Nuprl Lemma : list-max-imax-list
∀[T:Type]. ∀[f:T ⟶ ℤ]. ∀[L:T List]. (fst(list-max(x.f[x];L))) = imax-list(map(λx.f[x];L)) ∈ ℤ supposing 0 < ||L||
Proof
Definitions occuring in Statement :
list-max: list-max(x.f[x];L),
imax-list: imax-list(L),
length: ||as||,
map: map(f;as),
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
lambda: λx.A[x],
function: x:A ⟶ B[x],
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
pi1: fst(t),
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
list-max-property,
list-max_wf,
equal-wf-T-base,
int_subtype_base,
l_member_wf,
l_all_wf,
le_wf,
equal_wf,
less_than_wf,
length_wf,
list_wf,
imax-list-unique,
map_wf,
member_map,
equal-wf-base-T,
l_all_iff,
subtype_base_sq
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
independent_isectElimination,
cumulativity,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
productEquality,
intEquality,
setEquality,
lambdaFormation,
productElimination,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
universeEquality,
dependent_pairFormation,
independent_pairFormation,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} \mBbbZ{}]. \mforall{}[L:T List].
(fst(list-max(x.f[x];L))) = imax-list(map(\mlambda{}x.f[x];L)) supposing 0 < ||L||
Date html generated:
2017_04_17-AM-07_40_59
Last ObjectModification:
2017_02_27-PM-04_14_03
Theory : list_1
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