Nuprl Lemma : quasilinear-weighted-mean-properties
∀I,J:Interval. ∀f:{x:ℝ| x ∈ J} ⟶ {x:ℝ| x ∈ I} . ∀g:{x:ℝ| x ∈ I} ⟶ {x:ℝ| x ∈ J} .
(((∀x1,x2:{x:ℝ| x ∈ J} . ((x1 < x2) ⇒ ((f x1) < (f x2)))) ∨ (∀x1,x2:{x:ℝ| x ∈ J} . ((x1 < x2) ⇒ ((f x2) < (f x1)))\000C))
⇒ (∀x1,x2:{x:ℝ| x ∈ J} . ((x1 = x2) ⇒ ((f x1) = (f x2))))
⇒ (∀x1,x2:{x:ℝ| x ∈ I} . ((x1 = x2) ⇒ ((g x1) = (g x2))))
⇒ (∀x:{x:ℝ| x ∈ I} . ((f (g x)) = x))
⇒ weighted-mean-properties(I;quasilinear-weighted-mean(f;g)))
Proof
Definitions occuring in Statement :
quasilinear-weighted-mean: quasilinear-weighted-mean(f;g),
weighted-mean-properties: weighted-mean-properties(I;F),
i-member: r ∈ I,
interval: Interval,
rless: x < y,
req: x = y,
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
weighted-mean-properties: weighted-mean-properties(I;F),
and: P ∧ Q,
quasilinear-weighted-mean: quasilinear-weighted-mean(f;g),
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
cand: A c∧ B,
subtype_rel: A ⊆r B,
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
true: True,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
rge: x ≥ y,
sq_stable: SqStable(P),
rgt: x > y,
rfun: I ⟶ℝ,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
req_int_terms: t1 ≡ t2,
convex-comb: convex-comb(x;y;r;s),
rat_term_to_real: rat_term_to_real(f;t),
rtermAdd: left "+" right,
rat_term_ind: rat_term_ind,
rtermMultiply: left "*" right,
rtermDivide: num "/" denom,
rtermVar: rtermVar(var),
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
rleq_wf,
int-to-real_wf,
rless_wf,
radd_wf,
i-member_wf,
real_wf,
req_wf,
interval_wf,
convex-comb_wf,
convex-comb-same,
subtype_rel_sets_simple,
rneq_wf,
req_functionality,
req_weakening,
rleq-int,
istype-false,
rleq_weakening_equal,
trivial-rless-radd,
rless-int,
convex-comb-1-0,
rneq-int,
full-omega-unsat,
intformeq_wf,
itermConstant_wf,
istype-int,
int_formula_prop_eq_lemma,
istype-void,
int_term_value_constant_lemma,
int_formula_prop_wf,
rleq_weakening_rless,
rless_functionality,
req_inversion,
rless_functionality_wrt_implies,
radd_functionality_wrt_rleq,
sq_stable__rless,
radd_functionality_wrt_rless2,
convex-comb-strict-lower-bound2,
inverse-of-strict-increasing-function,
subtype_rel_dep_function,
sq_stable__i-member,
convex-comb-strict-upper-bound2,
inverse-of-strict-decreasing-function,
convex-comb-0-1,
nat_plus_properties,
convex-comb-strict-upper-bound,
convex-comb-strict-lower-bound,
rmul_wf,
rmul-nonneg-case1,
convex-comb-homog,
rmul_preserves_rless,
rless-implies-rless,
rsub_wf,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
sq_stable__and,
sq_stable__rleq,
radd_functionality_wrt_rless1,
convex-comb_wf1,
convex-comb_functionality,
rdiv_functionality,
rdiv_wf,
assert-rat-term-eq2,
rtermAdd_wf,
rtermMultiply_wf,
rtermVar_wf,
rtermDivide_wf,
rless_transitivity2,
convex-comb-rless2,
convex-comb-rless3,
convex-comb-rless1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
sqequalRule,
setIsType,
because_Cache,
productIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
setElimination,
rename,
inhabitedIsType,
functionIsType,
applyEquality,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
unionIsType,
dependent_functionElimination,
independent_functionElimination,
productEquality,
independent_isectElimination,
productElimination,
inrFormation_alt,
dependent_set_memberEquality_alt,
imageMemberEquality,
baseClosed,
approximateComputation,
dependent_pairFormation_alt,
isect_memberEquality_alt,
voidElimination,
equalityIstype,
sqequalBase,
imageElimination,
unionElimination,
setEquality,
int_eqEquality,
applyLambdaEquality
Latex:
\mforall{}I,J:Interval. \mforall{}f:\{x:\mBbbR{}| x \mmember{} J\} {}\mrightarrow{} \{x:\mBbbR{}| x \mmember{} I\} . \mforall{}g:\{x:\mBbbR{}| x \mmember{} I\} {}\mrightarrow{} \{x:\mBbbR{}| x \mmember{} J\} .
(((\mforall{}x1,x2:\{x:\mBbbR{}| x \mmember{} J\} . ((x1 < x2) {}\mRightarrow{} ((f x1) < (f x2))))
\mvee{} (\mforall{}x1,x2:\{x:\mBbbR{}| x \mmember{} J\} . ((x1 < x2) {}\mRightarrow{} ((f x2) < (f x1)))))
{}\mRightarrow{} (\mforall{}x1,x2:\{x:\mBbbR{}| x \mmember{} J\} . ((x1 = x2) {}\mRightarrow{} ((f x1) = (f x2))))
{}\mRightarrow{} (\mforall{}x1,x2:\{x:\mBbbR{}| x \mmember{} I\} . ((x1 = x2) {}\mRightarrow{} ((g x1) = (g x2))))
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} I\} . ((f (g x)) = x))
{}\mRightarrow{} weighted-mean-properties(I;quasilinear-weighted-mean(f;g)))
Date html generated:
2019_10_31-AM-06_25_32
Last ObjectModification:
2019_04_03-AM-00_24_27
Theory : reals_2
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