Nuprl Lemma : avoid-reals
∀L:ℝ List. ∀a,b:ℝ. ((a < b) ⇒ (∃a',b':ℝ. ((a ≤ a') ∧ (b' ≤ b) ∧ (a' < b') ∧ (∀x∈L.(x < a') ∨ (b' < x)))))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rless: x < y,
real: ℝ,
l_all: (∀x∈L.P[x]),
list: T List,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
or: P ∨ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
uimplies: b supposing a,
top: Top,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
lelt: i ≤ j < k,
sq_stable: SqStable(P),
squash: ↓T,
real: ℝ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
less_than: a < b
Lemmas referenced :
list_induction,
real_wf,
all_wf,
rless_wf,
exists_wf,
rleq_wf,
l_all_wf2,
l_member_wf,
or_wf,
list_wf,
rleq_weakening_equal,
l_all_nil,
nil_wf,
avoid-real,
rleq_transitivity,
l_all_cons,
cons_wf,
int_seg_wf,
length_wf,
rless_transitivity1,
select_wf,
sq_stable__less_than,
nat_plus_properties,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
rless_transitivity2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesis,
sqequalRule,
lambdaEquality,
because_Cache,
functionEquality,
hypothesisEquality,
productEquality,
setElimination,
rename,
setEquality,
independent_functionElimination,
dependent_pairFormation,
independent_isectElimination,
independent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
dependent_functionElimination,
productElimination,
natural_numberEquality,
unionElimination,
inlFormation,
addEquality,
imageMemberEquality,
baseClosed,
imageElimination,
int_eqEquality,
intEquality,
computeAll,
inrFormation
Latex:
\mforall{}L:\mBbbR{} List. \mforall{}a,b:\mBbbR{}.
((a < b) {}\mRightarrow{} (\mexists{}a',b':\mBbbR{}. ((a \mleq{} a') \mwedge{} (b' \mleq{} b) \mwedge{} (a' < b') \mwedge{} (\mforall{}x\mmember{}L.(x < a') \mvee{} (b' < x)))))
Date html generated:
2016_10_26-AM-09_33_40
Last ObjectModification:
2016_08_13-AM-11_40_27
Theory : reals
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