Nuprl Lemma : count-combinations
∀n,m:ℕ. Combination(n;ℕm) ~ ℕC(n;m)
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
combinations: C(n;m),
combination: Combination(n;T),
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
equipollent: A ~ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
combination: Combination(n;T),
cand: A c∧ B,
subtype_rel: A ⊆r B,
cons: [a / b],
guard: {T},
iff: P ⇐⇒ Q,
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
pi1: fst(t),
pi2: snd(t),
l_member: (x ∈ l)
Lemmas referenced :
nat_wf,
equipollent_wf,
combination_wf,
int_seg_wf,
subtract_wf,
combinations_wf_int,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
less_than_wf,
primrec-wf2,
all_wf,
combinations-step,
istype-false,
biject_wf,
list-cases,
length_of_nil_lemma,
nil_wf,
no_repeats_wf,
length_wf_nat,
set_subtype_base,
int_subtype_base,
product_subtype_list,
length_of_cons_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
non_neg_length,
int_seg_properties,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
subtype_rel-equal,
base_wf,
no_repeats_nil,
length_nil,
list_subtype_base,
lelt_wf,
decidable__equal_int,
subtype_base_sq,
eq_int_wf,
equal-wf-base,
bool_wf,
assert_wf,
bnot_wf,
not_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equipollent-zero,
combination-decomp,
subtype_rel_set,
list_wf,
equal-wf-base-T,
equal-wf-T-base,
subtype_rel_list,
product_subtype_base,
pi1_wf_top,
reduce_tl_nil_lemma,
reduce_tl_cons_lemma,
reduce_hd_cons_lemma,
cons_wf,
no_repeats_cons,
no_repeats-settype,
l_member_wf,
select_wf,
equipollent_functionality_wrt_equipollent,
equipollent_weakening_ext-eq,
ext-eq_weakening,
equipollent-subtract-one,
id-biject,
product_functionality_wrt_equipollent_dependent,
equipollent_same,
combination_functionality,
equipollent-multiply,
combinations_wf,
product_functionality_wrt_equipollent_right
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
universeIsType,
because_Cache,
rename,
setElimination,
sqequalRule,
functionIsType,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
hypothesisEquality,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
setIsType,
inhabitedIsType,
imageMemberEquality,
baseClosed,
productIsType,
equalityIsType4,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
intEquality,
promote_hyp,
hypothesis_subsumption,
imageElimination,
universeEquality,
instantiate,
applyLambdaEquality,
baseApply,
closedConclusion,
cumulativity,
equalityElimination,
equalityIsType1,
dependent_pairEquality_alt,
setEquality,
productEquality,
independent_pairEquality,
multiplyEquality
Latex:
\mforall{}n,m:\mBbbN{}. Combination(n;\mBbbN{}m) \msim{} \mBbbN{}C(n;m)
Date html generated:
2019_10_15-AM-11_16_03
Last ObjectModification:
2018_10_09-PM-02_12_24
Theory : general
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