Nuprl Lemma : rless-witness-property
∀[x,y:ℝ]. ∀[p:x < y]. ((x ≤ (y - (r1/r(rless-witness(x;y;p))))) ∧ ((x + (r1/r(rless-witness(x;y;p)))) ≤ y))
Proof
Definitions occuring in Statement :
rless-witness: rless-witness(x;y;p),
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
rless-witness: rless-witness(x;y;p),
so_lambda: λ2x.t[x],
so_apply: x[s],
pi1: fst(t),
cand: A c∧ B,
sq_type: SQType(T),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
squash: ↓T
Lemmas referenced :
rless-witness_wf,
rless_wf,
real_wf,
rleq_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
nat_plus_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_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_eq_lemma,
int_formula_prop_wf,
equal_wf,
radd_wf,
less_than'_wf,
squash_wf,
rless-implies-rleq,
all_wf,
exists_wf,
subtype_base_sq,
set_subtype_base,
less_than_wf,
int_subtype_base,
rleq-implies-rleq,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
req-iff-rsub-is-0,
sq_stable__and,
sq_stable__rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
independent_isectElimination,
inrFormation,
dependent_functionElimination,
because_Cache,
productElimination,
independent_functionElimination,
lambdaFormation,
applyLambdaEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
minusEquality,
instantiate,
functionEquality,
cumulativity,
independent_pairEquality,
axiomEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[x,y:\mBbbR{}]. \mforall{}[p:x < y].
((x \mleq{} (y - (r1/r(rless-witness(x;y;p))))) \mwedge{} ((x + (r1/r(rless-witness(x;y;p)))) \mleq{} y))
Date html generated:
2017_10_03-AM-09_06_22
Last ObjectModification:
2017_07_28-AM-07_42_14
Theory : reals
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