Nuprl Lemma : convex-comb-req
∀[x,y,r:ℝ]. ∀[s:{s:ℝ| r + s ≠ r0} ]. (convex-comb(x;y;r;s) = (((r1 - (s/r + s)) * x) + ((s/r + s) * y)))
Proof
Definitions occuring in Statement :
convex-comb: convex-comb(x;y;r;s),
rdiv: (x/y),
rneq: x ≠ y,
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
top: Top,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
rev_uimplies: rev_uimplies(P;Q),
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
squash: ↓T,
implies: P ⇒ Q,
sq_stable: SqStable(P),
all: ∀x:A. B[x],
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
member: t ∈ T,
convex-comb: convex-comb(x;y;r;s),
uall: ∀[x:A]. B[x]
Lemmas referenced :
real_term_value_minus_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rminus_functionality,
rmul-rinv3,
radd_functionality,
req_transitivity,
req_functionality,
itermMinus_wf,
itermConstant_wf,
rminus_wf,
req-iff-rsub-is-0,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
itermSubtract_wf,
rinv_wf2,
rsub_wf,
rmul_wf,
rdiv_wf,
rmul_preserves_req,
req_weakening,
sq_stable_rneq,
int-to-real_wf,
radd_wf,
rneq_wf,
real_wf,
set_wf
Rules used in proof :
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
equalitySymmetry,
equalityTransitivity,
productElimination,
independent_isectElimination,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
dependent_functionElimination,
rename,
setElimination,
because_Cache,
natural_numberEquality,
hypothesisEquality,
lambdaEquality,
sqequalRule,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[x,y,r:\mBbbR{}]. \mforall{}[s:\{s:\mBbbR{}| r + s \mneq{} r0\} ].
(convex-comb(x;y;r;s) = (((r1 - (s/r + s)) * x) + ((s/r + s) * y)))
Date html generated:
2017_10_04-PM-11_12_06
Last ObjectModification:
2017_07_29-PM-08_03_58
Theory : reals_2
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