Nuprl Lemma : choose_wf
∀[n:ℕ]. ∀[i:{0...n}]. (choose(n;i) ∈ ℕ)
Proof
Definitions occuring in Statement :
choose: choose(n;i),
int_iseg: {i...j},
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
choose: choose(n;i),
ycomb: Y,
int_iseg: {i...j},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
ifthenelse: if b then t else f fi ,
le: A ≤ B,
less_than': less_than'(a;b),
bfalse: ff,
subtract: n - m,
guard: {T},
subtype_rel: A ⊆r B,
cand: A c∧ B
Lemmas referenced :
int_iseg_wf,
nat_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
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,
less_than_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
bor_wf,
eq_int_wf,
bool_wf,
iff_transitivity,
equal-wf-T-base,
assert_wf,
or_wf,
iff_weakening_uiff,
eqtt_to_assert,
assert_of_bor,
assert_of_eq_int,
false_wf,
le_wf,
band_wf,
bnot_wf,
not_wf,
bnot_thru_bor,
eqff_to_assert,
assert_of_band,
assert_of_bnot,
int_iseg_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
int_subtype_base,
add_nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
extract_by_obid,
natural_numberEquality,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
unionElimination,
because_Cache,
equalityElimination,
baseClosed,
orFunctionality,
productElimination,
dependent_set_memberEquality,
productEquality,
impliesFunctionality,
applyEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[i:\{0...n\}]. (choose(n;i) \mmember{} \mBbbN{})
Date html generated:
2017_10_01-AM-08_18_48
Last ObjectModification:
2017_02_28-PM-02_03_40
Theory : rings_1
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