Nuprl Lemma : combinations-formula

∀[n,m:ℕ].  ((C(n;m) * (m - n)!) = (m)! ∈ ℤ supposing n ≤ m ∧ (C(n;m) = if n ≤z m then (m)! ÷ (m - n)! else 0 fi  ∈ ℤ))


This theorem is one of freek's list of 100 theorems



Proof



Definitions occuring in Statement :  combinations: C(n;m),  fact: (n)!,  nat: ℕ,  le_int: i ≤z j,  ifthenelse: if b then t else f fi ,  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  le: A ≤ B,  and: P ∧ Q,  divide: n ÷ m,  multiply: n * m,  subtract: n - m,  natural_number: $n,  int: ℤ,  equal: s = t ∈ T
Definitions unfolded in proof :  all: ∀x:A. B[x],  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],  and: P ∧ Q,  prop: ℙ,  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,  decidable: Dec(P),  or: P ∨ Q,  subtype_rel: A ⊆r B,  nat_plus: ℕ+,  bool: 𝔹,  unit: Unit,  it: ⋅,  uiff: uiff(P;Q),  bfalse: ff,  sq_type: SQType(T),  guard: {T},  bnot: ¬bb,  assert: ↑b,  nequal: a ≠ b ∈ T ,  squash: ↓T,  true: True,  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  top: Top,  cand: A c∧ B,  so_apply: x[s],  so_lambda: λ2x.t[x],  lelt: i ≤ j < k,  div_nrel: Div(a;n;q)
Lemmas referenced :  nat_properties,  full-omega-unsat,  intformand_wf,  intformle_wf,  itermConstant_wf,  itermVar_wf,  intformless_wf,  istype-int,  int_formula_prop_and_lemma,  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,  combinations-step,  istype-void,  istype-le,  minus-zero,  one-mul,  fact_wf,  decidable__le,  intformnot_wf,  itermAdd_wf,  int_formula_prop_not_lemma,  int_term_value_add_lemma,  add-zero,  istype-nat,  subtract-1-ge-0,  eq_int_wf,  eqtt_to_assert,  assert_of_eq_int,  intformeq_wf,  int_formula_prop_eq_lemma,  eqff_to_assert,  bool_cases_sqequal,  subtype_base_sq,  bool_wf,  bool_subtype_base,  assert-bnot,  neg_assert_of_eq_int,  mul-associates,  minus-one-mul,  add-commutes,  fact_unroll_1,  equal_wf,  squash_wf,  true_wf,  istype-universe,  combinations_wf_int,  itermMultiply_wf,  int_term_value_mul_lemma,  subtract_wf,  itermSubtract_wf,  int_term_value_subtract_lemma,  subtype_rel_self,  iff_weakening_equal,  minus-add,  minus-minus,  add-associates,  add-swap,  zero-add,  int_formual_prop_imp_lemma,  intformimplies_wf,  decidable__equal_int,  bnot_wf,  less_than_wf,  lt_int_wf,  assert_wf,  int_subtype_base,  le_wf,  set_subtype_base,  equal-wf-base,  le_int_wf,  uiff_transitivity,  assert_of_le_int,  assert_functionality_wrt_uiff,  bnot_of_le_int,  assert_of_lt_int,  satisfiable-full-omega-tt,  combinations_wf,  mul_bounds_1a,  div_unique2,  nat_plus_properties,  decidable__lt,  false_wf,  multiply-is-int-iff,  zero_ann_a,  istype-assert,  not_wf,  iff_transitivity,  iff_weakening_uiff,  assert_of_bnot
Rules used in proof :  cut,  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  lambdaFormation_alt,  introduction,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  thin,  hypothesisEquality,  hypothesis,  setElimination,  rename,  intWeakElimination,  natural_numberEquality,  independent_isectElimination,  approximateComputation,  independent_functionElimination,  dependent_pairFormation_alt,  lambdaEquality_alt,  int_eqEquality,  dependent_functionElimination,  Error :memTop,  sqequalRule,  independent_pairFormation,  universeIsType,  voidElimination,  axiomEquality,  functionIsTypeImplies,  inhabitedIsType,  dependent_set_memberEquality_alt,  addEquality,  unionElimination,  applyEquality,  equalityTransitivity,  equalitySymmetry,  because_Cache,  equalityElimination,  productElimination,  equalityIstype,  promote_hyp,  instantiate,  cumulativity,  minusEquality,  imageElimination,  universeEquality,  intEquality,  multiplyEquality,  imageMemberEquality,  baseClosed,  isectIsTypeImplies,  independent_pairEquality,  isect_memberEquality_alt,  isect_memberFormation_alt,  closedConclusion,  baseApply,  computeAll,  voidEquality,  isect_memberEquality,  lambdaEquality,  dependent_pairFormation,  dependent_set_memberEquality,  applyLambdaEquality,  pointwiseFunctionality,  inrFormation_alt,  functionIsType,  sqequalBase
Latex:
\mforall{}[n,m:\mBbbN{}].
    ((C(n;m)  *  (m  -  n)!)  =  (m)!  supposing  n  \mleq{}  m
    \mwedge{}  (C(n;m)  =  if  n  \mleq{}z  m  then  (m)!  \mdiv{}  (m  -  n)!  else  0  fi  ))


Date html generated: 2020_05_20-AM-08_15_43
Last ObjectModification: 2020_01_01-PM-02_21_53

Theory : general


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