Nuprl Lemma : fact-bound
∀n:ℕ. ((n)! ≤ n^n)
Proof
Definitions occuring in Statement :
fact: (n)!,
exp: i^n,
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x]
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,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
less_than': less_than'(a;b),
exp: i^n,
primrec: primrec(n;b;c),
fact: (n)!,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
int_term_value_mul_lemma,
itermMultiply_wf,
exp_preserves_le,
mul_preserves_le,
exp_step,
fact_unroll_1,
equal_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
nat_wf,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
le_wf,
false_wf,
nat_plus_wf,
fact_wf,
exp_wf2,
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 :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
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,
because_Cache,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
unionElimination
Latex:
\mforall{}n:\mBbbN{}. ((n)! \mleq{} n\^{}n)
Date html generated:
2016_05_15-PM-04_44_50
Last ObjectModification:
2016_01_16-AM-11_24_09
Theory : general
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