Nuprl Lemma : combinations_aux_linear
∀[n:ℕ]. ∀[b,m:ℤ]. (combinations_aux(b;n;m) = (b * combinations_aux(1;n;m)) ∈ ℤ)
Proof
Definitions occuring in Statement :
combinations_aux: combinations_aux(b;n;m),
nat: ℕ,
uall: ∀[x:A]. B[x],
multiply: n * m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
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,
combinations_aux: combinations_aux(b;n;m),
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,
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
squash: ↓T,
guard: {T}
Lemmas referenced :
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,
nat_wf,
decidable__equal_int,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
eq_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
bnot_wf,
not_wf,
value-type-has-value,
int-value-type,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
combinations_aux_wf_int,
le_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
callbyvalueReduce,
sqleReflexivity,
unionElimination,
because_Cache,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
equalityElimination,
productElimination,
impliesFunctionality,
dependent_set_memberEquality,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[b,m:\mBbbZ{}]. (combinations\_aux(b;n;m) = (b * combinations\_aux(1;n;m)))
Date html generated:
2018_05_21-PM-08_09_05
Last ObjectModification:
2017_07_26-PM-05_44_44
Theory : general
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