Nuprl Lemma : iseg_filter2
∀[T:Type]
∀L_1,L_2:T List. ∀P:{x:T| (x ∈ L_1)} ⟶ 𝔹.
(L_2 ≤ filter(P;L_1) ⇒ (∃L_3:T List. (L_3 ≤ L_1 ∧ (L_2 = filter(P;L_3) ∈ (T List)))))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
l_member: (x ∈ l),
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
istype: istype(T),
exists: ∃x:A. B[x],
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
nil: [],
it: ⋅,
guard: {T},
respects-equality: respects-equality(S;T),
cand: A c∧ B,
not: ¬A,
false: False,
top: Top,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
decidable: Dec(P),
or: P ∨ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
bool: 𝔹,
unit: Unit,
squash: ↓T,
true: True,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
list_induction,
all_wf,
list_wf,
l_member_wf,
bool_wf,
iseg_wf,
filter_wf5,
exists_wf,
equal_wf,
subtype_rel_dep_function,
subtype_rel_sets_simple,
iseg_member,
nil_wf,
cons_wf,
list-subtype,
subtype_rel_list_set,
respects-equality-list,
subtype-respects-equality,
istype-universe,
iseg_weakening,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
filter_nil_lemma,
istype-void,
iseg_nil,
assert_of_null,
equal-wf-T-base,
decidable__assert,
null_wf,
nil_iseg,
filter_cons_lemma,
cons_member,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
cons_iseg_not_null,
cons_iseg,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
assert_elim,
bnot_wf,
bfalse_wf,
bool_cases_sqequal,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
hypothesis,
functionEquality,
setEquality,
because_Cache,
Error :lambdaFormation_alt,
Error :universeIsType,
setElimination,
rename,
productEquality,
applyEquality,
Error :setIsType,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
Error :inhabitedIsType,
Error :functionIsType,
Error :productIsType,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
Error :dependent_set_memberEquality_alt,
instantiate,
universeEquality,
Error :dependent_pairFormation_alt,
independent_pairFormation,
voidElimination,
Error :isect_memberEquality_alt,
productElimination,
hyp_replacement,
applyLambdaEquality,
baseClosed,
unionElimination,
sqequalBase,
Error :inlFormation_alt,
cumulativity,
Error :inrFormation_alt,
equalityElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
Error :equalityIsType2,
baseApply,
closedConclusion,
Error :equalityIsType4,
Error :equalityIsType1,
promote_hyp
Latex:
\mforall{}[T:Type]
\mforall{}L$_{1}$,L$_{2}$:T List. \mforall{}P:\{x:T| (x \mmember{} L$_{1\mbackslash{}f\000Cf7d$)\} {}\mrightarrow{} \mBbbB{}.
(L$_{2}$ \mleq{} filter(P;L$_{1}$) {}\mRightarrow{} (\mexists{}L$_{3}\000C$:T List. (L$_{3}$ \mleq{} L$_{1}$ \mwedge{} (L$_{2}\mbackslash{}f\000Cf24 = filter(P;L$_{3}$)))))
Date html generated:
2019_06_20-PM-01_29_38
Last ObjectModification:
2018_11_23-PM-04_32_33
Theory : list_1
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