Nuprl Lemma : regular-less
∀[x,y:ℝ]. ∀n:ℕ+. ((x n) + 4 < y n ⇒ (∀m:ℕ+. ((x m) ≤ ((y m) + 4))))
Proof
Definitions occuring in Statement :
real: ℝ,
nat_plus: ℕ+,
less_than: a < b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
real: ℝ,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
nat_plus: ℕ+,
sq_stable: SqStable(P),
regular-int-seq: k-regular-seq(f),
squash: ↓T,
le: A ≤ B,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
false: False,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
guard: {T},
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
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,
subtype_base_sq,
bool_cases,
int_term_value_minus_lemma,
itermMinus_wf,
minus-is-int-iff,
not_wf,
bnot_wf,
assert_wf,
subtract-is-int-iff,
int_subtype_base,
lt_int_wf,
real_wf,
less_than'_wf,
nat_plus_wf,
false_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermSubtract_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
multiply-is-int-iff,
add-is-int-iff,
less_than_wf,
decidable__le,
nat_plus_properties,
absval_ifthenelse,
subtract_wf,
sq_stable__le,
mul_cancel_in_le,
mul_preserves_lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
addEquality,
applyEquality,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
independent_isectElimination,
hypothesis,
multiplyEquality,
independent_functionElimination,
dependent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
because_Cache,
productElimination,
dependent_set_memberEquality,
unionElimination,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_pairEquality,
axiomEquality,
instantiate,
cumulativity,
impliesFunctionality
Latex:
\mforall{}[x,y:\mBbbR{}]. \mforall{}n:\mBbbN{}\msupplus{}. ((x n) + 4 < y n {}\mRightarrow{} (\mforall{}m:\mBbbN{}\msupplus{}. ((x m) \mleq{} ((y m) + 4))))
Date html generated:
2016_05_18-AM-06_47_40
Last ObjectModification:
2016_01_17-AM-01_46_47
Theory : reals
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