Nuprl Lemma : expfact-property
∀k:ℕ. ∀n:ℕ+. ∃m:ℕ+. ((n * k^m) ≤ (m)!)
Proof
Definitions occuring in Statement :
fact: (n)!,
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
multiply: n * m
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
nat: ℕ,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
fact: (n)!,
primrec: primrec(n;b;c),
subtract: n - m,
le: A ≤ B,
sq_stable: SqStable(P),
uiff: uiff(P;Q)
Lemmas referenced :
fact-greater-exp,
nat_plus_subtype_nat,
nat_plus_wf,
nat_wf,
expfact_wf,
less_than_wf,
exp_wf2,
fact_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
nat_properties,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
exp0_lemma,
fact0_redex_lemma,
intformless_wf,
itermMultiply_wf,
int_formula_prop_less_lemma,
int_term_value_mul_lemma,
equal_wf,
squash_wf,
true_wf,
exp1,
iff_weakening_equal,
set_subtype_base,
false_wf,
le_wf,
set_wf,
sq_stable__le,
multiply-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
productElimination,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
multiplyEquality,
setElimination,
rename,
lambdaEquality,
independent_functionElimination,
unionElimination,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
setEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}k:\mBbbN{}. \mforall{}n:\mBbbN{}\msupplus{}. \mexists{}m:\mBbbN{}\msupplus{}. ((n * k\^{}m) \mleq{} (m)!)
Date html generated:
2018_05_21-PM-01_02_42
Last ObjectModification:
2018_01_28-PM-02_12_42
Theory : num_thy_1
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