Nuprl Lemma : member-decide-assert
∀[x:𝔹]. (if x then tt else inr (λx.⋅) fi ∈ Dec(↑x))
Proof
Definitions occuring in Statement :
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
decidable: Dec(P),
it: ⋅,
uall: ∀[x:A]. B[x],
member: t ∈ T,
lambda: λx.A[x],
inr: inr x
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
assert: ↑b,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top
Lemmas referenced :
bool_wf,
eqtt_to_assert,
it_wf,
subtype_rel_self,
equal-wf-base,
not_wf,
true_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
top_wf,
subtype_rel_union,
unit_wf2,
false_wf,
subtype_rel_dep_function
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
hypothesis,
thin,
extract_by_obid,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
equalityElimination,
sqequalRule,
isectElimination,
productElimination,
independent_isectElimination,
inlEquality,
applyEquality,
intEquality,
baseClosed,
because_Cache,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
inrEquality,
lambdaEquality,
voidEquality,
functionEquality,
functionExtensionality,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[x:\mBbbB{}]. (if x then tt else inr (\mlambda{}x.\mcdot{}) fi \mmember{} Dec(\muparrow{}x))
Date html generated:
2017_04_14-AM-07_39_02
Last ObjectModification:
2017_02_27-PM-03_10_51
Theory : equality!deciders
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