Nuprl Lemma : dec_iff_ex_bvfun
∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ]. (∀x,y:T. Dec(E[x;y]) ⇐⇒ ∃f:T ⟶ T ⟶ 𝔹. ∀x,y:T. (↑(x f y) ⇐⇒ E[x;y]))
Proof
Definitions occuring in Statement :
assert: ↑b,
bool: 𝔹,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s1;s2],
or: P ∨ Q,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
decidable: Dec(P),
true: True,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
not: ¬A,
false: False,
bfalse: ff
Lemmas referenced :
assert_wf,
iff_wf,
bool_wf,
exists_wf,
decidable_wf,
all_wf,
equal_wf,
bfalse_wf,
btrue_wf,
not_wf,
or_wf,
subtype_rel_self,
true_wf,
false_wf,
decidable_functionality,
decidable__assert
Rules used in proof :
universeEquality,
cumulativity,
functionEquality,
hypothesis,
applyEquality,
lambdaEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
isect_memberFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
instantiate,
unionEquality,
equalitySymmetry,
equalityTransitivity,
functionExtensionality,
dependent_pairFormation,
rename,
inlEquality,
applyLambdaEquality,
hyp_replacement,
natural_numberEquality,
inrEquality,
voidElimination,
productElimination
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(\mforall{}x,y:T. Dec(E[x;y]) \mLeftarrow{}{}\mRightarrow{} \mexists{}f:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}. \mforall{}x,y:T. (\muparrow{}(x f y) \mLeftarrow{}{}\mRightarrow{} E[x;y]))
Date html generated:
2019_06_20-PM-00_32_14
Last ObjectModification:
2018_10_15-PM-05_02_36
Theory : quot_1
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