Nuprl Lemma : deq-witness_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. (deq-witness(eq) ∈ ∀x,y:T. Dec(x = y ∈ T))
Proof
Definitions occuring in Statement :
deq-witness: deq-witness(eq),
deq: EqDecider(T),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
deq-witness: deq-witness(eq),
all: ∀x:A. B[x],
deq: EqDecider(T),
implies: P ⇒ Q,
exposed-it: exposed-it,
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,
eqof: eqof(d),
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A
Lemmas referenced :
bool_wf,
eqtt_to_assert,
safe-assert-deq,
not_wf,
equal_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lambdaEquality,
applyEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
extract_by_obid,
lambdaFormation,
unionElimination,
equalityElimination,
isectElimination,
productElimination,
independent_isectElimination,
cumulativity,
inlEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
inrEquality,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. (deq-witness(eq) \mmember{} \mforall{}x,y:T. Dec(x = y))
Date html generated:
2017_04_14-AM-07_39_09
Last ObjectModification:
2017_02_27-PM-03_10_54
Theory : equality!deciders
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