Nuprl Lemma : decidable__equal_union
∀[A,B:Type]. ((∀x,y:A. Dec(x = y ∈ A)) ⇒ (∀u,v:B. Dec(u = v ∈ B)) ⇒ (∀x,y:A + B. Dec(x = y ∈ (A + B))))
Proof
Definitions occuring in Statement :
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
not: ¬A,
false: False,
uimplies: b supposing a,
sq_type: SQType(T),
true: True
Lemmas referenced :
all_wf,
decidable_wf,
equal_wf,
not_wf,
subtype_base_sq,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
unionElimination,
thin,
unionEquality,
cumulativity,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
sqequalRule,
lambdaEquality,
hypothesis,
universeEquality,
dependent_functionElimination,
inlFormation,
inlEquality,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
inrFormation,
independent_functionElimination,
voidElimination,
because_Cache,
applyEquality,
natural_numberEquality,
instantiate,
intEquality,
independent_isectElimination,
equalityTransitivity,
promote_hyp,
inrEquality
Latex:
\mforall{}[A,B:Type]. ((\mforall{}x,y:A. Dec(x = y)) {}\mRightarrow{} (\mforall{}u,v:B. Dec(u = v)) {}\mRightarrow{} (\mforall{}x,y:A + B. Dec(x = y)))
Date html generated:
2016_10_21-AM-09_36_05
Last ObjectModification:
2016_07_12-AM-05_00_04
Theory : union
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