Nuprl Lemma : combinations-split
∀[m,n,k:ℕ]. C(n + k;m) = (C(k;m) * C(n;m - k)) ∈ ℤ supposing (n + k) ≤ m
Proof
Definitions occuring in Statement :
combinations: C(n;m),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
multiply: n * m,
subtract: n - m,
add: n + m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
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],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
less_than': less_than'(a;b),
le: A ≤ B,
btrue: tt,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
combinations_aux: combinations_aux(b;n;m),
combinations: C(n;m),
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
decidable: Dec(P),
so_apply: x[s],
so_lambda: λ2x.t[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
subtype_rel: A ⊆r B,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
int_upper: {i...},
subtract: n - m
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,
le_wf,
nat_wf,
subtract-1-ge-0,
subtract_wf,
false_wf,
combinations_wf_int,
decidable__le,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermAdd_wf,
intformeq_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
combinations-step,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
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,
add-commutes,
int_upper_properties,
add-associates,
minus-zero,
one-mul,
add-zero,
int_term_value_subtract_lemma,
itermSubtract_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
mul-associates
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
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,
addEquality,
because_Cache,
productElimination,
lambdaFormation,
multiplyEquality,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality,
computeAll,
voidEquality,
isect_memberEquality,
dependent_pairFormation,
unionElimination,
lambdaEquality,
intEquality,
cumulativity,
instantiate,
dependent_set_memberEquality_alt,
equalityElimination,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
promote_hyp,
equalityIsType1,
hypothesis_subsumption,
minusEquality
Latex:
\mforall{}[m,n,k:\mBbbN{}]. C(n + k;m) = (C(k;m) * C(n;m - k)) supposing (n + k) \mleq{} m
Date html generated:
2019_10_15-AM-11_16_19
Last ObjectModification:
2018_10_16-PM-03_13_36
Theory : general
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