Nuprl Lemma : convex-comb-strict-lower-bound2
∀r,s:{s:ℝ| r0 < s} . ∀x,y:ℝ. ((x < y) ⇒ (x < convex-comb(x;y;r;s)))
Proof
Definitions occuring in Statement :
convex-comb: convex-comb(x;y;r;s),
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: 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),
rgt: x > y,
rge: x ≥ y,
uiff: uiff(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
squash: ↓T,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
member: t ∈ T,
convex-comb: convex-comb(x;y;r;s),
implies: P ⇒ Q,
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,
radd_functionality,
req_transitivity,
rless_functionality,
radd_functionality_wrt_rless2,
rleq_weakening_rless,
rleq_weakening_equal,
rless_functionality_wrt_implies,
rinv_wf2,
req-iff-rsub-is-0,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
rmul_wf,
trivial-rless-radd,
set_wf,
real_wf,
rless_wf,
rdiv_wf,
rmul_preserves_rless,
radd_wf,
int-to-real_wf,
sq_stable__rless,
rsub_wf,
rless-implies-rless,
rmul_comm
Rules used in proof :
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
equalitySymmetry,
equalityTransitivity,
lambdaEquality,
productElimination,
inrFormation,
independent_isectElimination,
because_Cache,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
independent_functionElimination,
hypothesisEquality,
hypothesis,
natural_numberEquality,
isectElimination,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
rename,
thin,
setElimination,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}r,s:\{s:\mBbbR{}| r0 < s\} . \mforall{}x,y:\mBbbR{}. ((x < y) {}\mRightarrow{} (x < convex-comb(x;y;r;s)))
Date html generated:
2017_10_04-PM-11_13_04
Last ObjectModification:
2017_07_29-PM-10_15_01
Theory : reals_2
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