Nuprl Lemma : sine-approx-lemma-bad
∀a:{2...}. ∀N:ℕ. (∃k:ℕ [(N ≤ (a^((2 * k) + 3) * ((2 * k) + 3)!))])
Proof
Definitions occuring in Statement :
fact: (n)!,
exp: i^n,
int_upper: {i...},
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
sq_exists: ∃x:A [B[x]],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
int_upper: {i...},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
sq_type: SQType(T),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
rev_uimplies: rev_uimplies(P;Q),
exp: i^n,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
subtract: n - m
Lemmas referenced :
istype-le,
subtract_wf,
exp_wf2,
nat_properties,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
fact_wf,
istype-less_than,
primrec-wf2,
sq_exists_wf,
nat_wf,
le_wf,
istype-nat,
istype-int_upper,
mul_bounds_1a,
exp_wf4,
nat_plus_subtype_nat,
sq_stable__le,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
exp_add,
fact_add2,
iff_weakening_equal,
exp_preserves_le,
nat_plus_properties,
upper_subtype_nat,
istype-false,
le_functionality,
le_weakening,
multiply_functionality_wrt_le,
exp-nondecreasing,
mul_preserves_le,
intformless_wf,
int_formula_prop_less_lemma,
mul-associates,
mul_nat_plus,
decidable__lt,
itermSubtract_wf,
int_term_value_subtract_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
rename,
setElimination,
sqequalRule,
setIsType,
because_Cache,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
natural_numberEquality,
hypothesis,
multiplyEquality,
dependent_set_memberEquality_alt,
addEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
applyEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberFormation_alt,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
cumulativity,
intEquality,
equalityIstype,
applyLambdaEquality,
baseApply,
closedConclusion,
sqequalIntensionalEquality,
productElimination
Latex:
\mforall{}a:\{2...\}. \mforall{}N:\mBbbN{}. (\mexists{}k:\mBbbN{} [(N \mleq{} (a\^{}((2 * k) + 3) * ((2 * k) + 3)!))])
Date html generated:
2019_10_29-AM-10_32_57
Last ObjectModification:
2019_02_01-PM-08_46_17
Theory : reals
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