Nuprl Lemma : convex-comb-rless2
∀x:ℝ. ∀y:{y:ℝ| x < y} . ∀r:{r:ℝ| r0 < r} . ∀s:{s:ℝ| r0 < (r + s)} . ∀t:{t:ℝ| s < t} .
(convex-comb(x;y;r;s) < convex-comb(x;y;r;t))
Proof
Definitions occuring in Statement :
convex-comb: convex-comb(x;y;r;s),
rless: x < y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
rgt: x > y,
rge: x ≥ y,
top: Top,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
convex-comb: convex-comb(x;y;r;s),
squash: ↓T,
prop: ℙ,
uimplies: b supposing a,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
real_term_value_mul_lemma,
itermMultiply_wf,
rmul_comm,
rless_functionality,
rmul_preserves_rless,
radd_functionality_wrt_rless1,
rleq_weakening_equal,
rless_functionality_wrt_implies,
sq_stable__rless,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermSubtract_wf,
rsub_wf,
real_wf,
set_wf,
rless-implies-rless,
rmul_wf,
fractions-rless,
rless_wf,
rleq_weakening_rless,
rless_transitivity2,
radd-preserves-rless,
int-to-real_wf,
radd_wf,
sq_stable__rneq
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
lambdaEquality,
imageElimination,
baseClosed,
imageMemberEquality,
independent_isectElimination,
inrFormation,
sqequalRule,
productElimination,
independent_functionElimination,
natural_numberEquality,
hypothesisEquality,
hypothesis,
because_Cache,
isectElimination,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
rename,
thin,
setElimination,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}x:\mBbbR{}. \mforall{}y:\{y:\mBbbR{}| x < y\} . \mforall{}r:\{r:\mBbbR{}| r0 < r\} . \mforall{}s:\{s:\mBbbR{}| r0 < (r + s)\} . \mforall{}t:\{t:\mBbbR{}| s < t\} .
(convex-comb(x;y;r;s) < convex-comb(x;y;r;t))
Date html generated:
2018_05_22-PM-03_12_28
Last ObjectModification:
2018_05_20-PM-11_57_10
Theory : reals_2
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