Nuprl Lemma : iseg-mapfilter
∀[T:Type]
∀P:T ⟶ 𝔹. ∀[T':Type]. ∀f:{x:T| ↑(P x)} ⟶ T'. ∀L1,L2:T List. (L1 ≤ L2 ⇒ mapfilter(f;P;L1) ≤ mapfilter(f;P;L2))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
mapfilter: mapfilter(f;P;L),
list: T List,
assert: ↑b,
bool: 𝔹,
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],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
mapfilter: mapfilter(f;P;L),
top: Top,
or: P ∨ Q,
cons: [a / b],
uimplies: b supposing a,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
ge: i ≥ j ,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
squash: ↓T
Lemmas referenced :
list_induction,
all_wf,
list_wf,
iseg_wf,
mapfilter_wf,
assert_wf,
filter_nil_lemma,
map_nil_lemma,
filter_cons_lemma,
bool_wf,
nil_iseg,
nil_wf,
cons_wf,
list-cases,
product_subtype_list,
iseg_length,
length_of_cons_lemma,
length_of_nil_lemma,
eqtt_to_assert,
mapfilter_nil_lemma,
map_cons_lemma,
non_neg_length,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
cons_iseg,
equal-wf-T-base,
bnot_wf,
not_wf,
assert_elim,
not_assert_elim,
and_wf,
btrue_neq_bfalse,
uiff_transitivity,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
functionEquality,
functionExtensionality,
applyEquality,
setEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
because_Cache,
universeEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
independent_isectElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
instantiate,
baseClosed,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
imageMemberEquality,
imageElimination
Latex:
\mforall{}[T:Type]
\mforall{}P:T {}\mrightarrow{} \mBbbB{}
\mforall{}[T':Type]
\mforall{}f:\{x:T| \muparrow{}(P x)\} {}\mrightarrow{} T'. \mforall{}L1,L2:T List. (L1 \mleq{} L2 {}\mRightarrow{} mapfilter(f;P;L1) \mleq{} mapfilter(f;P;L2))
Date html generated:
2017_04_17-AM-08_50_07
Last ObjectModification:
2017_02_27-PM-05_07_04
Theory : list_1
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