Nuprl Lemma : imax-list-unique2
∀L:ℤ List. ∀a:ℤ. uiff((||L|| > 0) ∧ (imax-list(L) = a ∈ ℤ);(∀b∈L.b ≤ a) ∧ (a ∈ L))
Proof
Definitions occuring in Statement :
imax-list: imax-list(L),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
length: ||as||,
list: T List,
uiff: uiff(P;Q),
gt: i > j,
le: A ≤ B,
all: ∀x:A. B[x],
and: P ∧ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
gt: i > j,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
false: False,
cons: [a / b],
top: Top,
guard: {T},
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
not: ¬A,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
member-less_than,
length_wf,
imax-list-unique,
imax-list-eq-implies,
gt_wf,
equal-wf-base,
list_subtype_base,
int_subtype_base,
list-cases,
length_of_nil_lemma,
nil_member,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
nat_wf,
decidable__lt,
false_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,
equal_wf,
l_all_wf,
le_wf,
l_member_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
extract_by_obid,
isectElimination,
natural_numberEquality,
intEquality,
hypothesisEquality,
hypothesis,
independent_isectElimination,
axiomEquality,
rename,
dependent_functionElimination,
because_Cache,
productEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
unionElimination,
independent_functionElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
setElimination,
addEquality,
lambdaEquality,
minusEquality,
equalityTransitivity,
equalitySymmetry,
setEquality
Latex:
\mforall{}L:\mBbbZ{} List. \mforall{}a:\mBbbZ{}. uiff((||L|| > 0) \mwedge{} (imax-list(L) = a);(\mforall{}b\mmember{}L.b \mleq{} a) \mwedge{} (a \mmember{} L))
Date html generated:
2017_04_17-AM-07_40_08
Last ObjectModification:
2017_02_27-PM-04_13_46
Theory : list_1
Home
Index