Nuprl Lemma : accum_filter_rel_nil
∀[T,A:Type]. ∀[a,b:A]. ∀[P,f:Top]. uiff(b = accum(z,x.f[z;x],a,{x∈[]|P[x]});b = a ∈ A)
Proof
Definitions occuring in Statement :
accum_filter_rel: b = accum(z,x.f[z; x],a,{x∈X|P[x]}),
nil: [],
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s1;s2],
so_apply: x[s],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
accum_filter_rel: b = accum(z,x.f[z; x],a,{x∈X|P[x]}),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
top: Top,
or: P ∨ Q,
cons: [a / b],
subtype_rel: A ⊆r B
Lemmas referenced :
accum_filter_rel_wf,
nil_wf,
l_member_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
sublist_wf,
all_wf,
iff_wf,
list_wf,
equal_wf,
top_wf,
nil-sublist,
list_accum_nil_lemma,
list-cases,
product_subtype_list,
list_accum_cons_lemma,
cons_wf,
sublist_nil,
and_wf,
null_wf3,
subtype_rel_list,
null_cons_lemma,
bfalse_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
lambdaFormation,
setElimination,
rename,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
setEquality,
dependent_functionElimination,
axiomEquality,
productEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
universeEquality,
voidEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
dependent_set_memberEquality,
applyEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[a,b:A]. \mforall{}[P,f:Top]. uiff(b = accum(z,x.f[z;x],a,\{x\mmember{}[]|P[x]\});b = a)
Date html generated:
2016_05_15-PM-04_33_01
Last ObjectModification:
2015_12_27-PM-02_48_34
Theory : general
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