Nuprl Lemma : radd-list-rabs
∀L:ℝ List. (|radd-list(L)| ≤ radd-list(map(λx.|x|;L)))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rabs: |x|,
radd-list: radd-list(L),
real: ℝ,
map: map(f;as),
list: T List,
all: ∀x:A. B[x],
lambda: λx.A[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
istype: istype(T),
not: ¬A,
false: False,
absval: |i|,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
prop: ℙ,
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
uiff: uiff(P;Q)
Lemmas referenced :
list_induction,
real_wf,
radd_list_nil_lemma,
map_nil_lemma,
istype-void,
map_cons_lemma,
list_wf,
rleq_wf,
rabs_wf,
radd-list_wf-bag,
list-subtype-bag,
map_wf,
bag_qinc,
false_wf,
nat_wf,
absval_wf,
rleq-int,
iff_weakening_equal,
int-to-real_wf,
rabs-int,
true_wf,
squash_wf,
radd_wf,
cons_wf,
rleq_weakening_equal,
uimplies_transitivity,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rleq,
r-triangle-inequality,
rleq_functionality,
rabs_functionality,
radd-list-cons
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesis,
sqequalRule,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
rename,
universeIsType,
because_Cache,
hypothesisEquality,
lambdaEquality_alt,
applyEquality,
independent_isectElimination,
lambdaFormation,
independent_pairFormation,
setElimination,
productElimination,
universeEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
inhabitedIsType,
equalityIsType1
Latex:
\mforall{}L:\mBbbR{} List. (|radd-list(L)| \mleq{} radd-list(map(\mlambda{}x.|x|;L)))
Date html generated:
2019_10_29-AM-09_38_30
Last ObjectModification:
2018_11_10-PM-01_29_31
Theory : reals
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