Nuprl Lemma : i-approx-monotonic
∀I:Interval. ∀n,m:ℕ+. ∀x:ℝ. ((x ∈ i-approx(I;n)) ⇒ (x ∈ i-approx(I;m))) supposing n ≤ m
Proof
Definitions occuring in Statement :
i-approx: i-approx(I;n),
i-member: r ∈ I,
interval: Interval,
real: ℝ,
nat_plus: ℕ+,
uimplies: b supposing a,
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
uiff: uiff(P;Q),
interval: Interval,
i-approx: i-approx(I;n),
cand: A c∧ B,
guard: {T},
rneq: x ≠ y,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Lemmas referenced :
radd_functionality_wrt_rleq,
rsub_functionality_wrt_rleq,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
radd_wf,
rless_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
rless-int,
rdiv_wf,
rsub_wf,
int-to-real_wf,
rleq_transitivity,
member_rccint_lemma,
interval_wf,
nat_plus_wf,
le_wf,
real_wf,
i-approx_wf,
i-member_wf,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
itermConstant_wf,
itermMultiply_wf,
rleq-int-fractions,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_minus_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermMinus_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
rleq-int,
less_than'_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
lemma_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
minusEquality,
unionElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
because_Cache,
multiplyEquality,
inrFormation
Latex:
\mforall{}I:Interval. \mforall{}n,m:\mBbbN{}\msupplus{}. \mforall{}x:\mBbbR{}. ((x \mmember{} i-approx(I;n)) {}\mRightarrow{} (x \mmember{} i-approx(I;m))) supposing n \mleq{} m
Date html generated:
2016_05_18-AM-08_39_34
Last ObjectModification:
2016_01_17-AM-02_27_54
Theory : reals
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