Nuprl Lemma : regular-less-iff
∀[x,y:ℝ]. (∃n:{ℕ+| (x n) + 4 < y n} ⇐⇒ ∀b:{4...}. ∃n:ℕ+. ∀m:{n...}. (x m) + b < y m)
Proof
Definitions occuring in Statement :
real: ℝ,
int_upper: {i...},
nat_plus: ℕ+,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
sq_exists: ∃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 :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
real: ℝ,
so_apply: x[s],
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
int_upper: {i...},
le: A ≤ B,
guard: {T},
uimplies: b supposing a,
sq_exists: ∃x:{A| B[x]},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nat: ℕ,
ge: i ≥ j
Lemmas referenced :
int_upper_wf,
sq_exists_wf,
nat_plus_wf,
less_than_wf,
all_wf,
exists_wf,
less_than_transitivity1,
real_wf,
sq_stable__less_than,
int_upper_properties,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
mul_preserves_le,
nat_plus_subtype_nat,
decidable__lt,
intformless_wf,
itermMultiply_wf,
int_formula_prop_less_lemma,
int_term_value_mul_lemma,
multiply-is-int-iff,
int_subtype_base,
regular-consistency,
subtype_rel_sets,
le_wf,
absval_unfold,
subtract_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
add-is-int-iff,
itermSubtract_wf,
int_term_value_subtract_lemma,
false_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
itermMinus_wf,
int_term_value_minus_lemma,
absval_wf,
nat_wf,
int_upper_subtype_nat,
le_functionality,
multiply_functionality_wrt_le,
le_weakening,
add_functionality_wrt_le,
regular-less,
absval_ifthenelse,
subtract-is-int-iff,
assert_wf,
bnot_wf,
not_wf,
minus-is-int-iff,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
sqequalRule,
lambdaEquality,
addEquality,
applyEquality,
setElimination,
rename,
hypothesisEquality,
because_Cache,
dependent_set_memberEquality,
productElimination,
independent_isectElimination,
promote_hyp,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
multiplyEquality,
baseApply,
closedConclusion,
baseClosed,
setEquality,
applyLambdaEquality,
minusEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
lessCases,
sqequalAxiom,
imageMemberEquality,
imageElimination,
independent_functionElimination,
pointwiseFunctionality,
instantiate,
cumulativity,
impliesFunctionality,
dependent_set_memberFormation
Latex:
\mforall{}[x,y:\mBbbR{}]. (\mexists{}n:\{\mBbbN{}\msupplus{}| (x n) + 4 < y n\} \mLeftarrow{}{}\mRightarrow{} \mforall{}b:\{4...\}. \mexists{}n:\mBbbN{}\msupplus{}. \mforall{}m:\{n...\}. (x m) + b < y m)
Date html generated:
2017_10_02-PM-07_13_52
Last ObjectModification:
2017_07_28-AM-07_20_05
Theory : reals
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