Nuprl Lemma : frs-increasing-cons
∀p:ℝ List. ∀c:ℝ. (frs-increasing([c / p]) ⇐⇒ (0 < ||p|| ⇒ (c < p[0])) ∧ frs-increasing(p))
Proof
Definitions occuring in Statement :
frs-increasing: frs-increasing(p),
rless: x < y,
real: ℝ,
select: L[n],
length: ||as||,
cons: [a / b],
list: T List,
less_than: a < b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
l_all: (∀x∈L.P[x]),
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
sq_type: SQType(T),
sorted-by: sorted-by(R;L),
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
real: ℝ,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P)
Lemmas referenced :
less_than_wf,
length_wf,
real_wf,
sorted-by_wf,
l_member_wf,
rless_wf,
l_all_wf2,
select_wf,
false_wf,
sorted-by-cons,
cons_wf,
iff_wf,
frs-increasing-sorted-by,
frs-increasing_wf,
list_wf,
lelt_wf,
int_seg_wf,
int_seg_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
rless_transitivity2,
sq_stable__less_than,
decidable__le,
rleq_weakening_rless
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
productEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
because_Cache,
dependent_functionElimination,
functionEquality,
independent_isectElimination,
addLevel,
impliesFunctionality,
independent_functionElimination,
andLevelFunctionality,
dependent_set_memberEquality,
unionElimination,
imageElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
instantiate,
cumulativity,
addEquality,
applyEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}p:\mBbbR{} List. \mforall{}c:\mBbbR{}. (frs-increasing([c / p]) \mLeftarrow{}{}\mRightarrow{} (0 < ||p|| {}\mRightarrow{} (c < p[0])) \mwedge{} frs-increasing(p))
Date html generated:
2016_10_26-AM-09_33_00
Last ObjectModification:
2016_08_14-PM-01_14_23
Theory : reals
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