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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