Nuprl Lemma : fpf-join-dom
∀[A:Type]. ∀[B:A ⟶ Type].
∀eq:EqDecider(A). ∀f,g:a:A fp-> B[a]. ∀x:A. (↑x ∈ dom(f ⊕ g) ⇐⇒ (↑x ∈ dom(f)) ∨ (↑x ∈ dom(g)))
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fpf-dom: x ∈ dom(f),
fpf-join: f ⊕ g,
fpf: a:A fp-> B[a],
pi1: fst(t),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
bfalse: ff,
or: P ∨ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
false: False,
not: ¬A,
cand: A c∧ B,
true: True,
so_apply: x[s]
Lemmas referenced :
deq-member_wf,
pi1_wf_top,
list_wf,
bool_wf,
eqtt_to_assert,
assert-deq-member,
l_member_wf,
or_wf,
false_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
true_wf,
member_filter,
bnot_wf,
assert_wf,
filter_wf5,
iff_wf,
member_append,
append_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,
hypothesis,
productElimination,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
inlFormation,
productEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
inrFormation,
natural_numberEquality,
addLevel,
orFunctionality,
lambdaEquality,
setElimination,
rename,
setEquality,
impliesFunctionality,
orLevelFunctionality,
functionEquality,
applyEquality,
functionExtensionality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}eq:EqDecider(A). \mforall{}f,g:a:A fp-> B[a]. \mforall{}x:A. (\muparrow{}x \mmember{} dom(f \moplus{} g) \mLeftarrow{}{}\mRightarrow{} (\muparrow{}x \mmember{} dom(f)) \mvee{} (\muparrow{}x \mmember{} dom(g)))
Date html generated:
2018_05_21-PM-09_21_26
Last ObjectModification:
2018_02_09-AM-10_18_16
Theory : finite!partial!functions
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