Nuprl Lemma : divides-combinations
∀n:ℕ. ∀m:ℤ. ∀k:ℕ. (k | C(n;m)) supposing ((k ≤ m) and m - n < k)
Proof
Definitions occuring in Statement :
combinations: C(n;m),
divides: b | a,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
le: A ≤ B,
all: ∀x:A. B[x],
subtract: n - m,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
divides: b | a,
sq_type: SQType(T),
guard: {T},
squash: ↓T,
true: True
Lemmas referenced :
member-less_than,
subtract_wf,
less_than'_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
le_wf,
less_than_wf,
nat_wf,
combinations-step,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
eq_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-base,
int_subtype_base,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
decidable__equal_int,
equal_wf,
all_wf,
isect_wf,
divides_wf,
combinations_wf_int,
set_wf,
primrec-wf2,
subtype_base_sq,
itermMultiply_wf,
int_term_value_mul_lemma,
equal-wf-base-T,
decidable__lt,
squash_wf,
true_wf,
iff_weakening_equal,
mul-swap
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
isect_memberFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
natural_numberEquality,
hypothesis,
setElimination,
rename,
independent_isectElimination,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
voidElimination,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
unionElimination,
equalityElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
independent_functionElimination,
impliesFunctionality,
promote_hyp,
instantiate,
cumulativity,
multiplyEquality,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}m:\mBbbZ{}. \mforall{}k:\mBbbN{}. (k | C(n;m)) supposing ((k \mleq{} m) and m - n < k)
Date html generated:
2018_05_21-PM-08_09_59
Last ObjectModification:
2017_07_26-PM-05_45_35
Theory : general
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