Nuprl Lemma : fpf-join-range
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[df:x:A fp-> Type]. ∀[f:x:A fp-> df(x)?Top]. ∀[dg:x:A fp-> Type].
∀[g:x:A fp-> dg(x)?Top].
(f ⊕ g ∈ x:A fp-> df ⊕ dg(x)?Top) supposing
((∀x:A. ((↑x ∈ dom(g)) ⇒ (↑x ∈ dom(dg)))) and
(∀x:A. ((↑x ∈ dom(f)) ⇒ (↑x ∈ dom(df)))) and
df || dg)
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-compatible: f || g,
fpf-cap: f(x)?z,
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
top: Top,
prop: ℙ,
fpf-join: f ⊕ g,
fpf: a:A fp-> B[a],
pi1: fst(t),
fpf-dom: x ∈ dom(f),
fpf-cap: f(x)?z,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
false: False,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
true: True,
squash: ↓T,
cand: A c∧ B,
fpf-compatible: f || g
Lemmas referenced :
istype-universe,
assert_wf,
fpf-dom_wf,
subtype-fpf2,
fpf-cap_wf,
top_wf,
istype-void,
fpf-compatible_wf,
fpf_wf,
deq_wf,
deq-member_wf,
bnot_wf,
l_member_wf,
filter_wf5,
append_wf,
not_wf,
equal-wf-T-base,
bool_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf,
fpf-ap_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
fpf-join_wf,
subtype_rel-equal,
member_filter_2,
assert-deq-member,
member_append,
iff_weakening_equal,
fpf-join-ap,
true_wf,
squash_wf,
subtype_rel_wf,
subtype_rel_self,
ext-eq_weakening,
subtype_rel_weakening
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
functionIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
universeIsType,
applyEquality,
lambdaEquality_alt,
instantiate,
cumulativity,
universeEquality,
inhabitedIsType,
because_Cache,
independent_isectElimination,
lambdaFormation_alt,
isect_memberEquality_alt,
voidElimination,
equalityTransitivity,
equalitySymmetry,
isect_memberFormation_alt,
axiomEquality,
dependent_pairEquality_alt,
setEquality,
rename,
setElimination,
lambdaFormation,
lambdaEquality,
productElimination,
baseClosed,
voidEquality,
isect_memberEquality,
unionElimination,
equalityElimination,
independent_functionElimination,
dependent_functionElimination,
promote_hyp,
dependent_pairFormation,
imageMemberEquality,
natural_numberEquality,
imageElimination,
independent_pairFormation,
functionEquality
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[df:x:A fp-> Type]. \mforall{}[f:x:A fp-> df(x)?Top]. \mforall{}[dg:x:A fp-> Type].
\mforall{}[g:x:A fp-> dg(x)?Top].
(f \moplus{} g \mmember{} x:A fp-> df \moplus{} dg(x)?Top) supposing
((\mforall{}x:A. ((\muparrow{}x \mmember{} dom(g)) {}\mRightarrow{} (\muparrow{}x \mmember{} dom(dg)))) and
(\mforall{}x:A. ((\muparrow{}x \mmember{} dom(f)) {}\mRightarrow{} (\muparrow{}x \mmember{} dom(df)))) and
df || dg)
Date html generated:
2019_10_16-AM-11_25_33
Last ObjectModification:
2018_10_10-PM-01_06_52
Theory : finite!partial!functions
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