Nuprl Lemma : l_disjoint-fpf-dom
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[f:a:A fp-> Top]. ∀[L:A List].
uiff(l_disjoint(A;fst(f);L);∀[a:A]. ¬(a ∈ L) supposing ↑a ∈ dom(f))
Proof
Definitions occuring in Statement :
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
l_disjoint: l_disjoint(T;l1;l2),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
assert: ↑b,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
pi1: fst(t),
not: ¬A,
universe: Type
Definitions unfolded in proof :
fpf-dom: x ∈ dom(f),
l_disjoint: l_disjoint(T;l1;l2),
fpf: a:A fp-> B[a],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
top: Top,
iff: P ⇐⇒ Q,
cand: A c∧ B,
rev_implies: P ⇐ Q
Lemmas referenced :
l_member_wf,
assert_wf,
deq-member_wf,
pi1_wf_top,
list_wf,
subtype_rel_product,
top_wf,
all_wf,
not_wf,
and_wf,
uall_wf,
isect_wf,
deq_wf,
assert-deq-member
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
lemma_by_obid,
isectElimination,
hypothesisEquality,
lambdaEquality,
dependent_functionElimination,
because_Cache,
applyEquality,
functionEquality,
setEquality,
independent_isectElimination,
isect_memberEquality,
voidEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_pairEquality,
productEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f:a:A fp-> Top]. \mforall{}[L:A List].
uiff(l\_disjoint(A;fst(f);L);\mforall{}[a:A]. \mneg{}(a \mmember{} L) supposing \muparrow{}a \mmember{} dom(f))
Date html generated:
2018_05_21-PM-09_31_39
Last ObjectModification:
2018_02_09-AM-10_26_36
Theory : finite!partial!functions
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