Nuprl Lemma : count-unordered-combinations
∀[T:Type]
∀n,m:ℕ.
(T ~ ℕn
⇒ (UnorderedCombination(m;T) ~ ℕchoose(n;m) supposing m ≤ n ∧ UnorderedCombination(m;T) ~ ℕ0 supposing n < m))
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
unordered-combination: UnorderedCombination(n;T),
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n,
universe: Type,
choose: choose(n;i)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
member: t ∈ T,
le: A ≤ B,
not: ¬A,
false: False,
nat: ℕ,
prop: ℙ,
int_iseg: {i...j},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
unordered-combination: UnorderedCombination(n;T),
bag-no-repeats: bag-no-repeats(T;bs),
squash: ↓T,
bag-size: #(bs),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
inject: Inj(A;B;f),
no_repeats: no_repeats(T;l),
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
less_than'_wf,
le_wf,
member-less_than,
less_than_wf,
equipollent_wf,
int_seg_wf,
nat_wf,
unordered-combination_wf,
choose_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
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,
equipollent-choose,
equipollent-zero,
equipollent_functionality_wrt_equipollent,
unordered-combination_functionality,
equipollent_weakening_ext-eq,
ext-eq_weakening,
equal_wf,
bag-size_wf,
pigeon-hole,
select_wf,
int_seg_properties,
decidable__lt,
intformless_wf,
intformeq_wf,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
decidable__equal_int_seg,
int_seg_subtype_nat,
false_wf,
set_wf,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
because_Cache,
independent_isectElimination,
cumulativity,
natural_numberEquality,
universeEquality,
dependent_set_memberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
productEquality,
applyEquality,
independent_functionElimination,
imageElimination,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]
\mforall{}n,m:\mBbbN{}.
(T \msim{} \mBbbN{}n
{}\mRightarrow{} (UnorderedCombination(m;T) \msim{} \mBbbN{}choose(n;m) supposing m \mleq{} n
\mwedge{} UnorderedCombination(m;T) \msim{} \mBbbN{}0 supposing n < m))
Date html generated:
2016_10_25-AM-11_32_25
Last ObjectModification:
2016_07_12-AM-07_36_54
Theory : bags_2
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