Nuprl Lemma : extract-decider-of-decidable-prop
∀[T:Type]. ∀[P:T ⟶ ℙ]. ((∀t:T. ((P t) ∨ (¬(P t)))) ⇒ (∃f:T ⟶ 𝔹. ∀t:T. (↑(f t) ⇐⇒ P t)))
Proof
Definitions occuring in Statement :
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[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],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
isl: isl(x),
not: ¬A,
false: False,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True
Lemmas referenced :
all_wf,
or_wf,
not_wf,
isl_wf,
assert_wf,
assert_witness,
iff_wf,
and_wf,
equal_wf,
assert_elim,
bfalse_wf,
btrue_neq_bfalse
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
rename,
dependent_pairFormation,
independent_pairFormation,
comment,
because_Cache,
introduction,
independent_functionElimination,
productElimination,
promote_hyp,
unionElimination,
voidEquality,
inrEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
natural_numberEquality,
dependent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}t:T. ((P t) \mvee{} (\mneg{}(P t)))) {}\mRightarrow{} (\mexists{}f:T {}\mrightarrow{} \mBbbB{}. \mforall{}t:T. (\muparrow{}(f t) \mLeftarrow{}{}\mRightarrow{} P t)))
Date html generated:
2016_05_15-PM-03_14_26
Last ObjectModification:
2015_12_27-PM-01_02_15
Theory : general
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