Nuprl Lemma : rleq-implies
∀[x,y:ℝ]. ∀n:ℕ+. ((x (4 * n)) ≤ ((y (4 * n)) + 11)) supposing x ≤ y
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
real: ℝ,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
apply: f a,
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
rleq: x ≤ y,
rsub: x - y,
rnonneg: rnonneg(x),
rminus: -(x),
radd: a + b,
accelerate: accelerate(k;f),
has-value: (a)↓,
nat_plus: ℕ+,
squash: ↓T,
prop: ℙ,
real: ℝ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T},
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
le: A ≤ B,
nat: ℕ,
less_than': less_than'(a;b),
less_than: a < b,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
rev_implies: P ⇐ Q,
bfalse: ff
Lemmas referenced :
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
bool_cases,
not_wf,
bnot_wf,
assert_wf,
lt_int_wf,
int_term_value_minus_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
itermMinus_wf,
itermSubtract_wf,
itermAdd_wf,
intformle_wf,
minus-is-int-iff,
add-is-int-iff,
subtract-is-int-iff,
decidable__le,
absval_ifthenelse,
sq_stable__le,
set_wf,
nat_wf,
absval_wf,
rem_bounds_absval,
real_wf,
rleq_wf,
less_than'_wf,
nequal_wf,
mul_nat_plus,
div_rem_sum2,
false_wf,
mul_preserves_le,
l_sum_nil_lemma,
l_sum_cons_lemma,
map_nil_lemma,
map_cons_lemma,
iff_weakening_equal,
equal_wf,
int_subtype_base,
subtype_base_sq,
less_than_wf,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformless_wf,
intformand_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
intformeq_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
nat_plus_properties,
nil_wf,
nat_plus_wf,
cons_wf,
reg-seq-list-add-as-l_sum,
true_wf,
squash_wf,
le_wf,
int-value-type,
value-type-has-value
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
callbyvalueReduce,
sqleReflexivity,
dependent_functionElimination,
thin,
hypothesisEquality,
lemma_by_obid,
isectElimination,
intEquality,
independent_isectElimination,
hypothesis,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
minusEquality,
divideEquality,
functionEquality,
because_Cache,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality,
independent_pairFormation,
addLevel,
instantiate,
cumulativity,
independent_functionElimination,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
addEquality,
independent_pairEquality,
axiomEquality,
remainderEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
impliesFunctionality
Latex:
\mforall{}[x,y:\mBbbR{}]. \mforall{}n:\mBbbN{}\msupplus{}. ((x (4 * n)) \mleq{} ((y (4 * n)) + 11)) supposing x \mleq{} y
Date html generated:
2016_05_18-AM-07_04_40
Last ObjectModification:
2016_01_17-AM-01_50_37
Theory : reals
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