Nuprl Lemma : bunion-valueall-type
∀[A,B:Type]. (valueall-type(A ⋃ B)) supposing (valueall-type(B) and valueall-type(A))
Proof
Definitions occuring in Statement :
valueall-type: valueall-type(T),
b-union: A ⋃ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
b-union: A ⋃ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
valueall-type: valueall-type(T),
has-value: (a)↓
Lemmas referenced :
tunion-valueall-type,
bool_wf,
ifthenelse_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-base,
b-union_wf,
base_wf,
valueall-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
lambdaEquality,
instantiate,
hypothesisEquality,
universeEquality,
cumulativity,
independent_isectElimination,
lambdaFormation,
unionElimination,
equalityElimination,
because_Cache,
productElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
isect_memberEquality,
axiomSqleEquality
Latex:
\mforall{}[A,B:Type]. (valueall-type(A \mcup{} B)) supposing (valueall-type(B) and valueall-type(A))
Date html generated:
2017_04_14-AM-07_36_41
Last ObjectModification:
2017_02_27-PM-03_08_58
Theory : subtype_1
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