Nuprl Lemma : fpf-join-dom-sq
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[f,g:a:A fp-> Top]. ∀[x:A]. (x ∈ dom(f ⊕ g) ~ x ∈ dom(f) ∨bx ∈ 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),
bor: p ∨bq,
uall: ∀[x:A]. B[x],
top: Top,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fpf: a:A fp-> B[a],
fpf-dom: x ∈ dom(f),
fpf-join: f ⊕ g,
pi1: fst(t),
all: ∀x:A. B[x],
prop: ℙ,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
not: ¬A,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
cand: A c∧ B,
false: False
Lemmas referenced :
subtype_base_sq,
bool_wf,
bool_subtype_base,
iff_imp_equal_bool,
deq-member_wf,
append_wf,
filter_wf5,
l_member_wf,
bnot_wf,
bor_wf,
fpf_wf,
top_wf,
deq_wf,
member_filter,
assert_wf,
or_wf,
iff_wf,
member_append,
not_wf,
assert-deq-member,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_bnot,
equal-wf-T-base,
eqtt_to_assert,
eqff_to_assert,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
productElimination,
sqequalRule,
hypothesisEquality,
because_Cache,
lambdaEquality,
lambdaFormation,
setElimination,
rename,
setEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalAxiom,
isect_memberEquality,
universeEquality,
addLevel,
independent_pairFormation,
orFunctionality,
productEquality,
applyEquality,
impliesFunctionality,
andLevelFunctionality,
unionElimination,
inlFormation,
inrFormation,
equalityElimination,
baseClosed,
voidElimination
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f,g:a:A fp-> Top]. \mforall{}[x:A].
(x \mmember{} dom(f \moplus{} g) \msim{} x \mmember{} dom(f) \mvee{}\msubb{}x \mmember{} dom(g))
Date html generated:
2018_05_21-PM-09_21_33
Last ObjectModification:
2018_02_09-AM-10_18_19
Theory : finite!partial!functions
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