Nuprl Lemma : rless-iff4
∀x,y:ℝ. (x < y ⇐⇒ ∃n:ℕ+. ∀m:{n...}. (x m) + 4 < y m)
Proof
Definitions occuring in Statement :
rless: x < y,
real: ℝ,
int_upper: {i...},
nat_plus: ℕ+,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
real: ℝ,
int_upper: {i...},
le: A ≤ B,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
exists: ∃x:A. B[x],
rless: x < y,
sq_exists: ∃x:{A| B[x]},
sq_stable: SqStable(P),
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q)
Lemmas referenced :
le_wf,
false_wf,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermSubtract_wf,
itermConstant_wf,
itermAdd_wf,
intformless_wf,
subtract-is-int-iff,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
sq_stable__less_than,
nat_plus_properties,
int_upper_properties,
real_wf,
less_than_transitivity1,
less_than_wf,
int_upper_wf,
all_wf,
nat_plus_wf,
exists_wf,
rless_wf,
rless-iff-large-diff
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairFormation,
independent_functionElimination,
isectElimination,
sqequalRule,
lambdaEquality,
setElimination,
rename,
addEquality,
applyEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_isectElimination,
because_Cache,
introduction,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
imageElimination,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
dependent_set_memberFormation
Latex:
\mforall{}x,y:\mBbbR{}. (x < y \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}\msupplus{}. \mforall{}m:\{n...\}. (x m) + 4 < y m)
Date html generated:
2016_05_18-AM-07_03_54
Last ObjectModification:
2016_01_17-AM-01_50_05
Theory : reals
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