Nuprl Lemma : respects-equality-union
∀[T1,T2,S1,S2:Type].  (respects-equality(S1;T1) 
⇒ respects-equality(S2;T2) 
⇒ respects-equality(S1 + S2;T1 + T2))
Proof
Definitions occuring in Statement : 
respects-equality: respects-equality(S;T)
, 
uall: ∀[x:A]. B[x]
, 
implies: P 
⇒ Q
, 
union: left + right
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
implies: P 
⇒ Q
, 
respects-equality: respects-equality(S;T)
, 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
uimplies: b supposing a
, 
top: Top
, 
or: P ∨ Q
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
outl: outl(x)
, 
isl: isl(x)
, 
assert: ↑b
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
sq_type: SQType(T)
, 
true: True
, 
false: False
, 
outr: outr(x)
, 
bnot: ¬bb
, 
btrue: tt
Lemmas referenced : 
union-eta, 
subtype_rel_union, 
top_wf, 
istype-void, 
outl_wf, 
subtype_base_sq, 
int_subtype_base, 
subtype_rel-equal, 
equal_functionality_wrt_subtype_rel2, 
outr_wf, 
istype-base, 
respects-equality_wf, 
istype-universe
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :isect_memberFormation_alt, 
Error :lambdaFormation_alt, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
dependent_functionElimination, 
thin, 
equalityTransitivity, 
hypothesis, 
equalitySymmetry, 
applyEquality, 
isectElimination, 
hypothesisEquality, 
because_Cache, 
independent_isectElimination, 
Error :lambdaEquality_alt, 
Error :isect_memberEquality_alt, 
voidElimination, 
Error :universeIsType, 
sqequalRule, 
unionElimination, 
Error :inlEquality_alt, 
Error :dependent_set_memberEquality_alt, 
independent_pairFormation, 
Error :productIsType, 
Error :equalityIstype, 
baseApply, 
closedConclusion, 
baseClosed, 
sqequalBase, 
applyLambdaEquality, 
setElimination, 
rename, 
productElimination, 
promote_hyp, 
natural_numberEquality, 
instantiate, 
cumulativity, 
intEquality, 
independent_functionElimination, 
unionEquality, 
Error :inrEquality_alt, 
Error :unionIsType, 
Error :inhabitedIsType, 
universeEquality
Latex:
\mforall{}[T1,T2,S1,S2:Type].
    (respects-equality(S1;T1)  {}\mRightarrow{}  respects-equality(S2;T2)  {}\mRightarrow{}  respects-equality(S1  +  S2;T1  +  T2))
Date html generated:
2019_06_20-AM-11_20_02
Last ObjectModification:
2018_11_23-PM-02_15_26
Theory : union
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