Nuprl Lemma : sp-lub-is-bottom
∀[A:ℕ ⟶ Sierpinski]. (lub(n.A[n]) = ⊥ ∈ Sierpinski ⇐⇒ ∀n:ℕ. (A[n] = ⊥ ∈ Sierpinski))
Proof
Definitions occuring in Statement :
sp-lub: lub(n.A[n]),
Sierpinski: Sierpinski,
Sierpinski-bottom: ⊥,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
Sierpinski: Sierpinski,
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
cand: A c∧ B,
sp-lub: lub(n.A[n]),
Sierpinski-bottom: ⊥,
quotient: x,y:A//B[x; y],
or: P ∨ Q,
decidable: Dec(P),
false: False,
not: ¬A,
fun-equiv: fun-equiv(X;a,b.E[a; b];f;g),
true: True,
squash: ↓T,
bfalse: ff,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
istype-nat,
Sierpinski_wf,
sp-lub_wf,
Sierpinski-bottom_wf,
subtype-Sierpinski,
quotient-function-subtype,
nat_wf,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
bool_wf,
iff_wf,
equal-wf-T-base,
two-class-equiv-rel,
fun-equiv-rel,
squash_wf,
equiv_rel_squash,
subtype_rel_self,
quotient_wf,
subtype_rel_transitivity,
fun-equiv_wf,
equal-Sierpinski-bottom,
bfalse_wf,
quotient-member-eq,
istype-assert,
decidable__assert,
assert_wf,
decidable__not,
coded-code-pair,
assert_functionality_wrt_uiff,
coded-pair_wf,
iff_weakening_uiff,
code-pair_wf,
sp-lub_wf1,
equal_wf,
true_wf,
istype-universe,
iff_weakening_equal,
istype-void
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaFormation_alt,
hypothesis,
extract_by_obid,
equalityIstype,
universeIsType,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
baseClosed,
sqequalBase,
equalitySymmetry,
functionIsType,
because_Cache,
productElimination,
independent_pairEquality,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
independent_isectElimination,
intEquality,
natural_numberEquality,
functionEquality,
independent_functionElimination,
closedConclusion,
equalityTransitivity,
equalityIsType4,
equalityIsType1,
productIsType,
equalityIsType3,
pertypeElimination,
pointwiseFunctionalityForEquality,
voidElimination,
unionElimination,
rename,
spreadEquality,
promote_hyp,
imageElimination,
instantiate,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[A:\mBbbN{} {}\mrightarrow{} Sierpinski]. (lub(n.A[n]) = \mbot{} \mLeftarrow{}{}\mRightarrow{} \mforall{}n:\mBbbN{}. (A[n] = \mbot{}))
Date html generated:
2019_10_31-AM-06_36_16
Last ObjectModification:
2018_12_13-PM-03_00_18
Theory : synthetic!topology
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