Nuprl Lemma : locally-non-constant-via-rational
∀a,b,c:ℝ. ∀f:[a, b] ⟶ℝ.
(f[x] continuous for x ∈ [a, b] ⇒ locally-non-constant(f;a;b;c) ⇒ locally-non-constant-rational(f;a;b;c))
Proof
Definitions occuring in Statement :
locally-non-constant-rational: locally-non-constant-rational(f;a;b;c),
locally-non-constant: locally-non-constant(f;a;b;c),
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
rccint: [l, u],
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
locally-non-constant: locally-non-constant(f;a;b;c),
locally-non-constant-rational: locally-non-constant-rational(f;a;b;c),
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
continuous: f[x] continuous for x ∈ I,
i-approx: i-approx(I;n),
rccint: [l, u],
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
top: Top,
rneq: x ≠ y,
or: P ∨ Q,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
i-member: r ∈ I,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
uiff: uiff(P;Q),
nat_plus: ℕ+,
rev_implies: P ⇐ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B,
squash: ↓T,
rdiv: (x/y),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
cand: A c∧ B,
rsub: x - y,
rgt: x > y,
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
r-ap: f(x)
Lemmas referenced :
rccint-icompact,
rless_transitivity2,
rless_transitivity1,
rleq_weakening_rless,
icompact_wf,
rccint_wf,
member_rccint_lemma,
rleq_wf,
rless_wf,
locally-non-constant_wf,
continuous_wf,
i-member_wf,
real_wf,
rfun_wf,
small-reciprocal-real-ext,
rsub_wf,
r-ap_wf,
rleq_transitivity,
rless-implies-rless,
int-to-real_wf,
real_term_polynomial,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
radd-preserves-rless,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_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_wf,
radd_wf,
rmul_wf,
rinv_wf2,
sq_stable__and,
all_wf,
rabs_wf,
sq_stable__rless,
sq_stable__all,
sq_stable__rleq,
less_than'_wf,
nat_plus_wf,
squash_wf,
rless_functionality,
itermAdd_wf,
itermMultiply_wf,
real_term_value_add_lemma,
real_term_value_mul_lemma,
radd_functionality,
req_weakening,
rinv-as-rdiv,
rless-cases,
exists_wf,
int_subtype_base,
equal-wf-base,
int_formula_prop_eq_lemma,
intformeq_wf,
nequal_wf,
less_than_wf,
subtype_rel_sets,
int-rdiv_wf,
rmin_wf,
rationals-dense-ext,
radd-assoc,
req_inversion,
rmin_functionality,
rminus_functionality,
radd_comm,
radd-ac,
radd-rminus-assoc,
radd-rminus-both,
req_transitivity,
radd-zero-both,
rminus_wf,
rmul-distrib2,
rmul-identity1,
rminus-as-rmul,
radd-int,
rmul_functionality,
rmul-zero-both,
rmin_strict_ub,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
int-rdiv-req,
rleq_functionality,
uiff_transitivity,
radd-preserves-rleq,
rmin-rleq,
rsub_functionality,
rabs_functionality,
rabs-difference-bound-rleq,
trivial-rsub-rleq,
radd_functionality_wrt_rless2,
rleq_weakening,
radd_functionality_wrt_rleq,
radd_functionality_wrt_rless1,
rneq_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
sqequalRule,
dependent_set_memberEquality,
because_Cache,
introduction,
extract_by_obid,
independent_isectElimination,
isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
lambdaEquality,
applyEquality,
setElimination,
rename,
setEquality,
independent_pairFormation,
natural_numberEquality,
computeAll,
int_eqEquality,
intEquality,
inrFormation,
dependent_pairFormation,
functionEquality,
productEquality,
independent_pairEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
promote_hyp,
applyLambdaEquality,
levelHypothesis,
addLevel,
addEquality,
inlFormation
Latex:
\mforall{}a,b,c:\mBbbR{}. \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}.
(f[x] continuous for x \mmember{} [a, b]
{}\mRightarrow{} locally-non-constant(f;a;b;c)
{}\mRightarrow{} locally-non-constant-rational(f;a;b;c))
Date html generated:
2017_10_03-AM-10_30_55
Last ObjectModification:
2017_07_28-AM-08_12_12
Theory : reals
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