Nuprl Lemma : apply-partial
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[f:partial(a:A ⟶ B[a])]. ∀[a:A]. f a ∈ partial(B[a]) supposing value-type(B[a])
Proof
Definitions occuring in Statement :
partial: partial(T),
value-type: value-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
guard: {T},
so_lambda: λ2x.t[x],
uimplies: b supposing a,
top: Top,
prop: ℙ,
and: P ∧ Q,
exists: ∃x:A. B[x],
squash: ↓T,
has-value: (a)↓,
not: ¬A,
false: False,
true: True,
iff: P ⇐⇒ Q,
cand: A c∧ B,
rev_implies: P ⇐ Q
Lemmas referenced :
partial_wf,
pair-eta,
subtype_rel_product,
top_wf,
pi2_wf,
equal_wf,
pi1_wf,
partial-not-exception,
termination,
has-value_wf-partial,
function-value-type,
value-type_wf,
base-member-partial,
has-value_wf_base,
exception-not-value,
value-type-has-value,
is-exception_wf,
and_wf,
member_wf,
base-equal-partial,
termination-equality-base,
squash_wf,
true_wf,
subtype_rel_self,
subtype_rel_wf,
iff_weakening_equal,
equal-wf-base-T
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairEquality,
hypothesisEquality,
thin,
productEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
functionEquality,
cumulativity,
applyEquality,
functionExtensionality,
hypothesis,
lambdaFormation,
pointwiseFunctionality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
axiomEquality,
productElimination,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
dependent_pairFormation,
imageMemberEquality,
baseClosed,
baseApply,
closedConclusion,
callbyvalueApply,
applyExceptionCases,
imageElimination,
isectEquality,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
instantiate,
universeEquality,
natural_numberEquality,
hyp_replacement,
equalityElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:partial(a:A {}\mrightarrow{} B[a])]. \mforall{}[a:A].
f a \mmember{} partial(B[a]) supposing value-type(B[a])
Date html generated:
2017_04_14-AM-07_40_36
Last ObjectModification:
2017_02_27-PM-03_13_07
Theory : partial_1
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