Nuprl Lemma : q-max-exists
∀as:ℚ List. ∃a:ℚ. ((a ∈ as) supposing ¬↑null(as) ∧ (∀a'∈as.a' ≤ a))
Proof
Definitions occuring in Statement :
qle: r ≤ s,
rationals: ℚ,
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
null: null(as),
list: T List,
assert: ↑b,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
top: Top,
bfalse: ff,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j ,
decidable: Dec(P),
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
iff: P ⇐⇒ Q,
guard: {T},
nat: ℕ,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b)
Lemmas referenced :
rationals_wf,
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
list_wf,
int-subtype-rationals,
true_wf,
not_wf,
l_all_nil,
l_member_wf,
nil_wf,
l_all_wf2,
qle_wf,
subtype_rel_set,
qmax-list_wf,
cons_wf,
length_of_cons_lemma,
non_neg_length,
decidable__lt,
length_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
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,
false_wf,
qmax-list-member,
qmax-list-bounds,
length_of_nil_lemma,
nil_member,
btrue_wf,
and_wf,
equal_wf,
null_wf3,
subtype_rel_list,
top_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
length_wf_nat,
nat_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
cons_neq_nil,
qle_reflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
sqequalRule,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
natural_numberEquality,
applyEquality,
isect_memberFormation,
lambdaEquality,
rename,
independent_functionElimination,
independent_pairFormation,
productEquality,
isectEquality,
functionExtensionality,
independent_isectElimination,
setEquality,
addEquality,
int_eqEquality,
intEquality,
computeAll,
because_Cache,
setElimination,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
applyLambdaEquality,
minusEquality
Latex:
\mforall{}as:\mBbbQ{} List. \mexists{}a:\mBbbQ{}. ((a \mmember{} as) supposing \mneg{}\muparrow{}null(as) \mwedge{} (\mforall{}a'\mmember{}as.a' \mleq{} a))
Date html generated:
2018_05_22-AM-00_17_15
Last ObjectModification:
2017_07_26-PM-06_53_15
Theory : rationals
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