Nuprl Lemma : lower-right-endpoint-rless
∀a,b:ℝ. ∀n:ℕ+. ((a < b) ⇒ ((a < lower-right-endpoint(a;b;n)) ∧ (lower-right-endpoint(a;b;n) < b)))
Proof
Definitions occuring in Statement :
lower-right-endpoint: lower-right-endpoint(a;b;n),
rless: x < y,
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
lower-right-endpoint: lower-right-endpoint(a;b;n),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
int_nzero: ℤ-o,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
not: ¬A,
false: False,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
guard: {T},
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
rdiv: (x/y),
rge: x ≥ y
Lemmas referenced :
rless_wf,
nat_plus_wf,
real_wf,
int-rdiv_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
equal-wf-T-base,
nequal_wf,
radd_wf,
int-rmul_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
rmul_wf,
rmul_preserves_rless,
rinv_wf2,
rneq_functionality,
radd-int,
req_weakening,
rneq-int,
rless-implies-rless,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
req-iff-rsub-is-0,
rsub_wf,
rless_functionality,
int-rdiv-req,
rdiv_functionality,
radd_functionality,
int-rmul-req,
req_transitivity,
rmul_functionality,
rmul-rinv3,
req_inversion,
rless_functionality_wrt_implies,
rleq_weakening_rless,
radd_functionality_wrt_rless2,
rleq_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
independent_functionElimination,
inrFormation,
productElimination,
unionElimination,
addLevel,
levelHypothesis,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}n:\mBbbN{}\msupplus{}. ((a < b) {}\mRightarrow{} ((a < lower-right-endpoint(a;b;n)) \mwedge{} (lower-right-endpoint(a;b;n) < b)))
Date html generated:
2017_10_03-AM-09_32_11
Last ObjectModification:
2017_07_28-AM-07_50_34
Theory : reals
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