Nuprl Lemma : decidable-all-finite
∀[T:Type]. (finite(T) ⇒ (∀[P:T ⟶ ℙ]. ((∀t:T. Dec(P[t])) ⇒ Dec(∀t:T. P[t]))))
Proof
Definitions occuring in Statement :
finite: finite(T),
decidable: Dec(P),
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 :
guard: {T},
prop: ℙ,
false: False,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
subtype_rel_self,
istype-universe,
finite_wf,
decidable_wf,
istype-void,
decidable__not,
not_wf,
decidable-exists-finite
Rules used in proof :
productElimination,
Error :inrFormation_alt,
instantiate,
universeEquality,
voidElimination,
Error :functionIsType,
Error :dependent_pairFormation_alt,
Error :inlFormation_alt,
unionElimination,
because_Cache,
dependent_functionElimination,
Error :universeIsType,
applyEquality,
Error :lambdaEquality_alt,
sqequalRule,
independent_functionElimination,
Error :lambdaFormation_alt,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}[T:Type]. (finite(T) {}\mRightarrow{} (\mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}t:T. Dec(P[t])) {}\mRightarrow{} Dec(\mforall{}t:T. P[t]))))
Date html generated:
2019_06_20-PM-02_18_51
Last ObjectModification:
2019_06_12-PM-02_54_17
Theory : equipollence!!cardinality!
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