Nuprl Lemma : decidable__all_fset
∀[T:Type]. ∀eq:EqDecider(T). ∀[P:T ⟶ ℙ]. ((∀x:T. Dec(P[x])) ⇒ (∀s:fset(T). Dec(∀x:T. P[x] supposing x ∈ s)))
Proof
Definitions occuring in Statement :
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
true: True,
bfalse: ff,
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
fset-all: fset-all(s;x.P[x]),
guard: {T}
Lemmas referenced :
fset_wf,
all_wf,
decidable_wf,
deq_wf,
btrue_wf,
bfalse_wf,
equal_wf,
true_wf,
false_wf,
uiff_wf,
assert_wf,
isect_wf,
fset-member_wf,
decidable_functionality,
iff_weakening_uiff,
assert_witness,
fset-member_witness,
uall_wf,
fset-all_wf,
fset-all-iff,
decidable__assert,
fset-null_wf,
fset-filter_wf,
bnot_wf,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
rename,
dependent_pairFormation,
unionElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
introduction,
natural_numberEquality,
axiomEquality,
voidElimination,
productElimination,
allFunctionality,
independent_isectElimination,
inlFormation,
inrFormation,
isect_memberEquality,
because_Cache
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T)
\mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x:T. Dec(P[x])) {}\mRightarrow{} (\mforall{}s:fset(T). Dec(\mforall{}x:T. P[x] supposing x \mmember{} s)))
Date html generated:
2016_05_14-PM-03_41_25
Last ObjectModification:
2015_12_26-PM-06_40_36
Theory : finite!sets
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