Nuprl Lemma : strict-majority-property
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:T List]. ∀[x:T].
uiff(||L|| < 2 * ||filter(λy.(eq y x);L)||;strict-majority(eq;L) = (inl x) ∈ (T?))
Proof
Definitions occuring in Statement :
strict-majority: strict-majority(eq;L),
length: ||as||,
filter: filter(P;l),
list: T List,
deq: EqDecider(T),
less_than: a < b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
unit: Unit,
apply: f a,
lambda: λx.A[x],
inl: inl x,
union: left + right,
multiply: n * m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
prop: ℙ,
deq: EqDecider(T),
let: let,
strict-majority: strict-majority(eq;L),
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
member: t ∈ T,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
le: A ≤ B,
ge: i ≥ j ,
eqof: eqof(d),
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
implies: P ⇒ Q,
pi1: fst(t),
true: True,
less_than': less_than'(a;b),
subtract: n - m,
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
cand: A c∧ B,
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
nat: ℕ,
cons: [a / b],
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
exposed-bfalse: exposed-bfalse,
bfalse: ff,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
no_repeats: no_repeats(T;l),
int_seg: {i..j-},
lelt: i ≤ j < k,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
l_member: (x ∈ l),
nequal: a ≠ b ∈ T ,
isl: isl(x),
outl: outl(x)
Lemmas referenced :
istype-universe,
deq_wf,
list_wf,
member-less_than,
strict-majority_wf,
unit_wf2,
l_member_wf,
filter_wf5,
length_wf,
istype-less_than,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
non_neg_length,
length_of_nil_lemma,
istype-assert,
safe-assert-deq,
assert_wf,
not_wf,
l_all_iff,
filter_is_nil,
decidable__l_member,
decidable-equal-deq,
count-repeats_wf,
pi1_wf,
nat_plus_wf,
member-map,
member-count-repeats1,
member-count-repeats3,
le-add-cancel2,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-associates,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
sq_stable__le,
not-ge-2,
istype-false,
decidable__le,
int_subtype_base,
le_wf,
set_subtype_base,
multiply-is-int-iff,
length_wf_nat,
length_of_cons_lemma,
product_subtype_list,
list-cases,
hd_wf,
equal_wf,
nil_wf,
equal-wf-T-base,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
equal-wf-base-T,
assert_of_null,
eqtt_to_assert,
null_wf,
reduce_hd_cons_lemma,
null_cons_lemma,
pi2_wf,
lt_int_wf,
filter_is_singleton,
no_repeats-count-repeats1,
no_repeats-l_member!,
map-length,
select-map,
subtype_rel_list,
top_wf,
nat_properties,
nat_plus_properties,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
select_wf,
map_wf,
istype-top,
iff_weakening_equal,
map_select,
istype-void,
istype-nat,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
intformeq_wf,
itermMultiply_wf,
satisfiable-full-omega-tt,
decidable__lt,
assert_of_lt_int,
eqof_wf,
less_than_wf,
decidable__equal_int,
sum-count-repeats,
and_wf,
squash_wf,
lelt_wf,
int_seg_wf,
int_seg_properties,
isolate_summand,
neg_assert_of_eq_int,
assert_of_eq_int,
eq_int_wf,
bnot_wf,
uiff_transitivity,
iff_transitivity,
assert_of_bnot,
false_wf,
sum_lower_bound,
int_term_value_add_lemma,
itermAdd_wf,
btrue_neq_bfalse,
bfalse_wf,
isl_wf,
btrue_wf,
bool_cases,
outl_wf,
equal-wf-base,
subtype_rel_product,
member_filter,
cons_member,
nat_wf
Rules used in proof :
universeEquality,
instantiate,
independent_isectElimination,
because_Cache,
isectIsTypeImplies,
axiomEquality,
isect_memberEquality_alt,
independent_pairEquality,
productElimination,
inlEquality_alt,
unionIsType,
equalityIstype,
universeIsType,
inhabitedIsType,
setIsType,
rename,
setElimination,
applyEquality,
lambdaEquality_alt,
natural_numberEquality,
multiplyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesis,
sqequalRule,
independent_pairFormation,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
Error :memTop,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
voidElimination,
applyLambdaEquality,
productIsType,
equalityTransitivity,
dependent_set_memberEquality_alt,
equalitySymmetry,
hyp_replacement,
unionElimination,
lambdaFormation_alt,
dependent_functionElimination,
independent_functionElimination,
productEquality,
minusEquality,
imageMemberEquality,
addEquality,
intEquality,
imageElimination,
hypothesis_subsumption,
cumulativity,
promote_hyp,
baseClosed,
unionEquality,
equalityElimination,
functionIsTypeImplies,
functionIsType,
computeAll,
voidEquality,
isect_memberEquality,
dependent_pairFormation,
setEquality,
lambdaEquality,
lambdaFormation,
functionEquality,
dependent_set_memberEquality,
impliesFunctionality,
inlEquality,
inlFormation
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[L:T List]. \mforall{}[x:T].
uiff(||L|| < 2 * ||filter(\mlambda{}y.(eq y x);L)||;strict-majority(eq;L) = (inl x))
Date html generated:
2020_05_19-PM-09_52_28
Last ObjectModification:
2019_12_26-AM-11_47_50
Theory : decidable!equality
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