Nuprl Lemma : strict-majority-or-max-property
∀t:ℕ. ∀L:ℤ List.
((∀v:ℤ
(strict-majority-or-max(L) = v ∈ ℤ) supposing
(((t + 1) ≤ ||filter(λx.(x =z v);L)||) and
(||L|| = ((2 * t) + 1) ∈ ℤ)))
∧ (strict-majority-or-max(L) ∈ L) supposing ||L|| ≥ 1 )
Proof
Definitions occuring in Statement :
strict-majority-or-max: strict-majority-or-max(L),
l_member: (x ∈ l),
length: ||as||,
filter: filter(P;l),
list: T List,
nat: ℕ,
eq_int: (i =z j),
uimplies: b supposing a,
ge: i ≥ j ,
le: A ≤ B,
all: ∀x:A. B[x],
and: P ∧ Q,
lambda: λx.A[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
strict-majority-or-max: strict-majority-or-max(L),
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
subtype_rel: A ⊆r B,
ge: i ≥ j ,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
int-deq: IntDeq,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
sq_type: SQType(T),
guard: {T},
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q),
nequal: a ≠ b ∈ T ,
l_member: (x ∈ l),
less_than': less_than'(a;b),
strict-majority: strict-majority(eq;L),
let: let,
ifthenelse: if b then t else f fi ,
null: null(as),
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
count-repeats: count-repeats(L,eq),
list_accum: list_accum,
nil: [],
it: ⋅,
btrue: tt,
true: True,
cons: [a / b],
bfalse: ff
Lemmas referenced :
le_wf,
length_wf,
filter_wf5,
eq_int_wf,
l_member_wf,
equal-wf-base-T,
list_subtype_base,
int_subtype_base,
less_than'_wf,
ge_wf,
list_wf,
nat_wf,
strict-majority-property,
int-deq_wf,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
intformle_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
subtype_base_sq,
unit_wf2,
union_subtype_base,
unit_subtype_base,
strict-majority_wf,
equal_wf,
decidable__l_member,
decidable__equal_int,
filter_is_nil,
assert_wf,
select_wf,
int_seg_properties,
decidable__le,
int_seg_wf,
length_of_nil_lemma,
neg_assert_of_eq_int,
int_seg_subtype_nat,
false_wf,
less_than_wf,
equal-wf-T-base,
list-cases,
filter_nil_lemma,
null_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
filter_cons_lemma,
null_cons_lemma,
imax-list-member,
cons_wf,
add-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
isect_memberFormation,
introduction,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
addEquality,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
intEquality,
lambdaEquality,
setEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
independent_isectElimination,
multiplyEquality,
independent_pairFormation,
productElimination,
independent_pairEquality,
dependent_functionElimination,
voidElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
voidEquality,
computeAll,
instantiate,
cumulativity,
unionEquality,
independent_functionElimination,
imageElimination,
productEquality,
promote_hyp,
hypothesis_subsumption,
pointwiseFunctionality
Latex:
\mforall{}t:\mBbbN{}. \mforall{}L:\mBbbZ{} List.
((\mforall{}v:\mBbbZ{}
(strict-majority-or-max(L) = v) supposing
(((t + 1) \mleq{} ||filter(\mlambda{}x.(x =\msubz{} v);L)||) and
(||L|| = ((2 * t) + 1))))
\mwedge{} (strict-majority-or-max(L) \mmember{} L) supposing ||L|| \mgeq{} 1 )
Date html generated:
2017_04_17-AM-09_09_44
Last ObjectModification:
2017_02_27-PM-05_17_59
Theory : decidable!equality
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