Nuprl Lemma : sublist_filter_set_type
∀[T:Type]. ∀L1,L2:T List. ∀P:T ⟶ 𝔹. (L2 ⊆ L1 ⇒ L2 ⊆ filter(P;L1) supposing (∀x∈L2.↑(P x)))
Proof
Definitions occuring in Statement :
sublist: L1 ⊆ L2,
l_all: (∀x∈L.P[x]),
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sublist: L1 ⊆ L2,
cand: A c∧ B,
ge: i ≥ j ,
nat: ℕ,
true: True
Lemmas referenced :
assert_witness,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
list_set_type,
assert_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
l_all_filter,
l_all_wf,
sublist_wf,
list_wf,
sublist_filter,
equal_wf,
squash_wf,
true_wf,
subtype_rel_list,
non_neg_length,
lelt_wf,
length_wf_nat,
nat_properties,
iff_weakening_equal,
increasing_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
functionExtensionality,
cumulativity,
because_Cache,
setElimination,
rename,
hypothesis,
independent_isectElimination,
natural_numberEquality,
productElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
imageElimination,
setEquality,
functionEquality,
universeEquality,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
productEquality
Latex:
\mforall{}[T:Type]. \mforall{}L1,L2:T List. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. (L2 \msubseteq{} L1 {}\mRightarrow{} L2 \msubseteq{} filter(P;L1) supposing (\mforall{}x\mmember{}L2.\muparrow{}(P x)))
Date html generated:
2019_06_20-PM-01_25_25
Last ObjectModification:
2018_09_17-PM-06_53_49
Theory : list_1
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