Nuprl Lemma : combinations-choose
∀[m,n:ℕ]. C(m;n) = (choose(n;m) * (m)!) ∈ ℤ supposing m ≤ n
Proof
Definitions occuring in Statement :
combinations: C(n;m),
fact: (n)!,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
multiply: n * m,
int: ℤ,
equal: s = t ∈ T,
choose: choose(n;i)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
uimplies: b supposing a,
nat: ℕ,
not: ¬A,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
int_iseg: {i...j},
cand: A c∧ B,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
nat_plus: ℕ+
Lemmas referenced :
combinations-formula,
istype-le,
istype-nat,
le_int_wf,
assert_wf,
bnot_wf,
not_wf,
le_wf,
istype-assert,
istype-void,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
choose-formula,
mul_cancel_in_eq,
choose_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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_wf,
fact_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__equal_int,
multiply-is-int-iff,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
false_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_isectElimination,
setElimination,
rename,
because_Cache,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
sqequalRule,
functionIsType,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
unionElimination,
instantiate,
cumulativity,
independent_pairFormation,
lambdaFormation_alt,
multiplyEquality,
dependent_set_memberEquality_alt,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
universeIsType,
productIsType,
applyEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[m,n:\mBbbN{}]. C(m;n) = (choose(n;m) * (m)!) supposing m \mleq{} n
Date html generated:
2019_10_15-AM-11_21_41
Last ObjectModification:
2018_11_30-PM-01_15_46
Theory : general
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