Nuprl Lemma : separated-decider-not-extensional
∀a,b:ℝ. ((a < b) ⇒ (∀d:∀u:ℝ. ((a < u) ∨ (u < b)). (¬¬(∃x,y:ℝ. ((x = y) ∧ (↑isl(d x)) ∧ (↑isr(d y)))))))
Proof
Definitions occuring in Statement :
rless: x < y,
req: x = y,
real: ℝ,
assert: ↑b,
isr: isr(x),
isl: isl(x),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
so_apply: x[s],
and: P ∧ Q,
exists: ∃x:A. B[x],
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
bfalse: ff,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
isr: isr(x),
guard: {T}
Lemmas referenced :
no-real-separation-corollary,
assert_wf,
isl_wf,
rless_wf,
real_wf,
or_wf,
isr_wf,
not_wf,
exists_wf,
req_wf,
all_wf,
nat_plus_properties,
full-omega-unsat,
intformless_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
equal_wf,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
sqequalRule,
lambdaEquality,
hypothesisEquality,
hypothesis,
applyEquality,
functionExtensionality,
because_Cache,
independent_functionElimination,
voidElimination,
productEquality,
dependent_pairFormation,
unionElimination,
natural_numberEquality,
setElimination,
rename,
imageElimination,
productElimination,
independent_isectElimination,
approximateComputation,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidEquality,
equalityTransitivity,
equalitySymmetry,
inlFormation,
inrFormation
Latex:
\mforall{}a,b:\mBbbR{}.
((a < b) {}\mRightarrow{} (\mforall{}d:\mforall{}u:\mBbbR{}. ((a < u) \mvee{} (u < b)). (\mneg{}\mneg{}(\mexists{}x,y:\mBbbR{}. ((x = y) \mwedge{} (\muparrow{}isl(d x)) \mwedge{} (\muparrow{}isr(d y)))))))
Date html generated:
2017_10_03-AM-10_02_05
Last ObjectModification:
2017_06_30-PM-00_37_20
Theory : reals
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