Nuprl Lemma : list-max-property
∀[T:Type]
∀f:T ⟶ ℤ. ∀L:T List. let n,x = list-max(x.f[x];L) in (x ∈ L) ∧ (f[x] = n ∈ ℤ) ∧ (∀y∈L.f[y] ≤ n) supposing 0 < ||L||
Proof
Definitions occuring in Statement :
list-max: list-max(x.f[x];L),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
length: ||as||,
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
and: P ∧ Q,
function: x:A ⟶ B[x],
spread: spread def,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
list-max: list-max(x.f[x];L),
outl: outl(x),
list-max-aux: list-max-aux(x.f[x];L),
list_accum: list_accum,
prop: ℙ
Lemmas referenced :
list-max-aux-property,
member-less_than,
length_wf,
less_than_wf,
list_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
introduction,
natural_numberEquality,
independent_isectElimination,
rename,
productElimination,
sqequalRule,
functionEquality,
intEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} \mBbbZ{}. \mforall{}L:T List.
let n,x = list-max(x.f[x];L) in (x \mmember{} L) \mwedge{} (f[x] = n) \mwedge{} (\mforall{}y\mmember{}L.f[y] \mleq{} n) supposing 0 < ||L||
Date html generated:
2016_05_14-PM-01_43_13
Last ObjectModification:
2015_12_26-PM-05_31_41
Theory : list_1
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