Nuprl Lemma : fpf-union-join-member
∀[A:Type]
∀eq:EqDecider(A)
∀[B:A ⟶ Type]
∀f,g:a:A fp-> B[a] List. ∀R:⋂a:A. ((B[a] List) ⟶ B[a] ⟶ 𝔹). ∀a:A.
∀x:B[a]. ((x ∈ fpf-union-join(eq;R;f;g)(a)) ⇒ (((↑a ∈ dom(f)) ∧ (x ∈ f(a))) ∨ ((↑a ∈ dom(g)) ∧ (x ∈ g(a)))))
supposing ↑a ∈ dom(fpf-union-join(eq;R;f;g))
Proof
Definitions occuring in Statement :
fpf-union-join: fpf-union-join(eq;R;f;g),
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
prop: ℙ,
top: Top,
fpf-union: fpf-union(f;g;eq;R;x),
fpf-cap: f(x)?z,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
assert: ↑b,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
false: False,
cand: A c∧ B,
true: True,
not: ¬A,
iff: P ⇐⇒ Q
Lemmas referenced :
assert_witness,
fpf-dom_wf,
fpf-union-join_wf,
l_member_wf,
fpf-ap_wf,
list_wf,
bool_wf,
assert_wf,
fpf_wf,
deq_wf,
fpf-union-join-ap,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
false_wf,
subtype-fpf2,
top_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
member_append,
filter_wf5,
subtype_rel_dep_function,
subtype_rel_self,
set_wf,
member_filter,
or_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
hypothesis,
independent_functionElimination,
rename,
isectEquality,
functionEquality,
independent_isectElimination,
universeEquality,
voidElimination,
voidEquality,
unionElimination,
equalityElimination,
productElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
inlFormation,
natural_numberEquality,
independent_pairFormation,
productEquality,
inrFormation,
setEquality,
setElimination,
addLevel,
orFunctionality
Latex:
\mforall{}[A:Type]
\mforall{}eq:EqDecider(A)
\mforall{}[B:A {}\mrightarrow{} Type]
\mforall{}f,g:a:A fp-> B[a] List. \mforall{}R:\mcap{}a:A. ((B[a] List) {}\mrightarrow{} B[a] {}\mrightarrow{} \mBbbB{}). \mforall{}a:A.
\mforall{}x:B[a]
((x \mmember{} fpf-union-join(eq;R;f;g)(a))
{}\mRightarrow{} (((\muparrow{}a \mmember{} dom(f)) \mwedge{} (x \mmember{} f(a))) \mvee{} ((\muparrow{}a \mmember{} dom(g)) \mwedge{} (x \mmember{} g(a)))))
supposing \muparrow{}a \mmember{} dom(fpf-union-join(eq;R;f;g))
Date html generated:
2018_05_21-PM-09_23_33
Last ObjectModification:
2018_02_09-AM-10_19_21
Theory : finite!partial!functions
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