Nuprl Lemma : decidable-equality-implies-discrete
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ discrete-type(T))
Proof
Definitions occuring in Statement :
discrete-type: discrete-type(T),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
discrete-type: discrete-type(T),
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
eqof: eqof(d),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
true: True
Lemmas referenced :
deq-exists,
real_wf,
all_wf,
req_wf,
equal_wf,
decidable_wf,
int-discrete,
ifthenelse_wf,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
int_subtype_base,
assert_wf,
bnot_wf,
eqof_wf,
not_wf,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
hypothesis,
rename,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
applyEquality,
functionExtensionality,
dependent_functionElimination,
axiomEquality,
universeEquality,
setElimination,
intEquality,
natural_numberEquality,
because_Cache,
unionElimination,
equalityElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
voidElimination,
hyp_replacement,
applyLambdaEquality,
independent_pairFormation,
impliesFunctionality
Latex:
\mforall{}[T:Type]. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} discrete-type(T))
Date html generated:
2018_05_22-PM-02_13_55
Last ObjectModification:
2017_10_27-PM-05_04_16
Theory : reals
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