↓j ∈ Nat; Id(s(i), s(j))) Id(i, j)
noConfNat ∈ (↓i ∈ Nat; Id(0, s(i))) Empty
eqNat1 ∈ (i, j ∈ Nat) Plus(Id(i, j), Not(Id(i, j)))
eqNat ∈ DecR(Id())
Le ∈ (Nat; Nat) Set
ltLem1 ∈ (i ∈ Nat; Lt(i, 0)) Empty
ltLem2 ∈ (↓i, ↓j ∈ Nat; Lt(i, s(j))) Plus(Id(i, j), Lt(i, j))
accLtaux ∈ (i ∈ Nat; (j ∈ Nat; Lt(j, i)) Acc(Lt, j); k ∈ Nat; Lt(k, s(i))) Acc(Lt, k)
accLt ∈ (i ∈ Nat) Acc(Lt, i)
ltLem3 ∈ (m, n ∈ Nat; Lt(s(m), s(n))) Lt(m, n)
decRLt1 ∈ (m, n ∈ Nat) Plus(Lt(m, n), Not(Lt(m, n)))
decRLt ∈ DecR(Lt)
trichLt ∈ (m, n ∈ Nat) Plus(Lt(m, n), Plus(Lt(n, m), Id(n, m)))
totalLe ∈ (m, n ∈ Nat) Plus(Le(m, n), Le(n, m))
leSuccLem ∈ (m, n ∈ Nat; Le(m, n)) Lt(m, s(n))
ltSuccLem ∈ (n ∈ Nat) Lt(n, s(n))
assocAdd1 ∈ (x, y, z ∈ Nat) Id(add(add(x, y), z), add(x, add(y, z)))
commAddLem1 ∈ (x ∈ Nat) Id(x, add(x, 0))
commAddLem2 ∈ (x, y ∈ Nat) Id(s(add(x, y)), add(x, s(y)))
commAdd1 ∈ (x, y ∈ Nat) Id(add(x, y), add(y, x))
withUnitLAdd1 ∈ (x ∈ Nat) Id(add(0, x), x)
withUnitRAdd1 ∈ (x ∈ Nat) Id(add(x, 0), x)
assocAdd ∈ Assoc(Nat, add)
commAdd ∈ Comm(Nat, add)
withUnitLAdd ∈ WithUnitL(Nat, add)
withUnitRAdd ∈ WithUnitR(Nat, add)
addCommMonoid ∈ CommMonoid(Nat, add)
Ý
lists
List ∈ (A ∈ Set) Set
nil ∈ (↓A ∈ Set) List(A)
cons ∈ (↓A ∈ Set; a ∈ A; l ∈ List(A)) List(A)
map ∈ (↓A, ↓B ∈ Set; f ∈ (A) B; l ∈ List(A)) List(B)
append ∈ (↓A ∈ Set; List(A); List(A)) List(A)
ListAll ∈ (↓A ∈ Set; P ∈ (A) Set; List(A)) Set
la0 ∈ (↓A ∈ Set; P ∈ (A) Set) ListAll(P, nil())
la1 ∈ (↓A ∈ Set; ↓P ∈ (A) Set; ↓a ∈ A; ↓l ∈ List(A); P(a); ListAll(P, l)) ListAll(P, cons(a, l))
length ∈ (↓A ∈ Set; List(A)) Nat
listIdLem1 ∈ (↓A ∈ Set; ↓a ∈ A; ↓x ∈ List(A); Id(nil(), cons(a, x))) Empty
listIdLem2 ∈ (↓A ∈ Set;
↓a, ↓b ∈ A;
↓x, ↓y ∈ List(A);
Id(cons(a, x), cons(b, y))) Id(a, b)
listIdLem3 ∈ (↓A ∈ Set;
↓a, ↓b ∈ A;
↓x, ↓y ∈ List(A);
Id(cons(a, x), cons(b, y))) Id(x, y)
decIdList1 ∈ (↓A ∈ Set;
d ∈ (a, b ∈ A) Plus(Id(a, b), Not(Id(a, b)));
x, y ∈ List(A)) Plus(Id(x, y), Not(Id(x, y)))
decIdList ∈ (A ∈ Set; d ∈ DecR(Id())) DecR(Id())
assocAppend1 ∈ (A ∈ Set;
x, y, z ∈ List(A)) Id(append(append(x, y), z),
append(x, append(y, z)))
withUnitLAppend1 ∈ (A ∈ Set; x ∈ List(A)) Id(append(nil(), x), x)
withUnitRAppend1 ∈ (A ∈ Set; x ∈ List(A)) Id(append(x, nil()), x)
assocAppend ∈ (A ∈ Set) Assoc(List(A), append())
withUnitLAppend ∈ (A ∈ Set) WithUnitL(List(A), append())
withUnitRAppend ∈ (A ∈ Set) WithUnitR(List(A), append())
monoidAppend ∈ (A ∈ Set) Monoid(List(A), append())
appendLength ∈ (A ∈ Set;
x, y ∈ List(A)) Id(length(append(x, y)),
add(length(x), length(y)))
vec
Fin ∈ (Nat) Set
0Fin ∈ (n ∈ Nat) Fin(s(n))
sFin ∈ (↓n ∈ Nat; Fin(n)) Fin(s(n))
finToNat ∈ (↓n ∈ Nat; Fin(n)) Nat
emb ∈ (↓n ∈ Nat; Fin(n)) Fin(s(n))
addFin ∈ (↓m, ↓n ∈ Nat; Fin(m); Fin(n)) Fin(add(m, n))
Vec ∈ (Set; Nat) Set
nilVec ∈ (A ∈ Set) Vec(A, 0)
consVec ∈ (↓A ∈ Set; ↓n ∈ Nat; A; Vec(A, n)) Vec(A, s(n))
nth ∈ (↓A ∈ Set; ↓n ∈ Nat; Vec(A, n); Fin(n)) A
update ∈ (↓A ∈ Set; ↓n ∈ Nat; Vec(A, n); Fin(n); A) Vec(A, n)
insert ∈ (↓A ∈ Set; ↓n ∈ Nat; Vec(A, n); Fin(n); A) Vec(A, s(n))
delete ∈ (↓A ∈ Set; ↓n ∈ Nat; Vec(A, s(n)); Fin(n)) Vec(A, n)
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