Nuprl Lemma : union-deq_wf
∀[A,B:Type]. ∀[a:EqDecider(A)]. ∀[b:EqDecider(B)]. (union-deq(A;B;a;b) ∈ EqDecider(A + B))
Proof
Definitions occuring in Statement :
union-deq: union-deq(A;B;a;b),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
union-deq: union-deq(A;B;a;b),
deq: EqDecider(T),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
and: P ∧ Q,
prop: ℙ,
sq_stable: SqStable(P),
squash: ↓T,
sumdeq: sumdeq(a;b),
eqof: eqof(d),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
false: False,
sq_type: SQType(T),
guard: {T},
true: True
Lemmas referenced :
sumdeq_wf,
all_wf,
iff_wf,
equal_wf,
assert_wf,
deq_wf,
squash_wf,
sq_stable__all,
sq_stable__iff,
sq_stable__equal,
sq_stable_from_decidable,
decidable__assert,
assert_witness,
safe-assert-deq,
subtype_base_sq,
int_subtype_base,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaFormation,
because_Cache,
unionEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
universeEquality,
setElimination,
rename,
independent_functionElimination,
dependent_functionElimination,
productElimination,
independent_pairEquality,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
independent_pairFormation,
independent_isectElimination,
applyLambdaEquality,
inlEquality,
natural_numberEquality,
instantiate,
intEquality,
voidElimination,
promote_hyp,
inrEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[a:EqDecider(A)]. \mforall{}[b:EqDecider(B)]. (union-deq(A;B;a;b) \mmember{} EqDecider(A + B))
Date html generated:
2017_04_14-AM-07_39_20
Last ObjectModification:
2017_02_27-PM-03_10_59
Theory : equality!deciders
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