Nuprl Lemma : fset-some-iff-squash
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[P:T ⟶ 𝔹]. ∀[s:fset(T)]. uiff(fset-some(s;x.P[x]);↓∃x:T. (x ∈ s ∧ (↑P[x])))
Proof
Definitions occuring in Statement :
fset-some: fset-some(s;x.P[x]),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
fset-some: fset-some(s;x.P[x]),
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
fset: fset(T),
all: ∀x:A. B[x],
quotient: x,y:A//B[x; y],
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
fset-member: a ∈ s,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
cand: A c∧ B
Lemmas referenced :
l_member_wf,
l_exists_iff,
assert-deq-member,
decidable__assert,
decidable__l_exists,
set-equal_wf,
list_wf,
equal-wf-base,
list_subtype_fset,
all_wf,
not_wf,
deq_wf,
bool_wf,
fset_wf,
fset-member_wf,
and_wf,
exists_wf,
squash_wf,
fset-filter_wf,
fset-null_wf,
assert_wf,
fset-some_wf,
fset-some-iff2
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_pairFormation,
productElimination,
introduction,
independent_isectElimination,
imageElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
lambdaEquality,
applyEquality,
dependent_functionElimination,
voidElimination,
functionEquality,
universeEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
cumulativity,
because_Cache,
productEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
unionElimination,
setElimination,
rename,
setEquality,
dependent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:fset(T)].
uiff(fset-some(s;x.P[x]);\mdownarrow{}\mexists{}x:T. (x \mmember{} s \mwedge{} (\muparrow{}P[x])))
Date html generated:
2016_05_14-PM-03_41_11
Last ObjectModification:
2016_01_14-PM-10_40_59
Theory : finite!sets
Home
Index