Nuprl Lemma : rmax-ub-convex
∀a,b,t:ℝ. ((r0 ≤ t) ⇒ (t ≤ r1) ⇒ (((t * a) + ((r1 - t) * b)) ≤ rmax(a;b)))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rmax: rmax(x;y),
rsub: x - y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
prop: ℙ,
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T}
Lemmas referenced :
rleq-rmax,
rmul_preserves_rleq2,
rmax_wf,
rleq-implies-rleq,
rmul_wf,
rsub_wf,
int-to-real_wf,
rleq_functionality,
radd_wf,
rminus_wf,
rleq_wf,
real_wf,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
itermConstant_wf,
itermAdd_wf,
itermMinus_wf,
rleq_weakening,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rleq,
rleq_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
hypothesis,
independent_isectElimination,
natural_numberEquality,
because_Cache,
sqequalRule,
dependent_functionElimination,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}a,b,t:\mBbbR{}. ((r0 \mleq{} t) {}\mRightarrow{} (t \mleq{} r1) {}\mRightarrow{} (((t * a) + ((r1 - t) * b)) \mleq{} rmax(a;b)))
Date html generated:
2018_05_22-PM-01_32_32
Last ObjectModification:
2017_10_20-PM-04_50_09
Theory : reals
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