Nuprl Lemma : l_disjoint-fpf-join-dom
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[f,g:a:A fp-> Top]. ∀[L:A List].
uiff(l_disjoint(A;fst(f ⊕ g);L);l_disjoint(A;fst(f);L) ∧ l_disjoint(A;fst(g);L))
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf: a:A fp-> B[a],
l_disjoint: l_disjoint(T;l1;l2),
list: T List,
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
top: Top,
pi1: fst(t),
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
prop: ℙ,
guard: {T},
l_disjoint: l_disjoint(T;l1;l2),
fpf: a:A fp-> B[a],
top: Top
Lemmas referenced :
l_disjoint-fpf-dom,
fpf-join_wf,
top_wf,
fpf-join-dom,
assert_wf,
fpf-dom_wf,
l_member_wf,
pi1_wf_top,
list_wf,
l_disjoint_wf,
fpf_wf,
equal_wf,
deq_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
productElimination,
because_Cache,
cumulativity,
sqequalRule,
lambdaEquality,
hypothesis,
independent_pairFormation,
isect_memberFormation,
independent_isectElimination,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
inlFormation,
voidElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
inrFormation,
independent_pairEquality,
productEquality,
voidEquality,
universeEquality,
unionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f,g:a:A fp-> Top]. \mforall{}[L:A List].
uiff(l\_disjoint(A;fst(f \moplus{} g);L);l\_disjoint(A;fst(f);L) \mwedge{} l\_disjoint(A;fst(g);L))
Date html generated:
2018_05_21-PM-09_31_43
Last ObjectModification:
2018_02_09-AM-10_26_39
Theory : finite!partial!functions
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