Nuprl Lemma : convex-comb-rless1
∀x,y:ℝ. ∀r:{r:ℝ| r0 ≤ r} . ∀s:{s:ℝ| (r0 < s) ∧ (r0 < (r + s))} . ∀z:{z:ℝ| y < z} .
(convex-comb(x;y;r;s) < convex-comb(x;z;r;s))
Proof
Definitions occuring in Statement :
convex-comb: convex-comb(x;y;r;s),
rleq: x ≤ y,
rless: x < y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
and: P ∧ Q,
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),
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
squash: ↓T,
sq_stable: SqStable(P),
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
cand: A c∧ B,
implies: P ⇒ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
and: P ∧ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rmul-rinv3,
req_transitivity,
convex-comb-req,
rless_functionality,
req-iff-rsub-is-0,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
rinv_wf2,
rmul-zero-both,
equal_wf,
rmul_preserves_rless,
sq_stable__rless,
rdiv_wf,
rsub_wf,
rmul_wf,
rneq_wf,
subtype_rel_sets,
convex-comb_wf1,
rleq_wf,
radd_wf,
int-to-real_wf,
rless_wf,
real_wf,
set_wf,
radd-rminus-assoc,
rminus_wf,
radd-preserves-rless,
rmul_comm
Rules used in proof :
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
equalitySymmetry,
equalityTransitivity,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
dependent_functionElimination,
inrFormation,
productElimination,
setEquality,
independent_isectElimination,
applyEquality,
because_Cache,
rename,
setElimination,
natural_numberEquality,
productEquality,
hypothesisEquality,
lambdaEquality,
sqequalRule,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}x,y:\mBbbR{}. \mforall{}r:\{r:\mBbbR{}| r0 \mleq{} r\} . \mforall{}s:\{s:\mBbbR{}| (r0 < s) \mwedge{} (r0 < (r + s))\} . \mforall{}z:\{z:\mBbbR{}| y < z\} .
(convex-comb(x;y;r;s) < convex-comb(x;z;r;s))
Date html generated:
2017_10_04-PM-11_12_24
Last ObjectModification:
2017_07_29-PM-09_29_13
Theory : reals_2
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