Nuprl Lemma : fpf-union-contains
∀[A:Type]. ∀[B:A ⟶ Type].
∀eq:EqDecider(A). ∀f,g:x:A fp-> B[x] List. ∀x:A. ∀R:⋂a:A. ((B[a] List) ⟶ B[a] ⟶ 𝔹). f(x)?[] ⊆ fpf-union(f;g;eq;R;x)
Proof
Definitions occuring in Statement :
fpf-union: fpf-union(f;g;eq;R;x),
fpf-cap: f(x)?z,
fpf: a:A fp-> B[a],
l_contains: A ⊆ B,
nil: [],
list: T List,
deq: EqDecider(T),
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fpf-union: fpf-union(f;g;eq;R;x),
fpf-cap: f(x)?z,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
top: Top,
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 ,
band: p ∧b q,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False
Lemmas referenced :
fpf-dom_wf,
subtype-fpf2,
list_wf,
top_wf,
bool_wf,
eqtt_to_assert,
l_contains_append,
fpf-ap_wf,
filter_wf5,
subtype_rel_dep_function,
l_member_wf,
subtype_rel_self,
set_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
l_contains_weakening,
l_contains_nil,
nil_wf,
fpf_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
applyEquality,
lambdaEquality,
functionExtensionality,
hypothesis,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_functionElimination,
setEquality,
setElimination,
rename,
dependent_pairFormation,
promote_hyp,
instantiate,
independent_functionElimination,
isectEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}eq:EqDecider(A). \mforall{}f,g:x:A fp-> B[x] List. \mforall{}x:A. \mforall{}R:\mcap{}a:A. ((B[a] List) {}\mrightarrow{} B[a] {}\mrightarrow{} \mBbbB{}).
f(x)?[] \msubseteq{} fpf-union(f;g;eq;R;x)
Date html generated:
2018_05_21-PM-09_18_12
Last ObjectModification:
2018_02_09-AM-10_17_02
Theory : finite!partial!functions
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