Nuprl Lemma : select-filter-from-upto-increasing
∀[n,m:ℤ]. ∀[P:{n..m-} ⟶ 𝔹]. ∀[i,j:ℕ||filter(P;[n, m))||]. filter(P;[n, m))[i] < filter(P;[n, m))[j] supposing i < j
Proof
Definitions occuring in Statement :
from-upto: [n, m),
select: L[n],
length: ||as||,
filter: filter(P;l),
int_seg: {i..j-},
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
select-filter-from-upto-order-preserving,
int_seg_wf,
length_wf,
filter_wf5,
from-upto_wf,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
list_wf,
and_wf,
le_wf,
less_than_wf,
subtype_rel_self,
set_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
natural_numberEquality,
because_Cache,
applyEquality,
sqequalRule,
lambdaEquality,
setEquality,
intEquality,
independent_isectElimination,
setElimination,
rename,
lambdaFormation,
functionEquality
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[P:\{n..m\msupminus{}\} {}\mrightarrow{} \mBbbB{}]. \mforall{}[i,j:\mBbbN{}||filter(P;[n, m))||].
filter(P;[n, m))[i] < filter(P;[n, m))[j] supposing i < j
Date html generated:
2016_05_14-PM-02_01_13
Last ObjectModification:
2015_12_26-PM-05_12_56
Theory : list_1
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