Nuprl Lemma : reg-less_wf
∀[x:ℝ]. ∀[y:{y:ℝ| ∃n:{ℕ+| (x n) + 4 < y n}} ]. (reg-less(x;y) ∈ {n:ℕ+| (x n) + 4 < y n} )
Proof
Definitions occuring in Statement :
reg-less: reg-less(x;y),
real: ℝ,
nat_plus: ℕ+,
less_than: a < b,
uall: ∀[x:A]. B[x],
sq_exists: ∃x:{A| B[x]},
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
int_upper: {i...},
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
exists: ∃x:A. B[x],
sq_exists: ∃x:{A| B[x]},
so_lambda: λ2x.t[x],
real: ℝ,
so_apply: x[s],
reg-less: reg-less(x;y),
has-value: (a)↓,
uimplies: b supposing a,
squash: ↓T,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
top: Top,
true: True,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt
Lemmas referenced :
eqtt_to_assert,
add-swap,
add-associates,
bool_wf,
equal-wf-T-base,
all_wf,
iff_wf,
assert_wf,
assert_of_lt_int,
true_wf,
btrue_wf,
less_than_transitivity1,
iff_imp_equal_bool,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
subtype_rel_sets,
int_upper_wf,
le-add-cancel,
zero-add,
add-commutes,
add_functionality_wrt_le,
not-lt-2,
decidable__lt,
lt_int_wf,
find-ge_wf,
int-value-type,
function-value-type,
regular-int-seq_wf,
set-value-type,
value-type-has-value,
less_than_wf,
nat_plus_wf,
sq_exists_wf,
real_wf,
set_wf,
le_wf,
false_wf,
regular-less-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
productElimination,
independent_functionElimination,
dependent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
lambdaFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
addEquality,
applyEquality,
isect_memberEquality,
because_Cache,
callbyvalueReduce,
independent_isectElimination,
functionEquality,
intEquality,
imageMemberEquality,
baseClosed,
unionElimination,
voidElimination,
voidEquality,
dependent_pairFormation,
setEquality,
int_eqEquality,
computeAll,
addLevel,
impliesFunctionality,
productEquality,
levelHypothesis,
promote_hyp,
andLevelFunctionality
Latex:
\mforall{}[x:\mBbbR{}]. \mforall{}[y:\{y:\mBbbR{}| \mexists{}n:\{\mBbbN{}\msupplus{}| (x n) + 4 < y n\}\} ]. (reg-less(x;y) \mmember{} \{n:\mBbbN{}\msupplus{}| (x n) + 4 < y n\} )
Date html generated:
2016_05_18-AM-06_47_52
Last ObjectModification:
2016_01_17-AM-01_45_42
Theory : reals
Home
Index