Nuprl Lemma : factorial-greater-qexp
∀x:{q:ℚ| 0 ≤ q} . ∀p:ℚ. ∃n:ℕ. p * x ↑ n < (n)!
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qle: r ≤ s,
qless: r < s,
qmul: r * s,
rationals: ℚ,
fact: (n)!,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
subtract: n - m,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
int_upper: {i...},
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
true: True,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
qexpfact_wf,
false_wf,
le_wf,
qle_wf,
fact0_redex_lemma,
equal-wf-base,
int_subtype_base,
qexpfact-property,
minus-zero,
add-zero,
qexp_wf,
squash_wf,
true_wf,
nat_wf,
rationals_wf,
decidable__le,
int_upper_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
and_wf,
equal_wf,
qmul_wf,
qless_wf,
fact_wf,
subtype_rel_set,
less_than_wf,
int-subtype-rationals,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
hypothesis,
hypothesisEquality,
setElimination,
rename,
applyEquality,
because_Cache,
intEquality,
baseClosed,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
unionElimination,
independent_functionElimination,
voidElimination,
independent_isectElimination,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
imageMemberEquality,
productElimination,
setEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}x:\{q:\mBbbQ{}| 0 \mleq{} q\} . \mforall{}p:\mBbbQ{}. \mexists{}n:\mBbbN{}. p * x \muparrow{} n < (n)!
Date html generated:
2016_10_26-AM-06_36_15
Last ObjectModification:
2016_07_12-AM-07_58_03
Theory : rationals
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