Nuprl Lemma : deq-member-length-filter
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[L:A List]. ∀[x:A]. (x ∈b L ~ 0 <z ||filter(λy.(eq y x);L)||)
Proof
Definitions occuring in Statement :
length: ||as||,
deq-member: x ∈b L,
filter: filter(P;l),
list: T List,
deq: EqDecider(T),
lt_int: i <z j,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
natural_number: $n,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
lt_int: i <z j,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
iff: P ⇐⇒ Q,
nat_plus: ℕ+,
true: True,
rev_implies: P ⇐ Q,
assert: ↑b,
bfalse: ff,
bnot: ¬bb,
bor: p ∨bq
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
deq_member_nil_lemma,
filter_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
deq_member_cons_lemma,
filter_cons_lemma,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
testxxx_lemma,
length_of_cons_lemma,
bool_subtype_base,
iff_imp_equal_bool,
btrue_wf,
lt_int_wf,
length_wf,
filter_wf5,
l_member_wf,
add_nat_plus,
length_wf_nat,
eqof_wf,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
false_wf,
true_wf,
assert_of_lt_int,
assert_wf,
iff_wf,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
list_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
cumulativity,
applyEquality,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
equalityElimination,
setEquality,
imageMemberEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
addLevel,
impliesFunctionality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[L:A List]. \mforall{}[x:A]. (x \mmember{}\msubb{} L \msim{} 0 <z ||filter(\mlambda{}y.(eq y x);L)||)
Date html generated:
2017_09_29-PM-06_04_23
Last ObjectModification:
2017_07_26-PM-02_53_03
Theory : decidable!equality
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