Nuprl Lemma : choose-inequality1
∀n:ℕ. ∀i:ℕn. (choose(n;i) ≤ (n * choose(n - 1;i)))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
multiply: n * m,
subtract: n - m,
natural_number: $n,
choose: choose(n;i)
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
guard: {T},
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
int_iseg: {i...j},
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
sq_type: SQType(T),
nat_plus: ℕ+,
subtract: n - m,
true: True,
iff: P ⇐⇒ Q
Lemmas referenced :
decidable__le,
choose_wf,
subtract_wf,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
decidable__lt,
itermMultiply_wf,
int_term_value_mul_lemma,
int_seg_wf,
istype-nat,
mul_preserves_lt,
subtype_rel_sets,
lelt_wf,
le_wf,
istype-less_than,
mul_nat_plus,
fact_wf,
int_seg_subtype_nat,
istype-false,
choose-formula,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
multiply-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
false_wf,
nat_plus_wf,
set_subtype_base,
less_than_wf,
nat_plus_properties,
fact_unroll_1,
add-associates,
minus-one-mul,
add-swap,
add-commutes,
itermAdd_wf,
int_term_value_add_lemma,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
mul_preserves_le,
nat_plus_subtype_nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
multiplyEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
natural_numberEquality,
productElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
productIsType,
intEquality,
closedConclusion,
productEquality,
setIsType,
applyLambdaEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
instantiate,
cumulativity,
imageElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
baseClosed,
minusEquality,
addEquality,
imageMemberEquality,
universeEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}i:\mBbbN{}n. (choose(n;i) \mleq{} (n * choose(n - 1;i)))
Date html generated:
2019_10_15-AM-11_21_27
Last ObjectModification:
2018_10_18-PM-11_44_26
Theory : general
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