Nuprl Lemma : free-vs-dim-1-ext
∀S:Type. (S ⇒ (∀K:CRng. free-vs(K;S) ≅ one-dim-vs(K) supposing ∀x,y:S. (x = y ∈ S)))
Proof
Definitions occuring in Statement :
free-vs: free-vs(K;S),
vs-iso: A ≅ B,
one-dim-vs: one-dim-vs(K),
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
member: t ∈ T,
one-dim-vs: one-dim-vs(K),
mk-vs: mk-vs,
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
btrue: tt,
it: ⋅,
bfalse: ff,
infix_ap: x f y,
rng_car: |r|,
pi1: fst(t),
vs-lift: vs-lift(vs;f;fs),
vs-bag-add: Σ{f[b] | b ∈ bs},
vs-0: 0,
record-select: r.x,
vs-add: x + y,
vs-mul: a * x,
free-vs: free-vs(K;S),
formal-sum-mul: k * x,
bag-map: bag-map(f;bs),
map: map(f;as),
list_ind: list_ind,
cons: [a / b],
nil: [],
bottom: ⊥,
formal-sum: formal-sum(K;S),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
basic-formal-sum: basic-formal-sum(K;S),
bag: bag(T),
list: T List,
colist: colist(T),
corec: corec(T.F[T]),
nat: ℕ,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
less_than': less_than'(a;b),
true: True,
false: False,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
top: Top,
subtract: n - m,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
pi2: snd(t),
has-value: (a)↓,
colength: colength(L),
permutation: permutation(T;L1;L2),
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
length: ||as||,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
permute_list: (L o f),
mklist: mklist(n;f),
append: as @ bs,
select: L[n],
bfs-equiv: bfs-equiv(K;S;fs1;fs2),
least-equiv: least-equiv(A;R),
transitive-reflexive-closure: R^*,
or: P ∨ Q,
transitive-closure: TC(R),
bfs-reduce: bfs-reduce(K;S;as;bs),
bag-append: as + bs,
zero-bfs: 0 * ss,
rel_path: rel_path(A;L;x;y),
empty-bag: {},
formal-sum-add: x + y,
free-vs-inc: <s>,
single-bag: {x},
rng_plus: +r,
rng_times: *,
free-vs-dim-1,
free-vs-unique,
vs-iso_inversion,
unique-one-dim-vs-map,
free-vs-property
Lemmas referenced :
free-vs-dim-1,
free-vs-unique,
vs-iso_inversion,
unique-one-dim-vs-map,
free-vs-property
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}S:Type. (S {}\mRightarrow{} (\mforall{}K:CRng. free-vs(K;S) \mcong{} one-dim-vs(K) supposing \mforall{}x,y:S. (x = y)))
Date html generated:
2019_10_31-AM-06_30_49
Last ObjectModification:
2019_08_02-PM-04_37_53
Theory : linear!algebra
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