Nuprl Lemma : deq-implies
∀[T:Type]. (EqDecider(T) ⇒ {∀x,y:T. Dec(x = y ∈ T)})
Proof
Definitions occuring in Statement :
deq: EqDecider(T),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
all: ∀x:A. B[x],
member: t ∈ T,
deq: EqDecider(T),
decidable: Dec(P),
not: ¬A,
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
assert: ↑b,
ifthenelse: if b then t else f fi ,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
false: False,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
true: True
Lemmas referenced :
deq_wf,
bool_wf,
eqtt_to_assert,
iff_wf,
equal_wf,
true_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
false_wf,
it_wf,
equal_subtype,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
introduction,
hypothesisEquality,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesis,
universeEquality,
setElimination,
dependent_functionElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
functionExtensionality,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_pairFormation,
promote_hyp,
instantiate,
because_Cache,
voidElimination,
inlEquality,
intEquality,
natural_numberEquality,
baseClosed,
functionEquality,
inrEquality
Latex:
\mforall{}[T:Type]. (EqDecider(T) {}\mRightarrow{} \{\mforall{}x,y:T. Dec(x = y)\})
Date html generated:
2017_04_14-AM-07_39_08
Last ObjectModification:
2017_02_27-PM-03_10_45
Theory : equality!deciders
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