Nuprl Lemma : filter_iseg2
∀[T:Type]. ∀L2,L1:T List. ∀P:{x:T| (x ∈ L2)} ⟶ 𝔹. (L1 ≤ L2 ⇒ filter(P;L1) ≤ filter(P;L2))
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],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
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],
prop: ℙ,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
istype: istype(T),
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
cons: [a / b],
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
cand: A c∧ B,
squash: ↓T,
not: ¬A
Lemmas referenced :
list_induction,
list_wf,
l_member_wf,
bool_wf,
iseg_wf,
filter_wf5,
subtype_rel_dep_function,
subtype_rel_sets_simple,
iseg_member,
filter_nil_lemma,
istype-void,
nil_wf,
filter_cons_lemma,
cons_wf,
istype-universe,
iseg_nil,
assert_of_null,
sqequal-nil,
nil_iseg,
cons_member,
eqtt_to_assert,
list-cases,
product_subtype_list,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
cons_iseg,
equal_wf,
assert_elim,
bfalse_wf,
bnot_wf,
btrue_neq_bfalse,
not_assert_elim
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
functionEquality,
hypothesis,
setEquality,
applyEquality,
Error :setIsType,
Error :universeIsType,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
Error :functionIsType,
rename,
Error :inhabitedIsType,
instantiate,
universeEquality,
productElimination,
equalityTransitivity,
equalitySymmetry,
Error :inlFormation_alt,
Error :dependent_set_memberEquality_alt,
unionElimination,
equalityElimination,
Error :inrFormation_alt,
Error :equalityIstype,
promote_hyp,
hypothesis_subsumption,
Error :dependent_pairFormation_alt,
cumulativity,
hyp_replacement,
applyLambdaEquality,
independent_pairFormation,
setElimination,
imageMemberEquality,
baseClosed,
imageElimination,
Error :productIsType
Latex:
\mforall{}[T:Type]. \mforall{}L2,L1:T List. \mforall{}P:\{x:T| (x \mmember{} L2)\} {}\mrightarrow{} \mBbbB{}. (L1 \mleq{} L2 {}\mRightarrow{} filter(P;L1) \mleq{} filter(P;L2))
Date html generated:
2019_06_20-PM-01_29_10
Last ObjectModification:
2019_01_17-PM-04_24_47
Theory : list_1
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