Nuprl Lemma : imax-list-ub
∀L:ℤ List. ∀a:ℤ. a ≤ imax-list(L) ⇐⇒ (∃b∈L. a ≤ b) supposing 0 < ||L||
Proof
Definitions occuring in Statement :
imax-list: imax-list(L),
l_exists: (∃x∈L. P[x]),
length: ||as||,
list: T List,
less_than: a < b,
uimplies: b supposing a,
le: A ≤ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
cand: A c∧ B,
assoc: Assoc(T;op),
infix_ap: x f y,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
imax-list: imax-list(L)
Lemmas referenced :
member-less_than,
length_wf,
combine-list-rel-or,
imax_wf,
le_wf,
imax_ub,
all_wf,
iff_wf,
or_wf,
equal_wf,
squash_wf,
true_wf,
imax_assoc,
iff_weakening_equal,
less_than_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
intEquality,
hypothesisEquality,
hypothesis,
independent_isectElimination,
rename,
because_Cache,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
independent_functionElimination,
independent_pairFormation,
addLevel,
allFunctionality,
productElimination,
impliesFunctionality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}L:\mBbbZ{} List. \mforall{}a:\mBbbZ{}. a \mleq{} imax-list(L) \mLeftarrow{}{}\mRightarrow{} (\mexists{}b\mmember{}L. a \mleq{} b) supposing 0 < ||L||
Date html generated:
2017_04_14-AM-09_23_47
Last ObjectModification:
2017_02_27-PM-03_58_27
Theory : list_1
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