Nuprl Lemma : fpf-join_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[f,g:a:A fp-> B[a]]. ∀[eq:EqDecider(A)]. (f ⊕ g ∈ a:A fp-> B[a])
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fpf-join: f ⊕ g,
fpf: a:A fp-> B[a],
uall: ∀[x:A]. B[x],
member: t ∈ T,
pi1: fst(t),
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
guard: {T},
fpf-cap: f(x)?z,
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 ,
bfalse: ff,
fpf-dom: x ∈ dom(f),
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
or: P ∨ Q,
false: False
Lemmas referenced :
append_wf,
filter_wf5,
l_member_wf,
pi1_wf_top,
list_wf,
bnot_wf,
fpf-dom_wf,
subtype_rel_product,
top_wf,
subtype_rel_dep_function,
set_wf,
deq_wf,
subtype-fpf2,
bool_wf,
equal-wf-T-base,
assert_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf,
fpf-ap_wf,
member_append,
deq-member_wf,
assert-deq-member,
member_filter
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
dependent_pairEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
productElimination,
because_Cache,
lambdaEquality,
lambdaFormation,
hypothesis,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
applyEquality,
functionEquality,
setEquality,
functionExtensionality,
independent_isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
universeEquality,
baseClosed,
unionElimination,
equalityElimination,
independent_functionElimination,
dependent_functionElimination,
promote_hyp
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f,g:a:A fp-> B[a]]. \mforall{}[eq:EqDecider(A)]. (f \moplus{} g \mmember{} a:A fp-> B[a])
Date html generated:
2018_05_21-PM-09_20_58
Last ObjectModification:
2018_02_09-AM-10_18_02
Theory : finite!partial!functions
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