Nuprl Lemma : fact-greater-exp
∀k,n:ℕ. ∃m:ℕ. n * k^m < (m)!
Proof
Definitions occuring in Statement :
fact: (n)!,
exp: i^n,
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
multiply: n * m
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,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
subtract: n - m,
guard: {T},
uiff: uiff(P;Q),
sq_type: SQType(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than': less_than'(a;b),
fact: (n)!,
exp: i^n,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
less_than: a < b,
squash: ↓T
Lemmas referenced :
exp_add,
exp_wf_nat_plus,
mul_preserves_lt,
fact-positive,
mul_bounds_1a,
minus-zero,
minus-add,
condition-implies-le,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
not-equal-2,
not-lt-2,
add-is-int-iff,
decidable__lt,
primrec1_lemma,
decidable__equal_int,
set_subtype_base,
exp_step,
int_subtype_base,
subtype_base_sq,
false_wf,
multiply-is-int-iff,
exp_wf4,
nat_plus_subtype_nat,
add_nat_wf,
multiply_nat_wf,
general_arith_equation1,
mul_preserves_le,
add-commutes,
zero-add,
one-mul,
zero-mul,
mul-associates,
mul-commutes,
mul-swap,
add-associates,
mul-distributes-right,
mul-distributes,
int_term_value_mul_lemma,
itermMultiply_wf,
nat_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
le_wf,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__le,
equal_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
fact_unroll_1,
exp0_lemma,
exp_wf2,
nat_plus_wf,
nat_plus_properties,
fact_wf,
less_than'_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
introduction,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
productElimination,
independent_pairEquality,
applyEquality,
because_Cache,
multiplyEquality,
addEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
unionElimination,
minusEquality,
setEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
equalityEquality,
instantiate,
cumulativity,
imageElimination
Latex:
\mforall{}k,n:\mBbbN{}. \mexists{}m:\mBbbN{}. n * k\^{}m < (m)!
Date html generated:
2016_05_15-PM-04_05_57
Last ObjectModification:
2016_01_16-AM-11_06_10
Theory : general
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