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 :
or: P ∨ Q,
decidable: Dec(P),
fact: (n)!,
primrec: primrec(n;b;c),
exp: i^n,
less_than': less_than'(a;b),
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
le: A ≤ B,
prop: ℙ,
and: P ∧ Q,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
guard: {T}
Lemmas referenced :
int_term_value_mul_lemma,
itermMultiply_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,
full-omega-unsat,
nat_properties,
fact_unroll_1,
exp_step,
exp_preserves_le,
le_functionality,
multiply_functionality_wrt_le,
le_weakening,
le_transitivity
Rules used in proof :
multiplyEquality,
unionElimination,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
applyEquality,
because_Cache,
independent_pairEquality,
productElimination,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n:\mBbbN{}. ((n)! \mleq{} n\^{}n)
Date html generated:
2018_05_21-PM-01_05_00
Last ObjectModification:
2018_05_18-PM-04_16_45
Theory : num_thy_1
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