Nuprl Lemma : ifthenelse_wf-partial-base
∀T:Type
(value-type(T) ⇒ (∀c1,c2:partial(T). ∀b:Base. if b then c1 else c2 fi ∈ partial(T) supposing ¬is-exception(b)))
Proof
Definitions occuring in Statement :
partial: partial(T),
value-type: value-type(T),
is-exception: is-exception(t),
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
base: Base,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
base-partial: base-partial(T),
ifthenelse: if b then t else f fi ,
not: ¬A,
false: False,
squash: ↓T,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
partial: partial(T),
subtype_rel: A ⊆r B,
true: True,
per-partial: per-partial(T;x;y),
has-value: (a)↓,
uiff: uiff(P;Q),
cand: A c∧ B
Lemmas referenced :
not_wf,
is-exception_wf,
base_wf,
partial_wf,
value-type_wf,
equal-wf-base,
has-value_wf_base,
equal_wf,
has-value_wf-partial,
termination,
top_wf,
has-value-ifthenelse-type,
partial-not-exception,
per-partial-equiv_rel,
per-partial_wf,
base-partial_wf,
subtype_quotient,
quotient-member-eq,
true_wf,
squash_wf,
member_wf,
and_wf,
termination-equality-base,
istype-universe,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
Error :inhabitedIsType,
universeEquality,
because_Cache,
isectEquality,
productEquality,
independent_pairFormation,
baseClosed,
closedConclusion,
baseApply,
dependent_set_memberEquality,
lambdaFormation,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
unionEquality,
independent_isectElimination,
voidElimination,
imageMemberEquality,
imageElimination,
decideExceptionCases,
pointwiseFunctionality,
rename,
setElimination,
lambdaEquality,
applyEquality,
natural_numberEquality,
axiomSqleEquality,
productElimination,
applyLambdaEquality,
hyp_replacement,
instantiate,
Error :lambdaEquality_alt,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
Error :equalityIsType3,
Error :equalityIsType1
Latex:
\mforall{}T:Type
(value-type(T)
{}\mRightarrow{} (\mforall{}c1,c2:partial(T). \mforall{}b:Base.
if b then c1 else c2 fi \mmember{} partial(T) supposing \mneg{}is-exception(b)))
Date html generated:
2019_06_20-PM-00_34_34
Last ObjectModification:
2018_10_06-PM-03_44_00
Theory : partial_1
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