Nuprl Lemma : combinations-step
∀[n,m:ℕ]. (C(n;m) ~ if (n =z 0) then 1 else m * C(n - 1;m - 1) fi )
Proof
Definitions occuring in Statement :
combinations: C(n;m),
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
multiply: n * m,
subtract: n - m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
combinations: C(n;m),
combinations_aux: combinations_aux(b;n;m),
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
ge: i ≥ j ,
int_upper: {i...},
has-value: (a)↓,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
nat_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
false_wf,
le_wf,
nat_properties,
nequal-le-implies,
zero-add,
value-type-has-value,
int-value-type,
int_subtype_base,
subtract_wf,
combinations_aux_linear,
int_upper_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_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_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__equal_int,
multiply-is-int-iff,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
sqequalAxiom,
extract_by_obid,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
setElimination,
rename,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
hypothesis_subsumption,
dependent_set_memberEquality,
independent_pairFormation,
callbyvalueReduce,
intEquality,
multiplyEquality,
lambdaEquality,
int_eqEquality,
voidEquality,
computeAll,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[n,m:\mBbbN{}]. (C(n;m) \msim{} if (n =\msubz{} 0) then 1 else m * C(n - 1;m - 1) fi )
Date html generated:
2018_05_21-PM-08_09_16
Last ObjectModification:
2017_07_26-PM-05_44_57
Theory : general
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