Nuprl Lemma : iseg_filter_last
∀[T:Type]
∀P:T ⟶ 𝔹. ∀L_1,L_2:T List.
(0 < ||L_2||
⇒ L_2 ≤ filter(P;L_1)
⇒ (∃L_3:T List. (L_3 ≤ L_1 ∧ (L_2 = filter(P;L_3) ∈ (T List)) ∧ 0 < ||L_3|| ∧ (last(L_2) = last(L_3) ∈ T))))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
last: last(L),
length: ||as||,
filter: filter(P;l),
list: T List,
bool: 𝔹,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
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),
exists: ∃x:A. B[x],
and: P ∧ Q,
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
cons: [a / b],
bfalse: ff,
not: ¬A,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
iff: P ⇐⇒ Q,
top: Top,
length: ||as||,
list_ind: list_ind,
nil: [],
decidable: Dec(P),
cand: A c∧ B,
ge: i ≥ j ,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
true: True,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
iseg: l1 ≤ l2,
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3]
Lemmas referenced :
list_induction,
list_wf,
less_than_wf,
length_wf,
iseg_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
equal_wf,
last_wf,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
istype-void,
filter_nil_lemma,
nil_wf,
istype-less_than,
filter_cons_lemma,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
istype-universe,
iseg_nil,
cons_iseg,
decidable__lt,
cons_wf,
non_neg_length,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
squash_wf,
true_wf,
last_cons,
assert_elim,
null_wf,
bfalse_wf,
btrue_neq_bfalse,
istype-assert,
subtype_rel_self,
iff_weakening_equal,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
add-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
false_wf,
list_decomp,
reduce_tl_cons_lemma,
reduce_hd_cons_lemma,
filter_trivial,
l_all_cons,
assert_wf,
l_all_nil,
not_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
append_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
functionEquality,
hypothesis,
natural_numberEquality,
applyEquality,
because_Cache,
universeIsType,
setEquality,
setIsType,
independent_isectElimination,
setElimination,
rename,
productEquality,
applyLambdaEquality,
closedConclusion,
dependent_functionElimination,
unionElimination,
imageElimination,
productElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
Error :memTop,
independent_functionElimination,
inhabitedIsType,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation_alt,
equalityIstype,
instantiate,
cumulativity,
functionIsType,
productIsType,
universeEquality,
isect_memberEquality_alt,
independent_pairFormation,
addEquality,
approximateComputation,
int_eqEquality,
imageMemberEquality,
baseClosed,
equalityIsType1,
dependent_set_memberEquality_alt,
pointwiseFunctionality,
baseApply
Latex:
\mforall{}[T:Type]
\mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L$_{1}$,L$_{2}$:T List.
(0 < ||L$_{2}$||
{}\mRightarrow{} L$_{2}$ \mleq{} filter(P;L$_{1}$)
{}\mRightarrow{} (\mexists{}L$_{3}$:T List. (L$_{3}$ \mleq{} L$_{1}\mbackslash{}f\000Cf24 \mwedge{} (L$_{2}$ = filter(P;L$_{3}$)) \mwedge{} 0 < ||L$_{\000C3}$|| \mwedge{} (last(L$_{2}$) = last(L$_{3}$)))))
Date html generated:
2020_05_19-PM-09_42_41
Last ObjectModification:
2019_12_31-PM-00_12_27
Theory : list_1
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