Nuprl Lemma : decidable__inject-finite-type
∀[T:Type]
(finite-type(T)
⇒ (∀x,y:T. Dec(x = y ∈ T))
⇒ (∀[A:Type]. ((∀x,y:A. Dec(x = y ∈ A)) ⇒ (∀f:T ⟶ A. Dec(Inj(T;A;f))))))
Proof
Definitions occuring in Statement :
finite-type: finite-type(T),
inject: Inj(A;B;f),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
inject: Inj(A;B;f),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
finite-type-iff-list,
equal_wf,
l_member_wf,
all_wf,
l_all_iff,
l_all_wf,
iff_wf,
decidable_functionality,
inject_wf,
decidable__l_all,
decidable__implies,
set_wf,
decidable_wf,
finite-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
hypothesis,
independent_pairFormation,
cumulativity,
applyEquality,
functionExtensionality,
because_Cache,
sqequalRule,
lambdaEquality,
functionEquality,
addLevel,
impliesFunctionality,
allFunctionality,
dependent_functionElimination,
setElimination,
rename,
setEquality,
levelHypothesis,
allLevelFunctionality,
impliesLevelFunctionality,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[T:Type]
(finite-type(T)
{}\mRightarrow{} (\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} (\mforall{}[A:Type]. ((\mforall{}x,y:A. Dec(x = y)) {}\mRightarrow{} (\mforall{}f:T {}\mrightarrow{} A. Dec(Inj(T;A;f))))))
Date html generated:
2017_04_17-AM-07_47_42
Last ObjectModification:
2017_02_27-PM-04_22_08
Theory : list_1
Home
Index