Nuprl Lemma : ifthenelse_wf-partial
∀T:Type. (value-type(T) ⇒ (∀c1,c2:partial(T). ∀b:partial(𝔹). (if b then c1 else c2 fi ∈ partial(T))))
Proof
Definitions occuring in Statement :
partial: partial(T),
value-type: value-type(T),
ifthenelse: if b then t else f fi ,
bool: 𝔹,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
partial: partial(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
base-partial: base-partial(T),
uimplies: b supposing a,
bool: 𝔹,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
per-partial: per-partial(T;x;y),
uiff: uiff(P;Q),
sq_type: SQType(T),
guard: {T},
not: ¬A,
false: False,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
base-partial_wf,
bool_wf,
per-partial_wf,
equal_wf,
equal-wf-base,
partial_wf,
value-type_wf,
has-value_wf-partial,
union-value-type,
unit_wf2,
subtype_quotient,
per-partial-equiv_rel,
is-exception_wf,
subtype_base_sq,
bool_subtype_base,
base-partial-not-exception,
sqle_wf_base,
squash_wf,
true_wf,
base_wf,
iff_weakening_equal,
ifthenelse_wf-partial-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
because_Cache,
sqequalRule,
pertypeElimination,
productElimination,
thin,
equalityTransitivity,
hypothesis,
equalitySymmetry,
introduction,
extract_by_obid,
isectElimination,
rename,
setElimination,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
productEquality,
cumulativity,
universeEquality,
sqequalSqle,
divergentSqle,
independent_isectElimination,
applyEquality,
lambdaEquality,
instantiate,
sqleReflexivity,
voidElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}T:Type
(value-type(T) {}\mRightarrow{} (\mforall{}c1,c2:partial(T). \mforall{}b:partial(\mBbbB{}). (if b then c1 else c2 fi \mmember{} partial(T))))
Date html generated:
2017_04_14-AM-07_41_08
Last ObjectModification:
2017_02_27-PM-03_13_10
Theory : partial_1
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