Nuprl Lemma : product-deq_wf
∀[A,B:Type]. ∀[a:EqDecider(A)]. ∀[b:EqDecider(B)]. (product-deq(A;B;a;b) ∈ EqDecider(A × B))
Proof
Definitions occuring in Statement :
product-deq: product-deq(A;B;a;b),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
deq: EqDecider(T),
product-deq: product-deq(A;B;a;b),
proddeq: proddeq(a;b),
all: ∀x:A. B[x],
pi1: fst(t),
pi2: snd(t),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
eqof: eqof(d),
uiff: uiff(P;Q),
uimplies: b supposing a,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A
Lemmas referenced :
istype-universe,
assert_wf,
deq_wf,
eqof_wf,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
equal_wf,
bfalse_wf,
iff_transitivity,
assert_of_band
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
Error :dependent_set_memberEquality_alt,
Error :lambdaEquality_alt,
because_Cache,
Error :inhabitedIsType,
hypothesisEquality,
Error :productIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
Error :lambdaFormation_alt,
productElimination,
Error :functionIsType,
Error :equalityIsType1,
Error :universeIsType,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :isect_memberEquality_alt,
universeEquality,
independent_pairFormation,
applyLambdaEquality,
independent_pairEquality,
unionElimination,
equalityElimination,
setElimination,
rename,
independent_isectElimination,
Error :dependent_pairFormation_alt,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
productEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[a:EqDecider(A)]. \mforall{}[b:EqDecider(B)]. (product-deq(A;B;a;b) \mmember{} EqDecider(A \mtimes{} B))
Date html generated:
2019_06_20-PM-00_32_01
Last ObjectModification:
2018_10_06-AM-11_20_17
Theory : equality!deciders
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