Nuprl Lemma : list-max-property2
∀[T:Type]
  ∀f:T ⟶ ℤ. ∀L:T List.
    ∀n:ℤ. ∀x:T.  {(x ∈ L) ∧ (f[x] = n ∈ ℤ) ∧ (∀y∈L.f[y] ≤ n)} supposing list-max(x.f[x];L) = <n, x> ∈ (ℤ × T) 
    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]
, 
guard: {T}
, 
so_apply: x[s]
, 
le: A ≤ B
, 
all: ∀x:A. B[x]
, 
and: P ∧ Q
, 
function: x:A ⟶ B[x]
, 
pair: <a, b>
, 
product: x:A × B[x]
, 
natural_number: $n
, 
int: ℤ
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
guard: {T}
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
all: ∀x:A. B[x]
, 
uimplies: b supposing a
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
subtype_rel: A ⊆r B
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
sq_type: SQType(T)
, 
cand: A c∧ B
Lemmas referenced : 
list-max-property, 
member-less_than, 
length_wf, 
less_than_wf, 
list_wf, 
list-max_wf, 
equal-wf-T-base, 
int_subtype_base, 
subtype_base_sq, 
and_wf, 
equal_wf, 
l_member_wf, 
l_all_wf, 
le_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
cut, 
introduction, 
extract_by_obid, 
isect_memberFormation, 
hypothesis, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
lambdaFormation, 
dependent_functionElimination, 
natural_numberEquality, 
cumulativity, 
independent_isectElimination, 
rename, 
functionEquality, 
intEquality, 
universeEquality, 
lambdaEquality, 
applyEquality, 
functionExtensionality, 
productEquality, 
setEquality, 
productElimination, 
axiomEquality, 
independent_pairFormation, 
because_Cache, 
instantiate, 
independent_functionElimination, 
dependent_set_memberEquality, 
equalityTransitivity, 
equalitySymmetry, 
setElimination, 
hyp_replacement, 
Error :applyLambdaEquality, 
addLevel, 
levelHypothesis, 
independent_pairEquality
Latex:
\mforall{}[T:Type]
    \mforall{}f:T  {}\mrightarrow{}  \mBbbZ{}.  \mforall{}L:T  List.
        \mforall{}n:\mBbbZ{}.  \mforall{}x:T.    \{(x  \mmember{}  L)  \mwedge{}  (f[x]  =  n)  \mwedge{}  (\mforall{}y\mmember{}L.f[y]  \mleq{}  n)\}  supposing  list-max(x.f[x];L)  =  <n,  x> 
        supposing  0  <  ||L||
Date html generated:
2016_10_21-AM-10_10_33
Last ObjectModification:
2016_07_12-AM-05_29_14
Theory : list_1
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