Nuprl Lemma : filter_sublist
∀[T:Type]. ∀P:T ⟶ 𝔹. ∀L_1,L_2:T List. (L_1 ⊆ L_2 ⇒ filter(P;L_1) ⊆ filter(P;L_2))
Proof
Definitions occuring in Statement :
sublist: L1 ⊆ L2,
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
nil: [],
it: ⋅,
top: Top,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B
Lemmas referenced :
list_induction,
all_wf,
list_wf,
sublist_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
nil_wf,
filter_nil_lemma,
filter_cons_lemma,
eqtt_to_assert,
nil-sublist,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
cons_wf,
sublist_nil,
ifthenelse_wf,
cons_sublist_nil,
cons_sublist_cons,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
uiff_transitivity,
assert_of_bnot,
sublist_transitivity,
sublist_weakening,
sublist_tl2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
functionEquality,
applyEquality,
because_Cache,
setEquality,
independent_isectElimination,
setElimination,
rename,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
universeEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality,
inlFormation,
independent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L$_{1}$,L$_{2}$:T List. (L$_\000C{1}$ \msubseteq{} L$_{2}$ {}\mRightarrow{} filter(P;L$_{1}$) \msubseteq{} filter(P;L\000C$_{2}$))
Date html generated:
2019_06_20-PM-01_24_23
Last ObjectModification:
2018_09_17-PM-05_53_42
Theory : list_1
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