Nuprl Lemma : choose-formula
∀[n,m:ℕ]. (choose(n;m) * (m)! * (n - m)!) = (n)! ∈ ℤ supposing m ≤ n
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
fact: (n)!,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
multiply: n * m,
subtract: n - m,
int: ℤ,
equal: s = t ∈ T,
choose: choose(n;i)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
choose: choose(n;i),
ycomb: Y,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bor: p ∨bq,
ifthenelse: if b then t else f fi ,
subtract: n - m,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
less_than': less_than'(a;b),
int_upper: {i...},
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
cand: A c∧ B,
int_iseg: {i...j},
squash: ↓T
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
istype-nat,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
minus-zero,
mul-commutes,
fact0_redex_lemma,
one-mul,
fact_wf,
itermAdd_wf,
int_term_value_add_lemma,
add-zero,
itermMultiply_wf,
int_term_value_mul_lemma,
eqff_to_assert,
le_wf,
bool_subtype_base,
bool_cases_sqequal,
bool_wf,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
istype-false,
nequal-le-implies,
zero-add,
int_upper_properties,
minus-one-mul,
add-mul-special,
zero-mul,
nat_plus_wf,
equal-wf-T-base,
satisfiable-full-omega-tt,
fact_unroll_1,
minus-minus,
minus-add,
add-swap,
add-commutes,
minus-one-mul-top,
int_upper_wf,
mul-associates,
int_term_value_minus_lemma,
itermMinus_wf,
mul-distributes-right,
nat_wf,
add-associates,
add-is-int-iff,
iff_weakening_equal,
choose_wf,
mul_add_distrib,
true_wf,
squash_wf,
equal_wf,
mul-swap,
mul-distributes,
istype-universe,
add_functionality_wrt_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
productElimination,
because_Cache,
unionElimination,
applyEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
hypothesis_subsumption,
equalityElimination,
cumulativity,
intEquality,
multiplyEquality,
addEquality,
closedConclusion,
equalityIsType4,
baseApply,
baseClosed,
promote_hyp,
equalityIsType1,
minusEquality,
dependent_set_memberEquality,
computeAll,
voidEquality,
isect_memberEquality,
lambdaEquality,
dependent_pairFormation,
lambdaFormation,
imageMemberEquality,
productEquality,
universeEquality,
imageElimination
Latex:
\mforall{}[n,m:\mBbbN{}]. (choose(n;m) * (m)! * (n - m)!) = (n)! supposing m \mleq{} n
Date html generated:
2019_10_15-AM-11_21_10
Last ObjectModification:
2018_10_18-PM-11_44_54
Theory : general
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