Nuprl Lemma : decidable-exists-finite-type
∀[T:Type]. (finite-type(T) ⇒ (∀[P:T ⟶ ℙ]. ((∀t:T. Dec(P[t])) ⇒ Dec(∃t:T. P[t]))))
Proof
Definitions occuring in Statement :
finite-type: finite-type(T),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
finite-type: finite-type(T),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
not: ¬A,
false: False,
surject: Surj(A;B;f),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
decidable__exists_int_seg,
int_seg_wf,
all_wf,
decidable_wf,
finite-type_wf,
not_wf,
exists_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
instantiate,
lemma_by_obid,
dependent_functionElimination,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
independent_functionElimination,
unionElimination,
functionEquality,
cumulativity,
universeEquality,
inlFormation,
dependent_pairFormation,
inrFormation,
introduction,
voidElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination
Latex:
\mforall{}[T:Type]. (finite-type(T) {}\mRightarrow{} (\mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}t:T. Dec(P[t])) {}\mRightarrow{} Dec(\mexists{}t:T. P[t]))))
Date html generated:
2016_05_14-PM-01_51_22
Last ObjectModification:
2015_12_26-PM-05_37_35
Theory : list_1
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