Nuprl Lemma : free-1-normal-form
∀[S:Type]
∀s:S. ∀x:Point(free-vs(ℤ-rng;S)). ∃k:ℤ. (x = {<k, s>} ∈ Point(free-vs(ℤ-rng;S))) supposing ∀x,y:S. (x = y ∈ S)
Proof
Definitions occuring in Statement :
free-vs: free-vs(K;S),
vs-point: Point(vs),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
pair: <a, b>,
int: ℤ,
universe: Type,
equal: s = t ∈ T,
int_ring: ℤ-rng,
single-bag: {x}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
integ_dom: IntegDom{i},
crng: CRng,
rng: Rng,
vs-iso: A ≅ B,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
and: P ∧ Q,
vs-map: A ⟶ B,
prop: ℙ,
so_apply: x[s],
top: Top,
pi2: snd(t),
pi1: fst(t),
free-iso-int: free-iso-int(s),
implies: P ⇒ Q,
vs-point: Point(vs),
record-select: r.x,
int-vs: ℤ,
mk-vs: mk-vs,
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
bag: bag(T),
quotient: x,y:A//B[x; y],
basic-formal-sum: basic-formal-sum(K;S),
rng_car: |r|,
int_ring: ℤ-rng,
free-vs: free-vs(K;S),
formal-sum: formal-sum(K;S),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
free-iso-int_wf,
vs-point_wf,
int_ring_wf,
free-vs_wf,
istype-universe,
pi2_wf,
vs-map_wf,
int-vs_wf,
equal_wf,
pi1_wf_top,
istype-void,
subtype_rel_self,
single-bag_wf,
basic-formal-sum_wf,
rec_select_update_lemma,
formal-sum_wf,
subtype_quotient,
bfs-equiv_wf,
bfs-equiv-rel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
functionIsTypeImplies,
inhabitedIsType,
rename,
lambdaFormation_alt,
extract_by_obid,
isectElimination,
independent_isectElimination,
universeIsType,
applyEquality,
setElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
functionIsType,
equalityIstype,
instantiate,
universeEquality,
applyLambdaEquality,
productEquality,
functionEquality,
productElimination,
independent_pairEquality,
isect_memberEquality_alt,
voidElimination,
dependent_pairFormation_alt,
intEquality,
independent_functionElimination
Latex:
\mforall{}[S:Type]. \mforall{}s:S. \mforall{}x:Point(free-vs(\mBbbZ{}-rng;S)). \mexists{}k:\mBbbZ{}. (x = \{<k, s>\}) supposing \mforall{}x,y:S. (x = y)
Date html generated:
2019_10_31-AM-06_31_14
Last ObjectModification:
2019_08_15-AM-11_15_58
Theory : linear!algebra
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