Nuprl Lemma : separated-decider-not-extensional2
∀a,b:ℝ. ((a < b) ⇒ (∀d:∀u:ℝ. ((a < u) ∨ (u < b)). (¬(∀x,y:ℝ. ((x = y) ⇒ isl(d x) = isl(d y))))))
Proof
Definitions occuring in Statement :
rless: x < y,
req: x = y,
real: ℝ,
isl: isl(x),
bool: 𝔹,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
not: ¬A,
false: False,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
so_apply: x[s],
exists: ∃x:A. B[x],
and: P ∧ Q,
uimplies: b supposing a,
guard: {T},
uiff: uiff(P;Q),
isl: isl(x),
isr: isr(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff
Lemmas referenced :
separated-decider-not-extensional,
all_wf,
real_wf,
req_wf,
equal_wf,
bool_wf,
isl_wf,
rless_wf,
or_wf,
exists_wf,
assert_wf,
isr_wf,
assert_functionality_wrt_uiff,
false_wf,
true_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
voidElimination,
isectElimination,
sqequalRule,
lambdaEquality,
because_Cache,
functionEquality,
applyEquality,
functionExtensionality,
productElimination,
productEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
unionElimination
Latex:
\mforall{}a,b:\mBbbR{}. ((a < b) {}\mRightarrow{} (\mforall{}d:\mforall{}u:\mBbbR{}. ((a < u) \mvee{} (u < b)). (\mneg{}(\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} isl(d x) = isl(d y))))))
Date html generated:
2017_10_03-AM-10_02_20
Last ObjectModification:
2017_06_30-PM-00_38_04
Theory : reals
Home
Index