Nuprl Lemma : sp-lub_wf1
∀[B:f,g:ℕ ⟶ ℕ ⟶ 𝔹//fun-equiv(ℕ;a,b.↓a = ⊥ ∈ (ℕ ⟶ 𝔹) ⇐⇒ b = ⊥ ∈ (ℕ ⟶ 𝔹);f;g)]. (lub(n.B[n]) ∈ Sierpinski)
Proof
Definitions occuring in Statement :
sp-lub: lub(n.A[n]),
Sierpinski: Sierpinski,
Sierpinski-bottom: ⊥,
fun-equiv: fun-equiv(X;a,b.E[a; b];f;g),
quotient: x,y:A//B[x; y],
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
squash: ↓T,
member: t ∈ T,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
Sierpinski: Sierpinski,
uimplies: b supposing a,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
sp-lub: lub(n.A[n]),
all: ∀x:A. B[x],
so_apply: x[s],
prop: ℙ,
iff: P ⇐⇒ Q,
fun-equiv: fun-equiv(X;a,b.E[a; b];f;g),
uiff: uiff(P;Q),
squash: ↓T,
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
not: ¬A,
false: False,
guard: {T}
Lemmas referenced :
fun-equiv-rel,
nat_wf,
bool_wf,
squash_wf,
iff_wf,
equal-wf-T-base,
equiv_rel_squash,
two-class-equiv-rel,
Sierpinski-bottom_wf,
quotient_wf,
quotient-member-eq,
coded-pair_wf,
equal_wf,
equal-Sierpinski-bottom,
all_wf,
not_wf,
assert_wf,
fun-equiv_wf,
equal-wf-base,
code-pair_wf,
assert_functionality_wrt_uiff,
coded-code-pair
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesis,
functionEquality,
sqequalRule,
lambdaEquality,
hypothesisEquality,
baseClosed,
because_Cache,
independent_functionElimination,
isect_memberFormation,
pointwiseFunctionalityForEquality,
independent_isectElimination,
pertypeElimination,
productElimination,
dependent_functionElimination,
productEquality,
lambdaFormation,
spreadEquality,
independent_pairEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
addLevel,
allFunctionality,
imageElimination,
impliesFunctionality,
functionExtensionality,
imageMemberEquality,
levelHypothesis,
promote_hyp,
voidElimination,
axiomEquality
Latex:
\mforall{}[B:f,g:\mBbbN{} {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbB{}//fun-equiv(\mBbbN{};a,b.\mdownarrow{}a = \mbot{} \mLeftarrow{}{}\mRightarrow{} b = \mbot{};f;g)]. (lub(n.B[n]) \mmember{} Sierpinski)
Date html generated:
2019_10_31-AM-06_36_00
Last ObjectModification:
2017_07_28-AM-09_12_02
Theory : synthetic!topology
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