Nuprl Lemma : assert-fpf-is-empty
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[f:x:A fp-> B[x]]. uiff(↑fpf-is-empty(f);f = ⊗ ∈ x:A fp-> B[x])
Proof
Definitions occuring in Statement :
fpf-is-empty: fpf-is-empty(f),
fpf-empty: ⊗,
fpf: a:A fp-> B[a],
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
fpf: a:A fp-> B[a],
fpf-is-empty: fpf-is-empty(f),
pi1: fst(t),
fpf-empty: ⊗,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
top: Top,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
not: ¬A,
false: False
Lemmas referenced :
eq_int_wf,
length_wf,
assert_of_eq_int,
equal_wf,
squash_wf,
true_wf,
length_of_null_list,
nil_wf,
and_wf,
list_wf,
l_member_wf,
pi1_wf_top,
subtype_rel_product,
top_wf,
iff_weakening_equal,
assert_wf,
assert_witness,
equal-wf-T-base,
fpf_wf,
length_zero,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
null_wf3,
subtype_rel_list,
btrue_neq_bfalse,
set_wf
Rules used in proof :
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
natural_numberEquality,
independent_isectElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
intEquality,
dependent_set_memberEquality,
independent_pairFormation,
productEquality,
functionEquality,
setEquality,
setElimination,
rename,
applyLambdaEquality,
functionExtensionality,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination,
isect_memberFormation,
independent_pairEquality,
axiomEquality,
dependent_pairEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:x:A fp-> B[x]]. uiff(\muparrow{}fpf-is-empty(f);f = \motimes{})
Date html generated:
2018_05_21-PM-09_17_43
Last ObjectModification:
2018_02_09-AM-10_16_42
Theory : finite!partial!functions
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