Nuprl Lemma : qexpfact-property
∀[n:ℕ]. ∀[x:{q:ℚ| 0 ≤ q} ]. ∀[p:ℚ]. ∀[b:{b:ℤ| b = (n)! ∈ ℤ} ]. p * x ↑ qexpfact(n;x;p;b) - n < (qexpfact(n;x;p;b))!
Proof
Definitions occuring in Statement :
qexpfact: qexpfact(n;x;p;b),
qexp: r ↑ n,
qle: r ≤ s,
qless: r < s,
qmul: r * s,
rationals: ℚ,
fact: (n)!,
nat: ℕ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
int_upper: {i...},
nat_plus: ℕ+,
uiff: uiff(P;Q),
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
qexpfact: qexpfact(n;x;p;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
iff: P ⇐⇒ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
top: Top,
true: True,
has-value: (a)↓,
less_than: a < b,
less_than': less_than'(a;b),
subtract: n - m
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
qless_witness,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
sq_stable_from_decidable,
qless_wf,
qmul_wf,
qexp_wf,
qexpfact_wf,
subtract_nat_wf,
le_wf,
fact_wf,
int_upper_properties,
nat_plus_properties,
subtract-is-int-iff,
false_wf,
upper_subtype_nat,
sq_stable__le,
subtype_rel_set,
rationals_wf,
less_than_wf,
int-subtype-rationals,
decidable__qless,
q_less_wf,
eqtt_to_assert,
assert-q_less-eq,
iff_weakening_equal,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
sq_stable__equal,
qle_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
satisfiable-full-omega-tt,
qmul_one_qrng,
squash_wf,
true_wf,
qexp-zero,
value-type-has-value,
int-value-type,
rationals-value-type,
int_term_value_mul_lemma,
itermMultiply_wf,
add-subtract-cancel,
istype-void,
fact_unroll_1,
easy-member-int_seg,
exp_unroll_q,
qmul_ac_1_qrng,
istype-false,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
qmul_com,
equal-wf-base-T,
equal_wf,
int_upper_subtype_nat,
set_wf,
nat_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
isect_memberEquality_alt,
productElimination,
functionIsTypeImplies,
inhabitedIsType,
isectIsTypeImplies,
because_Cache,
unionElimination,
applyEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
promote_hyp,
hypothesis_subsumption,
isect_memberFormation_alt,
equalityIstype,
baseApply,
closedConclusion,
baseClosed,
intEquality,
sqequalBase,
pointwiseFunctionality,
imageMemberEquality,
imageElimination,
equalityElimination,
cumulativity,
setIsType,
addEquality,
lambdaFormation,
dependent_pairFormation,
lambdaEquality,
isect_memberEquality,
voidEquality,
computeAll,
universeEquality,
multiplyEquality,
callbyvalueReduce,
minusEquality,
hyp_replacement,
functionEquality,
isect_memberFormation,
dependent_set_memberEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x:\{q:\mBbbQ{}| 0 \mleq{} q\} ]. \mforall{}[p:\mBbbQ{}]. \mforall{}[b:\{b:\mBbbZ{}| b = (n)!\} ].
p * x \muparrow{} qexpfact(n;x;p;b) - n < (qexpfact(n;x;p;b))!
Date html generated:
2020_05_20-AM-09_26_37
Last ObjectModification:
2019_12_31-PM-04_57_23
Theory : rationals
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